7
`
`HARRY F.OLSON
`MUSIC, PHYSICS
`and
`ENGINEERING
`
`SECOND EDITION MZAudio, Ex. 2002, Page 1 of 13
`
`aa TITLEDMUSICAL ENGINEERING)
`
`ees
`
`MZ Audio, Ex. 2002, Page 1 of 13
`
`

`

`Music, Physics
`and. Engineering
`
`~
`
`HARRY . F. OLSON, E.E., PH.D., n.sc., HON.
`Sta.I! Vice President
`ilooustica,l · and Electromech.anioa,l Research.
`RCA Laboratories, Pri11oeto11, New Jersey
`
`SECOND EDITION
`
`DOVER PUBLICATIONS, INC.
`New York
`
`MZ Audio, Ex. 2002, Page 2 of 13
`
`

`

`Preface. ·to the Second Edition
`
`.,-
`
`Many and varied advances have been made in musical engineering in the
`13 years since the appearance of the first edition. The most important
`developments have been ma.de in the field of sound reproduction, subjec(cid:173)
`•. tive acoustics, and electronic music.
`The most recent, significant, and important advance in the field of sound
`reproduction is the large-scale commercialization of stereophonic sound in
`the consumer or mass market complex. Stereophonic sound provides audi(cid:173)
`tory perspective of the reproduced sound and thereby preserves a subjective
`illusion of the spatial distripution of the original sound sources. Three
`mediums for stereophonic sound reproduction have been commercialized
`on a large scale for the consumer's use in the home, namely, magnetic tape,
`disc and frequency-modulation radio.
`The· ma.in purpose of sound reproduction is to provide the listener with
`the highest order of artistic and subjective resemblance to the condition of
`the live rendition; To achieve this objective requires the utmost degree
`of excellence of the physical performance of the equipment as directed by
`the psychological factors involved. The state of the art in sound repro(cid:173)
`duction has advanced to a stage where a high order of physical performance
`can be obtained. Up until very recent times, the application of the psy(cid:173)
`chological characteristics as related to sound reproduction have lagged
`· .. behind the physical considerations. The commercialization of stereophonic
`sound in the consumer complex has hastened the work in the subjective
`aspects of sound reproduction.
`The properties of a musical tone described in the first edition of Musical
`Engineering remain the same. Once . a tone has been described it is pos(cid:173)
`sible to produce this tone by purely electronic means. Thus, it will be
`seen that any tone produced by a voice or musical instrument can be pro(cid:173)
`duced by an 'electronic system. In addition, electronic synthesizers pro(cid:173)
`vide the capability of creating musical tones which cannot be produced by
`V
`
`Copyright
`Copyright © 1952 by Harry F. Olson.
`Copyright © 1967 by Dover Publications, Inc.
`All rights reserved.
`
`Bibliographical Note
`This Dov.er edition, first published· in 1967, is a revised and
`enlarged version of the work first published by the McGraw-Hill
`Book Company, Inc., in 1952 under the title Musical Engineering.
`
`International Standard Book Number
`ISBN-13: 978-0-486-217690 7
`ISBN-JO: 0-486-21769-8 ·
`
`Library of Cong,-ess Catalog Card Number: 66-28730
`
`Manufactured in the United States by LSC Communications
`21769827
`2022
`www.doverpublications.com
`
`MZ Audio, Ex. 2002, Page 3 of 13
`
`

