Harvey E. WHITE
`
`DONALD H. WHITE
`
`The Science of Musical Sound
`
`MZ Audio, Ex. 2005, Page 1 of 9
`
`

`

`To Beverly
`
`_
`CoJJyright
`Copyright© 1980 by Saunders College/Holt, Rinehart and Winston
`·
`·
`· All rights reserved.
`·
`· ·
`·
`· ·
`
`-. Bibliographical Note
`This Dover edition, first published in 2014 is an .unabridged
`republication of the work originally published by Saunders · College/
`Holt, Rinehart and Wisnfon, Philadelphia, iri 1980. ·
`
`.
`Library of Congress Cataloging~in-Publication Data
`White, Harvey Elliott, 1902-1988, author.
`· Physics and_ music : the science of musical sound / Harvey E. White,
`Donald H. White.
`·
`·
`. ·
`.
`.
`pages crtt. -
`(Dover books on physics)
`. .
`. . .
`Repri_nt. Previous]~ published: Philadelphia : Saunders College/
`Holt, Rinehart and Wmston, cl 980.
`·
`· Includes bibliographical references and index.
`ISBN-13: 978-0-486-77934!8 (paperback)
`. ISBN-IO: 0-486-77934-3 (paperback) .
`1. Music-,--Acoustics and physics. 2. Musical instrumentS-:.(cid:173)
`Construction. 3. Sound---'-Recording and reproducing. 4. Architectural
`acoustics. I. White, Donald H., author. II. Title:
`·
`·
`·
`·
`_ML3805.W44 .2014
`78l.2-dc23
`
`2013049049
`
`Manufactured in the United States of America
`. .
`.
`77934311
`2022
`www.doverpublications.com
`
`Almost everyone enjoys mu
`book is written for the snide
`of music _to learn how music1
`The student does not need a I
`presented here. H6wever, he
`rure of music, for example, a
`. This text, intended .· to t
`related to the acoustics of m
`science of music, followed b
`arid the primary ways in whi<
`ter,stics common to a11 · inst
`mechanical and electronic re
`ic ~nd quadrophc,nic SOJ1nd, :
`Finally, we explore the aco,
`theaters in both perceptual 1
`able to analyze or modify t
`ment.
`We have tried ro make I
`
`MZ Audio, Ex. 2005, Page 2 of 9
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`

`

`• I
`
`•: .. ,
`:"\ ~ •
`~(;(:
`·\t,
`-·.•
`:,;.i{!i
`
`.Chapter Eight
`.
`.
`
`.
`
`HARMONICS AND
`. WAVE COMBINATIONS
`
`.
`
`,._
`
`.. .
`
`There ~e an infinite n~be~ of musical tones that can
`· be ·produced .V.:ith iI1struments and th~ _liuman voice,
`and yet each one can be_ described by specifying the
`_ _
`iritensicy or loucl.ness; the f'requency or pitch; and the waveform 9r timbre.
`From the standpoint of musk, the waveform or timbre of any. complex tone ia
`all-important an_d can .be described by ·specifying the relative amplitudes and
`phases of all the different frequencies of which it is composed: We wiU study
`these concepts in this chapter.
`-
`- .
`-
`
`· -{ · 8; 1. · -Waye analysis
`
`If a.wave is gtne,1'.ated by simple harmonic motion, it will be i. ;in11soielillora
`· sine wave.' See Section 3_.2 and Figure 3- 3; A sine wave is -ihdkative of one
`· well-defined and definite frequency. The analysis ofmo~unusical tones shows
`.
`that they lire composed of a nimiber of suc:h· components of vari<:>us frequen•
`. ,.
`. ' ,·
`.
`.
`.
`. :
`.
`. '
`. '
`c1es called partials. The proce_ss ofadding these co'mponents tp. produce any
`complex vibration or wave is called synthesis. The converse of this process,
`-breaking down any complex vibration or wave i_nto its components, is called
`,aJiaJysfs. _-· _ .
`•
`- • -
`·
`-·• · ·
`· -_
`·
`.
`· . · • - _
`· ·-· .
`F_igure 8 - 1 represents two common graph· 1or~s for die _ same -sound,
`DiagrlUJl (a) is a time graph repiesenting the vil>rations of a source emittin&
`sound viaves. Diagram (I;,) is a distanc.e graph, or; wave gra,ph, representins the
`.contour of the waves traveling to the righ,t through the air with a vefociiy V.
`.
`..
`.
`· . .
`.,
`
`. . ·
`
`'
`
`.
`
`.
`
`' •
`
`01asram (a) also cc
`atlcro_phone diaph.
`1f a wave grap
`it 910uid look exa
`~h (b), and vice
`;1 makes little diffc
`
`The simplest wavt
`harmonic motion.
`prongs of a tuning
`and the vibration 1
`of chis. See J,ligure
`monic motion in
`£erred to as a pure
`impure (see S~ctic
`strwnents, and ·ot:
`pure-tone frequen
`fundamental. -Al
`ferred to as upper
`In special cas,
`the fundamental a
`any fundamental t
`so on. If; for exam
`the first harmonic
`First hatino~
`Second·harmc
`
`MZ Audio, Ex. 2005, Page 3 of 9
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`

