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The fundamental mathematical and acoustical properties of woodwind and brass instruments Krush, Joseph Martin 1978

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In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I further agree that permission for extensive copying of th is thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be al1 owed without my written permission. Department of Music  The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date April 20, 1978 i i ABSTRACT The contents of this thesis belongs entirely to the area of instrumental acoustics. It is written in the style of a textbook for music students, and aims to serve an instructional and informative pur-pose. The f i r s t goal of the paper is to outline the manner in which woodx^ind and brass instruments func-tion. Similarities and differences between individ-ual instruments and families of instruments are dis-cussed. The basis of the thesis is a disclosure of the physical properties of sound waves in general, x>7aves in cylindrical tubes, waves in conical tubes, and xv'aves in both these types of tubes with certain boundary conditions imposed on them. These proper-ties are directly applied to the x-joodx-jind instru-ments, but they cannot be applied to brass instru-ments; consequently, a different procedure is adopt-ed for their discussion. The original goal of the author x??as' to discuss the most salient knox^ m, proven mathematical and acoustical pronerties of x<7oodwind and brass instru-ments. In the process of doing this, conclusions evolved which led to a claim and subsequent forma-tion of theories which are, to the best of the writer's knowledge, presented here for the f i r s t time, and x^hich s t i l l remain to be completely proven by complex mathematical methods. iv TABLE OF CONTENTS List of Tables for Testing the Equation v = f \ . . ., ". . . . . . . v i i List of Tables of Frequencies . . . . v i i i List of Illustrations x Preface x i i i Acknowledgements . . . . . . . . . . . xx I. Differential Equations of Wave-Motion . . 1 II. The Cylindrical Tube 14 Boundary Conditions for the Cylindrical Tube 14 Possible Frequencies 21 The Clarinet 21 Testing the Equation v = f \ 27 The Vent-hole and the Resonance Curves of the Clarinet 31 The Flute 38 Testing the Equation v = f\ 43 III. The Conical Tube 48 Boundary Conditions for the Conical Tube 56 V Possible Frequencies . 60 The Oboe 60 Testing the Equation v = f> . . . ' 65 Other Aspects of the Acoustics of the Oboe in Particular and Wind Instruments in General . . 69 The Bassoon . . . 76 Testing the Equation v = f> 88 The Saxophone 96 Testing the Equation v = n . . 103 IV. The Brass Instruments 109 The Mouthpiece 115 The Bell 117 The Trumpet . . . . . . . . . 123 The Valves 131 Testing the Equation v = f y 138 The French Horn 143 Use of the Right Hand . . 163 Testing the Equation v = f l . . . . . . . . . 166 The Trombone 178 The Slide 188 The F Attachment . . . . 191 Testing the Equation v = f i 196 v i The Tuba 201 Testing the Equation v = f) . . 217 V. Practical Application of Knowledge of the Acoustics of Woodwind and Brass Instruments 221 Final Summary . 232 Concluding Remarks 239 Footnotes 246 Selected Bibliography 255 Appendix 257 v i i LIST OF TABLES FOR  TESTING THE EQUATION v = f \ Clarinet . 30 Flute 4 5 Oboe • 6 8 Bassoon .94 Saxophone • • • • 1 0 7 Trumpet 140, 142 French Horn Horn in F 171, 176 Horn in B-flat 172, 177 Trombone 1 9 8> 2 0 0 Tuba 2 1 * . 22<> Summary of Results for Woodwind Instruments . 222 Summary of Results for Brass Instruments . . 225 A Theoretical Table for Woodwind Instruments 237 v i i i LIST OF TABLES OF FREQUENCIES The Trumpet Possible (Sounding) Frequencies . . . . . 126 Sounding Frequencies Produced 129 Chromatic Scale 130 Additional Accessible Frequencies . . . . 137 The French Horn F Side Possible (Written) Frequencies 150 Written Frequencies Produced 153 B-flat Side . . , Possible (Written) Frequencies . 155 Written Frequencies Produced . . . . . . . 157 Both Sides Sounding Frequencies Produced . . . . . . 158 Chromatic Scale . . . . . .161 The Trombone Possible Written (and Sounding) Frequen-cies 182 ix Written (and Sounding) Frequencies Pro-duced 185 Chromatic Scale 186-7 Possible Written (and Sounding) Frequen-cies With F Attachment 193 The Tuba Possible Written (and Sounding) Frequen-cies 210 Written (and Sounding) Frequencies Pro-duced . . . . . . . 212 Chromatic Scale 216 Frequencies of Pitches in the Equal-tempered Scale 257 X LIST OF ILLUSTRATIONS 1. A Regular Train of Waves . . 3 2. A Simple Harmonic Wave 7 3. A Standing Wave at t = 0, |, |, ^ . . 10 4. A Progressive Wave passing from Water to Air 11 5. Graph of velocity v and pressure p Against position x for the Closed Cylindrical Tube 15 6. Graph of pressure p Against position x for the Cylindrical Tube Closed at one end . 17 7. Graph of First Two Modes of Oscillation for the Cylindrical Tube Closed at one end 17 8. Graph of pressure p Against position x for the Cylindrical Tube Open at both ends . 19 9. Graph of First Two Modes of Oscillation for the Cylindrical Tube Open at both ends 19 10. Resonance Curve for the Lowest Note of the Clarinet, E3 31 11. Resonance Curve for a Plain Metal Tube Closed at one end 32 12. Harmonic Structure of an Internal Stand-ing Wave in a Plain Metal Tube Closed at one end 33 13. Weighted Harmonic Structure of an Internal Standing Wave in a Clarinet Sounding E3 . 34 14. Resonance Curve for the Highest Note Be-fore the Break of the Clarinet, Bb4 . . . 35 x i 15. Weighted Harmonic Structure of an Inter-nal Standing Wave in a Clarinet Sound-ing Bb4 . . " .35 16. Resonance Curve for the note D5 of the Clarinet 36 17. Weighted Harmonic Structure of an Inter-nal Standing Wave in a Clarinet Sound-ing D5 37 18. Cross-sectional View of the Essential Parts of a Flute 43 19. Illustration of Shifting of Overtones . . 46 20. Illustration of the Full Conical Tube . . 48 21. Graphs of y = cos x, y = sin x, and r., A sin kr c o f(p) = 52 22. Illustration of Truncated Conical Tube . . 54 23. Graph of pressure p Against position x for the Conical Tube Closed at one end . 59 24. Graph of First Two Modes of Oscillation for the Conical Tube Closed at one end . 59 25. Illustrations of the Conical Bore of the Oboe 66 26. Illustration of the Wing Joint of the Bassoon . . . . . . . . . . 76 2 7. Illustration of the Boot of the Bassoon . 77 28. Illustrations of Cross-fingerings . . . . 87 29. Illustrations of the Conical Bore of the Bassoon . . . . . . . . . . 89 30. Illustration of the Extrapolated Conical Bore of the Bassoon 91 x i i 31. Illustration of the Parts of a Saxophone 97 32. Illustration of the Conical Bore of the SaxoDhone 104 33. Illustration of the Extrapolated Conical Bore of the Saxophone 105 34. Illustration of the Parts of a Mouthpiece 115 35. Illustration of an "Exponential Horn" . 119 36. Illustration of the Mechanism of a Pis-ton Valve 131 37. Illustration of the Playing Positions of a Trombone 181 x i i i PREFACE Acoustics in general and the acoustics of in-struments in particular are aspects of music, math-ematics, and physics which continue to be generally neglected and unlearned by most students and profes-sors concerned with these disciplines. In general, the root of this problem l i e s in the fact that math-ematicians and physicists are most often primarily interested in aspects of their fields other than mu-sic, while musicians are interested in almost any aspect of music other than the mathematical and phy-s i c a l . From the point of view of a musician and mathematician (but not that of a physicistj), the writer sees this as most unfortunate. The three areas are so interrelated and enhance each other so greatly that i t is a wonder that so few cultures have grouped them into one discipline, and relatively few people bother to study these disciplines in re-lation to each other. From the point of view of a music theorist, the writer candidly admits that we as musicians have more to gain from studying mathematical and physical aspects of music than our counterparts in those two xiv disciplines have to gain by studying musical aspects of mathematics and physics. (It follows, therefore, that the converse is not true.) Why, then, are mu-sicians (some of whom spend a l l their lives studying music) so reluctant to study sound? One common answer to this question is that most musicians are understandably interested in instru-mental (or vocal) performance, and not in theoretical study. This, however, is perhaps a weak excuse for ignorance about how instruments operate and why they function the way they do. In the most p i t i f u l cases, musicians don't even understand the fundamental prop-erties of their own instrument. The area of musical acoustics is the one which best provides this infor-mation to musicians; this paper is directed towards that goal. The f i r s t purpose of this work is to present a study of the principles of operation of xroodwind and brass instruments. Since the object of the under-taking is to outline these properties in an informa-tive manner, (at an elementary level only), the text-book style of writing is employed throughout. For each instrument which is discussed, the reader may XV expect to find, in this thesis, a brief answer to the following three questions: A) How does this instrument operate? B) Why doesithis instrument function the way i t does? C) In what basic ways does the operation of this instrument compare and contrast with the operation of the other eight in-struments which are discussed? The best example of how these questions are answered is provided by the case of the clarinet. In this case, the explanation of fingering and overblowing becomes equivalent to the explanation of the contrast between the clarinet and a l l the other woodwind instruments. It is impossible to answer any of the above questions without f i r s t examining the physical prop-erties which govern the behavior of sound. This is why Section I does not deal with instruments, but with sound waves. For the same reason, Sections II and III begin with a discussion of tube shapes and the behavior of sound waves under specific boundary con-ditions. Since this paper is directed primarily at mu-sicians (and not at physicists or mathematicians), the amount of physics and mathematics has been re-xvi duced to a minimum. Section I, together with the introductory portions of Sections II and III, rep-resents this minimum, containing about 90% of a l l the mathematics in the paper. Consequently, these sections do not reflect the nature of the content of the remainder of the thesis. The second purpose of this work was not o r i -ginally anticipated, but came about only during the course of preparation of Section IV. With a glance at the Table of Contents, the reader w i l l notice that "Testing the Equation v = f *\ " permeates the entire body of the work. For the woodwind instru-ments, this experiment was carried out purely to satisfy curiosity. With the brass instruments how-ever, the original plan was to omit the experiment entirely, because i t is not possible to carry i t out in the same manner as for the woodwind instru-ments. When the writer discovered an alternate method of carrying out the experiment, several sur-prises axvTaited him. These are discussed in each subsection separately and summarized in Section V. Section V, therefore, represents the second purpose of this work: the compiling of results of the ex-x v i i periments, followed by the drawing of conclusions, which lead to a claim and formation of theories. A few words are in order about the originality of this thesis. With regard to this, the writer can categorically assert that he was unable to find any written material which attempts to answer a l l three of the questions A), B), and C) posed above. In fact, he was unable to find anything whatsoever which discusses question C) to any greater depth than, for example, that of a comparison of the trumpet to the french horn. What was to be found was discovered a l -most exclusively in "The Acoustical Foundations of Music" by Dr. John Backus of the University of South-ern California, the source which is quoted in this the-sis more than any other. The other written sources which are readily available are treatises of one type or another. They present an exhaustive study of one instrument, and do not compare or contrast even two families of instruments, let alone two individual in-struments. This thesis may well be considered the op-posite of a treatise in that, contrary to discussing any one instrument in depth, i t discusses nine instru-ments generally. One consequence of this is that the x v i i i corresponding minor instruments (bass clarinet, pic-colo, english horn, and contrabassoon) are not dis-cussed at a l l because the remarks made about the ma-jor instruments are applicable to these instruments as well. The situation just described led to the follow-ing consequences: 1) Relatively few written sources (besides that of Dr. Backus) were of much use in the writing of the thesis. This was especially true in the cases of the Saxophone and the Tuba (see Bibliography.) 2) Interviews with (and demonstrations by) students and professors of the nine instruments discussed were of the greatest importance. This was especially true with regard to the testing of the equation v = (see Acknowledgements.) 3) The main source of reference for the formulas T<?hich constitute the mathematical and physical basis for the thesis was not a written publication, but the teaching of Dr. Roger Howard of the Department of Physics, University of British Columbia (see Ac^ -. knowledg ement s.) As a point of cl a r i f i c a t i o n , i t should be ex-xix plained here that the system of pitch and octave designation used throughout this paper is the USA standard octave notation: 8 . This system was also chosen because i t is the one used by Prof. John Backus in his book, "The Acous-t i c a l Foundations of Music", which served as an in-valuable reference in the writing of this thesis. The frequencies of a l l pitches from CO to B8 in-clusively are given in the Appendix on page 257. XX ACKNOWLEDGEMENTS I hereby wish to acknowledge the aid of and express thanks and appreciation to the following musicians for assisting me in the experiments con-ducted for this thesis and for the valuable c r i t -iques and suggestions x-?hich they offered — and which I folloxtfedi Clarinettists Prof. Henry Ohlman, Mrs. Debbie Smith; Flautists Prof. Paul Douglas, Miss Barbara Kallaur; Oboists Prof. John Chappell, Mr. Bob Pritchard; Bassoonists Prof. Cortland Hultberg, Mrs. Elodie Kutama; Saxophonists Prof. Dave Branter, Mr. Paul Cram; Trumpeters Prof. Martin Berinbaum, Mr. James L i t -tleford; French Homists Prof. Brian G'Froerer, Miss Diana Bos; Trombonists Prof. R. Sharman King; Mr. Brian Gilbey; Tubists Prof. Dennis Miller, Mr. Rick Govier. In addition, I x^ould l i k e to express my deep thanks and gratitude to my thesis adviser, Prof. Cortland Hultberg of the Department of Music, UBC, for his optimistic attitude, scrupulous proofreading, xxi constructive criticism, invaluable suggestions, and overall guidance in the preparation of this thesis. Finally, I am greatly indebted to Dr. Roger Howard of the Department of Physics, UBC, who f i r s t introduced me to the area of acoustics of woodx^ind and brass instruments, and from whom the ideas for this work originated. I am especially grateful to Dr. Howard for his patient and thorough teaching of the physics which constitutes a solid basis for this paper. Without his voluntary willingness to sacrifice much extra time and trouble, this thesis could not have been written. Joseph M, Krush, The University of British Columbia, Vancouver, B.C., Canada, Apri l , 1978. I. DIFFERENTIAL EQUATIONS OF WAVE-MOTION 1 The study of acoustics of instruments is closely a f f i l i a t e d with the study of sound i t s e l f . Conse-quently, a practical point of departure in this study is an elementary examination of waves in general and sound waves in particular. A wave is defined as a progressive disturbance propagated from point to point in a medium or in space (without progress or advance of the points themselves). 2 There are many kinds of waves, but a l l may be classified as being of one or the other of the two basic types: the pulse or the regular train of waves. In the pulse, a single wave which has been created by a single event travels throughout a medium. An example of a pulse is seen when a stone is dropped on the surface of a tank of water. A regular train of waves is one in which there is a regular creation of successive pulses spaced at regular intervals. An example of this is seen when a floating cork is pushed up and down regularly on the surface of a tank of water. The nature of a prop-agation by the pulse or wavetrain may be a displace-ment, as in the above examples, or i t may be a density  fluctuation, as in the case of waves passing through -2-gases such as a i r . Waves may be divided according to the nature and direction of motion of the elements of the medium transmitting the wave. If a wave is transmitted by oscillations to and fro along the d i -rection of propagation of the wave, i t is called a longitudinal wave. The wave transmitted by a stretched string when i t i s plucked makes a lateral displace-ment of the elements of the string and is called a transverse wave. If a metal bar is gripped firmly at one end and then twisted at the other, upon re-lease of the tension the rod w i l l execute small c i r -cular arcs about i t s axis, transmitting torsional  waves. Sound waves f a l l into the general category of wave trains (a term synonymous with "regular train of waves"), and are longitudinal waves whose nature of propagation is through density fluctuation of air molecules. This study w i l l deal almost exclu-sively with waves of this nature. The behavior of sound waves w i l l be examined under the following d i f -ferent conditions: 1) waves traveling through space, 2) waves traveling through a medium (in particular an a i r column), which may be either a) a cylindrical -3-tube (open or closed), or b) a conical tube ( f u l l or truncated.) A regular train of waves traveling to the right along the x-axis with a velocity v may be represented graphically as follows: y / / / / \ \ \ \ \ > / / x — * The solid line represents the original position of the wave, and the dotted line represents i t s po-sition a short time later. The shape of the wave i t s e l f undergoes no change whatsoever. The above is represented analytically by the expression y = f(vt - x), (1) where f is some function of x (position) and t (time), and y is the magnitude of the condition which is changing. Suppose at some point (x + x^) and some subsequent time (t + T), y has the same value as in (1) above, then y =» f[v(t + T) - (x + x x)] . (2) Thus y w i l l remain unchanged provided x l v - If ( 3 ) because (1) y = f ^  - ^  - f ^  + x x - x - = y. (2) We may regard the singularity of magnitude y at (x, t) as having been conveyed to (x + x^) at time (t + T ) , that i s , of having traversed the distance x^ in the time T * . Thus v in equation (3) represents the ve-locity of the wave. Similarly, y = f(vt + x) rep-resents a wave traveling to the l e f t along the x-axis with velocity v. The most important type of sound wave i s the simple harmonic wave in which the periodic change f o l -lows the symmetrical sine or cosine law. Here, y has a maximum positive value "a" 1) at regular intervals of t i f we f i x x, 2) at regular intervals of x i f we f i x t. Also, as we proceed along the sine or co-sine wave, for each positive maximum of y we encounter two zero values, one corresponding to y increasing and the other to y decreasing. The equation for a sine wave is of the form y - Asin2T^ - $j (4) where: 1) i f we put x = 0, y passes through zero for y increasing at regular intervals of time T, and 2) i f we put t == 0, y passes through zero for y increasing at regular intervals of distance 'X. The quantities T and X are termed respectively the period and wave-length of the wave motion. The wave-length is defined as the distance between successive crests or successive troughs, or between alternate extrema (maxima and minima) of the wave-motion. Further, i f we write equation (4) in the form y = A s i n ^ ( ^ | - x) (5) and compare the coefficient of t with that in equa-tion (1), we see that the velocity v of propagation of the wave is given by v = . (6) The reciprocal of the period *X is termed the fre-quency f of the wave-motion, and hence equation (6) may be written v = f ^ . (7) This equation, connecting wave-length, frequency, and velocity of propagation, is of fundamental im-portance in wave-motion, and w i l l be used throughout this paper. If we rewrite equation (5) in the form -6-2TT y = A s l n ~ ( v t - x) (8) and differentiate (8) twice with respect to x, we obtain: 3 3y 2T 2TT , ^ v G = _ Acos — (vt - x) — £ = — ^ — Asin-rr-(vt - x) (-r—). But A s i n ^ C v t - x) = y, /. 2 j = i l l ! . y . (9) * a* 2 Ti Now we differentiate (8) twice with respect to t to obtain: ay 2ir A 2 i r , . x ^ ~ ~T~ ~y^  ~ x^v a 2y -2TT A . 2TT , _ w2TT * 2 — i = — - — Asin-s—(vt - x)(-s~)v . 3t A A A But Asin2X(vt - x) = y, = ihlljy. (10) * a t 2 V Now from equations (9) and (10) we obtain: ^ = v 2 ^ £ . (11) a t z dx z Equations (9), (10), and (11) — especially (11) — are called differential equations of wave-motion, and when we come across an equation of the type of equation (11), we know that a wave-motion must be -7-involved. Equation (11) has the important feature that the coefficient of the derivative on the right-hand side represents the square of the velocity of the wave; this means that we do not need to solve the equation to obtain the velocity of propagation. The coefficient "A" in equations (4), (5), and (8) is called the amplitude of the wave. The amplitude A is defined to be the absolute value of the maximum displacement from a zero value during one period of an oscillation. The following is a diagram of a simple harmonic wave: If instead of equation (8) we take y = Asin~-(vt - x) + Bsln^|-(vt + x), (12) we again obtain equation (11), which may represent a combination of two waves traveling i n opposite directions along Ox with velocity v. If more than one train of waves is present, the total displacement of the medium is obtained by superimposing the displacements which would be caused by each wave train acting alone. The fi n a l result w i l l depend upon whether the vibrations of the wave trains are in the same or in different planes. If the planes are the same, the waves in -terfere with each other. If two wave trains of the same wave-length and amplitude, and vibrating in the same plane, are in phase, that i s , i f two wave-crests, two wave-troughs, or two corresponding points on the wave surface, arrive at a given place simultaneously, the displacement of the medium is twice that which would occur with one wave train acting alone. Conversely, i f the wave trains are completely out of phase, that i s , i f a wave-crest from one wave train arrives simultaneously with a wave-trough from another, the two displacements of the medium are equal and opposite, and the total ef-fect is zero displacement; this is due to destructive  interference. The waves discussed thus far have been prog-ressive waves, i . e. waves spreading outwards con-tinuously from some source so that small portions of the medium oscillate up and down (transverse waves) or to and fro (longitudinal waves) as the waves pass by. On the other hand, i f the medium -9-is limited, for example by a r i g i d barrier, reflect-ed waves are produced which interfere with the on-coming waves. The effect of the interference is remarkable in that the progressive nature of the waves vanishes. At certain regular distances from the reflecting barrier the medium is perpetually at rest; these points are called nodes. There are also other regularly spaced points, called antinodes. at which the amplitudes have maximum, though fluc-tuating, values. Since the portion of the medium immediately in contact with the ri g i d barrier must be at rest, the amplitudes produced by the incident and reflected waves at that point must be equal and opposite. Reflection of waves at a denser or more ri g i d medium therefore results in a change of phase of T with respect to the incident wave. The forma-tion of stationary or standing waves is shown in the figure on the following page. The incident wave, represented by the blue line, travels from l e f t to right. It meets the r i g i d barrier on the right side and gives rise to a reflected wave represented by the dotted line traveling from right to l e f t . The total displacement of the medium or the standing -10-wave which results from the interference of the i n -cident and reflected waves i s shown by the black l i n e . The points N^, N^, (the nodes) ax-e ob-served to be perpetually at rest, whereas points midway between these points (the antinodes) undergo the maximum displacement. It is important to note that the wave-length is twice the distance between consecutive nodes. On the other hand, i f a progressive wave pas-ses from a denser to a less dense medium, the dis -placement of the less dense medium under an i n c i --11-cS en t compression is greater than the displacement of the denser medium. At the boundary, therefore, the denser medium moves into the less dense medium for a short distance, and thus a rarefaction is created in the denser medium close to the boundary. This results in a reflected wave of rarefaction. Since the boundary is free to move, the boundary be-comes an antinode and the waves are reflected with-out change of phase. An example of this sort of wave propagation is from water to air , as is i l l u s -trated below: Water Air The analytical examination of stationary waves pro-duced by reflection without change of phase is as follows. The total displacement y of the medium at -12-any point x from some arbitrary origin 0 is the sum of the two displacements y^ and y2 produced by the incident and reflected waves respectively. Hence y =» y^ + y2 - Asin^~-(vt - x) + A s i n ^ (vt + x) = A[(sin^-vt)(cos^y-x) - ( c o s ^ v t ) ( s i n ^ ~ x ) ] + K T f ^ 2IT fcW 2TT w , 2T ^ w . 27T A[ (sin-y-vt) (cos-^-x) + (cos-sj-vt) (sin-^px) ] = 2A • s i n ~ - v t • cos^~-x . (13) From equation (13) we want to find when the dis-placement y is zero for a l l values of t. We find, 2TT therefore, a l l values of x for which cos-^—x = 0. Therefore ~ ^ j " ~ x must equal (2n - 1)(^) for any integer n. Therefore x = (2n - 1)^ for any integer n. Conclusion: For a l l values of t, at distances given by_ x = (2n - l)-£, where n is an integer, the dis- placement y i s zero. These points clearly represent the nodes of the stationary wave system and the dis-tance between successive nodes is half the wavelength, In the case of waves propagated by a density fluctuation of a i r , the source of sound consists of some mechanically vibrating system such as a vibrat-ing strip. As the strip moves up, a i r which is in -13-contact with the strip on i t s upper surface is com-pressed. This compression causes a local rise in pressure which compresses in turn the a i r immediate-ly above i t . The process continues in this way, with the result that a compression pulse spreads through the a i r . As the strip moves downwards, a slight vacuum or rarefaction i s created i n the wake of the moving strip. Consequently, a i r moves from neigh-bouring points to occupy this vacuum, and therefore a rarefaction pulse spreads through the medium f o l -lowing the preceding compression pulse. Succeeding vibrations of the strip create a series of compres-sions and rarefactions following each other i n reg-ular succession. Such a series constitutes a sound  wave. It is detected by the ear, which possesses a delicate membrane which is set vibrating by the successive compressions and rarefactions which con-stitute the sound wave. If the frequencies of the vibrations are above or below the human hearing threshold, they are undetectable by the ear. -14-II. THE CYLINDRICAL TUBE Boundary Conditions for the Cylindrical Tube 1 There is a total of four possible boundary conditions for the cylindrical tube: 1) tube open at end one, closed at end two, 2) tube open at end two, closed at end one, 3) tube closed at both ends, 4) tube open at both ends. From the musician's standpoint, case 1 equals case 2 and case 3 has no application, because no sound is emitted from a tube closed at both ends and with-out any side holes. Only cases 1 and 4, therefore, w i l l be considered here. The two tubes w i l l be ta-ken to be of equal length ( i . e. L = L'); the clar-inet w i l l be used as an example of case 1, and the flute as an example of case 4. We start, now, with the general equations for pressure p and velocity v: p = Asin(kx) + Bcos(kx) (1) v |S = kAcos(kx) - kBsin(kx) (2) Case 1: We knox<r that i f we close one end of a tube, the molecular velocity v at that end equals zero. From the following graph, we may conclude, that with v = 0, the pressure p at that end is at a maximum, just as when p = 0, v is at a maximum: ^velocity v pressure p position x To impose boundary conditions, we f i r s t put x = 0 in equation (2) above, thus giving v cC |J£ = kAcos(kx). Therefore v oC ^  = kA, therefore A = 0. Equation (1) now becomes: p = Bcos(kx) (3) Next, we put x = L, where L represents the other (open) end of the pipe. Equation (3) nox<? becomes: p = Bcos(kL). We know that at the open end of a tube, p = 0, .*. Bcos(kL) = 0 .'. kL = (2n - 1)^- for any integer n (2n - 1)| .*. k = ^ for any integer n. (4) This is the value of k which is fixed by the boundary conditions. On the other hand, we ex-pect a simple harmonic wave to be in the form of: p = Bcos^p-x. (5) Therefore by definition in terms of wavelength, Now, combining equations (4) and (6) we get: (2n - 1)-^ L 3 • Therefore L = (2n - 1)-^ for any integer n, or L = —^— for odd n. Now we may find a l l values of x for which p = 0 by finding a l l values of x for which 2TT T -sjjj—x = (2n ~ 1)"2 f° r a n y integer n. Therefore n l Ti x = —£- for odd integers n, or x = (2n - 1)-^ for any integer n. This means that p = 0 when x = ^ , 3-\ 5*) 7^ —jr-, —^—, 4~J etc... Consequently, the general equation involving*\ of the cylindrical tube of length L closed at one end (here the clarinet) i s , n^ as stated above, L = —^— for odd n. In the diagram on the next page, n is assigned the value of 7. Cylindrical tube closed at one end: t •pressure antinode at mouthpiece position x Since L = —7— must be satisfied, a - 4L. As A * " T T I Q V 2ir max stated above, p = Bcos-^—x. Consequently, p = 0 beginning at 1| every 180°. 