`

`10
`
`MUSIC, PHYSICS, AND ENGINEERING
`
`SOUND WAVES
`
`11
`
`(1.1)
`
`1.5 VELOCITY OF PROPAGATION OF A SOUND WAVE
`The preceding examples have shown that a sound wave travels with a.
`definite finite velocity. The velocity of propagation, in centimeters per .
`second, of a sound wave in a gas is given by
`C = ✓i':o
`where '.Y. = _ratio of specific heats for a gas, 1.4 for air
`..
`Po = static pressure in the gas, in dynes per square centimeter
`p = density of the gas, in grams per square centimeter ·
`If the pressure is increased, the density is also increased. Therefore, .
`there is no · change in velocity due .to a change in pressure. But this is ·
`true only if the ·~mperature remains constant. · Therefore, the velocity
`can be · expressed in terms of the temperature. . The velocity of sound, ·
`in centimeters per second, in air is given by
`c = 33,W0 -Vl + 0.00366t
`where t = the temperature in degrees centigrade.
`1.6 . FREQUENCY OF A SOUND WAVE
`·· Referring to the Sec. 1.4 on Sourid Generators, it will be seen that these
`· genera.tors pr~duce similar ~ecurrent waves; A complete set of these
`recurrent waves constitute a. cycle . . These recurrent waves are propa-
`gated at .a definite velocity . . The number of recurrent waves or'cycles
`·which pass a certain observation point per second is te~med the frequency·
`of the sound wave.
`·
`
`(1.2)
`
`1;7 WAVELENGTH OF A SOUND WAVE
`The .wavelength of a sound wave is the distance the sound travels to
`complete one cycie; The frequency, of a sound wave is the number of
`cycles which pass . a. certain observation point per ~cond. · Thus it -will
`be seen that the velocity of propagation of a sound wave is the product of
`the wavelength and the frequency, which ·may be expressed as follows:
`C = }-f
`(1.3)
`where c = velocity of ,propagation, in centimeters per second
`>. = wavelength, in centimeters
`·
`.f = frequency, in cycles per second
`1.8 PRESSURE IN A SOUND WAVE
`A soun:d wave consists of pressures. above and below the normal undis,.
`turbed pressure in the gas (see Secs. 1.3 and 1.4) .
`The instantaneous sound pressure at a point is the total instantaneous
`
`preSBure at that point minus the static pressure, the static pressure being
`· the normal atmospheric pressure in the absence of sound. ·
`. The effective sound pressure at ,a point is the root-mean-square value
`of the instantaneous sound pressure over a complete cycle at that point.
`The unit is the dyne per square centimeter. The term "effective sound
`pressure II is frequently shortened to "sound pressure."
`·· The sound pressure in a spherical sound wave falls off inversely as the
`. ~ .
`distance frpm the sound source.
`·
`•
`1.9 PARTICLE DISPLACEMENT AND PARTICLE VELOCITY INA SOUNDWAVE
`• The passage of a sound "1ave produces a displacement of the particles
`or.molecules.in the gas froJ\l t;e norpial position, that is, tlie position in
`the absence of a. sound wave . • , 'Phe particle displacement in a normal
`sound wave in speech and music is a very small fraction of a millimeter .
`For example, in normal conversational speech at a distance of 10 feet
`from the speaker, the particle amplitude is of the order of one-millionth
`of an inch. The particle or molecule in the medium oscillates at the
`frequency of the sound wave. The velocity of a particle or molecule in
`the process of being displaced at the frequency of the sound wave is
`ternied the particle velocity. ·
`.. The relation between sound pressure and particle velocity is given by
`p = pCU
`(1.4)
`where p = sound .pressure, ii{ dynes per square centimeter
`p ;,,, density of air, in grams per square centimeter
`c ,;,, velocity of sound, in centimeters per second
`u = particle velocity: .in centimeters. per second
`The amplitude or dil!Jplacement of the particle from its position in the
`absence of a sound wave is given by
`u
`d = 21('!
`where d = particle amplitude, in centimeters
`u = particle velocity, in centimeters per second
`J = frequency, in cycles per second
`INTENSITY OR POWER IN A SOUND'WAVE
`l,10
`From the foregoing sections it is evident that energy is befog transmitted
`in a sound wave. The sound energy transmitted is termed the intensity
`of a; sound wave.·
`The intensity of a sound field, in a specified direction at a point, is the
`sound energy transmitted per unit of time in a specified direction through
`a unit area normal to this direction at t,he point. The unit is the erg
`per second per square centimeter.
`
`(1.5)
`
`MZ Audio, Ex. 2002, Page 4 of 13
`
`