`

`Dia,gram (a) also represents a time graph of the:vibradons of the eardrum, or a
`,
`microphone diaphr38ffi, detecting the sound.
`If a wave graph (b) were drawn traveling to the left instead of co the right,
`it would look exactly like graph ,(a). Graph (a) is just the mirror iiriage of
`graph (b), and vice versa Since a1t'three graphical representations look alike,
`it makes little, diff~rence whi,ch one is drawn .to represent a given sound,
`
`79
`Harm~nics and
`Wave Comoinations
`
`FiGURB 8-1
`(a) Time graph of
`the vibrations of a
`musical source of
`sound or of th_e
`vibrations imposed
`on the eardrum by
`incident sound
`waves; (b) A wave
`graph of the same
`sound as the waves
`travel with a speed
`V to the ,:ight.
`.
`
`"
`
`'
`
`'
`
`. 8.2 Partials and harmonics
`'
`The simplest' waveform is a sine ~ave, usuaily drawn as a time graph of simple
`harmonic motion. See Figures 2-4, 2-5,and 2-6. A time graph of o~ of the
`prongs of a tuning fork, the waves. trapsmitted through the air to an observer,
`and the vibration the waves impose upon .the eardturri serve as good examples
`of this. See Figure 3 - 5. Any vibrating body that rapidly executes simple har- .
`monic motion in air emits a sinusoidal sound wave, This sound wave is re(cid:173)
`ferred to as a pure tone, although the aural perception of even a p!,i.re tone is
`impure (see Section. 15.2). Actua11y, nearly all tones produced by musical in(cid:173)
`struments, and other sources in general, are not pure. tones but mixtures of
`pure-tone frequencies called partials. The lowest such frequency is called the
`fundamental~ .. All partials higher in frequency .than the fundamental are re~
`£erred to as upper partials, c;>r overtones.
`·
`· ·
`. In special cases, the frequencies of these overtones are exact multiples of
`the fundamental and are called harmonics. If we designate the frequency of .
`any fundamental by I, all higher harmo~ics are designated by 2/, 3/, 4/, 5/, and .
`so on. If, for example, we select a furidameotal frequency of2Q0 Hz and call fr
`the first harmonic, it and its higher harmonics are given by . ·..
`.
`•. ·
`.
`First harmonic: 1/ = 200 l;fa ·
`S!!cond harmonic: 2/ =.400 Hz
`
`Eight
`
`SAND
`LTIONS
`
`ones that can
`1uman voice,
`pecifying the
`m or timbre.
`np1ex tone ia
`1plitudes and
`Ve will study
`
`re analysis
`
`inusoidill or a
`cative of one
`l tones show,
`ious frequen·
`, produce any
`~ this process,
`~rits,-is called
`
`same sound,
`urce emittin&
`,resenting the
`, a velocity V,
`
`MZ Audio, Ex. 2005, Page 4 of 9
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`