3T second mode: x = - — 4 Case 4: (Using x f, p',^', k 1, and L' for x, p, "X, k, and L.) We know that at an open end of a tube, p' = 0. Consequently, to impose bound-ary conditions, we f i r s t put x* =0 in equation (1) above and get: p 1 = Bcos(k fx'), B = 0. Now equation (1) becomes: p 1 = Asin(k'x') . (7) We next put x' = L f, where L' represents the other open end of the pipe. Equation (7) now becomes: p f = Asin(k'L') .'. Asin(k fL') = 0 k'L' = mTV for any integer m, .'. k 1 = ^ Qy for any integer m. (8) This is the value of k* which is fixed by the bound ary conditions. On the other hand, we expect a sim pie harmonic wave to be in the form: p' = Bcos^r . (9) A Therefore by definition in terms of wavelength, k« = ^  . (10) Now, combining equations (8) and (10) we get: mT = 2]T_ L. y ,-• L' = — for any integer m. Now we may find a l l values of x f for which p' = 0 by finding a l l values of x' for which '~5JTX, = m ^ mlX ' for any integer m. Therefore x f = — ^ — for any integer m. This means that p* = 0 when x 1 =0, -V>v, 41' zV> 41'3^> Q > etc---The general equation involving^' of the cylindri-cal tube of length L* open at both ends (here the flute) is L 1 = — y ~ for some integer m. In the diagram belox<r, m is assigned the value of 7. Cylindrical tube open at both ends: I #pressure node at mouthpiece Since L* = ^  m ^ s t be satisfied, *V = 2L'. As . 2T max x' stated above, here p' = Asin ^ , . Consequently, p 1 = 0 beginning at 0 every 180°. f i r s t mode: x* = second mode: x f = ^ ' -20-Comparing the flute to the clarinet, we see that the fundamental frequency of the clarinet is half that of the flute, although the two in-struments are of nearly equal length: Clarinet Flute Length: 670 mm 618.5 mm Lowest pos- E3 (sounding D3, a C4 (a minor sev-sible pitch: major second lower) enth above D3) -21-Posslble Frequencies  The Clarinet Since the cylindrical tube closed at one end vibrates with modes that are odd multiples of the fundamental, the second mode of oscillation of the clarinet has a frequency three times that of the fundamental: y 31 2 p = 0 f i r s t at x = 7", next at x = 4 4 Musically, this results in a rise in pitch of an octave plus a perfect f i f t h , i . e. a perfect twelfth. This is why a clarinet overblows the twelfth, and not the octave; consequently, i t becomes necessary to f i l l in the gap between the seventh and the twelfth with additional holes and keys. By "addi-tional holes" is meant holes other than the orig-inal six tone holes which suffice to produce the f i r s t seven notes of the scale. The gap in the scale is f i l l e d in the follow-ing way: the lowest note of the Bb clarinet is E3, sounding D3, a whole tone lower. The lowest note of the basic scale is G3, a minor third above E3. By providing two additional holes and keys, the -22-two intervening semitones, F3 and F#3, are sup-plied. The basic scale is now ascended from G3 by opening the six holes in turn (with additional keys provided for sharps and f l a t s ) , until F4 is reached. This pitch range is known as the low (chalumeau) register: To " f i l l i n " the notes from F#4 to Bb4, addition-al holes, provided closer to the mouthpiece, must be opened. These additional holes (and keys) make clarinet fingerings somewhat more complicated than those of other woodxd-nds. This "linking" type of register on the clarinet is known as the throat register: Between Bb4 and B4 is found what is known as the f i r s t "break", i . e. the point at which the f i r s t (and most important) change in register oc-curs. To produce B4, a l l the holes must again be closed (so that the fingering is the same as for E3 plus the thumb key), and the instrument is over--23-blown to produce the next resonance mode a twelfth above the original. (This is accomplished by o-pening the vent-hole, located on the side of the clarinet about 15 cm down from the mouthpiece tip.) With the vent-hole open, the clarinet operates in the second mode, producing the clarion register: 3 The corresponding fingerings between the two reg-isters are as follows: A A A -e-The next "break" occurs between C6 and C#6; from C#6 up is found the extreme register of the clar-inet: \ 3 l We recall now, that p = 0 at x = |, x = — ^ j and -24-5\ 31 next at x = ""^  • Now just as resulted in a frequency three times that of the fundamental, similarly ~ p results in a frequency five times that of the fundamental, i . e. two octaves plus a major third above i t . But this may also be re-5 3l 31 garded as 3'("~^ —)» where —^— is the previous mode. It is in this way — by overblowing the corres-ponding pitches a major sixth below, that most of the pitches of the extreme register are produced. Thus, the following correspondences result: 4 . o o o o o • « • e • • • • • « o • • • o o o • o _o ... o o o o ** o A. o o -e-o o t I 1 I 1 t 3 C T • T • T» • • • • • • • • • • o • -- • • o o o • o o o o o o o o o o o It is important to realize that these p i t -ches, when produced by overblox^ing, are not a l -ways well in tune in equal temperament, because the ratio 5/3, when used to produce a pitch a ma-jor sixth higher by overblowing, produces the pitch in the just scale — not in the equal-tempered scale. For this reason, clarinettists do not usu-al l y use the fingering system shown above; instead, pitches which are in tune in the equal-tempered scale are produced by use of additional keys and holes. The fingerings above are so designated for purposes of examination of the clarinet from the acoustical standpoint. For the six highest pitches which may be produced on the clarinet, G6 to C7, the fingering (and thus the manner of producing the pitch) is not as consistent as i t is in the chalu-meau and clarion registers. Some modern writers give as many as 40 fingerings for these pitches; G6 alone has 10 or 12. For purposes of study from the acoustical standpoint, we examine here the f i n -gerings which come about through the method of derivation used hitherto. The next note, G6, can s t i l l be fingered in the same way as the previous six pitches, by overblowing the major sixth below. Just as this frequency is the result of i . e. five times the fundamental frequency, simi-2X r e s u l t s l n a f r e q u e n c y . e v e n t s . t h a t of the fundamental, i . e. theoretically two octaves and a minor seventh (= a twenty-first) above i t . We know, however, that in the harmonic series the seventh harmonic is always fl a t t e r than the cor-responding pitch in equal temperament. In this high a register, this is especially so, to the ex-tent that the twenty-first i s so f l a t that i t equals the equal-tempered twentieth. Thus, the Ab6 (= the fla t A^6) and the A6 (= the f l a t Bb6) and the Bb6 (= the fl a t B^6) may be produced by overblowing the tx«renty-first (= 2 octaves plus an extremely f l a t minor seventh): 5 7*) Finally, just as —-g- results in a frequency seven times the fundamental, similarly results in a frequency 9 times the fundamental, i . e. three octaves and a major second (= a twenty-third) above i t . It is by overblowing the twenty-third ( i . e. the twelfth twice over), that the two highest p i t -ches of the clarinet, B6 and C7, may be produced:6 -CL. It must be stressed here again, that the pitches produced by overblowing in this high a range are not always in tune in the equal-tempered scale. Often, clarinettists use alternate fingerings which result in better tuned pitches. The analogies which are made here are made in order to make an examina-tion of harmonics and overblowing from the acoustic-al standpoint. Testing the Equation v - fA 7 It is interesting to test the equation v = f\ in a practical situation involving the clarinet. We know that the velocity v of sound is approxim-ately 33145 cm/sec at 25° C , and that i t increases by 59 cm/sec for each degree C above 25° C.8 There-fore at body temperature, 37° C, the velocity v of sound is approximately 33145 + 59(12) = 33853 cm/sec. We know also that in the case of the clarinet of length L, L = must be satisfied, therefore L = 4L (see page 17). The total length of the max clarinet is 66,8 cm, therefore for the 5 max lowest pitch of the clarinet = 66.8(4) = 267.2 cm. Thus, from the equation v = f*\ , we get: 33853 cm/sec = f(267.2 cm), f = 126.7 cycles per second = 126.7 hertz (hz). The closest frequency to this in the equal-tempered scale is that of B2, whose frequency is 123.47 hz. In reality, the lowest pitch of the clarinet is the written E3, which sounds as D3, a minor third above B2. The explanation for this dis-crepancy is that not only is the end correction in-volved, but so is the entire length of the clarinet. Now the bore of the clarinet is not cylindrical throughout i t s whole length; the cylindrical section begins about 4 cm from the tip of the mouthpiece and extends about halfway down the bottom joint. From that point on, i t flares outward, assuming a conical shape in the b e l l . Thus i t is understand-able that pitches whose production involves the lower half of the bottom joint and the bell w i l l deviate from those which are derived by calcula-tions such as the above, because those calculations presuppose a perfectly cylindrical tube. For fur-ther comparison, three more similar calculations were made. In each case, the length L was mea-sured from the tip of the mouthpiece to the centre of the f i r s t open hole. The results are summarized in the table on the follox<ring page. We observe from the table, that the higher the pitch, the smaller the discrepancy. This is be-cause the clarinet i s , in fact, cylindrical in the top joint, which means that the practical case comes closer and closer to the ideal case as higher and higher pitches are examined. Finally, with a l l holes open at Bb4 (sounding Ab4), the two are very nearly identical. l a Examined Pitch: Written: E3 Sounding: D3 lb Known frequency of this pitch (hz): 146.83 Length of pipe (L) (cm): 66.8 ' A (where * A = 4L) (cm): m a x 267.2 Calculated frequency (33853 cm/sec -f- 4L) (hz): 126.7 2a Pitch whose frequency in equal temperament is nearest to that obtained above: B2 Known frequency of this pitch (hz): 123.47 Discrepancy of 2a with respect to l a (by in-terval): - 3rd lower Ratio of calculated frequency with res-pect to lb: .863 G3 F4 Bb4 F3 Eb4 Ab4 174.61 311.13 415.3 41.5 23.8 20.4 166.0 95.2 81.6 203.9 355.6 414.9 Ab3 F4 Ab4 207.65 349.23 415.3 - 3 r d + 2nd none hghr. hghr. 1.17 1.14 .999 -31-The Vent-Hole and the Resonance Curves of the c l a r i n g t - g As was mentioned earlier (see page 23), the vent-hole is located about 15 cm below the mouth-piece, and is operated by the thumb key. This is the key which enables overblowing by the twelfth by essentially destroying the lowest resonance. How this happens may be explained by examining the resonance curve for the lowest note of the clar-inet, E3: Relative excitation frequency Figure 1 When the vent-hole is opened, the lowest reson-ance moves from i t s position as shown by the solid line to that shown by the dotted line. The remaining higher resonances are essentially -32-unchanged. Let us now examine the resonance curve for a plain metal tube closed at one end: The resonance frequencies are quite regularly odd multiples of the fundamental. If a clarinet mouth-the resonances are again measured, the resulting curve is essentially identical to that of Figure 2 above, except for slight (relatively insignificant) changes in the positions of the high resonances (above the eleventh harmonic). These changes are due to the internal shape of the clarinet mouthpiece. If the reed is blown in the usual way to pro-duce a tone in the metal tube, an internal standing wave is obtained which contains mostly odd harmon-— S 1 \ 1—I—i \ \ \ i I i 1 ' l 11 f i l l 148 hz 2 3 4 5 6 7 8 9 11 13 17 21 Relative excitation frequency Figure 2 piece with reed is placed on this metal tube and ics, since their frequencies coincide with the resonance frequencies. The even harmonics have small amplitudes, since their frequencies l i e in between the resonance frequencies. The following figure shows the harmonic structure of this stand-ing wave. L O T T 0) • O O X ! x: o «H 0.5 DOpH •i-i u ex <u « e IS EC < 0- 1*1 5 7 9 11 Harmonic Number Figure 3 13 15 The amplitude of each harmonic has been multiplied by i t s number to give the weighted harmonic amp-litude shown above. For example, the weighted third harmonic amplitude is three times the mea-sured harmonic amplitude, etc... Since i t can be shown that the higher the frequency of the harmonic, the better i t radiates from the instrument, this weighting gives a more r e a l i s t i c representation of how the ear w i l l appreciate the actual harmonic structure. It also makes the harmonics easier to -34-see on the graph. If the same resonance measurements are made on the clarinet, i t is found that they are con-siderably distorted as compared with those in the cylindrical metal tube, as we see in Figure 1 above for the note E3 on the clarinet. The resonance fre-quencies depart more and more from the harmonic fre-quencies as we go to higher values; for the partic-ular note E3, the eighth harmonic, for instance, coincides more nearly with a resonance than either the seventh or ninth. If the clarinet i s blown and the harmonic structure of .the internal standing wave is analyzed, we find that the eighth harmonic is more prominent in this tone than the seventh or ninth, as is shown in the folloxtfing figure: I.OT T T I T T T T T 1 3 5 7 9 Harmonic Number Figure 4 11 As the scale is ascended on the clarinet, the res--35-onances become fewer in number; for the note Bb4 just before the break to the clarion register, there are only two good resonances, as shown in Figure 5 below: U V w « ft p-i o a x. 4-1 432 hz T r-Relative Excitation Frequency Figure 5 Consequently, the tone has very few strong harmonics, as is shown in the figure below, which shows the weighted harmonic structure of the internal standing wave in a clarinet sounding Bb4. 1.0' o u C u ( X o -w M E H •H ft D, <D cti E & SC <J 0 Figure 1 T T T T 1 3 5 Harmonic Number The lack of resonances in this region accounts for the poor quality of the throat register of the clarinet. In the extreme register, above the break, there is l i t t l e correspondence between the harmonic fre-quencies and the resonances, as may be seen in Fig-ure 7 below: Relative Frequency Figure 7 The third harmonic is the only one near a reson-ance, and i t is not very close. Consequently, the tone produced is mostly fundamental, but has both  even and odd harmonics, as shown in the following figure: -37-x: o 1-1 D O S H •H J-i D-. o) co e S K < 1.0 n 0.5 0 T T T T 1 2 3 4 Harmonic Number Figure 8 Figure 7 shows the resonance curve for the note D5 on the clarinet; figure 8 shows the weighted har-monic structure of the internal standing wave in a clarinet sounding this note. We see from these two figures why the quality of the clarinet tone is different in the higher register. The Flute Since the cylindrical tube open at both ends vibrates with modes that are integral multiples of the fundamental, the second mode of oscillation of the flute has a frequency twice that of the fun-damental : p* = 0 f i r s t at x' = 0, next at x' = ~~2~"> n e x t a t x ' = > . 1 0 Musically, this results in a rise in pitch of an octave, which is the basic interval by which the flute overblows. This D r o p e r t y , plus the fact that the flute has a pressure node (and not an antinode) at the mouthpiece, makes the construction and f i n -gerings of a flute much less complicated than are those of the clarinet. There are no "gaps" in the scale which need to be f i l l e d in due to overblow-ing by a txvrelfth; consequently, the flute has con-siderably less "additional" holes and keys. The basic scale of the flute is the six-hole, seven-note scale described earlier; i t starts at D4 with the six holes closed. Additional holes and keys are used to provide the sharps and flats and to extend the range down to C4. nC4 is produced with a l l holes closed; with a l l holes open, C#5 is produced. The normal range of the concert flute is from C4 to C7: The pitches from C4 to Eb5 have their own individ-ual fingerings: 1 2 The pitches which are normally produced by over-blowing the octave are the following: hx ho. zSti % (C> % Z) H & C5, C#5, D5, Eb5, plus a l l the pitches of the flute range higher than those shown above may be produced by overblowing the octave, but they not always are produced in this way, because there are other ways of producing them which result in better tuning. We recall now that p' = 0 at x' = 0, x' = 2~' x . = ( 2 ) ^ , x' = ( 3 ) ^ , *' - W - T - > x ' " (5>T~> x 1 = ( 6 ) — e t c . . . Thus the third mode of o s c i l --40-lation of the flute has a frequency of three times the fundamental, i . e. a tx<relfth above i t . But this may also be regarded as ^-C^ 1), where ^ is the previous mode. It is thus possible to pro-duce pitches on the flute by overblowing at the f i f t h , although this is rarely done. We continue now, in the same fashion, for three more modes of osci l l a t i o n : 1) x' = 2 V = frequency 4 times the fundamental 4 3V 3 l ' = (^)"~jf~ (where —^— is the previous mode^ This is the equivalent of overblowing by a perfect fourth (in the just scale). 2) x 1 = -—— = frequency 5 times the fundamental = r(2\ ') (where 2> is the previous mode). This is the equivalent of overblowing by a major third (in the just scale). 61' 3) x* = —7j— = frequency 6 times the fundamental 6 5 l f 5 l ' = (^~2~ (where is the previous mode). This is the equivalent of overblowing by a minor third (in the just scale). We may now consider the remainder of the p i t -ches above C#6. The next pitch, D6, may be pro--41-duced by overblowing D5, but i t is more in tune when produced by overblowing G5 as the third partial: _0_ 3 (The numbers indicate relative frequencies of the pitches.) The next pitch, Eb6, is produced by overblowing Eb4 at the double octave, as a fourth partial: Beginning with the next pitch, E6, we are back to octave doublings of the f i r s t series of pitches shown. This method of pitch production remains consistent right up to the top of the range of the flute, except for the pitches A6 and C7: Missing: A6 C7 As was the case with Eb6, each of these tones is -42-produced by overblowing at the double octave, i . e. each is the fourth partial of the note in brackets below i t . A6 is best produced as the f i f t h partial of F4: C7 is — surprisingly — best produced as the sixth partial of F4: There are several possible explanations for why these two pitches are best produced as shown above. F i r s t l y , the opposite end of the flute may influence a pitch in this high a range. Although the f u l l length of the flute is considered to be from the centre of the mouthhole to the remote tip of the flute, there is s t i l l molecular activity in the opposite (short) end on the other side of the mouthhole. Besides this, the total wavelength of -9~ 5 -r- 4 -43-the flute actually extends beyond the actual end of the tube, as shown in the following diagram. 1 3 adjusting screw mouth-hole cork / head joint tuning joint —'— / •—=-, L~ — 1 I I I molecular activity (distance =1.7 cm) Cross-sectional view of the essential parts of a flute (not drawn to scale). The distance marked "c" is known as the end cor-rection, and the effect of this end correction is the second possible explanation for producing A6 and C7 as .shown above. Testing the Equation v = f\ lk As was done with the clarinet, we now test the equation v = f ^ in a practical situation involving the flute. With a flute of length L, L = —t£- must be satisfied, therefore ~\ -It (see page 19). ' max v v & • The total length of the flute (from the mouth-hole/ to the open end) is 60 cm, therefore 'A = \ for r ' max the lowest pitch of the flute = 120 cm. With -44-v = 33853 cm/sec, substituting into v = f ^ , we get: 33853 cm/sec = f(120 cm) f = 282.1 hz. The closest frequency to this in the equal-tem-pered scale is that of C#4, whose frequency is 2 77.18 hz. In reality, the lowest pitch of the flute is C4, a semitone lower. For further com-parison, three more similar calculations were made. In each case, the length L was measured from the centre of the mouth-hole to the centre of the f i r s t open hole. The results are summarized in the table on the following page. The explanation for the discrepancies in the table is that the flute actually differs in a num-ber of ways from a simple tube open at both ends. For a l l but the lowest note on the instrument, one end of the vibrating a i r column is actually an ooen side hole on the tube. The key pad for this hole is close enough to have an effect on the a i r col-umn vibrations, and the remainder of the tube be-yond the open hole also has an influence. The ac-tual arrangement i s , therefore, more complicated acoustically than a simple open end. Also, the other end of the air column is the mouth-hole, l a Examined Pitch: C4 lb Known frequency of this pitch (hz): 261.63 Length of pipe (L) (cm): 60.0 "X (where ^ = 2L) (cm): m a x 120.0 Calculated frequency (33853 cm/sec 2L) (hz): 281.1 2a Pitch whose frequency in equal temperament is nearest to that obtained above: C#4 Known frequency of this pitch (hz): 277.18 Discrepancy of 2a with respect to l a (by in-terval): - 2nd hghr. Ratio of calculated frequency with res-pect to lb: 1.07 D4 293.66 52.0 104.0 325.5 E4 329.63 + 2nd hghr. 1.11 F4 349.23 43.0 86.0 393.6 G4 392.00 + 2nd hghr. 1.13 C#5 554.37 23.6 47.2 712.2 F5 698.46 + 3rd hghr. 1.29 -46-which is more or less covered by the l i p . In the portion of the flute between these two ends are a number of side holes covered with key pads; these add an additional volume in this region as compared to the plain tube. Furthermore, the head joint of the flute is not cylindrical at a l l , but i s , i n fact, conical, narrowing down to a diameter of 1.7 cm from the 1.9 cm diameter of the remaind-er of the flute. Thus, although a pipe 60 cm in length open at both ends should sound C#4 as its lowest pitch, in reality, C4 i s the lowest pitch of the f l u t e . 1 5 As is evident from the:table, the the higher the pitch, the greater the discrepancy between the calculated frequency and pitch with respect to the examined frequency and pitch. This exemplifies x^ hat is known as the "shifting of over-tones". Although ideally, harmonics should not be out of tune, in the equal-tempered scale, such an occurance as the follox<ring is not infrequent: Relative frequen-cies: 1 2 3 4 5 6 7 8 Ideal: | f I | [ | Actual! I I I I I I I I -47-In the ideal case, overblowing follows the log-arithmic scale precisely; in the actual case, the higher the harmonic, the fla t t e r i t becomes in comparison to the corresponding ideal pitch. Thus, by the time the highest register of the flute is reached, overblowing may not necessarily produce the pitch i t is "supposed to" in tune, and i t may well be possible to produce the desired pitch in tune through a more distantly related method of overblowing. This brings us back to the pitches A6 and C7 discussed previously. These two pitches serve as i l l u s t r a t i v e examples of the situation just described. -48-III. THE CONICAL TUBE 1 Two cases of the conical tube w i l l be con-sidered: 1) f u l l tube with peak, 2) the same tube with i t s peak cut off. Case 1: For large r and L, the lengths of r and L are The diagram above is not a cone according to the precise definition of the word, because the open end is not f l a t , but spherical. This is be-cause waves in a cone travel radially and not in parallel lines at right angles to the base of the cone: P D' taken to be equal. -49-Consequently, the distance PL is precisely equal to the length of the axis PO and to the distance r = PA. Clearly, i f the situation were as shown in the second of the two diagrams, the distance PL would be slightly larger than PA, although not by much. Another consequence of having this type of "base" for the cone is that this "base" is not an ordinary cir c l e with diameter DOD', because there is curvature involved, although i t is very slight, especially for small DOD1. For comparison, the cone of case 2 is drawn with a f l a t base. The three-dimensional co-ordinate system, called the spherical polar system, used here operates thus: A point "A" in the cone is defined by specifying r, ©, and f, where r = PA, © is the angle formed between the axis PO and PA, and f is the angle formed between OA and DOD1 (where DOD' is the " d i -ameter" of " c i r c l e " 0). The wave equation in this case is a generali-sation to three dimensions; thus i t involves three variables (x, y, z) with x = rsinBcos^f, y = rsin and z = rcos©. Now the general wave equation i s : In cartesian co-ordinates, that i s , in terms of x, y, and z, 2 2 2 V 2 d p , d p . d p , n s P = —2 —2 —2 * ( 2 ) dx" dy dz In this equation, a set of three numbers, x, y, and z, defines a point which can also be expressed in terms of r, © , and f* . In this case, when we transform the expression in x, y, and z to that in r, © , p , we find that each of the last two terms becomes equal to zero, because there is no varia-tion involved with either y or z ( i . e. with either © o r / 3 ) , since & represents radial waves in cross section and f represents circular waves. Noxtf from equation (2) above, through the transformation from cartesian co-ordinates to polar co-ordinates, we get: V 2p = ^ ( r 2 ^ [ f ) + \ a t e r m in 6 + a term in r ^a term in 0 + a term in 1 J2 2 , 2 u ; c dt The bracketed terms in 6 and f are omitted, because variations with 6 and f w i l l be associated with -51-small distances (of the order of the diameter), and hence with high frequencies. Consequently, they w i l l not be taken into consideration. The general solution for equation (3) in this case i s : A" sin kr .B cos kr , . . . x P = ^ + ^ (at any instant). (4) B cos kr The term ^ is omitted, because i t goes to A. s in ICTT i n f i n i t y at zero; therefore only ^ remains. This term remains f i n i t e , because for small x, sin x = x. The graph of this equation is the third graph on the following page. The cosine and sine graphs are given f i r s t for purposes of comparison. The following observations may now be made: 1) The graph of equation (4) is a type of combination of the cosine and sine graphs. It re-sembles the cosine graph in that i t reaches i t s maximum amplitude of 1 at 0. This means, that in practical application with an instrument, there xd.ll be a pressure antinode at the mouthpiece, as was the case x^ith the clarinet. It resembles the sine graph much more than i t does the cosine graph, because i t crosses zero on the x-axis at nT for any integer n (except for n = 0). Furthermore, -53-this curve ascends and descends comparatively with the sine curve. A l l this means that in practical application with an instrument, the modes of os-c i l l a t i o n w i l l be (both even and odd) integral mul-tiples of the fundamental. Consequently, overblow-ing w i l l occur at the octave (as in the case of the flute), and not at the twelfth (as in the case of the clarinet). 2 ) The amplitude a of the graph of equation (4) decreases as x increases: . r j _ _ 2A a z 2 ' a 3 T a t H a , 2A a t 2 , a a t I X a = 2A at 2 , a n . 9T _ 2A at 2 9 a ~*" gj 5 etc... This is because the flow of energy through a con-i c a l tube follows the inverse square law: E'<£ a 2 - ( A ) 2 , (5) ' » E<£•-—2 • r Conclusion: The closed conical tube has the same resonant frequencies as the cylindrical pipe open at both ends. -54-Case 2: f L 1 For large r and L, the lengths of r and L are taken to be equal. For purposes of calculation, P remains in the same position as in the f u l l conical tube. A cone with i t s peak cut off hehaves almost exact-ly as i f i t were a f u l l cone, because the distance designated "d" in the diagram is a very small fraction of the entire wavelength of the tube. Furthermore, the volume of the truncated portion of the cone is an even smaller fraction of the vol ume of the entire cone. In the case which we are considering here, we have a closed end at r — d: -55-Getting back now to equation (4) (as on page 51): - A sin kr B cos kr ° kr kr w For convenience in differentiation, we may change this to the equivalent expression: = C sin (kr + £)  p kr v ' dp C sin (kr + S) k C cos (kr + fc) , r \ ar k r 2 ^ * Nox^  at the closed end x^ hen r = d we require that |j| = 0, i . e. that the right-hand side of (6) = 0. /. - sin (kd + £) + kd cos (kd + £ ) =0. This condition does not have a simply expressed general solution, but i f kd and £ are both small ( i . e. i f their squares are much smaller than one), then we can make an approximation as follows: For small x, ( i . e. as x approaches zero), sin x ap-proaches x and cos x approaches 1. Thus for small kd and € the condition becomes: - (kd. + £) + kd • 1 = 0 c n C sin kr . . t = 0, p = . Thus the pressure equation is identical to that for the f u l l cone (see page 5l). The previous a--56-nalysis, therefore, can be used. The above i s , in essence, a mathematical consequence of the flatness of the curve of the third graph on page 52 when i t is near zero. Boundary Conditions for the Conical Tube As was the case with the cylindrical tube, there are theoretically four possible boundary conditions for the conical tube: 1) tube open at end one, closed at end two, 2) tube open at end two, closed at end one, 3) tube closed at both ends, 4) tube open at both ends. This time, from the musician's point of view, only case 1 is of interest, since i t represents the bore of the oboe (and the english horn) and the bassoon. Case 2 equals case 1; case 3 has no application, because no sound is emitted from a tube closed at both ends and without any side holes. Case 4 has no musical application either, because there is nothing to be gained from constructing an instru-ment with a conical tube open at both ends, since the cylindrical tube open at both ends has the same properties as the closed conical tube, as was shown in the previous section. Although i t was also shown in the previous section that the pressure expression for the truncated cone is identical to that for the f u l l cone, i t should be specified here that the bores of the oboe and bassoon are examples of the truncated cone. We start, now, with the general equations for pressure p and velocity v: A sin kr , B cos kr N p = — k ? — + — k F ~ ( 1 ) or A sin kr kA cos kr kB sin kr B cos kr , 0 v 2 ~ 2 ' ' kr kr kr kr Now for a tube closed at one end, we require, as before, that v = ^ = 0. Thus, after putting e-quation (2) equal to zero, we may multiply through 2 d by -kr to get: v = = A sin kr - kr A cos kr + + kr B sin kr + B cos kr = 0. (3) To impose boundary conditions, we f i r s t put r = 0 in equation (3) above, thus ending up with: v = l|2 = B cos kr = 0, .'