`

`12
`
`MUSIC, PHYSICS, AND ENGINEERING
`
`SOU.ND WAVES
`
`13
`
`2
`
`The intensity, in ergs per second per squu.re centimeter, of a plane
`sound· wave is
`
`= pu == pcu2
`
`(1.6)
`
`I= P
`pC
`where p = sound pressure, in dynes per square centimeter
`u = particle velocity, in centimeters per second
`c = velocity of propagation of sound, in centimeters per second
`p = density of the medium, in grams per cubic centimeter
`The intensity level, in decibels, of a sound is ten times·· the logarithm
`to the .base 10 of the ratio of the intensity of this sound to the reference
`intensity. Decibels will be described in the section which follows.
`
`TABLE 1.1; THE RELATION BETWEEN· DECIBELS
`AND PoWEl\ AND CUJtBENT OB VOLTAGE RATIOS
`
`Power ratio Decibels
`
`1
`0
`3.0
`2
`4.8
`3
`(.i.0,
`4
`7.0
`5
`7.8
`6
`8.5
`7
`9.0
`8
`9.5
`9
`10
`10
`20
`100
`30
`1,000
`40
`10;000
`50
`100,000
`1,000,000 I 60
`
`Current or Decibels
`voltage ratio
`~-··-
`0
`6.0
`9.5
`12.0
`14.0
`15,6
`16.1)
`IS.I
`11}. l
`20
`40
`60
`80
`100
`120
`
`1
`2
`3
`4
`5
`6
`7
`8
`9
`10
`100
`1,000
`10,000
`100,000
`1,000,000
`
`1,11 DECIBELS
`In acoustics the ranges of intensities, pressures, and particle velocities
`are so large that it is convenient to use .a condensed scale of smaller num(cid:173)
`bers termed decibels. The abbreviation db is used for the term decibel.
`The bel is the fundamental division of a logarithmic scale for expressing
`the ratio of two amounts of power, the number of bels denoting such a
`ratio being the logarithm to the base 10 of this ratio. The decibel is
`one-tenth of a bel. For example, with P1 and P 2:designating two amounts
`of power and n the number of decibels denoting their ratio, then
`n = 10 log10 ;: decibels
`
`(l.7)
`
`When the conditions a.re such that ratios of currents or ratios of voltages
`(or the analogous quantities such as pressures, volume currents, forces,
`and particle velocities) are the square roots of the co1Tesponding power
`ratios, the number of decibels by which the corresponding powers differ
`is expressed by the following formulas:
`n = 20 log10 ~ decibels
`i,
`•
`'b 1
`ei d
`~
`n = 20 log10 -
`ec1 e s
`e2
`where·i1/i2 and·e1/e2 are the given current and voltage ratios, respectively.
`For rele.tion between decibels al\d p9wer and current or voltage -ratios,
`see Table 1.1.
`·
`· · ·
`· .,.
`·
`
`(1.9)
`
`(1.8)
`
`1.12 DOPPLER EFFECT IN SOUND WAVES
`The Doppler effect is the phenomenon evidenced by the change in the
`observed frequency ofa sound wave in a transmission system caused by a
`time rate of change in the length of the path of travel between the source
`and the point of observation. The most common example of the Doppler
`effect is due to the relative motion of the source and observer. Examples
`are the change in the frequency of the tones emitted by horns and
`whistles of passing automobiles or locomotives.
`When the source and observer are approaching each other, the fre(cid:173)
`quency observed by the listener is higher than the actual frequency of
`the sound source. If the sour<;e and observer are receding from each
`other, the frequency is lower.
`The frequency atthe observation· point is
`.
`V -
`Vo
`!o =· --f.
`·v•
`where v = velocity of sound in the medium
`v0 = velocity of· the observer
`v8 = velocity of the source
`!@ = frequency of the source
`All the velocities must. be expressed in . the same units.
`No account is taken of the effect of wind velocity or motion of the
`In order to bring in the effect of the :wind, the
`medium.in Eq. (1.10).
`velocity v in the medium must be replaced by v + w, where w is the wind
`velocity in the direction in which the sound is traveling. Making this
`substitution in Eq. (1.10), the result is
`Jo = v + w - Vo I,
`V + W -
`ti•
`Equation (1.11) shows that the wind does not produce any change in
`
`V -
`
`(1.10)
`
`(l.ll)
`
`MZ Audio, Ex. 2002, Page 5 of 13
`
`