`

`80
`Htaring and
`#a"11ony
`
`Third harmonic: 3/= 600 Hz
`Fourth harmonic: 4/ == BOO Hz
`.
`.
`.
`• ..
`.
`and so forth.
`.·
`..
`.
`· If singing voices, or different musical instruments, sound notes of the
`same pitch and loudness., we recognize the pitch as that of the fundamental,
`but the timbre or quality of each note differs from the others by virtue of the
`relative amplitudes of its partials. In _most 'cases, particularly with the percus(cid:173)
`sion instruments, the upper partials (ovenones) are not exact multi1>les of the
`fundamental frequency. Such an overtone is called an inharmonic partial,
`and the combined tone is often unpleasant,
`.
`. Most musical tones are composed of harmonics. In tact, the entire musi(cid:173)
`cal scale, as played by most musicians today, is based on a scale of harmonics.
`(See Chapter 14.) With these prindples in mind; we begin o~ study with the
`· combination of two pure tones, combine them, and find their resultant wave-
`. fa~ '
`,'
`'
`'
`'
`
`cy and amplitude bu
`sram (a), the two sin
`right has a maximum
`phase; as shown at tr
`phase angle differen
`cremes.
`
`K3
`
`'
`
`'
`
`,Two pure tones in unison
`
`,.,
`
`. If two pure tones of the same frequency are sounded simultaneously, and both
`waves arrive at the listener's ears, the resultant vibrations will have the same
`frequency. Such sources are said to be vib;ating in unison .. This is illustrated
`in Figµre 8 - 2 by the coinbinatfon of two SHMs, each having a freq~ency of
`833 Hz and a period of 12 >< 10-4 s but with differentamplitudes,al = 8 X 10- 7
`m and,a3 = 6 X. 10-7 m, respectively. Vibration (a) has ari initial phase angle cf,11
`= 0°, and vibration (b) ha, an initial phase angle cf,0 = +90°. See Section 2.5
`and .Figure 2 - 6.
`'
`Since the frequencies are equal, the graph points Pi and p2 move ii.round
`their circles of reference in the same time, alw'ays keeping the same phase
`angle difference of 90° between them. As a consequence; their resultant am~
`plirude A always has the same magnitude of 10 x 10-7 m and an initial phase
`angle of cpo ~ 37°. The amplitudes al and "2 are added viciorial/y in the left(cid:173)
`hand side of diagram ( c). ·
`.
`· ·.
`·
`· · · ··
`.
`• •. · . •
`·
`Each of the gra:ph points Pi and P2;'as weH as theresuliant graph point JI, is
`seen to move once around its respective circle in the · same time, ~d the ·
`. SH.Ms along the y~axis trace out siriusgids with the period T. It will be ob(cid:173)
`served that the vertical JinesJrom Oto 12 sh9w that;at in points in tilIJe, the ..
`vertical displacements of curve (c) are always equal to the sum of the displace(cid:173)
`ments of curves (a) and (b). The three .time graphs are superposed in FigUre
`8;.. 3. We conch:iqe from this result that the combination of two SHMs of the
`same frtiquency will always give rise to a resultant SHM of .t_p.e sapie frequen(cid:173)
`cy, but with a resultant amplitude that depends upon the two amplitudes and
`their phase angle difference. . ·
`.
`· ·.
`.
`· , 1 .
`·
`·
`.· · ·
`This Sa.Ille principle is ilh,istrated for two vibrations ofthe sarrie freqµen(cid:173)
`',
`
`If the two amp
`tion will still have
`phase) will b.e equ
`tude (180° out ofp
`all other phase dill
`and foi.lnd by the m
`
`MZ Audio, Ex. 2005, Page 5 of 9
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`