• B = 0. Now equation (1) becomes: kr A s in kr /, \ P = w (4) -58-We next put r = L, where L represents the other (open) end of the pipe. Equation (4) now becomes: A sin kL P = kL * We know that at the open end of the tube, p = 0, A sin kL kL = 0, kL = m"TT , /, k = for any integer m, excluding m = 0, (5) because this would imply division by zero above. This is the value of k which is fixed by the bound-ary conditions. On the other hand, we expect a simple harmonic wave to be in the form: p = Bcos-^p-x. (6) Therefore by definition in terms of wavelength, Now, combining equations (5) and (7) we obtain: ml __ 2T L = ~ y ~ for any non-zero integer m. Now we may find a l l values of x for which p = 0 2 T TT by finding a l l values of x for which -~-x = mil for any non-zero integer m. Therefore x = for any integer m, m ^ 0. This means that p = 0 w h e n x - 'A , 2 "A , 3 *X , 4 "A , etc... The general equation involving'A of the conical tube of length L closed at one end (the oboe or the bassoon) is L = for some non-zero integer m. In the diagram below, m is assigned the value of 4. a, CD CO <D Conical tube closed at one end: .pressure node at mouthpiece position x Since L = ^Q- must be satisfied,*^ = 2L. As max stated above, here p = A sin 2TTr/'> 2Tr7\ Consequently, p = 0 beginning at IT every 180°. f i r s t mode: second mode: x ="Af -60-Possible Frequencies  The Oboe The oboe is basically a tube with a conical air column in x^hich the tip of the cone has been cut off and a double reed attached. The double reed consists of two halves of cane beating a-gainst each other. The oboe is constructed of three pieces of wood: the top joint, the bottom joint, and the b e l l . A l l three sections carry keys; this is in direct contrast to the clarinet, whose bell carries no keys. The oboe reed is attached to a conical piece of metal tubing called a sta-ple, which is inserted into the top joint. 2 Since the conical tube closed at one end v i -brates with modes that are integral multiples of the fundamental, the second mode of oscillation of the oboe has a frequency twice the fundamental: A *\ 3 o = 0 f i r s t at x = — • ; next at x = A . Musically, this results in a rise in pitch of an octave, which is the basic interval by which the oboe overblows. This property, plus the fact that the oboe has a pressure node (and not an antinode) at the mouthpiece, makes the construction and f i n --61-gerihgs of the oboe relatively simple, i . e. sim-i l a r to that of the flute and in contrast to that of the clarinet. The oboe has 10 to 16 "additional" holes and keys, which is considerably less than the clarinet, but is more than the flute. The basic scale of the oboe is the six-hole, seven-note scale, which starts on D4 with the six holes closed. Additional holes and keys on the lower joint and bell are used to provide the sharps and flats and to extend the range down to Bb3. The normal range of the oboe, then, is the following: The pitches from Bb3 to C5 have their own in -dividual fingerings: 1* The pitches Bb4, B4, and C5 may be produced by oc-tave overblowing, but this is not done, because i t is very d i f f i c u l t and because the tone quality suf-fers greatly. The Bb5 turns out sharp when pro-except^ional -62-duced in this way, but the B5 and C5 are f a i r l y well in tune. Between C5 and C#5 is where the f i r s t "break", i . e. the f i r s t change in register, occurs. The pitches which are normally produced by overblowing the octave are the following: half-hole thumb octave key side octave key conventional possible In each case, the fingering is identical to that used to produce the pitch an octave lox^er, except that one. of the three octave keys is added, as in-dicated above. x = ^2~3 x = 2^, etc... This means that the third mode of oscillation of the oboe has a frequency of three times the fundamental, i . e. a twelfth above i t . Of the pitches shown above, from F5 upwards, the pitches shown may be obtained by overblowing the twelfth, but this is not often done. If they -63-are produced in this way, these tones take on a peculiar character of their own. These special harmonics are a useful addition to the oboist's resources, particularly in solo work, but they require care in production and cannot be played forte. When used with discretion, their very in-dividual tone colour can be most effective. It should be added here that the oboe is undoubtedly at i t s best in this (medium) compass, i . e. approx-imately from F4 to B5 (varying with different mod-els), since in this range i t s tone is at i t s sweet-est, neither too reedy as i t sometimes tends to be lower dox<m, nor too thin, as i t may be above B5. Between C6 and C#6 is the point at t^hich the next break of the oboe occurs. From the practical standpoint, the fingerings for the seven highest pitches conventionally played on the oboe are quite individual: _, _ _ _ e T » T© T« T» o o (©) • • Ab • Ab o Ab o • • • B »Eb o Eb oEb • • o o o • • o • • • • o o o (•) • • • • o j | cj£> 4 * =£= * £ (y) M -64-From the acoustical standpoint, however, we find that i t is possible to produce these tones e i -ther as twelfths ( i . e. third harmonics) or as su-per-octaves ( i . e. fourth harmonics.) In the case of the super-octaves, this means that the oboe is in i t s fourth mode of oscillation, in x^hich the fre-quency is four times the fundamental (2*) as com-pared to —-^ 2". ) The tx^ o notes D6 and Eb6 require ad-ditional discussion. To finger D6, the oboist may, at w i l l , choose to close the holes indicated in brackets. To finger Eb6, these holes must be closed, and the C key must be replaced by the B key. Since these txvo notes may thus be fingered so similarly, i t is possible to think of Eb6 as being produced (from the acoustical and not from the practical standpoint) as an overbloxv'n D4 at the double oc-tave which is tuned to Eb6 by use of the B key. We must remember, that in this high a range, overblow-ing rarely results in pitches which are in tune in the equal-tempered system. That is why the finger-ings shown above are the ones that are, in fact, used — they result in better tunings than those achieved by overbloxvdng. When the player has -65-mastered both the conventional and the non-conven-tional fingerings, he w i l l soon notice minute d i f -ferences between them both in quality and in intona-tion. He may even discover additional or modified fingerings which suit his own instrument particularly well. The oboist should try to take advantage of such alternatives, as much as his a b i l i t i e s and mu-sicianship allow him to, because this is a l l part of the technique of oboe-playing. Such things trans-form a competent performance into an a r t i s t i c one and greatly increase the pleasure of both the player and his audience. 5 Testing the Equation v = fA 6 We nox>7 want to test the equation v = f\ in a practical situation involving the oboe. With an oboe of length L, L = must be satisfied, there-fore *\ = 2L (see page 59). We must f i r s t , how-max r to ' ' ever, calculate the theoretical length of the con-ic a l tube of the oboe by extrapolating the conical bore to the imaginary peak; -66-Now the measured dimensions (diameters and lengths) of the parts of the oboe are as follows: f / T 1 1.8 ~ mm 4^0 " 1 mm y.i mm 15 mm 25 + 27 52 mm 252 mm 257 mm 126 mm Reed + Staple Top Joint Bottom Joint , Bell 31 mm Since the Reed + Staple combination and the bell are the places at which the oboe tube deviates from its basically conical shape, we do not consider these in the extrapolation, and make use of only the top and bottom joints: r—= = — d^ = 4.0 mm 3 1 (9.1 mm) L = 512 mm r1 + L J d^ — 15 mm With d1 = K(r 1), let d x = 4 and d 2 - K(r 1 + L) = 15. Then KL = d 2 - d^^ = 11, .'. K = d2 " d l = = .0215, -67-d 15 Now r^ + L = __2 = 0215 = 698, . • = 186 mm. K To verify the value obtained for K, we can take the joints separately, as follows: Top Joint: Bottom Joint: let d x = K(r^) = 4 and let d1 = KCr^) =9.1 and d 2 = K(r x + 252) = 9.1 d2 = K ( r l + 2 5 7 ) = 1 5 Then K(252) = 5.1 Then K(257) = 5.9 K = = .0202 /, K = | ~ = .0230 . . , .0202 + .0230 n o 1 c , Thus K averages out to be 2" ~ »0216 by this method as compared to .0215 by the method used - above. For the four pitches whose corresponding wavelength was measured, the measurement was taken to the end of the top joint ( i . e. to the point at which the diameter of the tube is 4.0 mm) as shown in line "a" below. When r^ is added to each of these lengths, we get the results in line "b", which are then taken to be the length L in the table. Pitch: Bb3 B3 F#4 C5 a) Distance from top of top joint to f i r s t open hole (cm) : 52.0 46.5 27.5 15.2 r^ (cm): 18.6 18.6 18.6 18.6 b) L (cm) : 70.6 65.1 46.1 33.8 -68-We recall now, that the speed of sound at body temperature is approximately 33853 cm/sec. This is the last value which we need in order to use the equation v = f\ and to complete the table. IT) o m CM . CO CM m oo CO CO v O 1^. OO • o o m oo 0 0 CO o C CD CM |5 o I r-i m o> o> cr. • v O CO v O CM 00 VO CO v O co o c m o> co v O CM i n o CO o v D CM O CO v O v D CM CM X too i n o 00 0 0 o v O O o . • . CO CO o o CO CO X> CO <{• X> CO CM CM tt CM c O c co O o •• »J N O X o c ^ a 4-> CD ft i-l P x: O-, cr o 0) 4J 14-t TS ^ ft o CU iw a, c X •• ft C to 4-1 E g ft o x; 60 E cd C o q 4J w • 4 X 1 4J C CJ p i CM O CM P cu CD c 0) E cd ft a> CU II p cd X O •• cd X 4-> P T cr ^ *—s CM s-' cd X cr CU ^ N t—I X cd cu o O- o > O X iw cd p 4J E >-i Cl) IM E 4-1 O P ^ O r-< o •p-l f < IW CO CU Xi CU & •• --^  CU 4-> 4-> Cd P X >•> O cd X X! E 10 CO o o V 4-1 O Cl) cu o o r-t cu x; 0) 4J P O u i> eg ^  CU U ft cd 4J ^ c o a> cd co P cd p M-i CL Cv O r-l o 0) 4-1 X •• I - I i n •• cr cu cu cd P p co ^ x: a P cd p w M tx p o C " 4J ^ E O CO N o 4-1 3 ft o co U 1-t (1) o _ o t—i co x! 4J p to X) O X CO CU CU 4J u cu eg ^  W O Pi •H ft O P 4J « i-l >-( 4-> P Ra IW 0 . cd cd CM -69-Concluslonr When we substitute for r and for the pitch C5 in Kr = ^"*jpS w e obtain ^333^2)^* which = 1.73, which is not negligible in comparison to one. Since the results above come so close to the ones desired, we must conclude that for small changes from the ideal conical shape of the tube, we can work with equivalent volumes whether the lengths involved are equivalent or not. (The vol-ume of the reed + staple combination is approximate-ly equal to that of the truncated portion of the cone, which is approximately three times longer.) Other Aspects of the Acoustics of the Oboe in Par- ticular and Wind Instruments in General 7 As is the case with a l l wind instruments, the oboe has a generator or exciter and a resonator. In the oboe, the generator is the double reed, which may be regarded as a sort of valve which transforms a steady stream of a i r from the player's lungs in-to a pulsating one. This feeds periodic bursts of energy to the main a i r mass in the tube of the in-strument. Next comes the resonator, i . e. the tube or body of the instrument, or, better s t i l l , the column of a i r contained within that body. Taken -70-together, the generator and the resonator form a paired dynamic system, and the vibrations of the two are closely associated. The generator and res-onator should not be considered separately, because then one tends to neglect such details as the fact that in the oboe the mass of a i r in the staple and between the hollow reed-blades constitutes in i t -self a considerable proportion of the air column. In fact, under playing conditions, the generator and resonator greatly influence each other's be-havior and one cannot arrive at proper conclusions by examining either in isolation. Finally, there is one more common feature which can be noticed only during the act of playing. Then the generator is coupled both to the resonator and to the air cavities of the head, throat, and chest, and the a i r in them is also set in vibration. Consequently, during performance, the oboe-player and his instru-ment together form a coupled system of three elem-ents. In playing, the oboist can vary the length of the.resonator not only with his fingers, but also by adjusting the reed with his l i p s , pinching or relaxing as required, and by modifying the shape -71-and volume of his air cavities to some small ex-tent by the use of the chest and throat muscles. Here exactly l i e s the source of that flexible in-tonation so highly valued by musicians. As was concluded earlier, the truncated coni-cal tube closed at one end resonates to the entire series of harmonic overtones, i . e. the musical octave, tiv^elfth, fifteenth, etc... Unfortunately, musical instruments rarely coincide directly with theoretical calculations (and conclusions). These take no account of: 1) the way in which the sound waves have been excited, 2) the changes in mag-nitude, range, or pitch of overtones due to the coupling of the generator with the resonator, 3) altered frequencies due to the need for side-holes, and 4) many other features which practical use or construction impose on the instrument. Let us take, for example, one feature which seems f a i r l y obvious at sight. We know that our calcu-lations were greatly simplified by extending the bore of the oboe to i t s theoretical peak. But such an extension cannot be made in practice, and the reed + staple combination must substitute for the missing part as best i t can. Until quite recent-ly, this has been done without much understanding of the dimensions involved beyond that gathered from practical experience, t r i a l , and error. Re-cently, hottfever, i t has been shown that i f the in-ternal volume of the reed + staple combination slightly exceeds that of the missing t i p of the cone, a suitable interplay with other features of the rest of the air column (which is in i t s e l f a modified cone) can be brought about, and the re-sult is a f a i r l y good acoustic equivalent with mu-si c a l l y useful ratios between the different reso-nance frequencies. If we listen to an ascending scale played through the compass of the oboe, we distinguish changes in quality between successive notes and a progressive thinning of tone, although the general impression is one of homogeneity. Generally, the more interesting (although not always the most pleasing or aesthetically satisfying) tones to the ear are those containing many powerful overtones. These are more prominent in loud sounds than in soft. The sound radiated by an instrument also -73-generally gets less complex in the higher regis-ters. In general, each successive note produced on the oboe does not reproduce the same relative harmonic pattern simply based on a new fundamental as one hole after another is opened. As has already been mentioned, in spite of the audible differences between adjacent notes, a scale played on the oboe leaves the ear with an impression of homogeneity throughout the entire range. This leads one to suspect that there is really some com-mon factor running through the entire compass of the oboe, and that so far this factor has not yet been identified. At one time, (around 1837) i t was thought that the theory of formants provided the answer to this question. According to this theory, set forth by Charles Wheatstone and Hermann Helm-holtz, when a vowel is sounded by any human voice, regardless on which note i t is sounded, i t is i -dentified by certain vibration frequencies which are invariably present. This/phenomenon is conven-iently described by saying that the air-cavities of the head possess qualities of selective reson- ance. Around 1897, the physicist Harmann-Goldap -74-(who was the one to actually invent the xrord "for-mant" to describe the Wheatstone-Helmholz theory), assumed an analogy of the human voice to orchest-ral instruments, and showed a characteristic array of frequencies present throughout the compass. He, therefore, claimed that orchestral instruments, like the voice, possessed formants, and that this was the explanation for instrumental timbre. The idea of the formant frequency also gains support from experiments done x^ith the cor anglais. The data gathered are as follows: 1) A typical cor anglais bell shows as i t s chief characteristic a pronounced resonance in the region of 680 cycles per second. 2) Many people have confirmed the impression that this instrument in i t s lowest octave sings "aw". The characteristic frequency of this sound has been found to be 730 cycles. 3) It has been separately concluded, that each vowel shows two ranges of characteristic frequen-cies, a higher and a lower. 4) The nearest frequency in a recognized speech sound to 680 cycles is the lower one (704), typical of the shortened o as in the word "hot". Since the time of Hermann-Goldap, research has shown that while selective resonances are in-volved in both vocal and instrumental sound pro-duction, their sources are not comparable in gen-eral terms. The source of instrumental timbre is now known to l i e in the shape and dimensions of the bore and i t s array of note-holes, plus the ef feet of various properties of the reed. -76-The Bassoon The bassoon is essentially a conical tube with a total length of about 250 cm. Since this would be too long an instrument i f l e f t straight, i t is folded at a 180° angle, to bring i t down to a man-ageable length (about 122 cm.) The narrow end of the bassoon cone consists of a piece of tapered metal tubing called the crook or bocal, which has a diameter of approximately 4 mm at the point where the double reed is attached. The crook bends f i r s t upward and then down, and is inserted into the wing  joint. This joint (made of wood) has the top three holes of the basic scale, which are covered by the three fingers of the l e f t hand. These holes, i f bored straight through at the proper places, would be too far apart to be covered by the fingers of one hand. Consequently, the wood is thickened in this region, forming the "wing", and the holes are bored at a slant,, as shown below: -77-This type of construction allows the holes to be far enough along the bore as well as close enough together to be covered by the fingers. The wing joint is inserted into a section called the boot, also wooden, which contains two holes, one of these being the continuation of the bore of the wing joint. The boot carries the bottom three tone holes of the basic scale, for the right hand. A metal piece formed into a U-shaped tube is clamped to the bot-tom of the boot to connect i t s two conical holes so as to produce a continuous bore: -78-Into the boot is fitte d the long joint, which car-ries the conical bore up to the bel l ; this is the piece carrying the characteristic metal or ivory ring seen on the top of the bassoon. At the end of the bell the diameter of the conical bore is about 40 mm. 8 At this point, a comparison may be made between the bassoon and the traditional pipe organ. In the organ, each note of the scale has its own separate pipe, so that in a large organ several thousand pipes may be required, each of which w i l l have i t s own gen-erator and resonator. Thus i t becomes necessary to supply air to the pipes from a common source, and the wind pressure remains constant for each pipe. With the bassoon, the case is entirely d i f f e r -ent. The player must acquire a r t i s t i c virtuosity and adaptability to the numerous requirements of breath-control and s k i l l of the fingers. Bassoon playing requires complete mastery of a chromatic sequence in equal temperament over 3% octaves. In other words, about 44 notes must be produced from a single wooden pipe, and to do so, 8 fingers and 2 thumbs must control 18 tone or semitone holes -79-q as well as 3 harmonic holes. We recall now, that the conical tube closed at one end vibrates with modes that are integral multiples of the fundamental. Thus, the second mode of oscillation of the bassoon has a frequency twice the fundamental: *\ 10 p = 0 f i r s t at x = "~J~5 next at x = Musically, this results in a rise in pitch of an octave, which is the basic interval by which the bassoon overblows. The basic scale of the bassoon is the six-hole, seven-note scale, which starts on G2 with the six holes closed. For the nine chromatic notes below G2, a l l six holes, plus additional holes op-erated by keys, must be closed. The normal range of the bassoon, then, is the following: exceptional W The pitches from Bbl to F#3 have their own indiv-idual fingerings: 1 1 -80-With the basic scale being G2, A2, B2, C3, D3, E3, F3, the fingering for the "black notes" in between is always that of the previous "white note" plus an additional hole or key. Overblowing at the octave begins with the next note, G3. From G3 to D4, this is the conventional method of pitch production. to u \n In je- ^ ^ ^ From D#4 to F#4 overblowing at the octave is pos-sible, but there are other " a r t i f i c i a l " fingerings which result in better-tuned pitches. For over-blowing at the octave, fingering is identical to that used to produce the pitch an octave lower, ex-cept that the whisper key (which was used before) is not used. Furthermore, vent-holes are used. Since, however, the bassoon is so large, one vent-hole does not suffice, as i t does on the clarinet. The vent-hole must move up the instrument as the scale is ascended. This is accomplished for the lowest notes in the higher register by half-holing the top hole of the basic six-note scale, which is one of the complications of bassoon playing. For the higher notes, other vent-holes are provided, covered with pads and opened by keys operated by the l e f t thumb.12 We recall now, that p = 0 f i r s t at x = ~2» next at x = A , and next at x = —5jp~« This means that the third mode of oscillation of the bassoon has a frequency of three times the fundamental, i . e. a twelfth above i t . It is by overblowing the twelfth that the next five pitches of the bassoon may be produced: 6.) fry' a) These pitches are not always fingered this way, because again other fingerings may be found x^hich bring superior results in tuning and tone quality. -82-For the remainder of the range of the bassoon, pitch production by further overblowing is theo-re t i c a l l y possible, but this is almost never done. Instead, individual, unrelated fingerings are used. This is not surprising when one remembers that the bassoon is very seldom called upon to produce pit-ches above B4. Generally speaking, the bassoon is in every way the most irregular of the woodwind, instruments. Because of i t s complicated physical construction, in that i t is bent over on i t s e l f , there results a complicated acoustical construction, in that the lateral subdivision of the tube length is irregu-la r . Since the tone holes for F, G#, A, c, and c#, i f placed in their natural position, would occur in the socket or tenon or in the U-bend of the boot, they must be displaced. To correct the intonation i t i s , therefore, necessary to bore oblique holes with varying depth of penetration and diameter. Thus, additional resonance-holes are bored in the wide-bore of the butt to assist tone-holes in the narrow bore. The result a l l this has on bassoon playing is that overblowing succeeds in practice -83-only as far as D4. After that, as a result of the compromises in placing the tone-holes, con-siderable differences in intonation and tone-qual-it y can emerge. These differences can be compen-sated by auxiliary fingerings. Normally, the low tone-holes are opened or closed as far as the a-v a i l a b i l i t y of the thumbs and l i t t l e fingers per-mits. Another factor which greatly influences the acoustics of the bassoon is the taper of the bore of the crook and of the wing joint, because there the cone is the narrowest. The crook has, in fact, been called the soul of the bassoon. There is a wide variety of crooks, each having one or another advantage for the particular instrument.to which i t is f i t t e d . One crook may f a c i l i t a t e the pro-duction of low notes, x^hile another w i l l suit only for producing the highest notes. Even the slight-est deviations in the course of the bore result in the alteration of the theoretical length of the tube and, accordingly, upset i t s pitch. Theorists and even bassoonists often underestimate the in-fluence which the construction of the tube has upon -84-both the intonation and tone-colour of the bassoon. Another decisive factor in determining the quality of the bassoon tone is the reed. A skilled player usually makes his own reeds or has these made by a reed-maker to suit him. Such reeds may nevertheless be unsuitable for another player, be-cause he may have a different embouchure, different breath control, and different breath volume. The quality of a reed cane, i t s place of origin, and its age a l l play an important part. Finally, a player should have a sound knowledge of the acous-t i c a l conditions under which the reed sets in v i -bration the air-column in the instrument. With long experience of wind technique, a skil l e d player a l -most unconsciously produces with the reed a l l fre-quencies necessary to obtain a l l the notes of the 3%-octave range of the bassoon. There is really no generally suitable model reed, but one general-ization may be made: the frequency of the reed i t s e l f in making i t "croak" should produce a pitch somewhere between F3 and. G#3. More may be said about the tone colour of the bassoon. The lengthening of the bassoon tube pro--85-duces the result that although the fundamental is always heard, i t has only a fraction of the power of the entire sound volume. Measurements have been taken of the strength of the partials in the indiv-idual registers and i t was found that, for example, from Bbl to G2 the f i f t h harmonic has about 60% of the power, while the fundamental has only 2%. From G#2 to E3 the third harmonic has 60% of the whole pox^er, and the fundamental only about 4%. From C#2 to C4 the second harmonic has 95% and the fundamen-tal 3%; from C#4 to C5 the f i r s t harmonic ( i . e. the fundamental) about 90% and the other harmonics the remainder of the power. The conclusion is that the prominent partials with the greatest power l i e between 350 and 500 cycles per second. Tonal spec-tra have shown that the bassoon has a particularly strong formant precisely in this region — at about 1 3 500 cycles per second (hertz.) It i s , therefore, obvious that the tone-colour of the bassoon is actually different in the three octaves. Measurements of resonances in the bassoon show that for the low tones of the instrument, sev-eral resonances appear, lying reasonably close to -86-the harmonic frequencies. As the scale is ascend-ed, the resonances become fewer in number and de-part more from the harmonic frequencies. Above the break in the bassoon there is again l i t t l e cor-respondence between the resonances and the harmonic frequencies. In the bassoon, that portion of the instrument below the f i r s t open tone hole has a considerable influence on the resonances of the portion of the instrument producing the tone, much more so than in the other woodwinds. Closing or opening a tone hole in the lower part of the bassoon can make quite a change in the quality of a tone higher up on the instrument. For this reason bassoonists have a num-ber of fingerings available to improve the quality of certain bassoon tones,lw*The most common form of auxiliary fingering used is that of cross-fingering. Many experiments have been performed in the area of cross-fingerings. These began with most accur-ate measurements both as to pitch and dimensions in good existing instruments — in the f i r s t place with the clarinet. Starting about the middle of the compass with the f i r s t four holes closed (A), -87-i t was found that closing the sixth hole (B) would appreciably flatten the note sounded with A. Closing the seventh hole (C) and reopening the sixth produced much less flattening, while closing the eighth hole (D) had almost negligible effect: A t-t • • • a o n o o o f ) B • • • • o « o o o D C o » » » o o « o Q ~ D D • • • • o o o « o~Q The conclusion is that the flattening due to clos-ing a lower hole decreases the further away i t is from the topmost open hole. 1 5Unfortunately, tones produced in this way are not always of good qual-ity , and often adjustments must be made through the use of other holes and keys. The reason for the poor quality of the tones produced by cross-finger-ing is the effect on the higher modes of the por-tion of the tube below the f i r s t open hole. These higher modes of the a i r column are pushed even fur-ther away from the harmonic frequencies than they -88-are normally, so the harmonic content of the tone is further reduced; this results in poorer tone quality. Much more work needs to be done on the bassoon as well as the other woodwinds to work out the re-lationships between the structure of the instru-ment, the positions of the resonances, and the har-monic structure of the tone. At present, we are not only ignorant of these relationships, but we do not even know in physical terms the difference between a "good" and a "bad" tone. 1 6 Testing the Equation-v = f ^ 1 ? As was done with the other three instruments, we nox>7 want to test the equation v = f\ in a prac-t i c a l situation involving the bassoon. We must f i r s t , however, calculate the (theoretical) length of the conical tube of the bassoon by extrapola-tion of the bore to i t s imaginary peak. F i r s t l y , we "straighten out" the tube, and find that i t looks as follows: -89-Now the measured dimensions (lengths and diameters) of the parts of the bassoon are as follows: Double Reed 5 |mm 305 mm fcrook Main Wooden Section 1834 mm 12 mm 475 mm Tenor Joint D O 15 mm 794 mm Boot 25 mm 35 mm 565 mm Long Joint 40 mm 359 mm Bell As was done with the oboe, we w i l l f i r s t use only the most uniform, main wooden section for our ex-trapolation, and ignore the bell and the crook. d i -12 mm! Tenor Joint d 2 = 15 md Boot d3 " 25 mm 1834 mm — * L Long Joint inn) d4 " 35 mm r1 + L--90-W it h d 1 = K C ^ ) , let d 1 = 12 and d 4 = K(r x + L) = 35. d, - d, _ 23 Then KL = d. - d- =23, /. K = ~4 ~1 = 1834 = .0125 To verify the value obtained for K above, we can take the joints individually, as follox<rs: Tenor Joint: let d x = K(r x ) = 12 and d 2 = K(r x + 475) = 15 Then K(475) = 3 ,*, K = ^ = .0063 Boot: let d 2 = K(r 2) =15 and d 3 = K(r 2 + 794) =25 Then K(794) = 10 t • K = yg^ ~ •012 6 Long Jo int: let d 3 = K(r 3) = 25 and d 4 = K(r 3 + 565) = 35. Then K(565) = 6^ - d 3 = 10 /, K = ~ | = .0177. Thus K for the main wooden section averages out to be 475(.Q063) 4- 794C.0126) + 565(.Q177) 1834 2.9925 + 10.0044 + 10.0005 _ n 9_ 1834 U.I/D, i . e. identical to the value obtained above, but shown not to be uniform throughout the three joints. Now taking K = .0125, for d^ = K(r^) we obtain: 12 12 = ,0125r^ ' ' ri ~ 5 = ^60 mm, -91-and for d 4 = K(r x + L) we get: 35 = .0125(r + L) 35 r l .0125 L = 2800 - 1834 = 966 mm. Thus r^ averages out to be approximately 963 mm (including the crook). This leads us to believe that this extrapolation is not appropriate, since the theoretical length of the tube comes out to be too great in proportion to i t s practical length. We must, therefore, assume that, contrary to the case with the oboe, the bell and the crook of the bassoon play a decisive role in determining the the-oretical length of its cone. We now, therefore ex-pand the previous diagram to include the entire bas-soon cone, which we w i l l use for the extrapolation: Double Reed Croow* r0 = ? 