`

`200
`
`MUSIC, PHYSICS, AND EMGINEHIHG
`
`Wind Chest. A wind chest is a reservoir for supplying air under pres-
`sure to the pipes or reeds of an organ;
`·
`·
`Wrest Pin. Wrest pins are the turnable pins around which the ends of
`the strings are wound in the harp and piano. The strings can be tightened
`or loosened by turning the wrest pin .
`
`. ~/ .. ~
`
`..
`
`CHAPTER stx ··
`
`Characteristics of Musical Instruments
`
`6,1 INTRODUCTION
`The two· characteristics of music· which are principally a function of the
`musical instrument are the tonal and dynamic. The tonal aspect depends
`upon the pitch and the timbre of the instrument.. The pitch is in general
`determined by the fundamental·· frequency .and fundamental-frequency
`range of the instrument. The timbre is the instantaneous-aco~ical
`spectrum . of the instrument. · The timbre · invoives the . frequencies and
`amplitudes of both the fundamental and the overtones, The dynamic
`aspect depends upon the absolute intensity level produced by the instru-
`ment and the dynamic range or intensity range.
`·
`Musical instruments and the voice produce fundamental frequencies·
`and overtones of fundamental frequencies. · The overtone structure is
`one of the characteristics which distinguishes various instrwnents and
`voices.
`If musical instruments produced the fundamental.without over- ·
`tones, each instrument would produce a pure sine wave, · and it would,
`therefore, be the same as the output ofall other instruments except for the
`possibility of a difference in frequency and intensity. The fundamental
`frequency is the lowest frequency component in a complex wave. When
`a musician speaks of the range of a voice or musical instrument, he usually
`means the frequency range of . the fundamental frequency.
`In other
`words, the fundamental-frequency range of a musical instrument com(cid:173)
`mands a certain section of the musical scale.
`. In addition, each · instru(cid:173)
`ment or voic!l produces harmonics or overtones of the fundamental fre(cid:173)
`quency. In . general, the overtones cover a tremendous frequency range
`or section of the musical scale. ·
`·
`·
`·
`The dynamic aspect of music depends upon the intensity. The inten(cid:173)
`sity ranges of musical instruments involve the absolute value of the upper
`and lower intensity ranges and the resultant dynamic range.
`·
`The intensity and timbre of the sound produced by a musical instni-
`201
`
`MZ Audio, Ex. 2002, Page 6 of 13
`
`