`

`structure of the same note played by .two violinists, for exainple, will not be
`84
`Hearing tznd · identical, for .various reasons. No two instruments.are exactly alike sttuctural-
`ly, and no two musicians will bow their strings in exactly the same way. While
`Htzrm~ny .
`tb.e harmonic sttuetures will ail be slightly different, each one will, of course,
`sound like a violin. The sound specuum of each note will have a fundamental,
`as well as the appropriate harmonics, but will vary slightly from one instru-
`ment to another.
`.
`·
`. ·
`.. •
`·
`We have seen in the pre~ious section that th~ .fundamentals of a group of
`violins will not have the same phase angles and thatin general they will be
`random, It is also reasonable to assume that all musicians will not produce
`· exactly the same frequency. This means that beat notes of different frequen(cid:173)
`cies will be produced between funi:lainentals, between se~ond harmonics,
`between third harmonics, and so on, and · these will make the .overall wave(cid:173)
`form fro¢1 the .!µ'<>Up of viplins more complex. The. sound quality produced by
`the combined frequencies from · a number of instruments of the same kind,
`playing the same note, is called the chorus effect Although the primary pur~
`. -·pose.of using a number of violins in the string section of a symphony orches(cid:173)
`tra,, for . example, is to obtain . a loudness · balance with the _other orchestral
`inst:rWllents, the chorus · effect contributes to ·the overall richness of the musi-
`cal sound .. ·
`.
`.
`
`R6 Composition of first . and second harmonics
`
`"J
`.
`.
`·,
`. ·
`.
`:
`.
`..
`: Let tis assume that a. musical instrument sounds a tone in which the first and
`second_ harmonics, and no others, are present. The same resultant vibrations.at
`. 'the .ear C:an be produced by sounding of one of the . pure tones by one . in•
`
`suwnent arid the otl
`with frequencies in
`giver} graphically in
`(Iighdiries) is show
`Since /2 has twice tl
`
`PIGURES-5 ·
`Time graphs for
`the generation of
`two SHMs with
`initial phase angles
`c/,o = O": (a)
`·
`frequency/ and
`amplitude 4, (b)
`frequency 2/ ilnd
`amplitude 3 . . ··
`
`, ,
`
`+y ~T2---:---i
`
`·•~-~~,~-~~~~~~~~~~-~~~~~~
`
`{bl
`
`M
`
`~
`
`:
`
`· 0
`
`10, 4 ~ - s:,,-- =:_ y
`
`MZ Audio, Ex. 2005, Page 6 of 9
`
`

`

`ill not be
`,rrucniral(cid:173)
`ay. While
`of course,
`damental,
`ne inlitru-
`
`1groupof
`!Y will be
`t produce
`t freqtien.
`iarmonics,
`tall wave(cid:173)
`oduced by
`ame kind,
`4naeypur(cid:173)
`ny orches(cid:173)
`orchestral
`:the musi-
`
`:rmonics
`
`1e first and
`brationsat
`by one in-
`
`la)
`
`- · l
`-----1
`
`(bl
`
`·•.
`.
`.
`stcU.lllent and the other pure tone by a separate instrument; These two SHMs,
`. with frequencies in · the ratio of 1 to 2 and initial phase angles both zero, are
`given graphically in Figure 8- ~- The su.ui of the two displacements J; and /2
`(light liries) is shown by the resultant vibration R. (heavy ·line) in Figure 8-6.
`Since/; has twice the frequency ofJ;, the graph points p2 and p1 (on the circles
`.
`.
`
`.
`
`~
`
`.
`
`.
`
`.
`
`.
`
`FIGURE 8-6
`
`85
`.Harm~niis and
`Wave CombinationI
`Composition of the .
`two ~HMs ·in ..
`·Figure g.:..5 show111g
`the resultant R in ·
`· relation to the
`· ampUtudes a, · and
`a2 of the separate
`com pone no.
`
`FIGURE 8-7
`Time graphs
`combining· the · first
`and second;
`harmonics of a
`fundameiitaI ·
`frequency/,. to
`form ii resultant.
`Both frequencies/.
`and /2 have (a) the
`sanie initial phase
`angles but different
`amplitudes, (b)
`different initial
`phase angles and
`different ·.
`·
`. amplitudes, (c)
`· different initial
`phase angles but
`· equal amplitudes;
`and (d) different
`initial phase angles
`and different .
`amplitudes. All
`four res11lta~ts
`(heavy lines) ha"e
`different shapes but
`reveal the same .
`two frequen<;ies Ii
`and/2.
`
`'
`
`t ➔
`
`_t~.
`OA--+-->AA--H~,--+~'r--+H..,_~__.,11'!"---,~r--
`
`MZ Audio, Ex. 2005, Page 7 of 9
`
`