5 mm h -305 mm K-Main Wooden Section' 12 mm 1834 mm L = 2498 mm-r 0 + L 35 mml Bell 40 wd 359 mm -92-To find the value of K for the crook, we l e t d Q = K(r Q) = 5 and d± = K(rQ + 1 ) = 12. Then K l x = d x - d Q = 7, /. K = d l ~ d0 = 3 ~ = .0229. From above, we know that K for the Main Wooden Section averages out to be .0125. To find the val-ue of K for the b e l l , we let d 5 = K ( r $ ) = 4 0 a n d d4 = K ( r 4 + = 3 5 • T h e n K l 2 = d 5 - d 4 = 5, K = d5 " d4 = = .0139. 12 Thus K for the entire bassoon tube averages out to: 305(.Q229) + 1834C.0125) + 359(.0139) 2498 6.985 4- 22.925 + 4.9901 n i , 2491 = - 0 1 4 ' Now, taking K = .014, for d^ = K(TQ) we get: 5 = .014(rQ), '. = 357 mm, and for d,- = K(r^ + L) we get: 40 = .014(r Q + 2498), rQ = - 2498 = 359 mm. r^ averages out to be 358 mm, which is an ap-propriate proportion to the practical length of the tube. Now rQ + L, the theoretical length of the bassoon cone, becomes 2854 mm. -93-Now we may proceed with testing the equation v = in the case of the bassoon. With a bas-n'k soon tube of theoretical length L, L = must be satisfied, therefore A = 2L (see page 59). 5 max v to For the four pitches whose corresponding wavelength was measured, the measurement was taken to the top of the tenor joint ( i . e. to the point at which the diameter of the tube is 12 mm), as shox«m in line "a" below. When we add to each of these lengths 305 mm for the crook plus 358 mm for r0> we get the results in line "b", which are then ta-ken to be the length L in the table on the next page. Pitch: Bbl G2 F3 C4 a) Distance from too of tenor joint to f i r s t open hole (cm): 195.9 99.4 23.8 7.3 Crook length (cm): 30.5 30.5 30.5 30.5 r Q (cm): 35.8 35.8 35.8 35.8 b) L (cm): 262.2 165.7 90.1 73.6 Conclusion: When we substitute for r and\ for the 2 IT r 2 IT (Tvft') pitch C4 in Kr = w e § e t 736 > w h i c h equals approximately 1.53, which is not negligible in com-parison to one. Since the results above come so l a Examined Pitch: Bbl lb Known frequency of this pitch (hz): 58.27 Length of pipe (L) (cm): 262.2 (where 'A = 2L) (cm): m a x 524.4 Calculated frequency (33853 cm/sec -4- 4L) (hz): 64.56 2a Pitch whose frequency in equal temperament is nearest to that obtained above: C2 Known frequency of this pitch (hz): 65.41 Discrepancy of 2a with respect to l a (by in-terval): + 2nd hghr. Ratio of calculated frequency with res-pect to lb: 1.108 G2 97.99 165.7 331.4 102.15 G#2 103.83 - 2nd hghr. 1.042 F3 174.61 90.1 180.2 187.86 F#3 185.00 - 2nd hghr. 1.076 C4 261.63 73.6 147.2 229.98 Bb3 233.08 + 2nd lower .879 -95-close to the ones desired, we must conclude, as we did in the case of the oboe, that for small changes from the ideal conical shape of the tube, we can work with equivalent volumes whether the lengths involved are equivalent or not. -96-The Saxophone The saxophone i s one of the few instruments which was "invented", as opposed to being evolved. Its in-ventor, Adolphe Sax, who was famous for construction of both brass and woodwind instruments, wished to cross the two families and come up with a brass clar-inet with a conical bore and using simple fingerings. He did this by combining the single reed of the clar-inet with a brass conical bore body and with woodwind-type fingering mechanisms.18The basic design of the instrument has never been changed, although many im-provements have been made, one of which has been a considerable widening of the original range. Theoret-i c a l l y , a l l saxophones except the soprano consist of four sections: the Goose Neck (an L-shaped tube to which the mouthpiece and reed are attached), the Main Body, the Bell, and a U-shaped tube connecting the Bell to the Main Body. Practically, though, there are only two sections, because the U-shaped tube is permanently attached to both the bell and the body, thus these three pieces form one long sec-tion. In this discussion, the U-shaped tube w i l l be considered as a portion of the Main Body. -97-Mouthpiece Holes and keys are found throughout the entire length of the instrument. The family of saxophones in use today consists of five instruments of various sizes: Soprano in Bb, Alto in Eb, Tenor in Bb, Baritone in Eb, and Bass in Bb. A l l members of the saxophone family have the same fingering system — the structural difference is mainly one of size. Because of the great simi-l a r i t y of these instruments to one another, only one wi l l be considered here: the Eb Alto. Since the conical tube closed at one end. v i --98-brates with modes that are integral multiples of the fundamental, the second mode of oscillation of the saxophone has a frequency of twice the fundamental: Musically, this results in a rise in pitch of an octave, which is the basic interval by which the saxophone overblows. This property, plus the fact that the saxophone has a pressure node (and not an antinode) at the mouthpiece, makes the construction and fingerings of the saxophone relatively simple. The saxophone has 12 "additional" holes and keys, which is about the same number as the oboe. The basic scale of the Eb Alto saxophone is the six-hole, seven-note scale, which starts on D4 with the six holes closed. This D4 sounds as F3, a major sixth lower than written. Additional holes and keys on the bell are used to extend the range down to (written) Bb3. The normal range of the saxophone, then, is the following: p = 0 f i r s t at x = 1 next at x 1 9 2' Sounding: exceptional -99-The written range shown here is that of a l l saxo-phones; the sounding range shown is that of the Eb Alto saxophone. The pitches from Bb3 to C#5 have The basic scale oroceeds: D4, E4, F4, G4, A4, B4, C#5, and the chromatic notes in between are produced through the use of additional holes and keys. There are two ways of producing C#5. The f i r s t , which is conventionally used, is with a l l holes open, i . e. as the last note of the basic scale. The sec-ond (and worse) method is by overblowing C#4 at the octave. The pitches Bb4, B4, and C5 may also be pro-duced by overblowing at the octave, but the tone quality is inferior, so this is not done. The f i r s t break of the saxophone occurs between C#5 and D5. Beginning with D5, octave overblowing is used for a l l pitches up to C#6: their own individual fingerings: .20 -100-In each case, the fingering is identical to that ,. used to produce the pitch an octave lower, except that the octave key is added. We recall now, that p = 0 f i r s t at x = next at x = A , and next at x = —^—. This means that the third mode of oscillation of the saxophone has a frequency of three times the fundamental, i . e. a twelfth above i t . In the case of the saxophone, how-ever, there is only a limited range in which over-blowing by the twelfth is possible. From F5 to A5 this can be done quite readily, but from Bb5 to C6 i t is more d i f f i c u l t : {\ o fn n fe- ¥• ^ ^ L W L ^ W ( i i j M i p sf exceptional Furthermore, since a l l these pitches are best pro-duced through the second mode of oscillation, as shown above, overblox^ing by the twelfth is a resource which is seldom used on the saxophone. The last three pitches of this register, however, may be produced by use of the fourth mode of oscillation, (where :;; -101-x = 2*X ), i . e. by overblowing at the double oc-tave: n - e -Between C#6 and D6 is where the next break of the saxophone occurs. From here on up, the best method of pitch production from the acoustical standpoint is derived from the overtone series on Bb3, B3, and C4, the three lowest pitches of the in-strument: ^ ^ 5 ( 4) "ST- ( 3) 4 (2) ) When we continue the previously given series of values for which p = 0, we get the following: P - 0 when x = 1 , ^  , l X , 2^, ^ L , 3^ X , i X , 4^, etc... When x = 5-A the instrument- is operating in i t s f i f t h mode of oscillation, i . e. with a fre-quency of. five times the fundamental, or two octaves plus a major third above i t . This is how the f i r s t -102-three Ditches of this register, D6, D#6, and E6, may be produced. This pattern i s repeated f o r the six t h mode of o s c i l l a t i o n , i . e. fo r x = 3^ , where the res u l t i n g p i t c h i s s i x times the fundamental, or two octaves and a perfect f i f t h above i t . This i s the case for the next three pitches, F6, F#6, and G6. When we come to the seventh mode of o s c i l l a t i o n (where x = we encounter an inevitable problem: the r e s u l t i n g Ditches turn out f l a t t e r than the de-si r e d ones because of the nature of the harmonic se-r i e s . Rather than sounding two octaves plus a minor seventh above the fundamental, the res u l t i n g p i t c h frequently sounds at an in t e r v a l of two octaves plus a sharp major sixth. Consequently, we are forced to seek another method of producing Ab6, A6, and Bb6. For Ab6 and A6, the best solution i s to return to the s i x t h mode of o s c i l l a t i o n and overblow Db4 and D4 by two octaves and a perfect f i f t h : -103-For Bb6, our problem has a much simpler solution. We simply group i t with the next two pitches, B6 and C7, and produce a l l three of these pitches through the eighth mode of oscillation, where x = 4^ , i . e. by overblowing at the tri p l e octave, as shown in the previous example. It must be stressed here, that this method of pitch production is valid from the acoustical stand-point, but from the practical standpoint, saxophon-ists often find their own " a r t i f i c i a l " fingerings, which either suit them better, or suit their instru-ment better, or simply give more satisfactory re-sults. One should not be surprised, therefore, to find many different usable fingerings for the pitches in the highest register of the saxophone. Testing the Equation v = f) 2 1 We now want to test the equation v = f X in a practical situation involving the saxophone. With a saxophone of length L, L = must be satisfied (see page 59). We must f i r s t , however, calculate the theoretical length of the conical tube of the saxophone by extrapolating the conical bore to the -104-imaginary peak. F i r s t l y , we "straighten out" the tube, and find that i t looks as follows: Now the measured dimensions (diameters and lengths) of the saxophone are as follows: Reed-Since the bell is permanently attached to the main body, the diameter of the bore at that point had to be estimated as follows: The external circumference was measured and found to be 173 mm. 173 ~- IT gave 55 mm. The thickness of the tube wall was estimated at 3 mm, therefore 6 mm were subtracted for a final -105-es timate of 49 mm. Since the bell of the saxophone flares out much more rapidly than the remainder of the bore, we shall not consider i t in the extrapolation, and use only the greater part of the tube: d l = 13 mm (24 mm) •L = 770 mm r1 + L 3> d2 -49 mm With d± = K(r 1), let d1 = 13 and d 2 = K(r ] [ + L) =49. Then KL = d 2 - d^ = 36, K Now r. + L = d2 = -Jtto = 1047, . . r. = 2 77 mm. 1 — .0468 ' 1 To verify the value obtained for K, we can take the two sections separately, as follows: Goose Neck: Body: let d^ = K(r^) =13 and d 2 = K(r x +175) = 24 Then K(175) = 11 11 let d x = K(r x) = 24 & K = 175 .0629 d 2 = K(r 1 +595) Then K(595) = 25 25 49 . K 595 = .0420 -106-mi „ _ , 175(.0629) + 595(.0420) Thus K averages out to be * T70 11.0075 + 24.990 35.9975 n / C Q . = JJQ = —JJQ— = .0468, I . e. iden-t i c a l to the value obtained above, but shown not to be uniform throughout the tube. For.the four pitches whose corresponding wave-length was measured, the measurement was taken to the top of the Main Body ( i . e. to the point at which the diameter of the tube is 24 mm) as shown in line "a" below. When we add to each of these lengths 175 mm for the Goose Neck plus 2 77 mm for r^, we get the results in line "b", which are then taken to be the length L in the table on the next page. Pitch: Written: Bb3 F4 A4 C#5 Sounding: Db3 Ab3 C4 E4 (a) Distance from top of Main Body to f i r s t open hole (cm): 77.0 39.7 23.3 10.8 Length of Goose Neck (cm): 17.5 17.5 17.5 17.5 r-j, (cm): 27. 7 27.7 27. 7 27.7 (b) L (cm): 122.2 84.9 68.5 56.0 Conclusion: When we substitute for r and *\ for the l a Examined Pitch: Written: Bb3 Sounding: Db3 lb Known frequency of this pitch (hz): 138.59 Length of pipe (L) (cm): 122.2 "X (where *A = 2L) (cm): " 244.4 Calculated frequency (33853 cm/sec -r~ 2L) (hz): 138.51 2a Pitch whose frequency in equal temperament is nearest to that obtained above: Db3 Known frequency of this pitch (hz): 138.59 Discreoancy of 2a with respect to l a (by in-t erval): none Ratio of calculated frequency with res-pect to lb: .999 F4 Ab3 207.65 84.9 169.8 199.37 G3 196.00 - 2nd lower .960 A4 C4 261.63 68.5 137.0 247.10 B3 246.94 - 2nd lower .944 C#5 E4 329.63 56.0 112.0 302.26 D4 293.66 + 2nd lower .917 -108-u nun • v 21T r _ 2 T (2 77) , , Ditch C#5 in Kr = ~ — , we get H20 which = 1.55, which is not negligible in comparison to one. Since the results come so close to the ones desired, we must conclude that for small changes from the i -deal conical shape of the tube, we can work with e-quivalent volumes whether the lengths involved are equivalent or not. One further observation may be made from the results obtained. Since the discrepancy between the calculated frequency and the known frequency is systematic rather than uniform, this means that r^ has an increasingly greater effect on this discrep-ancy for increasingly higher pitches. This implies that i t would be possible to make a better choice of the value of r^, so that the discrepancy would, instead, be constant for the entire range of the instrument. Thus, although i t would be slightly greater for the lowest Ditches, i t would be far smaller for a l l the higher pitches than i t is pres-ently. -109-IV. THE BRASS INSTRUMENTS 1 Like the woodwinds, the brass instruments have their origin in prehistoric times. Originally, these instruments were nothing more than seashells or hollow animal horns, later, x»?ith the development of techniques and s k i l l s of metal working, these shapes were reproduced in copper or brass. As time went on, alterations and improvements were made ac-cording to musical requirements, until the instru-ments acquired the shapes which they have today. Acoustically, the brass instruments might be classified with the woodwinds as brass wind Instru-ments, however, they di f f e r in enough important re-spects from the woodwinds to be considered a separ-ate family. Surprisingly, the least important di f -ference i s that which is inherent in the names of the two groups of instruments, which emphasize the material of construction — metal as opposed to wood. We must remember, that one of the modern woodwinds, the flute, is not wooden at a l l , but metal, and an-other woodwind, the clarinet, may also be construct-ed of materials other than wood. The fundamental differences, then, are the following: -110-First, the vibrations of the air column in the brass instruments are maintained by the vibra-tions of the player's l i p s , instead of by air streams or reeds. Since the mass of the lips is considerably greater than that of woodwind reeds, the lips can influence the a i r column vibrations more easily. Second, the brass instruments use many more resonance modes of the air column than the woodwinds do. Some brass instruments use only resonance modes; the majority, however, have a complete scale, which is made available by f i l l i n g in the gaps between modes. Third, the principle on which brass instruments are constructed is the exact opposite of the side-hole system of the woodwinds. Instead of effective-ly shortening the a i r column, i t is progressively lengthened by the addition of extra lengths of tub-ing. This is achieved by means of either a movable telescopic section in the main tube, or by a number of spring-loaded valves, each of which, when depressed, switches in a preadjusted length of supplementary tubing and cuts i t out again when released. The t e l --111-escopic system is found today only in the slide trombone, but i t is the older and simpler of the two systems. Fourth, since there are no open holes in the sides of brass instruments, a l l the sound must come out of the be l l . Consequently, the size and shape of the bell play a decisive role in the acoustics of brass instruments, whereas in the woodwinds the influence of the bell is not that great. Fifth, there is a fundamental difference in the acoustical foundations of the two groups of instru-ments. We know now that the woodwind instruments u t i l i z e the well-defined, harmonically related res-onance frequencies of cylindrical and conical tubes. The brass instruments, on the other hand, cannot be divided so distinctly into two such groups. Instead, they are a l l partly cylindrical and partly conical, which brings us back to the point mentioned above, stressing the acoustical importance of the construc-tion of the b e l l . The sixth and final difference between the two families of instruments lies in the significance of their deviations from basic cylindrical and conical -112-shapes. In woodwind instruments, the presence of covered tone holes and other factors results in unavoidable deviations of the bores from perfect cylindrical or conical shapes. These deviations shift the resonance frequencies away from the har-monic frequencies, which leads to intonation prob-lems and the necessity of getting around these by a r t i f i c i a l means. In short, unavoidable deviations from the original shapes of the tube are a basic source of problems in the construction and finger-ings of the woodwind instruments. On the other hand, in the brass instruments, the opposite is true. Distortions from the simple shapes are in-troduced deliberately and in a large degree in order to produce a musically useful series of resonance modes. We w i l l now go on to consider how this is done. If the lips are placed against a smooth ring and put under some tension, they can be made to v i -brate by blowing air through them from the lungs, thus producing a buzzing sound. Since the mass of the lips is relatively large, the vibration frequen-cy produced is relatively low. If the tension of -113-the lips is increased, they w i l l vibrate at a higher frequency, just as vibrating strings do. Now i f we attach a (cylindrical) tube to this ring, the vibrating lips w i l l build up oscillations of that particular resonance mode of the a i r col-umn which has a frequency near that of the l i p s . The mechanism for producing vibrations in this sys-tem is then basically the same as that for the reed instruments, except that here the lips play the role of the reed. The important acoustical consequence of this similarity is that there is a pressure anti-node in the air column at the l i p end, where energy is being supplied. The frequency of the combination of l i p and air column w i l l be primarily determined by the fre-quency of the particular mode excited. However, be-cause of the relatively large mass of the l i p s , which are here playing the role of the reed, the in-fluence of the lips on the vibration frequency is much more considerable than i t is in the woodwinds. For this reason, increasing the l i p tension can ex-cite the higher modes of vibration of the air column f a i r l y easily; thus the vent-holes needed on the -114-woodwinds are not necessary on brass instruments. This makes i t possible to use many more of the res-onance modes of the air column than may be used in woodwinds. Since there is a oressure antinode at the l i o end of the column, this end behaves like a closed end. Consequently, the general equation involving 'A for such a tube of length L is L = |, which means 1 3I 5 l l\ that p = 0 when x = |, ^g-, - L^ -i etc... This tube then vibrates with modes that are odd multi-Dies of the fundamental, as does the clarinet. The conclusion is that such a tube does not have a musically useful series of modes. From the musical standpoint, the tube has an-other fault which so far has not been mentioned. The tone i t produces when blown with the lios is muffled and of poor quality — nothing similar to a good brass tone. For these two reasons, the sim-ple cylindrical tube needs much alteration and im-provement to make i t into a useful musical instru-ment. Two basic alterations have been made which succeed in achieving this goal: a mouthpiece has -115-been added to the l i p end of the tube to replace the ring, and the opposite end of the cylindrical tube has been flared into a b e l l . The precise shaoe of both is of crucial importance. The Mouthpiece 2 The mouthpiece consists of a small cup, with a rim designed to f i t against the l i p s . The cup is joined at the throat to a tapering tube of consider-ably smaller diameter than the rest of the instru-ment; this small tube is called the back-bore.• The following diagram shows a section through a mouthpiece: Rim^  The width of the rim is a matter of comfort, but an excessively wide rim w i l l reduce f l e x i b i l i t y and may even disturb tone quality because of the un-natural position which i t enforces upon the embou-chure. A large mouthpiece tends to produce a bigger tone, but not every player's embouchure is strong enough to use such a mouthpiece. The depth of the -116-cup is of great importance; the deeper the cup, the darker and more solid the tone. However, a deep cup also makes the high register more d i f f i c u l t to play in. Also, the shape of the cup, whether i t is a real cup or even a funnel, w i l l affect quality and ease of playing. The bore is of great impor-tance; a large bore offers a minimum of resistance and a maximum of tone, but not a l l players have the physical stamina required by such a mouthpiece; fur-thermore, such a mouthpiece is d i f f i c u l t to work with in extreme piano passages. The back-bore also has a bearing on quality and playing characteristics; a cone with a high K factor in the back-bore tends to f a c i l i t a t e playing, but may cause the instrument to become rough in the forte passages, while a more cylindrical back-bore offers greater resistance, re-sulting in a more solid tone which w i l l not speak quite so readily. If the mouthpiece shown above is attached to the tube we have been considering and the cup is closed off by the l i p s , the tube w i l l effectively be lengthened. At low frequencies, where the values of ^ are large in comparison to the size of the -117-mouthpiece, i t turns out that the amount of length-ening is exactly that of a piece of the tube having the same volume as the mouthpiece. The higher res-onances, however, do not remain unchanged when the mouthpiece is added. The combination of cup + tube has a certain resonance frequency, and at this fre-quency i t behaves the same as a tube closed at one end that has this as its fundamental frequency. The result is that the mouthpiece lengthens the tube by an amount that increases as the frequency increases. Consequently, the tube with the mouthpiece attached has the same lowest resonance frequency as before, but the high modes are shifted dox<mward from their original values, a desirable result. In general, the mouthpiece is probably the greatest single factor in tone quality and proper tone production in brass instruments. A below-ave-rage instrument can be made to sound f a i r l y good with a good mouthpiece, and an excellent instrument can be made to sound inferior with a bad one. The Bell Just as we wanted to shift the high modes of the plain cylindrical tube down, our second desirable -118-iraorovement is that i t s low modes move up. This is accomplished by flaring the open end of the tube by increasing i t s diameter as the open end is approached, thus forming the familiar bell of brass instruments. At the same time, in order to leave the higher res-onances unchanged, the total length of the instru-ment must be properly adjusted. The actual amounts by which the lower modes w i l l come up depends on the detailed shape of the b e l l . With a properly shaped bell (and mouthpiece), the resonance frequencies w i l l be shifted into those modes we conventionally asso-ciate with a brass instrument. Adding the bell to a brass instrument has an-other very important effect: the tone i t produces is much louder and clearer than that produced by the cylindrical tube without the b e l l ; this is our third desirable improvement. There are two reasons for this effect. First , as might be expected, the amount of sound which radiates from the open end of a v i -brating air column depends on the area of the open end. Thus the bell of a brass instrument serves the very important function of increasing the sound out-put of the instrument. Second, the production of -119-harmonics in the tone is facilitated. When we sound a note in the simple tube, none of i t s harmonics li e s very close to a resonance, therefore their pro-duction is poor. For a trumpet, on the other hand, the harmonics of a note a l l coincide very well and give the tone i t s characteristic brass timbre. Let us now take a closer look at the actual con-struction of the b e l l . A l l modern brass instruments are provided with bells ending in a short section or "skirt", which expands much more rapidly than the tapered portions of the main tube. The main taper of the majority of brass instruments develops in what is called "exponential" form, in x^hich the ratio of successive radii at unit distances apart along the tube is a constant. The following figure shows the section along an "exponential horn" in which this constant quantity is 17/14: This ratio has been called the "flare coefficient", -120-which can be regarded as a measure of the expan-sion of such a tube. The following table shows approximate flare coefficients for four modern brass instruments. It is remarkable that three of them are identical. Instrument Flare Coefficient French Horn 1:25 Cornet 1:25 Trumpet 1:25 Trombone 1:3 This seems to be hardly accidental and implies that the coefficient of 1:25 has been arrived at by man-ufacturers through practical experience. Although at f i r s t sight the french horn appears to have a very wide b e l l , the exponential flare i s , in fact, continuous from one end of the instrument to the other. 5 The flare coefficient of the trombone varies so drastically because the trombone is the only in-strument employing the sliding-valve orinciDle, which demands that i t be perfectly cylindrical for the greater part of its length. To compensate for this, the oart that flares out does so at a high ra-t i o . Having considered the b e l l , let us now look -121-briefly at muting. It is a commonly held view that the sole purpose of a mute is to reduce the volume of sound eminating from an instrument. This, however, is not the whole story, because although reduction in volume does occur, the real function of the mute is to modify a complex tone by disturb-ing the normal distribution of i t s partials. By muting, some partials may be reduced in relative intensity while others (often the more dissonant ones) are made more orominent. The mute does this by altering the physical character of the b e l l , which governs the sound to a large extent. The insertion of the olayer's hand into the bell has a similar effect. Unlike primitive mutes, modern mutes do not, as a rule, raise the pitch. A short discussion of timbre is also appropriate at this point. The factors which influence the tim-bre of brass instruments have been recognized for a long time. There are basically five, the f i r s t two of which have already been mentioned: 1) The construction of the mouthpiece 2) The curvature and size of the bell 3) The ratio of the diameter of the tube to i t s -122-length, i . e. the scale of the tube 4) The shape of the tube (cylindrical, conical, or mixed) 5) The thickness of the tube walls. -123-The Trumpet The trumpet consists essentially of three par-a l l e l lengths of straight brass tubing connected with two bends, which together form one long oval coiled one and a half times around. The three sec-tions may conveniently be called the leadpipe (to the end of which the mouthpiece is attached), the middlepipe, and the bellpipe. The f i r s t two of these sections are cylindrical and the last conical. The flare of the bellpipe actually begins somewhere in the second bend. Since the bore of the trumpet, like that of a l l brass instruments, is neither entirely cylind-r i c a l nor entirely conical, but a combination of the two, the trumpet does not behave like an instru-ment belonging to either the cylindrical or the con-ic a l group. It does, however, lend i t s e l f easily to overblowing and to operation in at least 8 — and in the hands of good players up to 16 — modes of oscillation. More importantly, i t s successive modes of oscillation coincide with the harmonic series beginning with the second harmonic. For practical purposes, i t is best to take" the eighth -124-harmonlc as the last accessible one. Consequently, the trumpet, as described thus far, is naturally capable of producing the following pitches: Written: Sounding: A word of explanation is in order here regard-ing the absence of the f i r s t harmonic. As was ex-plained in the previous section, when the end of a cylindrical tube is flared into a b e l l , the low modes of the tube are moved up. In the case of the trumpet, the second, third, and fourth modes end up in the proper places, corresponding to the respect-ive members of a harmonic series. The fundamental, however, is not shifted up far enough, and ends up being more than a perfect fourth below the position of the f i r s t harmonic of the series. For this rea-son, the f i r s t mode of osci l l a t i o n of the trumpet is not used. 8 In order to provide the trumpet with a com-plete scale, the intervals between the notes shown above must be f i l l e d in. The largest interval which -125-needs to be filled- in is the f i r s t one, that of a perfect f i f t h , which requires six additional semi-tones to f i l l the space. These are provided by lengthening the air column through the use of valves. Six additional lengths are sufficient; in addition to f i l l i n g in the space between the f i r s t two modes, they allow the trumpet to play six semitones below the lowest pitch shown above, and are more than adeauate to f i l l in the spaces between the higher modes. 