`

`202
`
`MUSIC, PHYSICS, AND EN~INEERING
`
`ment are also governed by the directivity pattern.. The directivity pat(cid:173)
`tern deptcts the sound output as a function of the angle with respect to
`some reference axis of the instrument.
`In general, the directivity pattern
`is complex, in j;hat it is a function of both the angle and the frequency.
`Under these conditions, both the intensity and the timbre will vary aa
`the frequency or angle of orientation of the instrument is altered.
`The growth', decay, and steady-state characteristics and the duration of
`a tone produced by a musical instrument influence the tonal and dynamic
`characteristics of the · tone.
`It is the purpose of this chapter to consider the fundamental and over(cid:173)
`tone frequency ranges, the acoustic spectrums, the intensity ranges, the
`directional patterns, and the growth, decay, steady-state,· and duration
`characteristics of various musical instruments;
`
`6.2 FUNDAMENTAL AND OVERTONE FREQUENCY RANGES OF MUSICAL
`INSTRUMENTS
`Musical instruments and the voice produce fuµdamental frequencies and
`overtones of fundamental frequencies. The overtone structure is one or
`the characteristics which distinguishes various instruments and voices.
`If musical instruments produced the fundamental without overtones, each
`instrument would produce a pure sine wave and woµld, therefore, be the
`same as the output of all other instruments except for the poBSibility of a
`difference in frequency and intensity. The fundamental frequency is the
`lowest {requency component in a complex sound wave. When a musician
`speaks of range of a voice or musical instrument, he usually means the
`frequency range of the fundamental frequencies which the voice or instru(cid:173)
`ment is capable of producing. · The fundamental-frequency ranges1 of
`voices and various musical instruments are shown in Fig. 6.1. It will
`be seen that the fundamental-frequency range of each musical instrume'lt
`commands a certain section of the musical . scale. There may be some
`variations, from the frequency ranges sh9w,µ in Fig. 6.1, among vario\18
`instruments and voices, but, in general'; these frequency :canges are typi-.
`cal. Comparing the frequency ranges of ihe !undamental of Fig. 6.1
`with the entirefreq_uency spectrum1 of Fig. 6.2, it.will be se'en that the
`frequency ranges of the overtones of the instruments extend the frequency
`ranges- by a factor of two or more octaves, Referring to l'ig. 6.2; it will
`be seen that the low-frequency ranges of some instrument extend to a
`lower frequency than the fundamental-frequency ranges of Fig. 6.1.
`
`1 Olson, H. F., Acoustical Engineering, D. Yan Nostrand Company, Inc., Princeton,
`1957.
`1 Snow, W. B., Jour. Acoust. Soc, Amer., Vol. 3, No. 1, Part 1, p. 155, 1931.
`
`CHARACTERISTICS OF MUSICAL INSTRUMENTS
`203.
`This is due to the fact that some instruments generate noises and sub(cid:173)
`harmonics having frequency components below the true fundamental(cid:173)
`frequency range. . It will also be seen that the limits of the low-frequency
`range of some instruments in Fig. 6.2 are actmtlly higher than those of
`Fig. 6.1. This is due to the fact that in the case of some of the lowest
`tones produced by some instruments the fundamental is so weak that it ·
`ean be eliminated without discerning any change in the character of the
`t-0ne. These facts . relating to the low-frequency range are relatively
`unimportant. The important factor in the comparison between Figs.
`6.1 and 6.2 is the great extension of the frequency range when the har-
`monics are included.
`·
`·
`·
`
`\
`
`80l'IIANO \/OIC:l
`ALTO 'VOIC:l ·
`YR!fE
`T.
`SAR1ron VOICE
`BAH vo,~~
`
`I
`
`VIOLIN
`VIOL"
`cu.1..0
`I
`IIA&~ vtt L
`HAIII'
`GUfTM
`.... 0
`UKULI.L~
`
`J,ANO
`I. · TAUMl'CT
`T-IIONE
`ua11· TIIOM90Nt
`,111MCHHOIIN
`■AM TUU I
`
`I
`
`'
`
`.... ,.;
`
`IAH SAXOPHONE
`
`1.
`
`,ic:;cOLO
`ll'L.UTI.
`
`I l
`SOPRANO CLARIICiT
`Al.TO C:LAAIN!T
`' I
`IIAU C.LUINtT
`1 ao~liAICII &A'KOPHONt
`ALTD . . vnpUQ,i~
`TtNOR ~,HOME
`JlfflJMI
`
`l
`l . . o■o!
`
`t;NC:LIAH HORN
`1"9110011 I
`REEi> ' ORGAN
`Accoa1>10N
`ICA
`HAIi
`
`~l,C OIIIGiN
`, !LI.CTIUC OAGA-.
`
`-
`-
`-
`• .
`Ill ,o~
`fOiC
`~
`""
`•
`ii
`rRE:QUtNCY
`IN CYCLts PtR StC0N0
`Fia. 6.1. Frequency ranges of the funds.mental frequencies of voices and various
`muaical instrumenta . . (After Ol80tl,, ACOU8ticat Engineering, D. Van N0&lrand Company,
`l11C., Prmuton, 1857.)
`.
`
`Tf-uf
`
`I
`XYLOPHONE
`I
`
`10
`
`C
`
`4
`
`•. J()41
`
`MZ Audio, Ex. 2002, Page 7 of 13
`
`