`

`86 of reference) rotate with frequencies in the ratio 2 to 1. The second harmonic
`Hearing and makes two vibrations for every one of the first harmonic. For this example, the
`periods are assumed robe 12 X 10-• sand 6 X 10-• s, corresponding ro fre(cid:173)
`Harmony
`quencies of 833 Hz and 1666 Hz, respectively. The resultant R in Figure 8-6
`is obtained by adding the vertical displac_ements of J; and /2. at each instant
`of time and drawing a smooth curve through them.
`If the relative amplitudes are chaaged without changing the initial phase
`angles, we obtain curves of a different shape. Changing the relative ampli(cid:173)
`tudes and the initial phase angles also changes the resultant curve. Typical
`graphs with such changes are shown in Figure 8 - 7. It should be pointed out
`that these are but a few of the infinite number of resultant vibration patterns
`that can be drawn. See Figure 15 - 5 for ochers.
`
`8. 7 Two, three, and four harmonics
`
`Suppose we sound a pure t0ne of any given frequency, and then, one after an(cid:173)
`other, we add the second, third, and fourth harmonics. The quality of each
`combination will depend upon the relative amplitudes, while the resultant
`vibration pattern becomes progressively more complex and, in many cases,
`more pleasant to hear. (Two consonant notes sounded together are called a
`dyad, three notes a triad, and four notes a letrad.)
`As an example, let w choose a first harmonic, or fundamental, of 8 3 3 Hz,
`followed by the second, third, and fourth harmonics. Let the relative ampli(cid:173)
`tudes of the four harmonics be a1 = 8, a2 = 6, a3 = 4, and a4 = 6 X 10-1 m, and
`let the initial phase angles be ,J,1 = 90°, ,f,2 = 45°, q,8 = -90°, and </>4 = -45°.
`Graphs of these combinations are given in Figure 8- 8. It can be seen that, as
`harmonics are added, the resultant vibration curve becomes more and more
`complex, and, in general, the tone becomes richer in quality.
`
`8.8 Wave generation
`
`The separation of any sound into its various components can be accomplished
`by mechanical or electronic devices called analyzers, and any set of compo(cid:173)
`nents can be recombined tO produce the original sound by similar mechanical
`or electrical devices called synthesizers. 2 In 1622 the French mathematician
`Fourier showed that it was possible to break down any complex periodic
`curve into a series of sin~soids whose frequencies are harmonically related.
`Stated another way, any periodic waveform can be constructed by combining a
`sufficient number of sine waves. This is called Fourier's theorem. This means
`'
`that any periodic sound wave of arbitrary waveform will act acoustically as a
`combination of pure tones. While we will not go into the mathematics, we
`will graphically add, or synthesize, a number of SHMs to form several special
`vibration forms used by electronic engineers in the development of oscil-
`
`MZ Audio, Ex. 2005, Page 8 of 9
`
`

`

`loscope$ and television receivers, by audio engineers in their development of
`electronic music devices for special effects, and by manufacturers in produc(cid:173)
`ing musical instruments of various k(nds: -. ·
`
`·
`
`87
`H;",.,,,oni~s am/
`Wa11t Combinations
`
`FIGURE 8-8 . . · ·
`Ti'me graphs for
`the addition of the
`first four humonics
`of a given
`fundamental ·
`frequency/.
`Vibration modes
`for (a) the
`fundamental alone,
`(b) the first and
`second harmonics
`together, (c) the
`· sum of the first,
`second; and· third.
`harmonics together;
`and (d) all four . · · :
`harmonics together.
`,.
`
`t
`
`A
`
`f + 2( + 31 + 4f
`
`. (d)
`
`The four simplest waveforms, or vibration forms, i!l common use i11 syn(cid:173)
`thesizers today are called (a) sine waves, (b) sawtooth waves, (c) square
`waves;and (d) triangular waves. See figure 8-9, Diagrartis (b) and (d) be(cid:173)
`long to a family of straight-line forms called ramp waves: _All four of the~e
`wavefo.i:ins can be produc;ed with relatively simple electronic ci~cuits. Sinc,e
`the analysis of complex waves can be broken down into sine waves, and the
`synthesis of a number of si~e waves (harmonics) _can be compounded to pro(cid:173)
`duce complex waveforms, we can apply Fourier analysis-that is, a series of
`
`MZ Audio, Ex. 2005, Page 9 of 9
`
`