9 Three valves are used in the B-flat trumpet. The middle valve adds tubing which lowers the original pitch by one semitone. The valve clos-est to the player (valve 1) lowers i t by two semi-tones, and the farthest valve (valve 3) lowers i t by three semitones. Consequently, when valves are combined, the resulting shift of frequency down-ward (in semitones) is as follows: 1 0 Valves Number of semitones 1 + 2 3 2 + 3 4 1 + 3 5 1 + 2 + 3 6 -126-Through the use of these valves, the follow-ing pitches may be produced on the trumpet as i n -dicated: 1 1 Valves Possible (Sounding) Frequencies Mode: 2 3 4 5 6 7 8 -127-Thls is so because each fundamental which results from a new combination of valves has i t s own over-tone series, the members of which are accessible in each case by overblowing. In general, for the fourth series shown, valves 1 + 2 are used as op-posed to valve 3, for reasons to be explained later. Although there are seven rows containing seven notes each, only 31 of these 49 notes are d i f f e r -ent. These 31 notes have been subdivided into three categories as follows: 1) 16 pitches which appear only once ~ unnumbered, 2) 12 pitches which appear twice each — indicated with the same number in both places, 3) 3 pitches which appear three times each — in-dicated with the same number underlined in a l l three places. The next step is to determine which valves the trumpeter w i l l , in practice, use to produce the 31 pitches of the range of his instrument. Clearly, for the 16 pitches for which only one fingering is possible he has no choice. These pitches are indi-cated in whole notes in the table on page 129. For the remainder of the pitches, the process of elimi--128-nation i s used to determine which fingering w i l l give the best r e s u l t . F i r s t l y , the p o s s i b i l i t i e s presented i n the sixth column can be safely elim-inated, because these are seventh harmonics, tv Thich are considerably f l a t t e r than the corresponding note i n the equal-tempered scale, and for thi s rea-son would, be the worst choices to make. Elimina-t i o n of the sixth column y i e l d s two r e s u l t s : 1) It determines the only possible fingering re-maining for pitches 5, 10, 13, and 14. These p i t -ches are indicated i n half notes i n the table on page 129. 2) It reduces the number of possible fingerings for pitches 3, 4, and 9 from three to two. Now there are two choices of fingerings f o r each of the eleven remaining pitches. For each of these, we choose the one which comes f i r s t , (reading from top to bottom); t h i s way no more than one valve w i l l have to be used to produce any one of these Ditches. This i s , i n general, better than the use of two or more valves, f o r reasons to be given l a t e r . These l a s t eleven pitches are indicated i n quarter notes i n the following table. -129-Valves Sounding Frequencies Produced This table clearly shows that the use of more than one valve is necessary for at least 8, but at -130-most 13 of the 31 pitches of the normal range of the trumpet. The complete chromatic scale of the trumpet may now be written out, showing the f i n -gerings (by valve numbers) and the modes of o s c i l -lation used. As was explained earlier, the f i r s t mode of oscillation is not used, so we begin with the second: *2 -131-The Valves Generally, two types of valve systems are in use today: rotary valves and piston valves. Since piston valves are by far the more common ones on this continent, l e t us examine In detail just hox«7 they operate. The mechanism of a piston valve 13 is. shown in the following figures: A c y l i n d r i c a l piece of metal called the piston i s arranged to move up and down inside a piece of tub-ing permanently attached to the cylindrical- portion of the main tube. The piston i s normally held i n the "up" position by a spring, but can be depressed with the finger. In the "up" position, a hole in the piston called the valve port (Indicated "1" i n -132-the figures) carries the a i r column, as shown in the f i r s t figure above. In the "down" position, two other valve ports in the piston (indicated "2" and "3" in the figures) divert the a i r column, as shown in the second figure above, so that i t passes through an additional piece of tubing. Pushing the valve down thus lengthens the a i r column; the amount of increased length depends on the length of the added valve tubing. This is made adjustable for tuning purposes by providing i t with a movable U-shaped portion called the valve-slide. As we have seen, three valves are used in the B-flat trumpet. While these valves work well when used individually, they are not magic, and when they are combined, a fundamental d i f f i c u l t y arises. If the f i r s t and second valves were designed to insert tubing of the correct length to lower the pitch by a tone and a semitone respectively, then the two valves together would not add enough tubing to lower the pitch by three semitones. Further, i f a l l three valves were used together, they would not add enough tubing to lower the pitch by six semitones. This problem is clearly illustrated by working -133-out some actual numbers. Suppose that the original length of the trumpet is 150 cm. To lower the pitch produced by this tube a semitone in the equal-tem-pered scale ; requires that the frequency be divided by a factor of "VT, or approximately 1.05946. This can be done by multiplying the equivalent length of the a i r column by the same factor, which amounts to an increase in length of 5.946%, which may be round-ed off to 6%. Thus, to lower the pitch of the tube by a semitone, valve 2 would have to add 9 cm of length to the tube, giving i t a total length of 159 cm. To lower the pitch by another semitone would require adding 6% of this, or 9.54 cm. This means that valve 1 would have to add 9 + 9.54 = 18.54 cm to go down two semitones. Now the effective length of the tube would be 168.54 cm. The next step down would require an addition of 6% of this length, or 10.1124 cm, which may be rounded off to 10.11 cm. But the f i r s t valve adds only 9 cm, so the combina-tion of valves 1 and 2 is 1.11 cm short of the amount necessary to lower the pitch by a minor third. Now i f we wanted to sound the pitch lying a minor third below this one, we would have to add 6% of -134-178.65 cm, the present length, three times over, that is ([ (178.65) (1067c) ][ 106%f),(106%), which, round-ed off, equals 212.77 cm. But the basic length + valve 2 + valve 1 + valve 3 give a total of 150 + 9 + 18.54 + 28.65 = 206.19 cm, therefore 6.58 cm less than needed. l k Such a system i s , in fact, unsatisfactory, and requires modification in order to make i t into a usable arrangement. It is possible to use a system in which extra pieces of tubing are automatically ad-ded to the regular valve tubing whenever the third valve is used in combination with either of the other two. In such a system, only the two pitches produced with a l l three valves together are slightly out of tune. Unfortunately, a l l the extra pieces of tubing which are required by this arrangement add consider-ably to the complication and expense of manufacture. Consequently, this system is not commonly employed. Fortunately, there is another way of reducing the errors inherent in the valve system. If the third valve tube is made slightly longer than i t s correct acoustical length, then when this valve is used in combination with either of the f i r s t two, -135-the total length of the tubing is much nearer to it s correct value. Such a compromise closely re-sembles the compromise of equal temperament i t s e l f , in that the errors of intonation are reduced by be-ing spread more widely. With the third valve slide modified in this way, a l l pitches produced with the open tube, or with the f i r s t and second valve alone, w i l l be cor-rectly in tune. The f i r s t and second valves togeth-er w i l l give notes approximately one tenth of a semitone sharp; the third valve used alone w i l l give these same notes approximately one f i f t h of a semitone sharp. This is why the use of the f i r s t and second valves together is better than the use of the third valve alone. On the other hand, pitches produced with the second and third valves together are now very nearly in tune, and the combination 1 + 3 gives notes capable of correction by the play-er. The combination of a l l three valves together gives pitches which are about a third of a semitone sharp. 1 5 It is now clear why, when choosing fingerings, valve combinations were avoided in a l l cases except -136-those involving the combination 1 + 2 . Although pitches involving this combination are now well in tune, we s t i l l require improvement in intonation of pitches involving valve 3. This is especially noticeable when a l l three valves are used together. For this reason, on the modern trumpet, the valve slide of the third valve may be moved with the l i t t l e finger of the l e f t hand while the instrument is being played. By lengthening the slide when a l l three valves are down, the discrepancy may be removed. Finally, the player can do some correcting of discrepancies with his l i p s , sometimes as much as 3/4 of a semitone. Lip control is also necessary to compensate for intonation problems caused by d i f -ferences in temperature. If, however, l i p correc-tion is too great, the tone suffers, so i t is most advisable for the player to tune his instrument to the best of his a b i l i t y . 1 6 Having determined the best fingerings for a l l the notes of the normal range of the trumpet, we can now explore other p o s s i b i l i t i e s , which so far have not been considered. Taking Bb5 to be the top limit of the trumpet range, i t is possible to continue the -137-harmonic s e r i e s b u i l t on E3, F3, F#3, G3 and Ab3 f a r t h e r t h a n a t f i r s t , and s t i l l r e m a i n w i t h i n t h e range v V a l v e s N o r m a l l y A c c e s s i b l e A d d i t i o n a l Sounding. Frenuenc1es A c c e s s i b l e F r e q u e n c i e s  W i t h Bb5 Taken as TOD L i m i t Mode: 2 3 4 5 6 7 8 2 + 3 1 + 3 -& CL -e- -©-3z: 3x: ice -Q--e-1 + 2 + 3 XT - e — ^ 9 10 11 -O-Thus i t is possible to gain three more f i n -gerings for Bb5, two more fingerings for A5 and -138-G#5, and one more fingering for each of G5 and F#5. Of these, only one must be rejected, the A#5 which is the eleventh harmonic of E2, because eleventh harmonics, like sevenths, are considerably f l a t t e r than the corresponding note in the equal-tempered scale. But this s t i l l leaves eight additional f i n -gerings, which may at times be useful in simplify-ing a d i f f i c u l t passage, as may a l l the alternate fingerings given earlier. Testing the Equation v =* f ^ The testing of the equation v = f \ in prac-t i c a l situations involving the brass instruments w i l l have to be carried out by a method quite di f -ferent from that employed with the woodwind instru-ments. 1 7The reason for this is simply that we do not have an equation relating ^ m a x to L for a pipe which is partly cylindrical and partly conical. Neither ^ _ = 2L nor ^ = 4L holds true. What we shall max max do, therefore, is substitute known values of v and f into the equation, which w i l l give us *\, which we w i l l then compare to L. To begin with, we need to know the lengths of -139-a l l the parts of the B-flat trumpet. These lengths were measured and found to be as follows: Part Length (in cm) Mouthpiece 6.3 Leadpipe 33.0 First Bend 16.5 Middlepipe 20.7 Bell Section 63.5 Basic Length (equals total of above lengths) 140.0 Tubing added by valve 1 20.3 Tubing added by valve 2 12.2 Tubing added by valve 3 31.7 Tubing added by valves 1 + 2 32.5 Tubing added by valves 2 + 3 43.9 Tubing added by valves 1 + 3 52.0 Tubing added by valves 1 + 2 + 3 64.2 From these values we may easily find, by referring back to the table on Dage 129 and by addition, the total length of the pipe (L) used to produce the seven lowest pitches of the trumpet. We also need to include in the table the known frequencies of these pitches. Now, using v = 33853 cm/sec, we calculate ^ for each of the seven pitches. Finally, we compare 1 to L by solving for u the equation max ^ = uL for each of the seven values of ^ which max were just calculated. The completed table, then, is found on the following page. Ideal results would have been obtained i f the l a Examined Pitch Written: F#3 G3 Sounding: E3 F3 lb Known freauency of this pitch (hz): 164.81 174.61 Length of pipe (L) (cm): 204.2 192.0 'X (= 33853 cm/sec -r- f) (cm): 205.4 193.9 u (= "\ v -T- L): 1.01 1.01 max G#3 F#3 185.00 183.9 183.0 .995 A3 G3 196.00 172.5 172.7 1.00 Bb3 Ab3 207.65 160.3 163.0 1.02 B3 A3 152.2 153.9 1.01 C4 Bb3 140.0 145.2 1.04 220.00 233.08 -141-value of u turned out identical for each examined pitch. The values obtained, however, range from .995 to 1.04 and average out to 1.014. Now i t is more reasonable to assume that the average error in the calculations is .014 than to take the value of u to be precisely 1.014. If, therefore, we take *\ to be equal to (1)L, we can test the equation max v - f\ by the same method that was used for the woodwind instruments. The results thus obtained are excellent, as shown in the table on the follow-ing page. l a Examined Pitch Written: Sounding: lb Known frequency of this pitch (hz): Length of pipe (L) (cm): ^ (where = L) (cm): Calculated Frequency (33853 cm/sec -4- L) (hz): 2a Pitch whose fre-quency in equal temperament is nearest to that obtained above: Known frequency of this pitch (hz): Discrepancy of 2a with respect to la (by interval): F#3 G3 E3 F3 164.81 174.61 204.2 192.0 204.2 192.0 165.78 176.31 E3 F3 164.81 174.61 none none Ratio of calculated frequency with respect to lb: 1.01 1.01 G#3 F#3 185.00 183.9 183.9 184.08 F#3 185.00 none .995 A3 G3 196.00 172.5 172.5 196.25 G3 196.00 none 1.00 Bb3 Ab3 207.65 160.3 160.3 211.19 Ab3 207.65 none 1.02 B3 A3 220.00 152.2 152.2 222.42 A3 220.00 none 1.01 C4 Bb3 233.08 140.0 140.0 241.81 B3 246.94 - 2nd hghr. 1.04 -143-The French Horn It is not possible, for acoustical reasons, to construct a single horn of either true treble or true bass range which has real horn tone quality. 1 8 For this reason, the horn in common use today is a double instrument, consisting essentially of the combination of a horn in F and a horn in B-flat. The same mouthpiece and bell are used for each horn by being switched through two separate channels of different lengths by means of a special valve oper-ated by the l e f t thumb. The three main valves each have two sets of valve ports and two sets of valve slides, one for each channel, so the same three valves are used for both horns. The length of the main a i r column in the F horn is extremely great ~ about 370 cm; to keep i t s physical length to rea-sonable proportions, i t is coiled into a c i r c l e . Aside from the differences in timbre and range (which w i l l be discussed later), there are two fun-damental differences between the horn and the trum-pet. Fi r s t , while the trumpet rarely plays above the eighth resonance mode, the horn uses the series up to an octave higher, as far as the sixteenth. -144-Since partials which l i e above the seventh are a l l a diatonic whole step or less apart, the problem of accuracy on the horn is somewhat more serious than on the trumpet. Use of partials as high as the sixteenth requires that the high resonances of the instrument be pronounced and distinct. The shape of the a i r column i n the horn has been de-rived by experiment and experience to accomplish this; the leadpipe is in the form of a long cone with a gradual taper, the middlepipe is almost c y l -indrical, and the end flares out rapidly into a large b e l l . The second fundamental difference between the horn and the trumpet li e s in the construction of the mouthpiece. The horn mouthpiece differs from that of the trumpet in that i t has a deeper cup with a more gradual transition to the back bore. The acoustical reasons for this difference have s t i l l not been determined. Since the horn plays in a range where the modes are close together (as w i l l be shown in detail later), i t has more available notes and thus more f l e x i b i l i t y than, for example, the bugle. Nevertheless, i t s t i l l g i v e s o n l y the notes which correspond to i t s modes. For this reason, the length of the horn was origin-a l l y chosen to give a series of modes whose frequen-cies were in the key of the composition being played. This was done by using a single horn to-gether with interchangeable pieces called crooks, which could be inserted at the mouthpiece end of the horn to add to the length of the instrument and put i t in the desired key. This usually required the horn player to carry around with him several pounds of crooks. The development of valves and their application to horns eased the horn player's problems tremen-dously, because i t became possible to play i n d i f -ferent keys without having to change, crooks«. The additional length of tubing brought into use by a valve plays the same role- formerly played by a crook — that of putting the horn into a different key. The principle — that of inserting additional tub-ing — i s the same as for the trumpet, however, horn valves are constructed differently, being based on rotational rather than sliding action* 1 9 The working compass of the horn i s as follows: -146-Written: Sounding: exceptional I (on B-flat horn only) exceDtional {on B-flat horn only) Although this range is wide on paper, in prac-tice i t is obtainable only by the most outstanding players, because of the d i f f i c u l t y which arises in forming an embouchure capable of producing pitches in both extremes with certainty and comfort. Con-sequently, orchestral olayers specialize in either the higher or lower Dart of the range. Generally, the f i r s t and third players play parts written in the upper part of the compass, and the second and fourth players play the lower parts. Between them, the two pairs of players divide the complete normal range into two overlapping portions, as shown on the following page: -147-Horns 1, 3: v re-Horns 2, 4: W r i t t e n : Sounding: P W r i t t e n : Sounding: t r It goes without saying that these limits are not s t r i c t l y applied, especially i f , as is often the case, a l l four players are required to play in unison. Like most instruments, the horn has i t s "com-fortable" range, in which i t s tone is superior. This region is approximately from Ab3 to G5: Written: Sounding: As w i l l be shown l a t e r , t h i s i s t h e range o f t h e e i g h t h mode o f o s c i l l a t i o n o f t h e two h o r n s . Below t h e l o w e r l i m i t o f t h i s range the tone can e a s i l y be-come somewhat h a r s h and c o a r s e . Above t h e uooer l i m -i t t h e tone becomes b r i g h t , and a n i a n i s s i m o i s v e r y h a r d t o a c h i e v e t h e r e . I n i t s medium r e g i s t e r t h e h o r n i s a t i t s most c h a r a c t e r i s t i c t i m b r e . 20 Although the a i r column of the horn most r e -sembles a cone, i t s great l e n g t h of tubing makes the K f a c t o r v ery s m a l l . In f a c t , the tube i s .al-most c y l i n d r i c a l , erceot f o r some f l a r e i n the le a d p i p e and at the o m o s i t e end beyond the poi n t at which i t begins to f l a r e out i n t o a b e l l . Con-sequently, the horn does not behave l i k e an i n s t r u -ment belonging to e i t h e r the c y l i n d r i c a l o r the c o n i c a l groun. I t does, however, lend i t s e l f e a s i l y to overblowing and to o p e r a t i o n i n uo to 16 modes o f o s c i l l a t i o n , which correspond to the harmonic se-r i e s . Of these 16, modes 7, 11, 13, and 14 are not used because the r e s u l t i n g p i t c h e s are not i n tune i n e^ual -temperament.- Furthermore, the f i r s t mode i s not used because i t i s d i f f i c u l t to Produce owing to the g r e a t l e n g t h and narrow bore o f the i n s t r u -ment. Thus we are l e f t ^ T i t h 11 useable r e s u l t i n g p i t c h e s f o r each fundamental. The n a t u r a l harmonic s e r i e s of the F horn i s based on w r i t t e n C2, sound-ing F l , a p e r f e c t f i f t h lo r- 7er, and the useable mem-bers of i t s overtone s e r i e s are the f o l l o w i n g : P " = g CD J (7) (11) (13)(14) -149-In order to provide the horn with a complete scale, the intervals between the notes shown above must be f i l l e d in. The largest interval which needs to be f i l l e d in is the f i r s t one, that of a perfect f i f t h , which requires six additional semitones to f i l l the space. These are provided by lengthening the a i r column through the use of valves. Six addi-tional lengths are sufficient; in addition to f i l l i n g in the space between the f i r s t two modes, they allow the horn to play six semitones below the lowest pitch shown above, and are more than adequate to f i l l in the gaps between the higher modes — at least on paper. As in the trumpet, three valves are used in the horn. They function in the same way as trumpet valves do, shifting the original frequency — produced with-out any valves — downward. The following table shows the resulting shift of frequency (in semitones) for each valve combination: Valve(s) Number of semitones 2 1 3 1 + 2 2 + 3 1 + 3 1 + 2 + 3 1 2 3 3 4 5 6 ""--150-Through the use of these valves, the following pitches may be produced on the F horn as indicated: -151-This is so because each fundamental which results from a new combination of valves has i t s own over-tone series, the members of which are accessible in each case by overblowing. Although there are seven rows containing eleven notes each, only 43 of these 77 notes are different. These 43 notes have been subdivided into three categories as follows: 1) 16 pitches which appear only once — unnumbered, 2) 20 pitches which appear twice each ~ indicated with the same number in both places, 3) 7 pitches which appear three times each — i n -dicated with the same number underlined in a l l three places. The next step is to determine which valves the horn player would, in practice, use i f he wanted to produce these 43 pitches on the F side of his in-strument. Clearly, for the 16 pitches for which only one fingering is possible he has no choice. These pitches are indicated in whole notes in the table on page 153. For the remaining 27 pitches, the process of elimination is used to determine which fingering w i l l give the best result. Although -152-the fifteenth mode of oscillation may be used, pit-ches produced by i t are not as well in tune in equal temperament as pitches produced through modes 12 and 16. Since every note in this column may also be produced through the twelfth or sixteenth mode of oscillation, this entire column can be eliminated. This means that pitches 9, 18, 23, and 26 w i l l be produced through the sixteenth mode, i . e. by over-blowing, at the triple octave, and pitches 8, 17, and 22 w i l l be produced through the twelfth mode. For the 20 other remaining pitches, the one w i l l be cho-sen which comes f i r s t , (reading from top to bottom), because this way a l l but five of these pitches w i l l be produced through the use of one valve only. This i s , in general, better than the use of two or more valves, because the pitches are produced as lower harmonics from a higher fundamental, rather than as higher harmonics from a lower fundamental which would be more d i f f i c u l t . A l l 2 7 pitches are indicated in half notes in the table on the next page. For pi t -ches 6 and 15, the alternative use of the tenth mode of oscillation is also good; these alternatives are indicated in bracketed half notes. Valves Written Frequencies Produced Mode: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 + 3 1 + 2 + 3 The B - f l a t side of the horn w i l l now be exam-ine d . The n a t u r a l harmonic s e r i e s of t h i s horn i s based on sounding B b l , a p e r f e c t f o u r t h higher than the corresponding p i t c h o f the horn i n F. Ordinar-i l y , f o r a B - f l a t instrument, t h i s would be w r i t t e n as C 2 , a major second higher than i t sounds. I f , however, t h i s n o t a t i o n were used f o r the B - f l a t side of the french horn, these p i t c h e s would sound a ma-j o r second lower than w r i t t e n , w h i l e the remainder sound a p e r f e c t f i f t h lower than w r i t t e n . This prob-lem i s solved by tra n s p o s i n g a l l D i t c h e s w r i t t e n f o r the B - f l a t side of the hoi-n by a p e r f e c t f o u r t h up — as i f they were a l s o w r i t t e n f o r horn In F. Thus, the sounding p i t c h Bbl 1' = : : : : xrould be notated as psy. ^ Ws / n o y F2 -cr r a t h e r than as C2 This does not a l t e r the f i n g e r i n g s used, since any g i v e n sounding p i t c h i s f i n g e r e d the same way regard-l e s s of where i t i s notated. The n a t u r a l harmonic se-r i e s of the horn, then, i s based on w r i t t e n F2, and the highest useable member of i t s overtone s e r i e s i s the t w e l f t h harmonic: © Q = — . HE »7 _ - HE (1) ^ ~°~ (7) (11) --1.55-When we proceed, as before, to orovide this horn with a complete scale by f i l l i n g in a l l the intervals be-tween the notes shown above, we obtain the following possible frequencies: cn 4) i-l O c u ll X> T-l (A 0) CO + CM CM + * + + + O I-I CM r-l -156-Althotigh there are seven rows containing nine notes each, only 38 of these 63 notes are different. These 38 notes have been subdivided into three cat-egories as follows: 1) 18 pitches which appear only once — unnumbered, 2) 15 pitches which appear twice each — indicated with the same number in both places, 3) 5 pitches which appear three times each — in-dicated with the same number underlined in a l l three places. As before, the next step is to determine which valves the horn player would, in practice, use i f he wanted to produce these 38 pitches on the B-flat side of his instrument. Clearly, for the 18 pitches for which only one fingering is possible he has no choice. These pitches are indicated in whole notes in the table on the following page. For the remain-ing 20 pitches, the one*: is chosen which comes f i r s t , reading from top to bottom, for the same reasons as before. For reasons to be shown later, exceptions w i l l be made for pitches 3 and 10. The alternate fingerings w i l l be used for these two pitches, a l -though the fingerings which would normally be chosen -157-may also be used. A l l 20 pitches are indicated in half notes in the table below. For pitches 7 and 14, the use of the twelfth mode is also good; these a l -ternatives are indicated in bracketed quarter notes. u c <u g| or u c o; 4J 4J •H u s ON •D + U o CO + CM en + + CM + -158-Th e final tables for each of the two horns may now be superimposed, showing the sounding pitches for each written pitch shown before. For each pair of notes, the top one is produced on the B-flat side of the horn; the bottom one is produced on the F side. r-l CO ft) •o CO + CM CO co CO CM + U + + + o ft CM i - l r-l -159-Although there are 81 notes, only 43 of them are different. These 43 notes have been subdivided into two categories as follows: 1) the 5 lowest pitches which appear only once — indicated in quarter notes and unnumbered, 2) the remaining 38 pitches which appear twice each indicated in whole notes and with the same number in both places. The above table shows that of the entire range of 43 notes of the whole horn, 38 can theoretically be produced on either horn. In practice, however, a l l the pitches of the lower half of the range are produced on the F horn, and those of the upper half on the horn in B-flat. This, of course, is the whole purpose of having two horns — so that low harmonics are used rather than high ones. Several more specific observations can be made from the table: 1) The use of the sixteenth mode of the horn in F is really never necessary; i t only provides the player with optional fingerings for the top five pitches of the range of the instrument. 2) Only three of the pitches in the twelfth column -160-(pitches 8, 20, and 30) cannot be produced through a lower mode of oscillation. 3) Only two of the pitches in the tenth column (pitches 7 and 19) cannot be produced through a lower mode of oscillation. 4) Only two of the pitches in the ninth column (pitches 6 and 18) cannot be produced through a lower mode of oscil l a t i o n . The conclusion to be reached from these obser-vations is that the use of modes (of either horn) higher than the eighth is only necessary for the top 7 of the 43 pitches of the instrument^ normal range: Written: Sounding: The complete chormatic scale of the entire horn may now be written out, showing the fingerings (by valve numbers) and the modes of oscillation used. A l l the notes are written as for horn in F, therefore a l l sound a perfect f i f t h lower than written. When the B-flat horn is to be used, this is indicated by the letter "T" for "thumb key". 