`

`204
`
`MUSIC, PHYSICS·, AND EN G,Um ll
`
`CHARACTERISTICS OF MUSICAL INSTRUMENTS
`
`205
`
`AVtRA<.t
`TOTAL .
`PRESSVRC
`
`AVtRAC:t
`DISTANCE TOTAL
`PRtsSURt
`
`01$TANCC
`
`-"VCAACC
`TOTAL
`Pllt~SUAt
`
`.
`PIANO
`
`015TANC.C
`
`•OFT.
`
`~~-4.6 1iP1tct ORCHtsTAA ISF~:7.9 ~SPIECC A' ·- S7A-" IQFT~:-2.e
`-
`-
`-
`.... ~
`-
`-
`~ ~
`L - ~IWl~l■f 1
`i " ffllUI ~ M" -
`~·-~'HIil
`::m•
`-r
`:-Ill"·-·" .,,.]lllfflf fflfflHffl
`ill II llftffl-llHlffflD
`"~' un.:~_ 1.•11
`
`,~,-T,
`
`PIPlt --- OR(iAN
`
`"II
`
`~-•
`~ a
`-a,
`!
`IC(cid:173)
`,-
`
`~ · : ....L..LI.....L..L...1.L...L....i,_.J...J .. 'NQ
`
`...
`
`~a-0
`
`-I
`
`I
`
`-eo
`~
`
`_
`
`- 14
`
`I .
`
`-IQO
`
`-1'4
`
`-
`
`1
`
`..,MO
`
`-
`
`-
`
`~•I
`
`a:.~1,4
`
`-60
`
`l
`i .. ~~
`
`.. , .. o.
`
`~-,.
`
`- I
`
`'
`
`TU .. ,ANJ
`
`6A&& DRUM
`
`I
`
`!tkARt ORUM
`
`1,4,t• C'YNI.AL&
`
`9A$'!i VIOL
`
`CI.U,0
`1
`
`PIAN0
`
`YIOl.tN
`
`USS TUBA
`1
`TROMIIOHC
`
`rRtNCH OIOIIN
`
`TIIUMPET
`
`DASS
`IAllOPIIOME
`IIA$BOON I
`•A&! C.LA.RtNt.T
`kLAIIIN&T I
`SOPRANO 5AXOPM0Nt
`
`oaot ' I
`
`ACCQROIOII
`
`!HARMONICA
`rLurt' I
`PICCOLO I
`PIP£
`DR-GAil
`
`M.M, .. £
`~PEl'.CH
`I
`r1u,•L~ 5'PC£.C.H
`' .
`~OOT STl."5
`
`HAND CLAPPING
`11
`IIOOM NOIH
`
`I
`
`•·
`
`.
`..
`
`I
`l
`
`. i
`
`tt£'f JIMCLIHC.
`
`"
`4
`' 101
`4
`I 101
`~
`l
`IN CYCLES PER SECOND
`r11EQU£NCY
`FIG. 6.2. Frequency. ranges required for- speech, musical instruments, aod not.
`(After Snow.)
`tha.t no frequency discrimination will be a.pparent;
`
`I
`
`e 10'
`
`i
`
`•
`
`,
`
`t:: H 21N.
`
`.... ~
`l.
`
`•IAO
`
`-14
`
`2
`
`e.3
`
`r t MA C !iP
`
`611 - -
`_,_ --
`
`-
`
`'"'I
`
`·.
`
`-
`
`-
`
`..,
`
`4
`
`•l
`
`0
`
`100
`
`1000
`
`10000 ·••~o IOO
`10000·•
`1000
`~
`PCA
`IN CVC~£S
`FAtQU£NCV
`5£1:0NO
`FIG, 6.3. Ratio of the average pressure per cycle to the average total pressure of the
`entire spectrum for speech, various musical instruments, and orchestras. The dis-(cid:173)
`tance and average total pressure,· in dynes per aqua.re centimeter, are shown · above
`e&oh graph,
`(After Bivian, Dunn, and White.)
`
`4
`
`100
`
`1000
`
`10000
`
`The ratios of the average· sound pressure3 per cycle to the av,
`total pressure of the spectrum for speech, musical instruments,
`orchestras are shown in Fig. 6.3.
`The ratios of the peak pressures' to the average pressure of the t111lia
`spectrum for· speech, musical instruments, and orchestras are shown la:
`Fig. 6.4.
`i, 4 Sivian, Dunn, and White, Jour. Acouat. Boe, Amer., Vol. 2, No, 3, p, 830, 1111.
`
`The average sound-pressure characteristics give the average sound
`pressure per cycle over a long period of time. The peak-pressure charac(cid:173)
`teristics give the ratio of the maximum peak of sound pressures that occur
`in a frequency band to the total -average sound pressure. These bands
`&re indicated by the steps 30 to 60, 60 to 120, 120 to 240, 240 to l>00, 500 to
`l,000, 1,000 to 1,400, etc. The peak levels may occur for- only a small
`fraction of the time. From Fig. 6.3 it will be seen that the peak levels
`&re many times as great as the average level.
`It is _the peak pressure
`that must be considered in the design of sound-reproducing· equipment.
`
`MZ Audio, Ex. 2002, Page 8 of 13
`
`

`

`206
`
`MUSIC, PHYSICS, AND ENGINEEl04G
`
`AVE.RAGt .
`1OTAL
`PAC$$UR£
`
`.
`
`AYtRAGt
`Ol$1ANCE TOTAL
`PRt$~URt
`
`AYtRAG~
`DlnANCt TOTAL
`.
`PAti-.URE
`
`OISTANCt
`
`-
`
`•30~
`30 2 ,1
`
`.
`
`P1Pt. ORGAN
`
`1zrT. 30 ,,.o BASS· OJUJM
`
`.,f f l § f f l f f i •lO~
`l rr. 301 1 114.e
`5NARC OR\.lt.4
`'rT.
`
`-
`
`'"'"""' m. •·
`
`0
`
`~-J~
`
`•lO~
`
`•lO
`
`·:_·· '"~ ·~·~-~-
`
`l:oo~ooirl~ T·):-
`:.Ifflll.1111.~.
`~llllllJII
`t• _ll[III
`"'."M ,.,,,_ ~.,
`>o_ ~•-
`~ Jffffj@ffl 0 ~
`- ~-
`10000 •l'j!o
`
`1000
`
`-J~a
`
`100
`
`IOOO
`100
`IOOOO · •~ o~ o c i - ~ lllOOO
`rA£QUENCY
`IN CYC:t.ES
`F£A StCONO
`FIG. 6.4. Ratio of the .peak pressure to the average total pressure of tlie eni.n
`spectrum for -speech, musical instruments; and orchestras. The distance and average
`.total pressure, in dynes per square centimeter, are shown above each -graph.
`(After
`Bivian, D~nn, and -White,)
`The equipment must be capable of · handling the peak power .without
`overloading or distorting.
`· ·
`.The average power gives the general impression of the balance of the
`music. The peak levels produce the dramatics and dynamics in musical
`renditions;
`
`6.3 TIMBRE OF MUSICAL INSTRUMENTS _
`A sound wave may be represented by me.ans of a graph in whfoh the ordi(cid:173)
`nates ·represent pressure or particle velocity in the sound wave and the
`
`CHARACTERl$TICS OF MUSICAL INSTRUMENTS
`
`207
`
`abscissas represent time {Sec. 1.3). - This type of representation of a
`sound wave may be obtained by the combination of a microphone ampli(cid:173)
`fier and cathode-ray oscillograph (Fig. 6.5). The microphone converts
`the variations in pressure or particle velocity -into the corresponding
`electrical variations (Sec. 9.2). The electrical variations are amplified
`by the vacuum-tube amplifier and applied to the vertical deflection sys(cid:173)
`tem of the cathode-ray oscillograph (Sec. 9.5) . The vertical deflections
`of the·electren 'beam in the oscillograph correspond to the amplitudes of
`the pressure or particle velocity in the original sound wave. The hori(cid:173)
`zontal deflection of the- electron beam is produced by an oscillator which
`drives the electron beam at a constant rate in a horizontal direction.
`· The electron beam impinges upc>n -the fluorescent screen and produces a
`visible trace which depicts the sound wave in graphical form. This wave
`If the
`gives the cross section of the tone over a certain interval of time.
`wave on the oscillograph is produced by a musical instrument, the tonal
`structure of the musicaUnstrument may be obtained from an analysis of
`this wave, because a complex wave may be resolved into components
`consisting of the fundamental and harmonics or overtones. The struc(cid:173)
`ture of complex waves will be described in the section which follows.
`
`M 1eROflHONE
`
`AMPLIF I ER
`
`CATHODf; RAY
`OSC ILI..OG RA PH
`Flo. 6;5. Apparatus for depicting the wave shape of a sound wave.
`
`A. Representatlon of Sound Waves
`A complex wave may be considered to be composed of the fundamental
`and harmonica or overtones in the proper amplitude and phase relations.
`The composition of a complex wave is illustrated in Fig. 6.6. The wave(cid:173)
`form of the fundamental is shown in Fig. 6.6A. The waveform of the
`combination of the fundamental and the second harmonic, or first over(cid:173)
`tone, is shown in Fig. 6.6R . The waveform of the combination of the
`fundamental and the second and third harmonics, or the first and second
`overtones, is shown in Fig. 6.6C. The waveform of the combination of
`the fundamental and the second, third, and fourth harmonica is shown
`in Fig. 6.6D. The waveform of the combination of the fundamental and
`
`MZ Audio, Ex. 2002, Page 9 of 13
`
`