2 1 -161-'Iz^^^^jdjzn^jp^^ -..i!.:'"T - ± L__ second 1 I t h i r d F: 1 1 2 1 1 2 0 1 1 2 1 1 2 0 2 3 3 2 2 3 3 2 3 3 r,^4o4^-b no '--ft \ . —y- _____ . • • . t ^ r ^ _ _ 0 . _ T € t _ ^ ^ _ , -y-I — f o u r t h 1 L- f i f t h - J I — s i x t h - J F: 2 1 1 2 0 1 1 2 0 1 2 0 3 2 2 Bb: 2 3 , T 5~ i , . ~ JZfZjBJZ L F: 2 3 0: 1 2 T A l t e r n a t e ( f i f t h n o d e ) ; seldom used. 1 2 0 2 1 1 2 0 T T T 3 2 T T T T T s i x t h j i — e i g h t h - e i g h t h 1 1 2 2 0 - J X Bb: 2 T 0 T I n i n t h - 1 \-U Q ~ 2 3 2 T 1 2 0 T I t e n t h — A l t e r n a t e ( t w e l f t h mode); seldom used, 1 T 2 0 T T 1 t w e l f t h 1 ho. -fyi. z&z -162-The overlap from Ab4 to C5 represents the rec-ommended area for switching horns. This is why the alternate fingerings for the pitches Ab4 and A4 (ori-ginally numbered 3 and 10) were chosen. This way, the fingerings for these pitches are easy to remem-ber because they are identical for the two horns. The regular fingerings may also be used, as may the other alternatives shown above. The choice is often made according to the key in which a piece of music is written; certain fingerings being superior to others in specific keys. In practice, the side of the horn to be used to produce these five pitches depends on the context in which they appear. When approached from below, they are played on the F side, and when approached from above, they are played on the B-flat side. In certain cases, a l l pitches up to C5 may be produced exclusively on the F horn. It may be added here, that i t is one of the horn player's obligations to learn to match the sound of the two sides of his instrument to each other, especially in passages where changing sides is involved. -163-Use of the Right Hand Prior to the development of valves, when crooks were s t i l l being used, i t was discovered that i f the horn were shaped so that the hand could be placed in the be l l , the pitch of the instrument could be low-ered by an amount that depended on how far the hand was inserted. Placing the hand in the horn bell re-stricts i t s area; this increases i t s acoustic mass at the end of the resonant air column, and lowers the freouencies of the horn. Furthermore, the hand acts as a reflector and sends the resonances back into the horn. It fixes the notes into position, which facil i t a t e s accuracy in playing by making the notes easier to land upon. The technique of hand closure was developed to the point where i t was pos-sible to play a complete chromatic scale over a part of the valveless instrument's range. However, the hand-closed tones differed in quality from the open tones, and much practice was required to get a rea-sonably even scale.* 1 With the invention of the valve, hand stopping for the purpose of obtaining chromatic notes became unnecessary but the practice was retained as a -164-means of varying tone colour and f a c i l i t a t i n g play-ing. Nowadays, the horn is s t i l l played with the right hand in the b e l l , so that the pitch is always slightly f l a t t e r than the length of the tubing should produce. For this reason, horns are commonly built slightly sharp. 2 2 Generally speaking, i f the horn is played with-out the hand in the b e l l , i t s tone i s somewhat coarse and brassy. Placing the hand in the bell has a re-fining effect on the tone, giving i t a slightly veiled and quite characteristic colour unlike that of any other instrument. The position of the hand in the bell should be, generally speaking, such as to allow almost unob-structed passage of the sound between the palm of the hand and the inner side of the b e l l . The exact position of the hand can only be determined by t r i a l and error; almost everything depends upon the size of the hand, the bore of the instrument, and the tone quality which is desired. The more the bell is covered, the darker the tone w i l l be, and the fl a t t e r the pitch w i l l be. Conversely, the less the bell is covered, the brighter the tone w i l l be, and -165-the sharper the p i t c h . 2 3 If the bell is closed off as much as possible with the hand, a tone of quite different quality, commonly called a stopped tone, is produced. The pitch apparently rises by about a semitone when this is done, so the player must transpose down. What happens during hand stopping is an extension of the hand-closure effect described above. The acoustic mass at the end of the horn is increased so much that the frequencies of a l l the modes are pulled down to where each mode has moved to about a semi-tone above the original frequency of the mode below i t . A given note is then played in the next higher mode when the horn is stopped, the one originally used for the note having gone much too f l a t . This can be demonstrated by blowing a note on the horn while the right hand slowly closes off the bell and. reaches stopping position. If no attempt is made to keep the note in tune by using the l i p s , the mode can be followed smoothly down in frequen-cy to its final position, below its original val-ue. If the player tries to hold the pitch, the note jumps abruptly to the next (higher) mode as -166-the hand is inserted, and ends up higher in frequen-cy than i t s original value. A given stopped note is thus played in a higher mode than the unstopped note. Testing the Equation v = f V 5 To test the equation v = f*\ in a practical s i t -uation involving the horn, the same method must be used that was used for the trumpet. Again, since the horn is partly cylindrical and partly conical, nei-ther =» 2L nor *X = 4L holds true. Conse-max max quently, we must substitute known values of v and f into the equation, which w i l l yield ^, which we wi l l then compare to L. It might be pointed out here, that even i f \^ „ = 2L or ^ = 4L were true, another problem max max ' r would arise i f we tried to adopt exactly the same procedure that was used for the woodwind instruments. As was the case with the other instruments of the conical bore family, we would here also have to ex-trapolate the cone to i t s theoretical peak. Since, however, we are dealing with two horns in one, we would have to make two extrapolations: one for each horn. This would result in two different values of length (previously denoted by r^) and of volume for the missing portion of the cone, which i s reolaced by the mouthpiece. Consequently, we would conclude that a different mouthpiece must be used for each side of the horn — the volume of each one being equivalent to the volume of the missing portion of the cone. Since we know that the same mouthpiece is used for both sides of the horn, we know that this approach would be incorrect simply because i t is impossible for one mouthpiece to be of two di f -ferent volumes. We proceed now with the testing of the equation. It w i l l be best to work on the two sides of the horn separately, treating them as individual instruments. Let us f i r s t test the horn in F. To begin with, we need to know the lengths of a l l the parts of the horn. These lengths were measured and found to be as follows Part Length (in cm) Mouthpiece 4.6 Leadpipe 81.7 Middlepipe 148.2 Bellpipe 124.0 Bell 14.0 Basic Length (equals total of above lengths) 372.5 -168-Tubing added by valve 1 Tubing added by valve 2 Tubing added by valve 3 Tubing added by valves 1 + 2 Tubing added by valves 2 + 3 Tubing added by valves 1 + 3 Tubing added by valves 1 + 2 + 3 44.0 19.0 72.5 63.0 91.5 116.5 135.5 (The terms "leadpipe, middlepipe, and bellpipe" are used for convenience only. Actually, the tube is quite continuous, and contains many bends rather than consisting of three distinct sections.) Although this has no direct application to the testing of the equation, one additional observation may be made at this point. The table above shows that the length of tubing added by valve 3 is ac-tually greater than that added by valves 1 + 2 . (This is also true of the horn in B-flat.) With the trumpet, the opposite was the case. This is be-cause modification of the third valve slide of the horn must meet different requirements than the cor-responding modification in the trumpet. Because the higher harmonics of the horn are used more, the third valve is used less often than in the trumpet. Returning now to the experiment, from the val-ues obtained by measurement we may easily find, by referring back to the fingering chart on page 153 -169-and by addition, the total length of the pipe (L) used to produce the 7 lowest pitches of the horn. We also need to include in our table the known fre-quencies of these pitches. Now, using v = 33853 cm/sec, we calculate^ for each of the 7 pitches. Finally, we compare to L by solving for w the equation 'A = wL for each of the 7 values of \^ - max which were just calculated. The completed table, then, is found on page 171. Before interpreting the results, let us test the B-flat side of the horn. Again, we start by measuring the lengths of a l l the parts of the horn. Part Length (in cm) Mouthpiece 4.6 Leadpipe 81.7 Middlepipe 55.1 Bellpipe 124.0 Bell 14.0 Basic Length (equals total of above lengths) , 279.4 Tubing added by valve 1 35.0 Tubing added by valve 2 15.0 Tubing added by valve 3 56.0 Tubing added by valves 1 + 2 50.0 Tubing added by valves 2 + 3 71.0 Tubing added by valves 1 + 3 91.0 Tubing added by valves 1 + 2 + 3 106.0 It was not possible to determine by observation the length of tubing cut out by the thumb key. With -170-the basic length of the F horn being 372.5 cm, the basic length of the B-flat horn was assumed to be 372.5(3/4) = 279.4 cm, i . e. 93.1 cm less. Thus, the length of the middlepipe was estimated to be 148.2 - 93.1 - 55.1 cm. From the values above, we again find, by refer-ring to the fingering chart on page 157 and by ad-dition, the total length of the pipe (L) used to produce the 7 lowest possible pitches of the B-flat side of the horn. (In practice these pitches are produced on the F side of the horn.) We also i n -clude in our table the known frequencies of these pitches. Now, using v = 33853 cm/sec, we calculate *\ for each of the 7 pitches. Finally, we compare *\ to L by solving for x the equation *A = xL ^ max J e ' max for each of the 7 values of ^ which were just calcu-lated. The completed table, then, is found on page 172. Looking at the two tables together, we can make the following observations: 1) The values of w range from 1.04 to 1.08 and ave-rage out to 1.054; the values of x range from 1.04 to 1.07 and average out to 1.049. In each case, Horn In F l a Examined Pitch Written: F#2 G2 Ab2 A2 Bb2 B2 C3 Sounding: Bl C2 Db2 D2 Eb2 E2 F2 lb Known frequency of this pitch (hz): 61.735 65.406 69.296 73.416 77.782 82.407 87.307 Length of pipe (L) (cm): 508.0 489.0 464.0 435.5 416.5 391.5 372.5 'X (= 33853 cm/sec - i - f) (cm): 548.4 517.6 488.5 461.1 435.2 410.8 387.7 w (= 7 -r- L): 1.08 1.06 1.05 1.06 1.04 1.05 1.04 max i Horn in B-flat l a Examined Pitch Written: B2 C3 C#3 D3 Eb3 E3 F3 Sounding: E2 F2 F#2 G2 Ab2 A2 Bb2 lb Known frequency of this pitch (hz): 82.407 87.307 92.499 97.999 103.83 110.00 116.54 Length of pipe (L) (cm): 385.4 370.4 350.4 329.4 314.4 294.4 279.4 (= 33853 cm/sec -j - f) (cm): 410.8 387.7 366.0 345.4 326.0 307.7 290.5 N> I 1.07 1.05 1.04 1.05 1.04 1.05 1.04 -173-A deviates from 1.00 the most for the lowest max pitch ~ the one whose production involves the use of a l l three valves. This is consistent with what was discussed about the nature of the valve system in the chapter on the trumpet. When a l l three valves are used together, the total length of tub-ing they add is insufficient for production of the desired pitch. This was sufficiently noticeable with the trumpet, but with the french horn i t is even more noticeable, because greater lengths of tubing are involved. In practice, horn players move the third valve slide out with the l i t t l e finger when using a l l three valves, and thus remove the discrepancy. In the calculations, however, nothing more was added to the basic length than the amount of tubing added by the three valves when used in-dividually. This showed up very clearly in the re-sults. 2) In view of the above, i f we exclude from con-sideration the lowest pitch of each of the horns, the values of w and x average out to 1.05 and 1.045 respectively. If we want to claim that, in reality, A = (1)L in both cases, these deviations must -174-be accounted for. In fact, the discrepancy evident in these results was already predicted on page 164, where the use of the right hand was discussed. Gen-erally, in any open tube, the terminal antinode is formed slightly beyond the open end, which makes the tube's acoustical length slightly longer than i t s physical length. This phenomenon was already en-countered with the second instrument which was test-ed, the flute. The difference between the two lengths, known as the end correction, is especially great when the open end is obstructed; this is ex-actly the case when the hand is inserted into the b e l l . For the horn in F, the end correction has turned out to be 57» of the basic length, i . e. 18.6 cm, and for the horn in B-flat, i t appears to be 4.57, of the basic length, I. e. 12.5 cm. Thus i t seems that the longer the tube, the greater percent-age of i t s basic length must be added as end correc-tion to determine the actual acoustical length of the tube. Clearly, the small amount of testing which has been done here is insufficient for making general conclusions. The difference between the two values (.57.) could well be attributable to noth--175-ing but experimental error. In any event, i f we take ^ to be equal to J ' max M (1)L plus end correction, we can test the equation v = fA by the same method that was used for the woodwind instruments. The results thus obtained are excellent for both horns, as shown by the ta-bles on the following two pages. l a Examined Pitch Written: Sounding: lb Known frequency of this pitch (hz): Length of pipe plus end correction (L+) (cm): ^ (where 'a = L+) (cm) max Calculated frequency (33853 cm/sec -r L+) (hz): 2a Pitch whose frequency in equal temperament is nearest to that obtained above: Known frequency of this pitch (hz): Discrepancy of 2a with respect to la (by interval): Ratio of calculated frequency with respect to lb: Horn in F F#2 G2 Ab2 A2 Bb2 B2 C3 Bl C2 Db2 D2 Eb2 E2 F2 61.735 65.406 69.296 73.416 77.782 82.407 87.307 526.6 507.6 482.6 454.1 435.1 410.1 391.1 526.6 507.6 482.6 454.1 435.1 410.1 391.1 64.286 66.692 70.147 74.550 77.805 82.548 86.558 C2 C2 Db2 D2 Eb2 E2 F2 65.406 65.406 69.296 73,416 77.782 82.407 87.307 - 2nd hghr, none none none none none none 1,04 1.02 1.01 1,02 1.00 1.00 ,991 l a Examined Pitch Written: Sounding: lb Known frequency of this pitch (hz): Length of pipe plus end correction (L+) (cm): '"a (where 7 = L+) (cm) max Calculated frequency (33853 cm/sec -f- L+) (hz): 2a Pitch whose frequency in equal temperament is nearest to that obtained above: Known frequency of this pitch (hz): Discrepancy of 2a with respect to la (by interval): Ratio of calculated frequency with respect to lb: Horn in B-flat B2 E2 82.407 397.9 397.9 C3 F2 87.307 382.9 382.9 C#3 F#2 92.499 362.9 362.9 D3 G2 97.999 341.9 341.9 Eb3 Ab2 103.83 326.9 326.9 E3 A2 110.00 306.9 306.9 F3 Bb2 116.54 291.9 291.9 85.079 88.412 93.285 99.014 103.56 110.31 115.98 F2 F2 F#2 G2 Ab2 A2 Bb2 87.307 87.307 92.499 97.999 103.83 110.00 116.54 - 2nd hghr. none none none none none none 1.03 1.01 1.01 1.01 .997 1.00 .995 -178-The Trombone The body of the modern slide trombone consists of two straight cylindrical tubes, the mouthpipe and the middlepipe, plus a conical bellpipe which flares out much more rapidly than does, for example, the bellpipe of the trumpet. The three sections are par-a l l e l to each other and are connected by two semicir-cular bends. The f i r s t of these two bends is cylind-r i c a l , but the modern bell-bow is conical, so that the expansion of the bell actually begins somewhere in the bend. Sometimes the bell and the bow are man-ufactured as one piece. The greater part of the mouthpipe and the middlepipe form the inner of two pairs of closely f i t t i n g telescopic tubes, the outer pair of which is connected by the other bend which results in the U-shaped slide. The slide is freely movable, and by pulling i t out, the overall length of 2 6 the body may be extended by nearly two-thirds. The family of slide trombones in use today con-sists of five instruments of various sizes: the Sop-rano, normally in B-flat (practically nonexistent), the Alto, normally in E-flat or F (rare), the Tenor in B-flat, the Bass in B-flat (convertible to F; -179-sometimes called the Tenor-Bass Trombone), and the Contrabass, normally in B-flat. A l l members of the slide trombone family are constructed in basically the same way — the structural difference is mainly one of size. The bore diameters of modem Tenor Trombones vary from 1.27 cm to 1.39 cm; that of the Bass Trombone is approximately 1.42 cm. Because of the great similarity of the five instruments to each other, only one w i l l be considered here: the Tenor in B-flat. Although two-thirds of the trombone bore is cyl-indrical, the trombone does not behave like an instru-ment belonging to the cylindrical group. It does, however, lend i t s e l f easily to overblowing and to op-eration in at least 12 — and in the hands of good players up to 16 — modes of oscillation. Further-more, the successive modes of oscillation coincide with the complete harmonic series. For practical pur-poses, i t i s best to take the twelfth harmonic as the last accessible one. Consequently, the trombone, as described thus far, is naturally capable of producing the following pitches: -180-Written and Sounding: In order to provide the trombone with a com-plete scale, the intervals between the notes shown above must be f i l l e d i n. For the moment, we w i l l not try to f i l l in the notes forming the f i r s t (and widest) interval of the octave, since this would re-quire 11 additional semitones to f i l l the space. Instead, we xvill aim to f i l l in the next interval of a perfect f i f t h , which requires only 6 additional semitones. This is accomplished by lengthening the ai r column by pulling the slide out an appropriate distance. Six additional lengths obtained in this way w i l l be sufficient for the present goal; in addition to f i l l i n g in the space between the second and third harmonics, they allow the trombone to play six semi-tones below the second harmonic, and are more than adequate to f i l l in the gaps between the higher modes. The trombone; slide, then, can be placed in seven d i f -ferent useable positions, as shown on the next page: -181-P l a y i n g p o s i t i o n s of t h e t r o m b o n e . Through the use of these p o s i t i o n s , the pi t c h e s shown i n the t a b l e on the f o l l o w i n g page may be produced as i n d i c a t e d . This i s so because each fundamental which r e s u l t s from a new s l i d e p o s i t i o n has i t s OVTU over-tone s e r i e s , the members of which are a c c e s s i b l e i n each case by overblowing. Although there are seven rows c o n t a i n i n g twelve notes each, o n l y 45 of these 84 notes are d i f f e r e n t . These 45 have been subdivided i n t o f o u r c a t e g o r i e s as f o l l o w s : 1) 23 p i t c h e s which appear only once —unnumbered, 2) 10 p i t c h e s which aonear twice each — i n d i c a t e d w i t h the same number i n both p l a c e s , 3) 7 o i t c h e s which appear three times each — i n d i -cated w i t h the same number u n d e r l i n e d i n a l l three p l a c e s , 4) 5 p i t c h e s which appear four times each — i n d i -cated w i t h the same number c i r c l e d i n a l l f o u r places. -183-The next step is to determine which positions the trombone player w i l l , in practice, use to pro-duce each of these 45 pitches of the range of his in-strument. Clearly, for the 23 pitches for which only one position is possible he has no choice. For the remaining 22 pitches, the process of elimination is used to determine which positions w i l l give the best result. In general — although not always — t h e shortest available position, i . e. the one which comes f i r s t , (reading from top to bottom) w i l l be chosen. This is because a trombonist usually prefers to use the shorter positions rather than the longer ones, for reasons to be shox-m later. Consequently, we be-gin by eliminating a l l "third choice" and "fourth choice" positions for the 12 pitches for which these choices are theoretically available. While i t is true that the trombone player may very accurately adjust the tube length for any pitch he wishes to produce, i t is also true that for-sev-enth and eleventh harmonics the normal positions w i l l always yield pitches which w i l l deviate slightly from the ones desired, and shortening of the tube w i l l a l -ways be necessary. Since i t is physically impossible -184-to shorten f i r s t position, the highest pitches of this position in these two modes (Ab4 and Eb5) are automatically eliminated. Furthermore, since the other pitches of mode 11 are easily obtainable in other modes, this entire mode is eliminated. The eliminations which have been made yield the following results: 1) The 23 pitches x^hich originally appeared only once are unaffected. 2) Only one choice of position remains for pitches 5, 8, 9, and 17. 3) Two choices of position are l e f t for the remain-ing 18 pitches of the trombone range: pitches 1, 2, 3, 4, 6, 7, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, and 22. In the table on the follox^ing page, the most common choice for each of these pitches is in-dicated with a quarter note, and the second choice xtfith a bracketed quarter note. Although "third choice" and "fourth choice" positions have not been included in the table on page 185, they do exist, and should not be totally ruled out. Some trombonists, especially jazz play-ers, do make use of these positions. At any rate, -186-i t is clear from this table that the use of positions 6 and 7 is unavoidable for production of only 6 of the 45 pitches of the range of the trombone. It was decided previously not to begin by trying to f i l l in the 11 semitones between the f i r s t two harmonics, Bbl and Bb2. As is now evident, 6 of these have meanwhile been f i l l e d in through the 6 additional positions which have just been examined. Thus, only 5 semitones — Bl, C2, C#2, D2, and Eb2 — s t i l l need to be f i l l e d in. These 5 semitones are not obtainable unless the instrument is fitt e d with what is called an "F attachment". This device w i l l be discussed further in a separate section. The chromatic scale of the tenor trombone may now be written out, showing the positions and the modes of oscillation used. 2 8 f i r s t Not available with-out F attachment ~7T 7 6 5 4 3 2 1 -187-- second- 1 r •third -e- ° 7 6 5 w U o I rr 1 n tit) -a—e-5 4 3 2 1 7 6 5 4 3 2 1 Alternate: (fourth mode): ; 7 6 fourth- -fifth- i r sixth 32: H E 3 2 ~0~ 1 5 4 3 2 1 Alternates: 7 6 5 (f i f t h mode) 7 6 5 4 (sixth mode) 6 5 4 (seventh mode) p-seventh-jp-eighth—| (-ninth-] (-tenth-] (— twelfth—j \>n jo __ 5 2 3 2 1 Alternates: 3 2 5 4 (eighth mode) 4 3 (ninth mode) 4 3 (tenth mode) -188-The Slide The use of a slide is the simplest method avail-able for lengthening a tube. A slide, however, demands that the tube be perfectly cylindrical for the entire length over which i t is to operate. The trombone is the only instrument which uses this system. As was shown by the diagram on page 181, the slide arrange-ment of the trombone provides six additional lengths to give the additional six semitones. Going down a semitone amounts to approximately a 6 percent decrease in frequency, corresponding to about a 6 percent in-crease in length. Since the total length of the tube increases as the scale is descended, so does the 6 percent of the total length which must be added in or-der to descend another semitone. Consequently, each additional increase in length is slightly greater than the previous one, as the diagram on page 181 shows. The trombonist learns the precise locations of these 29 positions through thousands of repetitions. The slide system of changing tube length has one basic disadvantage. While a shift of one or two posi-tions may be made quickly and accurately enough, a shift of five or six positions is hard to execute -189-quickly and accurately, and takes up an amount of time which is not negligible. In this respect the valve system is clearly the superior one, because valve action is instantaneous, so the time taken up in depressing valves is negligible. In the case of the trombone, one major d i f f i c u l t y arises as a con-sequence of the nature of the system: many passages become practically impossible to play because of the awkward changes in positions involved. The following example shows one such passage which trombonists would rather not see. 3 0 Allegro /'fry i j 3 J ^ J". ^  |J-|J i ^JTj 1 f t\ 1 7 2 6 7 1 7 2 5 The conclusion to be drawn is that the trombone is at i t s best in the lowest positions. Thus, not only are wide shifts (such as 1 to 6 or 7) eliminat-ed, but also the distances between adjacent positions are shorter. As can be seen from the diagram on page 181, the distance between positions 1 and 2 is smal-ler than that between 6 and 7, so the shift takes -190-proportionately less time. Aside from this, produc-tion of sound is easier from a shorter tube than from a longer one. Furthermore, sounds produced as lower-numbered partials of fundamentals based on a shorter tube length are somewhat brighter In colour than the corresponding higher-numbered partials from a longer tube. This is why, in determining which position is generally used to produce any given pitch, the shortest available position was chosen. Exceptions, however, do occur. In a passage in which positions 5, 6, and 7 predominate by necessity, i f one note appears which may be produced using a high or a low position, the trombonist would choose the high position, simply because of convenience. In the following passage, for example, F3 could be produced in f i r s t position, but because of the context, sixth position would be the better choice. Allegro 6 6 7 6 6 6 2 4 6 (1) In general, because of the greater length of tubing -191-involved, a richer tone results when a pitch is pro-duced in a higher position. It goes x<dthout saying that the trombonist should be aware of a l l the positions in which the pitches of the range of his instrument may be produced, and have these at his disposal as practical alternatives in special situations. The F Attachment The F attachment is an extra piece of tubing which is built in at the bend connecting the bellpipe to the middlepipe. It is activated by a trigger us-ing a rotary valve operated by the l e f t thumb. When this is done, the result is similar to that obtained by using the thumb key on the french horn: the instru ment becomes a trombone in F. Consequently, the p i t -ches which are available in f i r s t position are the following: Written and Sounding: With the extra length of tubing having been ad--192-ded, the distances between adjacent slide positions are greater than they were before. Accordingly, the slide is now only long enough to accomodate six positions, of which only the f i r s t is in the same location as that of the tenor trombone. The remain-ing five positions relate to the original locations as follows: 3 1 Tenor-Bass Trombone Tenor Trombone 2 \> 2 3 \> 3 4 #5 5 [ 6 6 1?7 As was done previously, the entire table of theoretically possible frequencies is presented on the next page. In examining this table, we are i n -terested primarily in: 1) The six pitches which are obtainable on the tenor trombone only in positions 6 or 7, 2) Any new pitches which are not obtainable on the tenor trombone. By comparing the pitches in this table to those in the table on page 185, we see that a l l the pitches originally produced through positions 6 and 7 are now available in positions 1 and \l re-r Position Possible Written (and Sounding) Frequencies with F Attachment With Trigger Mode: 1 2 3 4 5 6 7 '8 9 10 11 12 -• - .• • .'  •'if'" _ _ ' .. _ X .. 8--J U > l l " ^ " t o ^ L — =J* o ^ 8—' -t^FlJ l" T ^ to ^~ — o — 8 > -194-spectively. Regarding 1) above, the six pitches in question are indicated in half notes in the above table. (The seven alternatives indicated in bracketed notes in positions 6 and 7 in the table on page 185 may noxvT be regarded as alternatives in positions 1 and \/2 respectively, although in practice they would be used in this way very rarely.) Regarding 2) above, we find that eight new p i t -ches appear: four of the five which formed the "gap" between the f i r s t two positions of the tenor trom-bone, and the lower octaves of these four pitches. These eight pitches are indicated in quarter notes in the above table. A T\?ord of explanation is in order regarding the only remaining pitch from the original "gap" which is s t i l l not obtainable: Bl. This is due directly to the fact that the seventh position has been lost be-cause of the great lengths of tubing involved. A further consequence of this factor is that the low C's (Cl and C2) are almost certain to be too sharp, because the slide is not even long enough for a true sixth position. In practice, i t is usually possible -195-to pull the tuning slide out of the attachment to provide the extra length necessary for production of these three pitches. The other alternative is to provide the instrument with a second attachment, tvrhich is operated by a second trigger. This is com-monly found on bass trombones, but not on tenor trom-bones. 3 2 As can be seen from the table above, the F at-tachment orovides more alternative positions for many pitches other than those which have already been dis-cussed. If the trombonist is aware of a l l the alter-natives the F attachment brings with i t , he w i l l have these at his disposal in special situations. Just how valuable a resource the F attachment actually is can be clearly demonstrated by quoting once more the passage given on page 189. With the use of the F attachment, the passage becomes very sim-3 3 pie to play: Allegro \l 1 b2 2 1 \l 1 b2 2 5 \>2-T T T T T T -196-Testing the Equation v = f"\ 3 k As was done with the trumpet and the horn, we now wish to test the equation v = in a practical situation involving the trombone. As mentioned pre-viously, the trombone tube is cylindrical for about two-thirds of i t s length. The proportion of cylind-r i c a l to conical length, however, Increases with the positions used. A l l this means that the trombone does not behave like an instrument belonging to e i -ther the cylindrical or the conical group. Conse-quently, we cannot use ^ = 2L or 1 = 4L, and J 3 max max ' must again substitute known values of f and v into the equation, which w i l l give us ^ , which we w i l l then compare to L. The parts of a trombone in f i r s t position were measured and found to be as follows: Part Length (in cm) Mouthpiece 5.5 First Slide Stock 7.2 Slide 142.3 Second Slide Stock 8.2 Bellpipe 113.8 Basic Length (equals total of above lengths) 277.0 Now we need only to measure the amount of tubing ad-ded by each subsequent position of the slide to find, -197-by addition, the total length of the tube for each position. Length of tubing Ad- Total Length Position ded to Basic Length of.Pipe (cm) 2 10.0 287.0 3 29.6 306.6 4 49.6 326.6 5 67.2 344.2 6 90.8 367.8 7 110.2 387.2 Referring back to the table on page 185, we may find the seven lowest pitches of the trombone (called the pedal notes) which correspond to the seven positions. We also include in our table the known frequencies of these pitches. Now, using v = 33853 cm/sec, we calculate *\ for each of the seven pitches. Finally, xce comoare ' A to L by solving J ' max J b for y the equation ^ = yL for each of the seven J max J values of ^ which were just calculated. The complet ed table, then, is found on the following page. Ideal results would have been obtained i f the value of y turned out to be exactly 2.0 for each examined pitch. In each case, however, the value of y is 2.1. If we want to claim that, in reality, A = 2L, this deviation must be accounted for. max ' By comparison to the corresponding deviations for la Examined Pitch: El Fl lb Known frequency of this pitch (hz): 41.203 43.654 Length of pipe (L) (cm): 387.2 367.8 \ (= 33853 cm/sec -S- f) (cm): 821.6 775.5 y ( = ^max - L> : 2 ' X F#l 46.249 344.2 732.0 2.1 Gl 48.999 326.6 690.9 2.1 Abl 51.913 306.6 652.1 2.1 Al 55.00 287.0 615.5 2.1 Bbl 58.270 277.0 581.0 2.1 -199-the two sides of the french horn, we see that this deviation is nearly identical: Horn in F (average discrepancy of w) 5.0% Horn in B-flat (average discrepancy of x) 4.5% Tenor Trombone (average discrepancy of y) 5.0% Consequently, i t is most reasonable to assume that this discrepancy is attributable to end correction (and experimental error), as was the case with the flute and the french horn. If we take ^ to be equal to 2L plus 5% end max 3 r correction, we can test the equation v = by the same method that was used for the woodwind instru-ments. The results thus obtained are excellent, as shown by the table on the following page. l a Examined Pitch: El F l lb Known frequency of this pitch (hz): Length of pipe plus end correction (L+) (cm): A (where 1 = 2L+)(cm): max Calculated frequency (33853 cm/sec -f- 2L+) (hz): 2a Pitch whose frequency in equal temperament is nearest to that obtained above: Known frequency of this pitch (hz): Discrepancy of 2a with respect to la (by interval): Ratio of calculated frequency with respect to lb: 41.203 43.654 406.6 386.2 813.2 772.4 41.630 43.828 El Fl 41.203 43.654 none none 1.01 1.00 F#l 46.249 361.4 722.8 46.836 F#l 46.249 none 1.01 Gl 48.999 342.9 685.8 49.363 Gl 48.999 none 1.01 Abl 51.913 321.9 643.8 52.583 Abl 51.913 none 1.01 Al 55.000 301.4 602.8 56.160 Al 55.000 none 1.02 Bbl 58.270 290.1 580.2 58.347 Bbl 58.270 none 1.00 -201-The Tuba The tuba is the lowest—lying member of the brass family. Since i t is capable of producing pit-ches which are even loxvjer than AO (2 7.5 hz), the lowest note of a piano, i t s basic length is extreme-ly great — about 550 cm. To reduce i t to manageable proportions i t is coiled four times around. Although the tube is actually quite continuous, for conven-ience in description, i t may be considered to consist of four sections: the leadpipe, middlepipe, b e l l -pipe, and b e l l . The middlepipe by i t s e l f is very short, but is lengthened through the use of valves, which add tubing to the basic length. This section is largely cylindrical; the other three sections are conical, but their K factor is not constant through-out. The two most common tubas in use today are in BB-flat and C. (Two less common models are in E-flat and F.) The BB-flat tuba is used in bands and opera orchestras; the C tuba is the symphonic instrument on this continent. For consistency with the other brass instruments, only the BB-flat tuba w i l l be con-sidered here. -202-Since the bore of the tuba, like that of a l l brass instruments, is neither entirely cylindrical nor entirely conical, the tuba does not behave like an instrument belonging to either the cylindrical or the conical group. It does, however, lend i t s e l f easily to overblowing and to operation in at least 12 modes of oscillation. Of these 12, modes 7 and 11 are not used because the resulting pitches are not in tune in equal temperament. Furthermore, the entire f i r s t mode is seldom used because i t is d i f -f i c u l t to produce owing to the low frequency of i t s pitches and to the great length of the instrument. Thus we are l e f t with nine useable resulting pitches for each fundamental. The natural harmonic series of the BB-flat tuba is based on BbO, and the useable members of i t s overtone series are the following: Written and Sounding: 2£L. -G~ -©-(1) fa (7) ( I D In order to provide the tuba with a complete -203-scale, the intervals between the notes shown above must be f i l l e d in. The largest interval which needs to be f i l l e d in is the f i r s t one, that of a perfect f i f t h , which requires six additional semitones to f i l l the space. These are provided by lengthening the a i r column through the use of valves. Thus far, the situation described is directly comparable to that encountered with the trumpet and french horn. At this point, however, a fundamental difference arises. While with the trumpet and french horn three valves were sufficient to obtain a l l the desired pitches, this is not so easy with the tuba. It is theoretically possible with the use of the valve slide on the third valve, but even then certain valve combinations result in pitches which are sharp-er than the ones desired. For this reason, a fourth valve is always found on professional models. The necessity of the fourth valve is due directly to the deficiency inherent in the valve system, which was discussed on pages 131-136. Just as in the case of the french horn the valve combination 1+2, when used in combination with other valves, does not add enough tubing to lower the pitch by three semitones, -204-thus necessitating the use of valve 3, similarly with the tuba the valve combination 1 + 3 , even when used alone, does not add enough tubing to lower the pitch by five semitones, thus necessitating the construc-tion and use of a fourth valve. This is because in the low range of the tuba the (physical, not propor-tional) length of the tubing which must be added every time a pitch is to be loxvTered is so great that the same valves which individually serve their purpose well f a i l to do so in combination. For the same rea-son a f i f t h valve is often added to the tuba. If used in the middle or high register (where i t is sel-dom used), i t would lower a given pitch by approxim-ately 2% semitones. When used in the low register, i t lowers a given pitch by two semitones. This sys-tem is made workable by slide adjustment, which is f a i r l y common even during playing. The following table shows the resulting shift of frequency downward (in semitones) for each valve combination. (Combinations involving the f i f t h valve are not considered, because this valve is de-signed to be used only in the low register, where shifts totaling more than six semitones are made.) -205-Valve(s) Number of semitones 2 1 3 1 + 2 2 + 3 4 1 + 3 * 2 + 4 1 + 2 + 3 * 1 2 3 3 4 5 (5) (6) 6 The combinations marked * are almost never used be-cause valve 4 is substituted for the combination 1 + 3 . With five valves at his disposal, the tuba play-er can descend more than 6 semitones below the pitch sounded with no valves. The following valve combina-tions have been derived by experiment and experience for each successive shift of frequency downward (in semitones): Valves Number of semitones In comparing the two tables above, several observa-tions may be made. If the f i f t h valve is taken to lower a pitch by exactly two semitones, then, ac-cording to the f i r s t table: 4 + 5 2 + 3 + 4 1 + 3 + 4 2 + 3 + 4 + 5 1 + 2 + 3 + 4 + 5 7 8 9 10 11 -206-1) "7 semitones" follows from the combination "4 + T" because "5 semitones" follows from "4" alone, and "6 semitones" follows from "2 + 4". There is no con-f l i c t i n g situation here. 2) "8 semitones" does not follow from "2 '+ 3 + 4", because 1 + 3 + 5 = 9 semitones, and because "2 + 4" = "6 semitones". One would expect the combination "2 + 4 + 5" to be correct here, but in reality this combination produces a pitch which is much too sharp. 3) "9 semitones" does not follow from "1 + 3 + 4", because 2 + 3 + 5 = 10 semitones, and because "4 + 5" = "7 semitones". One would expect the combination "1 + 4 + 5" to be correct here, but in reality this combination produces a pitch which is again much too sharp. 4) "10 semitones" does not follow from "2 + 3 + 4 + 5", because 1 + 3 + 5 + 2 = 11 semitones. If, however, "2 + 3 + 4" is taken to result in "8 semitones", then this combination follows from "8 semitones", but i t does not follow directly from "9 semitones" ("1 + 3 + 4".) One would expect that the combination "1 + 2 + 3 + 4" would be correct here, and that the use of the f i f t h valve would not be necessary, just as i t was -207-not necessary to lower the pitch by 1 semitone from "8 semitones". In reality, the combination "1 + 2 + 3 + 4" produces a pitch which is much sharper than the desired one, and the f i f t h valve must be used in-stead of valve 1. 5) Finally, the combination " 1 + 2 + 3 + 4 + 5 " for "11 semitones" does not follow from anything. It does not follow from the f i r s t table, because 2 + 1 + 3 + 5 + 2 = 1 3 semitones, and i t does not follow from the second table, because given that "9 semitones" follows from "1 + 3 + 4", and that "10 semitones" follows from "2 + 3 + 4 + 5", one would expect the proper combination for "11 semitones" to be "1 + 3 + 4 + 5". In reality, this combination produces a pitch which is almost a semitone too sharp, and the further addition of valve 2 is required. The following conclusions may be made from these observations: A) Numbers 2), 3), and 4) above are consistent with respect to each other, in that each deviates by one semitone from the result one would expect by addi-tion. They are not consistent with respect to num-bers 1) and 5), because 1) does not deviate from the -208-expected result at a l l and 5) deviates from the ex-pected result by two semitones. B) The major reason for a l l the discrepancies is the fundamental shortcoming ( l i t e r a l l y ) of the valve system, especially in a range where great lengths of tubing are involved. Since the deficiencies of the valve system have already been discussed (on pages 131-136), a l l that remains to be added here is that the tuba provides the most vivid i l l u s t r a t i o n of the problem because i t s length is the greatest of a l l the brass instruments. C) The second reason for the discrepancies is the compromising principle upon which the f i f t h valve is designed. (In fact a l l valves are designed upon a compromising principle, but this shows up most clear-ly in the case of the f i f t h valve, because of the low range in which i t operates.) This is best seen by comparing the f i f t h valve to valves 1 and 3. For "7 semitones" and "10 semitones" the f i f t h valve is just right; i f valve 1 were used instead, the resulting pitch would be too sharp, and i f either valves 1 + 2 or 3 were used, itowould be too f l a t . For "8 semi-tones", the f i f t h valve is too sharp i f i t replaces -209-valve 3. For "9 semitones", the f i f t h valve is too f l a t i f i t replaces valve 1 and too sharp i f i t re-places valve 3. Finally, as was already mentioned, for "11 semitones" i t does not suffice to add the f i f t h valve to the valves used for "9 semitones" (1 +3 + 4); valve 2 must also be added. A l l of this shows that the f i f t h valve is not designed so much to be a substitute for either valve 1 or valve 3, but to be an alternate for one of them, x^hich sometimes yields better results, while at other times i t yields worse results. Furthermore, i t is incorrect to state categorically that the f i f t h valve lowers a pitch by any fixed number of semitones. On the contrary, i t s effect is different for each combination to which i t belongs. Also, i t must be remembered that slide ad-justment during playing is what really makes this system workable, as was mentioned earlier. For a l l these reasons, the valve combinations in the second table above do not follow any fixed pattern. Through the use of the five valves in the manner indicated in the two tables above, the following p i t -ches may (theoretically) be produced on the tuba: Possible T J r i t t e n (and Sounding) Fren-gencies Valves Mode: 1 2 3 4 5 6 7 8 9 10 11 12 11 33 18 30 28 -211-This is so because each fundamental which results from a new combination of valves has i t s own over-tone series, the members of which are accessible in each case by overblowing. Although there are twelve rows containing nine notes each, only 43 of these 108 notes are different. These 43 notes have been subdivided into four cate-gories as follows: 1) 10 pitches which appear only once — unnumbered, 2) 11 pitches which appear twice each — indicated with the same number in both places, 3) 12 pitches which appear three times each — indi-cated with the same number underlined in a l l three olaces, 4) 10 pitches which appear four times each — in-dicated with the same number circled in a l l four places. The next step is to determine which valves the tuba player would, in practice, use to produce the 43 different Ditches on his instrument. Clearly, for the 10 pitches for which only one fingering is pos-sible there is no choice. These pitches are indicat-ed in whole notes in the table on page 212. For the -213-remaining 33 pitches, the one is chosen which comes f i r s t (reading from top to bottom), for the same reasons as in the cases of the trumpet and french horn. In the table on page 212, these pitches are indicated as follows: 1) If the choice x^ as made from two alternatives, the pitch is indicated with a half note, 2) If the choice was made from three alternatives, the pitch is indicated with a quarter note, 3) If the choice was made from four alternatives, the pitch is indicated xvTith an eighth note. These pitches are so designated in order to shox<? that although the alternatives have not been indicat-ed on the table, they do exist, and should not be totally ruled out. While the pitches x^hich are in-dicated are f i r s t choices, the tuba player may xvrish to use some of the alternatives in at least txro spe-c i a l situations: 1) in t r i l l i n g or in fast passages, XvThere technical ease may be greater with the use of an alternative, 2) in certain keys, where intonation of some of the notes may be better x<d.th the use of an alternative. Consequently, i t is to the tuba player's advantage -214-to be aware of the alternate fingerings of the pi t -ches of his instrument, and thus to have these at his disposal. For two of the pitches (C#3 and D3) the second choices (in mode 6) are just as good, and often bet-ter. This is because the f i f t h mode is the "Achil-les heel" on many tubas, and is better avoided i f possible. The two pitches have been indicated in bracketed eighth notes in the table. The table on page 212 clearly shows that the use of the f i f t h valve is necessary for only three of the 43 pitches of the normal range of the tuba. Expert players can extend the lower limit of this range considerably — down to EO (20.602 hz) — a perfect fourth below the lowest note of the piano. (It follows that these players can produce pitches through the f i r s t mode of oscillation.) The extra notes which belong only to the exceptional range of the tuba are indicated in bracketed whole notes in the table on page 212. In summary, then, the compass of the tuba is the following: -215-High Register: Middle Register: Low Register: Extra-low Register (exceptional): Naturally, the tuba is at it s best in the middle reg-ister, where i t is the easiest to play and i t s tone is the most characteristic. In the high register, the attacks are less sure and the extreme dynamic ranges (pp and f f ) are more d i f f i c u l t to achieve. Similar problems are encountered in the lox<? register, where the problem of air supply becomes quite serious, and speed of articulation is hampered. In the extra-low register, these additional problems are compound-ed, and a different embouchure is often used. The complete chromatic scale of the normal range of the tuba may now be written out, showing the f i n -gerings (by valve numbers) and the modes of o s c i l l a -tion used. 36 -216--e-1 2 3 4 5 2 3 4 5 second-^ ^e- ^ ^ 1 3 4 2 3 4 4 5 2 4 1 ^ = ^ ^ 2 1 1 3 2 0 r third fourth 2 0 sixth-2 3 ^ ^ - f©= 2 4 2 4 3 , f i f t h -1 1 2 1 2 1 2 eighth -o #o w: 0 IjO <> f<» 1 1 2 0. 2 Alternate: 2 1 3 2 (sixth mode) 0 2 1 3 2 0 j-ninth—, r— tenth• twelfth-EE 0 2 0 0 -217-z\ 37 Testing the Equation v = f A As was done with the other brass instruments, we now wish to test the equation v = f'X in a prac-t i c a l situation involving the tuba. Again, since the tuba is partly cylindrical and partly conical, nei-ther \^ = 2L nor ^ = 4L holds true. Conse-max max quently, we must substitute known values of v and f into the equation, which w i l l yield , which we w i l l then compare to L. To begin, we need to know the lengths of the parts of the tuba. These lengths were measured and found to be as follows: Part Length (in cm) Mouthpiece 6.5 Leadpipe 63.0 Middlepipe 55.0 Bellpipe 354.0 Bell 76.0 Basic Length (equals total of above lengths) 554.5 Tubing added by valve 1 77 Tubing added by valve 2 38 Tubing added by valve 3 124 Tubing added by valve 4 207 Tubing added by valves 1 + 2 115 Tubing added by valves 2 + 3 162 Tubing added by valves 2 + 4 245 Tubing added by valves 2 + 3 + 4 369 Tubing added by valves 1 + 3 + 4 408 From these values we may easily find, by referring -218-back to the fingering chart on page 212 and by ad-dition, the total length of the pipe (L) used to produce seven of the pitches of the second mode of the tuba. We also need to include in the table the known frequencies of these pitches. Now, using v = 33853 cm/sec, we calculate 1\ for each of the seven pitches. Finally, we comoare 0 to L by solving for z the equation *\ = zL for each of the seven ^ max values of *\ which were just calculated. The com-pleted table, then, is found on the following page. Ideal results would have been obtained i f the value of z turned out identical for each examined pitch. The values obtained, however, range from .961 to .989 and average out to .973. Now i t is more rea-sonable to assume that the average error in the cal-culations is .02 7 than to take the value of z to be precisely .973. Since this ratio is so close to 1.00, the "L+" factor does not need to be introduced. If we take ^ to be equal to (1)L, we can test the max • equation v = f\ by the same method that was used for the woodwind instruments. The results thus obtained are excellent, as shown in the table on page 220. Examined Pitch: F l F#l Known frequency of this oitch (hz): 43.654 46.249 Length of pipe (L) (cm): 799.5 761.5 (= 33853 cm/sec -f* f) (cm): 775.49 731.97 = -f- L): .970 .961 max Gl 48.999 716.5 690.90 .964 Abl Al 51.913 55.000 669.5 631.5 652.11 615.51 .974 .975 Bbl Bl 58.270 61.735 592.5 554.5 580.97 548.36 .981 .989 l a Examined Pitch: lb Known frequency of this pitch (hz): Length of pipe (L)(cm): 0 (where Ok,, = L)(cm): max Calculated frequency (33853 cm/sec -f- L)(hz): 2a Pitch whose frequency in equal temperament is nearest to that obtained above: Known frequency of this pitch (hz): Discrepancy of 2a with respect to la (by interval): Ratio of calculated frequency with respect to lb: Fl F#l 43.654 46.249 799.5 761.5 799.5 761.5 42.343 44.455 El F l 41.203 43.654 - 2nd - 2nd lower lower .973 .982 Gl 48.999 716.5 716.5 47.248 F#l 46.249 - 2nd. lower .979 Abl 51.913 669.5 669.5 50.565 Abl 51.913 none .974 Al 55.000 631.5 631.5 53.607 Al 55.000 none .975 Bbl 58.270 592.5 592.5 57.136 Bbl 58.270 none .981 Bl 61.735 554.5 554.5 61.051 Bl 61.735 none .989 -221-V. PRACTICAL APPLICATION OF KNOWLEDGE OF THE ACOUS-TICS OF WOODWIND AND BRASS INSTRUMENTS 1 Having tested the equation v = f\ and summa-rized the results for each instrument individually, the results w i l l now be examined collectively and analyzed. Since the situation for the brass instru-ments is completely different from that of the wood-winds, the two families w i l l be treated separately. The table on the following page is a summary of the results which were obtained for each woodwind instrument. Column one does not consist of experi-mental results, but of formulas which were derived in Section II and used as premises in the testing of the equation. Regarding the discrepancies in column two, explanations for these were already giv-en at the end of each respective section, therefore w i l l not be repeated here. Column three represents more than just an alternate means of presenting the statistics given in column two. It is much more ac-curate than the former, simply because a semitone is too large a unit of measurement for these experiments (being subject to relative error of +3% of the actual frequency in hertz.) Instrument (see indicated page for com-plete table) 1) Known value r. 'Xmax ° f L 2) Average discrepancy of 2a with respect to la (in semitones) 3) Average ratio of calculated frequency with respect to lb Clarinet (p. 30) 4 - 3 + 3 + 2 + 0 c 4 1.04 Flute (p. 45) 2 1 + 2 + 2 + 3 4 1.15 Oboe (p. 68) 2 0 + 1 + 0 - 1 4 U 1.01 Bassoon (p. 94) 2 2 + 1 + 1 - 2 4 , ! > 1.03 SaxoDhone (p.107) 2 0 - 1 - 1 - 2 4 1 Overall average discrepancy: 0.4 .955 Overall average ratio: 1.04 -223-In summary, i t should be emphasized that for the woodwind instruments these experiments were conducted solely out of curiosity — to see how the values of f obtained by experiment would compare to those obtained by calculation. With the brass instruments, the situation is so different, that s t r i c t l y speaking, i t is not correct to say that the equation v = was "tested", a l -though this term was used to designate the corres-ponding section of the discussion of each instru-ment. If the equation were to be tested as i t was for the woodwinds, the mathematics involved would be extremely complicated. Since in the case of the woodwinds the mathematics necessary for a f i r s t ap-proximation could be done f a i r l y simply, testing for validity of this f i r s t approximation could be car-ried out. With the brass instruments, any discrepancies which result are not genuine discrepancies, because there is no known norm from which values can be ob-served to deviate. Furthermore, with instruments such as the tuba, where the bore diameter becomes extremely large, dimensions other than length have -224-an effect on the behavior of sound waves within the tube; these were not taken into consideration. That brass instruments do not behave in the man-ner that closed tubes are expected to behave is prov-en by the simple fact that a l l of them (even the trombone, which is approximately two-thirds cylind-r i c a l ) produce both even and odd harmonics, although, being closed tubes, they should produce only odd har-monics. In this way, the construction of brass in-struments can be considered similar to the construc-tion of instruments employing conical tubes: alter-ations from the natural conical shape are made which result in capability of production of the same even and odd harmonics which are naturally produced by an open tube of equal length, as was explained in Sec-tion II. The table on the following page is a summary of the results obtained for each brass instrument. Each column w i l l be examined and analyzed individu-a l l y . Column One: Contrary to the case with the wood-wind instruments, here the values of u, w, x, y, and z were not only unknown, but they were also complete-Instrument (see indicated sage for com-alete table) 1) Average calculated _ Amax Imax value of ' L or L + 2) Average discrepancy of 2a with respect to l a (in semitones) 3) Average ratio of calculated frequency with respect to lb Trumpet (pp. 140, 142) Average of u = 1.01 = [.995 + 1.00 + 1.01(3) + 1.02 + 1.04] -J- 7 1 + 0(6) _ , A 7 1.01 French Horn; F side (pp. 171, 176) Average of w = 1.05 -[1.04(2) + 1.05(2) + 1.06(2) + 1.08] -r- 7 0(6) 7 + 1 - .14 1.01 French Horn; B-flat (PP. 172, 177*)' Average of x = 1.05 = [1.04(3) + 1.05(3) + 1.07] -J- 7 0(6) 7 + 1 - .14 1.01 Trombone (pp. 198, 200) Average of v = [2.1(7)] 4- 7 = 2.1 0(7) 7 = o 1.01 Tuba (pp. 219, 220) Average of z = .973 .973 " • ° 3 0(4) +1(3) . > 4 3 Overall average discrepancy: .17 .979 Overall average ratio: 1.00 -226-ly unpredictable, because there was no basis upon which to make any predictions. At this point, i t is easy to explain why the average value of y is the only one which is approximately equal to 2, while the others are approximately equal to 1: the trom-bone is the only brass instrument on which the com-plete f i r s t mode can be produced. If, for the other four instruments, there were an easy method of ex-citing the tube so that the fundamental pitches could be produced, then the f i r s t mode (rather than the second) would have been used for the calculations. (This could easily have been done in the case of the tuba.) Consequently, the values of \ would have been twice as large and the values of u, w, x, and z would have turned out to be 2.02, 2.1, 2.1, and 1.95 re-spectively. By comparing the table for the trombone to that of any of the other brass instruments, i t can be clearly seen that: 1) this is a t r i v i a l case of working in a different octave, and 2) this has no significant bearing on the fin a l results. Upon observing that the discrepancy from the ideal result is at most 5% (in fact exactly 5% in the three cases where end correction was involved), = 2L IS A FUNDAMENTAL MATHEMATICAL AND ACOUS-max -227-there is basis upon which to make the following claim: max TICAL PROPERTY OF BRASS INSTRUMENTS. A l l discrepancies are attributable to exactly the same three factors which attributed to the dis-crepancies of f in the cases of the woodwind instru-ments (where the values of — ^ — are known.) These three factors are: 1) experimental error, 2) de-viations from the "perfect" shape of the tube, and 3) the imperative simultaneous operation of two systems: the generating (or excitation) system and the resonating system, which was discussed on pages 69 - 71. It must be clearly understood that the above claim is not the equivalent of claiming that ^ m a y . = 2L is a property of an a i r column which is partly cylindrical and partly conical. Such a claim, i f made, would have to be proven mathematically, and this proof would be a landmark in the f i e l d of acous-t i c s . The conclusion to x-?hich a l l of the above points is the following: -228-THE VITAL CHARACTERISTIC OF A BRASS INSTRUMENT IS NOT THE ACTUAL SHAPE OF ITS AIR COLUMN, BUT THAT THIS SHAPE ~ WHATEVER IT BE — POSSESS THE PROPERTY 0> max _ 0 L~ ~ L ' The theory to which the above claim and conclusion give birth is the following: GIVEN ANY BRASS INSTRUMENT, THE NEARER THE VALUE OF •^-—2 FOR THAT INSTRUMENT APPROACHES 2, THE MORE ACOUSTICALLY PERFECT IS THAT INSTRUMENT. It must be added, that i f an instrument were specif-i c a l l y designed so that i t s value for — ^ — equalled exactly 2, i t may prove to have different, undesir-able properties, but that would then be a separate problem. On the other hand, i t is also possible that such an instrument would be acoustically superior in a l l other respects. It would certainly be worthwhile, therefore, to manufacture a brass instrument which would possess this property, with no regard for any of i t s other characteristics. The next step would be the elimination, one by one, of any undesirable properties which this instrument may have without altering the value of — £ — , i . e. not on a compromis--229-ing principle. Such a task would be a formidable one indeed, and would probably take forever to a-chieve, but i t could result in a next-to-perfect in-strument. Column Two: Out of the seven pitches which were tested for each brass instrument, only one pitch deviated by a semitone for three of the instru-ments tested, and none deviated in the case of the trombone. In every case except those of the trum-pet and tuba, this is directly attributable to the fact that "L+" rather than "L" was used in the cal-culations. Since i t was not necessary to use ,,L+M for the trumpet and the tuba, we can speculate about why this was necessary for the french horn and the trombone. The french horn is the only instrument employing hand insertion into the b e l l , and the trom-bone is the only valveless instrument. This means that the manner of operation of these two instruments differs considerably from that of the standard prin-ciples of operation of the trumpet and the tuba. The most noteworthy point i s , however, that no de-viation is greater than one semitone, and that there is no deviation at a l l (in semitones) for the great -230-majority of pitches. Column Three: The remarkable (and least ex-pected) result obtained here is that the overall average ratio is exactly 1.00* From this result and those in the f i r s t two columns, the following con-clusion may be reached: THE EQUATION V => F^ CAN BE SHOWN TO HOLD FOR BRASS INSTRUMENTS IF AND ONLY IF THE PROPER RELATIONSHIP OF ^  TO L IS FIRST ESTABLISHED. The " i f " part of this conclusion is proven by the results in the table. The "only i f " part can easily be proven by substituting values for u, w, x, y, and z other than those which were actually used. This conclusion is the strongest evidence there is to substantiate the claim, conclusion, and theory presented earlier. One more observation may be made by examining the instruments individually: the trombone turns out to be by far the most acoustically consistent instrument of a l l . It is the only instrument in which there is no deviation in the calculated value of —j-—, and the only instrument in which there is no discrepancy in semitones of 2a with respect to l a , -231-not even for a single pitch. This leads to the following theory: COMPLETE CONTROL OVER NOTHING MORE THAN TUBE LENGTH ALONE IS ENOUGH TO MAKE AN INSTRUMENT THAT IS ACOUS-TICALLY NEXT-TO-PERFECT. This theory is actually very similar to the theory presented above. The important difference is that i t follows from different conclusions. The results obtained for the trombone also serve to justify the introduction of the concept of "L+", which is dis-cussed in greater detail below. At this point, every claim, conclusion, and theory which has been presented must be qualified by pointing out that the small amount of testing which has been done here is highly insufficient for generalizations or for claiming universal applica-tion of the statements made. The statements are based on nothing more and nothing less than the ex-perimentation which was performed, and should be re-garded in this light. A large amount of further testing is necessary to either strengthen or weaken the validity of these statements. -232-Ftrial Summary In order to t i e together everything that has been done, i t is necessary to examine in greater de-t a i l the variables v, L, f, A , and =— ind i v i -' ' ' "max* L dually and with respect to each other. To begin with v, we recall that the velocity of sound is approximately 33145 cm/sec at 25° C, and that i t increases by 59 cm/sec for each degree C above 25° C. For both woodwinds and brasses, i t was decided to use the velocity of sound at body temper-ature (37° C), which is why v was taken to be 33145 + 59(12) = 33853 cm/sec. There was no account made of the fact that at the end of the instrument, v could have been as low as the velocity of sound at room temperature (20° C), therefore 33145 - 59(5) = 32850 cm/sec, i . e. approximately 3% lower. Also, within the instrument v could have been anywhere be-tween these two values. Since, however, v was treat ed as a constant for both woodwinds and brasses, the situation did not change when the second family of instruments was tested. What must be realized, how-ever, is that in view of the above, a l l resultant discrepancies are almost certainly due, in part, to -233-the fact that v is not, in reality, the constant 33853 cm/sec that i t was taken to be. Looking next at L, we recall that L was always taken to be the measured length of the instrument, therefore a constant, although a different constant in each case. For the woodwind instruments, where the relationship between L and *\ is known, these r max txro variables were assumed to be in direct proportion to each other (4:1 or 2:1.) The greatest d i f f i c u l t i e s arise when we begin dealing with the remainder of the variables. From the conclusion given on page 230 i t follows that in order to test the equation v , i t is not s u f f i -cient to know the values of v and L; the relationship between the open air value of "X and L must also be knox^ m. Since this relationship _is known for woodwind instruments, only one variable remained to be solved for: f. Testing the equation was, therefore, very straightforward; the results shox-yed considerable de-viation between calculated ( i . e. predicted) values and those obtained by experiment. Taking the same approach for the brass instru-ments, since the relationship between the open a i r -234-value of ^  and L is not known, we reached a dead end. For this reason, as mentioned before, i t is not entirely correct to say that the equation v = f ^ was "tested". More accurately, this equation was used in an attempt to find the correct values of two unknowns: — and A. Since there were already two unknox<ms, i t was necessary to treat f as a con-stant. This, however, was a reluctantly made, forced decision, because the experiments for the woodwinds showed just hoxvT great the discrepancy be-tween the calculated values of f and the actual val-ues of f can be. Consequently, \<re were forced by circumstances to provide a "standing invitation" for discrepancy. At any rate, the next step was to find the val-ue of ~ — . To do this, the equation v = fA x>?as, in essence, substituted by the equation v = f(kL), and this equation xras solved for,each instrument in-stead, (k x^ as replaced by u, w, x, y, and z respect-ively.) This resulted in a simple relationship be-tween ^ and uL and zL, but not between and max ' max wL, xL, and yL. For these three instruments, end correction entered the picture (5%, 4.5%, and 5% re--235-spectively.) Consequently, the original assumption that ' A is directly Droportional to L (0 = kL) max J r max was replaced by the assumption that ' A = kL+, and J max thus the concept of "L+" was introduced. The "+" part of "L+", however, was nothing more than a cal-culated guess. Since the equation v = f)\ was used (or misused) , in this way, the resultant discrepancy _ "max of A from max the desired value of 2 (or 1) cannot be considered to be a discrepancy of 0 alone, but of v, r J max max and f (which was assumed to be fixed, although i t really was not.) There is no precise way of deter-mining exactly how much of the total discrepancy is attributable to discrepancy in f alone, but i t is a reasonable assumption that this discrepancy is approx-imately equal to what i t was with the woodwind instru-ments. In the final analysis, one question remains: What was the justification for choosing those values for the part of "L+" which were chosen? In con-junction with what was said about the trombone above, the results obtained for this instrument point to the conclusion that the choices for the part of "L+" were well made. A much stronger and far more sig-nificant justification, however, is provided by the results in column three. The choices which were made so that the values of w, x, and y would approach 2 (or 1) turned out to yield results in column three which were identical, and extremely close to 1.00. It is precisely this justification which is offered to substantiate the claim made on page 22 7. One more argument may be presented in defence of the choices for the "+" part of ,T_+" which were made. Let us suppose that the relationship of 0 max to L was unknown for woodwinds, and use the same ap-proach that was used for brasses. The resulting ta-bles for each instrument are given on the following page. We see that the averages of the variables a, b, c, d, and e are as folloxvs: Average of a'(clarinet) 4.28 Ideal value of a 4 Average of b (flute) 2.30 Ideal value of b 2 Average of c (oboe) 2.02 Ideal value of c 2 Average of d (bassoon) 2.12 Ideal value of d 2 Average of e (saxophone) 1.91 Ideal value of e 2 12.63 12 Average discrepancy: 12.63 -4- 12 = 1.0525. Thus we see that i f i t would have been necessary to introduce the concept of "L+" for the woodwinds, the -237-Clarinet (original table on page 30) Frequency (f) (hz): 146.83 174.61 311.13 415.3 ^ (= 33853 cm/sec^. 260.56 193.88 108.81 81.51 Length of pipe (L): 66.8 41.5 23.8 20.4 L a x - L ( = a ) : 3 ' 9 0 4- 6 7 A ' 5 7 4.00 Flute (original table on page 45) Frequency (f) (hz): 261.63 293.66 349.23 554.37 ^ ( = 33853^cm/sec); 1 2 9 > 3 9 1 1 5 # 2 8 96.94 61.07 Length of pipe (L): 60.0 52.0 43.0 23.6 0t -j- L (= b): 2.16 2.22 2.25 2.59 max Oboe (original table on page 68) Frequency (f) (hz): 233.08 246.94 369.99 523.25 "\ (= 33853 cm/sec^. 1 4 5 # 2 4 137.09 91.50 64.70 Length of pipe (L): 70.6 65.1 46.1 33.8 "X-...,. -r L (= c): 2.06 2.11 1.98 max 1.91 Bassoon (original table on page 94) Frequency (f) (hz): 58.27 97.99 174.61 261.63 ^ (= 3 3 8 53 fcm/sec ) ; 6 4 7 # 6 6 3 4 5 , 4 7 1 9 3 . 8 8 129.39 Length of pipe (L): 262.2 165.7 90.1 73.6 ^ -+ L (= d): 2.47 2.09 2.15 1.76 max Saxophone (original table on page 107) Frequency (f) (hz): 138.59 207.65 261.63 329.63 ^ (= 33853^cm/sec); 2 4 4 # 2 7 1 6 3 # 0 3 129.40 102.70 Length of pipe (L): 122.2 84.9 68.5 56.0 1 -r L (= e): 2.00 1.92 1.89 1.83 max -238-part of "L+" which xrould have been chosen would have been, on the average, 5.25% of the basic length, which is strikingly close to the average for the brasses (4.83%.) Finally, a word of explanation is in order re-garding the question of accuracy of figures. The accuracy of the figures in this paper i s acceptable from the physicist's point of view, but not from the musician's. An error of, for example, a semitone is a small error mathematically, (being in the order of +3%) but musically i t is a greater error than one of a perfect octave, which, in turn, is a large mathem-atical error, being of a factor of 2. Thus, from a musician's standpoint, measurements which result in a pitch deviation of even a quarter-tone are not ac-curate enough to be useable for practical purposes. On the other hand, a physicist would be delighted with such great accuracy. This difference in prac-t i c a l interpretations must be kept in mind when deal-ing with the mathematics of music. -239-CONCLUDING REMARKS In the previous section, a rather strong effort was made to support a claim which cannot be support-ed by a mathematical proof: that — ^ — = 2 is a fundamental property of brass instruments. This claim was based largely on the observation that brass instruments, as they are constructed today, already seem to possess the property ^ ™ a x 2 — for what-ever reasons. This suggests that manufacturers have derived through experiment and experience a result which a theorist would now like to arrive at and to prove correct by using formulas and eouations. But this exact situation represents a historical trend. In general, the regular order of incidents is that f i r s t a professional in a practical art deter-mines empirically that a particular model is the most perfect one available, then (in some cases many years later) a theorist or a mathematician or a physicist appears on the scene to say to him: "You are correct; I can prove i t j " [To make an analogy from outside the f i e l d of acoustics — the writer was once told that when s c i -entists assigned themselves the task of determining -240-the shape of a structure which was the most resist-ant to wind and to environmental conditions in gen-eral, they ultimately found that the structure which they arrived at through highly sophisticated methods was in the exact shape of a tree. "The Creator was correct; they can prove i t j " ] A l l of this may be summed up by one basic ques-tion regarding construction and manufacturing: "How does the method of formulas and equations compare to that of the " t r i a l and error + experiment and exper-ience" method? The writer posed precisely this question to the following seven manufacturers of woodwind and brass instruments: 1) Larilee Woodwind Manufacturers, 1700 Edwardsburg Rd., Elkhart, Indiana 46514, USA 2) Selmer Brass and Woodwind Manufacturers, Box 310, Indiana 46514, USA 3) G. Leblanc Corporation, 7019 30th Ave., Kenosha, Wisconsin 53140, USA 4) Norlin Musical Instruments Ltd., 51 Nantucket Boulevard, Scarborough, Ontario, M1P 2N6, CANADA 5) Fox Products Corporation, South Whitley, Indiana 46787, USA -241-6) Renold Schilke, Schllke Music Products Inc., 529 S. Wabash Ave., Chicago, I l l i n o i s 60605, USA 7) Yamaha Nippon Gakki Co., Ltd., P.O. Box 1, Hamamatsu, 430,. JAPAN The following are (excerpts from) the replies received from the f i r s t five manufacturers: 1) No reply 2) No reply 3) Mailed an article by Arthur H. Benade entitled: "The Physics of Brasses". The arti c l e was quite tech-nical, and included discussions of such topics as res-onance peaks of a trumpetlike instrument, geometry of horn flare, the trombone b e l l , two different imped-ance-measuring apparatuses, impedance patterns, re-gimes of oscillation, the tone color of a trumpet, and the extension of the range of the french horn by hand insertion. 4) "As a general principle i t may be stated that a l l reputable manufacturers employ a combination of theory and "bench-test" prior to the production of a prototype. It in turn is field-tested until the f i -nal design is settled upon. This ap-plies to everything from mousetraps to jetliners." "...the application of an established formula does not mean that the design -242-of the finished instrument is immutable and incapable of improvement, and this is where the manufacturer's experience and ingenuity come in, to produce an instrument one or more s t e D s better than his competitors - perhaps freer-blowing, better intonation, and what-have-you. (signed) W. C. Duncan, Manager, Customer Service" 5) "...the bore dimensions and tone hole sizes of a l l our instruments are deter-mined by achieving an understanding of of the behavior of those sections and shaping them to achieve the desired re-sult. There is no math or theoretical formulas involved and to my knowledge, which includes frequent contact with A. H. Benade at Case Institute in Cleveland, there are none which s u f f i -ciently deal with the problem to achieve the results that we are seek-ing. You may find some more practical theo-retic a l approaches i f you explore the brass instrument area, which is some-what simpler than the woodwind area,... (signed) Alan H. Fox, President" This last statement is a very interesting oDinion in-deed. 6) "In brass instruments as well as xrood-wind instruments, i t is impossible to figure out your nodal patterns math-ematically and the best way I found to find your pressure points and points of rarefaction is by the use of the con--243-tact mike u t i l i z i n g a high speed camera and an oscilloscope to pick up the pat-tern i t s e l f . In this way you can ob-tain exact patterns and i t is the only way that i t is possible. I have worked on the theory of u t i l i z i n g the speed of sound, and dividing i t by the number of vibrations per second and worked on this theory for several years and came up with the complete fallacy as the accur-acy of the points of rarefaction in ac-cordance with the pressure points and in no case was the rescinding note at a l l accurate. I have worked with the top physicists in the world, including the most out-standing one of them a l l , Dr. Aebi from Bergdorf, Switzerland and I have found that this is the only approach that is at a l l accurate and can lead to fur-thering the success of whatever instru-ment you are working on in woodwinds or brass, or for that matter the stringed instruments. In fact, anything that produces sound can be analyzed in this manner. Enclosed please find a copy of The  Physics of Inner Brass, and while this is not going into i t very deep i t was a lecture I gave for educators and as a result I had to avoid going into too deep of technical terms. The fact that you are working with c y l -indrical as well as tapered tubes, con-ic a l and none of them definite but a l l as french curves, i t is definitely im-possible to use a mathematical formula for determining the length of an instru-ment, but the method that I suggested to you of using contact microphones in the manner I suggested is the only ac-curate way of doing i t , because the ac-tual density of the metal as i t changes w i l l give you definitely different lengths of tubing for each instrument. (signed) Renold Schilke" -244-The article enclosed was very general in scope and in-cluded discussions of such topics as pressure points, the K factor, materials of manufacture, plating of instruments, and mouthpieces. The reply from the Yamaha Company in Japan, which is quoted below, was the most interesting. The two portions of the last two paragraphs which have been underlined below are particularly priceless — they sum up the entire situation exquisitely. 7) "Basically, we also use " t r i a l & error plus experiment & experience" method in designing or improve our instruments. This principle is employed through our production process by way of the stri c t inspection for quality control and rou-tine feed-back system to our design, which gives us the opportunities to un-derstand the causal relation between design and result. We of course experience copying the most desirable instrument available for study purpose, however i t is only a close re-semblance and incorporating the result of our experience and findings, we can make such resemblance better than the original in many cases. We can find some theory of acoustics of wind instruments as by Benade, Cremer, Coltman, Elder, Backus, and so forth, hox<rever most of them are just theory and do not do much for practical purpose. Many of such books even include rather wrong or false story. For instance, many books say that the pitch of a horn is decided by i t s total length of tubing, but in reality, the form or taper of -245-tubing influences the pitch of horn as well as harmonic complex. As you wi l l realize, the mensuration of actual horn is neither conical or ex-ponential. We actually use more than  one hundred data to specify the exact  shape of tubing and i t is nearly impos-sible to get resonant frequency ana— l y t i c a l l y . Also, the boundary condi-tion of wind instruments at the mouth-pipe or lead pipe is neither '^closed" nor "open". Brass instruments are in the "closed" side, however i t is also subjective to performer or performance style. In conclusion, we x-jould say that only  " t r i a l and error" method can be used  in actual designing of instruments and from this point, we can continuously seek for an evolution of design. Some scien t i f i c system or equipment such as "wave equation" and "computer machine" can be used as an assistance for ana-lysi s or working out some data, how-ever they are just an equipment to help us save time and do never take i n i t i a t i v e role in designing instru-ments. (signed) S. Hasegawa, Manager, Export Dept." -246-FOOTNOTES I. DIFFERENTIAL EQUATIONS OF WAVE-MOTION ''"Unless otherwise stated, a l l the information found in Section I is based on the follox<ring source: F. C. Champion, University Physics, Part Four, Wave-Motion and Sound (London.. and Glasgow: Blackie and Son Limited, 1948), op. 1-8. It is presented in a simplified version as taught by Dr. Roger Howard of the Deoartment of Physics, Uni-versity of British Columbia. 2 Jess Stein, editorial director, The Random House  College Dictionary, (New York: Random House, Inc., 1975), p. 1488. is the designation for "partial derivative of y with respect to x". This represents a ratio of small changes of y and x under the condition that a l l other factors remain constant. .2 —Z is the designation for "second partial deriva-dx^ tive of y with respect to x". These concepts are introduced and explained in calculus textbooks. -247-II. THE CYLINDRICAL TUBE Boundary Conditions for the Cylindrical Tube *This subsection (to cage 20) is based entirely on Section I. It consists of application of the raw material presented, in Section I. A l l the equations and graphs presented on these pages are the gener-ous contribution of Dr. Roger Howard of the Depart-ment of Physics, University of British Columbia. Possible Frequencies The Clarinet ^This result was obtained on page 16. 3 Most of the discussion of the clarinet up to this point is based on the following source: John Backus, The Acoustical Foundations of Music, (New York: W. W. Norton and Company Inc., 1969), p. 191. 4 The fingering chart is quoted from: Keith Stein, The Art of Clarinet Playing, (Evanston, I l l i n o i s : Summy-Birchard Company, 1958), p. 24. The writer also consulted clarinettists, who demon-strated alternate fingerings for certain pitches — especially those in the extreme register. ^The writer makes this inference from the subjective discussion found in: Geoffrey Randall, The Clarinet, (London: Ernest Benn Limited, 1960), p. 37. See footnote 5 above. ^The idea of testing this equation in pracitcal s i t -uations involving the woodwind instruments is that of Dr. Roger Howard of the Department of Physics, University of British Columbia. No additional sourc-es were necessary to carry out this experiment. This -248-applies to every experiment involving a woodwind instrument. The instruments which were used were chosen at random in every case. 8 The velocity of sound may be considered common knowledge of physics. 9 This entire subsection (to page 37) consists of a paraphrase of the discussion found in: John Backus, Ibid., pp. 75, 203-206. The Flute ^ T h i s result was obtained on pages 18-19. '''"'"John Backus, Ibid., p. 188. 12 The discussion of fingering is based on the f o l -lowing source: Edx^in Putnik, The Art of Flute Playing, (Evanston, I l l i n o i s : Summy-Birchard Company, 1970), p. 27. The writer also consulted flute players, who dem-onstrated alternate fingerings for certain D i t c h e s . 13 This diagram is adapted from: John Backus, Ibid., p. 187. 14 See footnote 7. ^John Backus, Ibid., p. 188. -249-III. THE CONICAL TUBE ''"The f i r s t part of this section (to page 59) is a continuation of pages 1-20. As was the case with the material presented on pages 14-20, a l l the e-quations and graphs oresented on these pages are the generous contribution of Dr. Roger Howard of the Department of Physics, University of British Columbia. Possible Frequencies The Oboe 2 This paragraph contains a summary of information obtained from two sources: oboists consulted by the writer and the following book: Philip Bate, The Oboe: An Outline of i t s History,  Development and Construction^ (New York:W. W. Norton and Co. Inc., 1975), pp. 3-4. 3 This result was obtained on page 59. ^The discussion of fingering is based orimarily on demonstrations by oboists whom the writer consulted. The following source was also used: Robert Sprenkle, The Art of Oboe Playing, (Evanston, I l l i n o i s : Summy-Birchard Company, 1961), pp. 36-37. 5 P h i l i p Bate, Ibid., pp. 184-185. ^See footnote 7 under Section II. 7This entire subsection (to page 75) is a loose par-aphrase of the following source: Philip Bate, Ibid., pp. 123-129, 150-151. The Bassoon The discussion of the bassoon up to this point is based on the following source: John Backus, Ibid., pp. 195-196. -250-9 The last two paragraphs are a paraphrase of an excerpt from the following source: Lyndesay G. Langwill, The Bassoon and. Contrabassoon, (New York: W. W. Norton and Co., Inc., 1975), p. 145. *^This result was obtained on page 59. 1 1The discussion of fingering is based primarily on demonstrations by bassoonists whom the writer con-sulted. The following source was also used: Julius Weissenborn, Practical Method for the Bassoon. (New York: Carl Fischer Inc., 1958), Supplement. 1 2John Backus, Ibid., pp. 196-197. 13 The last four paragraphs consist of a loose, par-aphrase of an excerpt from the following source: Lyndesay G. Langwill, Ibid., pp. 146-148. ^John Backus, Ibid., P. 206. 1 5 P h i l i o Bate, Ibid., pp. 141-142. 16 John Backus, Ibid., p. 88. 1 7See footnote 7 under Section II. The Saxophone 18 John Backus, Ibid., p. 197. 19 This result was obtained on page 59. 20 The discussion of fingering is based primarily on demonstrations by saxophonists whom the writer con-sulted. The following source was also used: Larry Teal, The Art of Saxophone Playing, (Evanston, I l l i n o i s : Summy-Birchard Company, 1963), pp. 68-69. -251-21 See footnote 7 under Section II, IV. THE BRASS INSTRUMENTS ''"This discussion (to page 115) consists largely of a loose paraphrase of an excerpt from the following source: John Backus, Ibid., pp. 215-219. 2 With the exception of one paragraph, this subsec-tion (to page 117) is based on the following source: James H. Winter, The Brass Instruments, (Boston: Allyn and Bacon Inc., 1964), pp.'11-12. 3John Backus, Ibid., pp. 219-220. ^The discussion of the bell up to this point is based on the following source: John Backus, Ibid., pp. 220-222. 5 P h i l i D Bate, The Trumpet and Trombone, (London: Ernest Benn Ltd., 1972), pp. 25-26. 6Ibid., p. 29. ''ibid., pp. 15-16. The Trumpet o John Backus, Ibid., p. 221. ^Ibid., p. 223. 1 0 I b i d . , pp. 224-225. -252-The format used in indicating pitches for the brass instruments is adapted from: Paul Hindemith, The Craft of Musical Composition:  Book I: Theoretical Part, (New York: Associated Music Publishers, Inc., 1937), p. 20. 12 This table of fingerings was prepared in col-laboration with performers. The same holds true for the corresponding tables for the other brass instruments. Arthur H. Benade, Horns, Strings, and Harmony, (New York, Doubleday and Co. Inc., 1960), p. 181. ^The discussion of the valves up to this point is based to a large extent on the following source: John Backus, Ibid., pp. 225-226. ^The preceding three paragraphs consist largely of a paraphrase of an excerpt from the following source: Robin Gregory, The Horn, (New York: Praeger Pub-lishers, 1969), pp. 68-69. 16 John Backus, Ibid., p. 227. detailed discussion of the entire situation and of the problems involved in "testing" this equation for the brass instruments w i l l be present-ed in Section V. Robin Gregory, Ibid., p. 22. 19 The principal source of reference for this sub-section up to this point was: John Backus, Ibid., pp. 228-231. 20 The principal source of reference for the ore-ceding two pages was: -253-Robin Gregory, Ibid., pp. 23-24. Performers were consulted by the writer for verif-ication of details. 21 This fingering chart represents the culminating point of the step-by-step derivation and determin-ation of pitches and fingerings which began on page 148. For verification, the writer consulted per-formers, as well as the following two sources: Robin Gregory, Ibid., pp. 70, 75, William C. Robinson, An Illustrated Method for  French Horn Playing, (Bloomington, Indiana: Wind Music Inc., 1968), pp. 64-65. 22 Robin Gregory, Ibid., p. 50. 2 3 I b i d . , pp. 60-61. John Backus, Ibid., p. 231. 25 See footnote 17. The Trombone Philip Bate, The Trumpet and Trombone, (London: Ernest Benn Ltd., 1972), p. 47. 27 John Backus, Ibid., o. 224. 2 8 This chart represents the culminating point of the step-by-step derivation and determination of pitches and positions which began on page 181. For verification, the writer consulted performers, as well as the following two sources: Philip Bate, The Trumpet and Trombone, (London: Ernest Benn, Ltd., 1972), p. 202. Robin Gregory, The Trombone (New York: Praeger Publishers Inc., 1973), p. 63. 29 John Backus, Ibid., pp. 223-224. -254-Kent Kennan: The Technique of Orchestration, (New York: Prentice-Hall, Inc., 1952), p. 138. 31 Robin Gregory, The Trombone, (New York: Praeger Publishers Inc., 1973), p. 86. 3 2 I b i d . , pp. 88-89. 33 Adopted from Kent Kennan, Ibid., p. 138. 3^See footnote 17. The Tuba 35 This information was obtained upon consultation with performers. This chart represents the culminating point of the step-by-step derivation and determination of pitches and positions which began on page 209. For v e r i f i -cation, the writer consulted performers, as well as the following source: Donald Wesley Stauffer, A Treatise on the Tuba, (Rochester, New York: University of Rochester Press, 1942), p. 64. 37 See footnote 17. V. PRACTICAL APPLICATIONS OF KNOWLEDGE OF THE  ACOUSTICS OF WOODWIND AND BRASS INSTRUMENTS ^This section was prepared with the generous aid of Dr. Roger Howard of the Deoartment of Physics, University of British Columbia. -255-SELECTED BIBLIOGRAPHY Backus, John: The Acoustical Foundations of  Music; W. W. Norton and Co., Inc.; New York; 1969. Bate, Philip: The Flute; Ernest Benn Ltd.; London; 1969. Bate, Philip: The Oboe: An Outline of i t s His-tory, Development and Construction; Third Edition; W. W. Norton and Co. Inc.; New York; 1975. Bate, Philip: The Trumpet and Trombone; Ernest Benn Ltd.; London; 1972. Benade, Arthur H.: Fundamentals of Musical Acous-ti c s ; Oxford University Press; New York; 1976. Benade, Arthur H.: Horns, Strings, and Harmony; Doubleday and Co. Inc.; New York; 1960. Boehm, Theobald: The Flute and Flute Playing in  Acoustical, Technical, and A r t i s t i c Aspects; Dover Publications; New York; 1964. Champion, F. C.: University Physics, Part Four: Wave-Motion and Sound; Blackie and Son Lim-ited; London and Glasgow; 1948. Coar, Birchard: The French Horn; De Kalb, I l l i -nois; 1947. Gregory, Robin: The Horn; Praeger Publishers, Inc.; New York; 1969. Gregory, Robin: The Trombone; Praeger Publishers Inc.; New York; 1973. Hindemith, Paul: The Craft of Musical Composition:  Book I: Theoretical Part; Associated Music Publishers, Inc.; New York; 1937. -256-Kell, Reginald: The Kell Method for Clarinet; Book One; Boosey and Hawkes; Oceanside, New York; 1968. Kennan, Kent: The Technique of Orchestration; Prentice-Hall, Inc.; New York; 1952. Langwill, Lyndesay G.: The Bassoon and Contra-bassoon; W. W. Norton and Co., Inc.; New York; 1975. Maxted, George: Talking About the Trombone; John Baker Publishers, Ltd.; London; 1970. Morley-Pegge, R.: The French Horn; Ernest Benn Limited; London; 1960. Putnik, Edwin: The Art of Flute Playing; Summy-Birchard Company; Evanston, I l l i n o i s ; 1970. Randall, Geoffrey: The Clarinet; Ernest Benn Ltd.; London; 1971. Robinson, William C : An Illustrated Method for  French Horn Playing; Wind Music, Inc.; Bloomington, Indiana; 1968. Sprenkle, Robert: The Art of Oboe Playing; Surnmy-Birchard Company; Evanston, I l l i n o i s ; 1961. Stauffer, Donald Wesley: A Treatise on the Tuba; University of Rochester Press; Rochester, New York; 1942. Stein, Keith: The Art of Clarinet Playing; Summy-Birchard Company; Evanston, I l l i n o i s ; 1958. Teal, Larry: The Art of Saxophone Playing; Summy-Birchard Company; Evanston, I l l i n o i s ; 1963. Winter, James H.: The Brass Instruments; Allyn and Bacon Inc.; Boston; 1964. Wood, Alexander: The Physics of Music; Methuen and Co. Ltd.; London; 1947. -257-APPENDIX Frequencies of Pitches in the Equal-tempered Scale. CO 16.352 C3 130.81 C6 1046.5 c#o 17.324 C#3 138.59 C#6 1108.7 DO 18.354 D3 146.83 D6 1174.7 D#0 19.445 D#3 155.56 D#6 1244.5 EO 20.602 E3 164.81 E6 1318.5 FO 21.827 F3 174.61 F6 1396.9 F#0 23.125 F#3 185.00 F#6 1480.0 GO 24.500 G3 196.00 G6 1568.0 G#0 25.957 G#3 207.65 G#6 1661.2 AO 27.500* A3 220.00 A6 1760.0 A#0 29.135 A#3 233.08 A#6 1864.7 BO 30.868 B3 246.94 B6 1975.5 CI 32.703 C4 261.63** C7 2093.0 C#l 34.648 C#4 277.18 C#7 2217.5 Dl 36.708 D4 293.66 D7 2349.3 D#l 38.891 D#4 311.13 D#7 2489.0 El 41.203 E4 329.63 E7 2637.0 Fl 43.654 F4 349.23 F7 2793.8 F#l 46.249 F#4 369.99 F#7 2960.0 Gl 48.999 G4 392.00 G7 3136.0 G#l 51.913 G#4 415.30 G#7 3322.4 Al 55.000 A4 440.00 A7 3520.0 A#l 58.270 A#4 466.16 A#7 3729.3 Bl 61.735 B4 493.88 B7 3951.1 C2 65.406 C5 523.25 ***C8 4186.0 C#2 69.296 C#5 554.37 C#8 4434.9 D2 73.416 D5 587.33 D8 4698.6 D#2 77.782 D#5 622.25 D#8 4978.0 E2 82.407 E5 659.26 E8 5274.0 F2 87.307 F5 698.46 F8 5587.7 F#2 92.499 F#5 739.99 F#8 5919;9 G2 97.999 G5 783.99 G8 6271.9 G#2 103.83 G#5 830.61 G#8 6644.9 A2 110.00 A5 880.00 A8 7040.0 A#2 116.54 A#5 932.33 A#8 7458.6 B2 123.47 B5 987.77 B8 7902.1 *Lowest note of 88-key piano **Middle C ***Highest note of 88-key piano 

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