`

`MUSIC; PHYSICS, AND ENGIMEfllNG
`
`CHARACTERISTICS Of · MU&ICAL INSTRUMENTS
`
`209
`
`208
`
`·~ FUNDAMENTAL
`
`8 .,..
`
`¾.
`
`_.e
`
`X
`
`.,,
`
`I
`
`FUNDAMENTAi.
`
`Df ' - : / "'c..J
`
`-.........
`
`FUNDAMENTAL
`
`r==>
`<"'- ~ .
`c::.;, ~~
`
`E-
`
`C
`
`.
`RESULTANT
`Fta. 6.6. A. Fundamental ·sine wave. B. Fundamental, second hamionic, and tb,
`resultant. C. Fundamental, . second and third harmonics, and the : resultant. D
`Fundamental, second, third, and fourth harmonics, and the resultant. F, Funda(cid:173)
`mental, second; third, fourth, and fifth harmonics, and the resultant.
`·
`
`the second, third, fourth, and fifth harmonic is shown in Fig. 6.6E.
`If 111
`infinite number of components with the appropriate amplitude and ph
`are used, the resultant will be a saw.;tooth wave. This will be described
`later.
`A complex wave may be expressed mathematically, as 13h.own in Fi.
`6.7 . . The first five components of the resultant wave of Fig; 6.6E a.re 11
`follows:
`
`·-
`
`··
`
`· .,.
`
`P1 =sin"''
`P, =½sin 2"'t
`Pa=¼ sin 3"'t
`P4 == ¼ sin 4"'t
`P. =¼sin 5"'t
`where P1 "" fundamental
`Ps =-. second harmonic
`Pa =. tbir<\:harmonic
`P 4 = fourth harmonic
`P 6 .,. fifth harmo:uic
`I,) :_ 2,rf
`I~ frequency
`t = time
`The resultant wave is given by
`Pa = P1 + Ps +Pa+ P4 + P•
`This is the sine series
`Pa = sin "'t + ½ sin 2"'t + ¼ sin 3"'t + ¼ sin 4"'t + % sin 5"'t
`(6.7)
`The structure of a sound wave produced by a musical .instrument may
`be depicted by a spectrum graph. The spectrum of the resultant wave
`is shown in Fig. 6. 7 A. The l.'elative amplitudes of the components of the
`resultant wave are depicted as a function of the frequency. The heights
`of the verticallines are proportional to the amplitudes of the fundamental
`and harmonics. The position. along the abscissa determines the fre(cid:173)
`quency. For example; the spectrum shows that the frequency of the
`second harmonic is tw<t times that of the fundamental and the amplitude
`is one-half that of the fundamental, the frequency of the third harmonic
`is three times that of the fundamental and the amplitude is one-third
`that of the fundamental, etc.
`.
`The first three components and the resultant of the combination of
`the first three components of the odd sine series are shown in Fig. 6.7B.
`The first three components of this wave are as follows: .
`P1 =sin"'' ·
`Pa=¼ sin 3"'t
`. P, = ¾ sin 5'.it
`The resultant wave is given by
`Pa = Pi + Pa + Pa
`This is the odd sine series
`Pa = sin "'t + ¼ sin 3"'t + ~i sin 5'.it
`(6.12)
`It will be seen that the wave of Fig. 6. 7 A is of the saw-tooth type, while
`the wave of Fig. 6. 7 B_ is of the -rectangular type.
`
`(6.1)
`(6.2)
`(6.3)
`(6.4)
`(6.5)
`
`(6.6)
`
`(6.8)
`(6;9)
`(6.10)
`
`(6.11)
`
`MZ Audio, Ex. 2002, Page 10 of 13
`
`

`

`210
`
`MUSIC, PHYSICS, AND EMGIMEEIIHG
`
`CHARACTERISTICS OF MUSICAL INSTRUMENTS
`
`211
`
`A
`
`B
`
`A
`
`C:YC~l --1
`
`B
`
`~ -------
`r:::::,,,
`~
`~~ c : : : :> c::.,
`;I
`~
`~ ~=--=,,e.=---.::r=""-"-"'= ....... ~ ~
`J
`J
`__ __ _
`_
`1
`1
`,:.:.fsi"•"'t
`~ ~...,,.;1.:00...:,~ .... ~= .... ,,,,,. ........ ~ < r - - -~~
`Pa=f si"suit
`
`w
`
`Q :, s a r,ME ! \
`
`CL
`::I
`<
`
`t=-co.awt •.
`+ .7COIU•t
`+ .,c:au411t
`
`1
`
`., .
`
`w
`0
`
`cYc:1.c7
`s"' - - ~ ..
`CL I
`
`IJ.= slnwt
`- .sslu•t
`+,1111111..t
`-,assia,on
`
`Pcf1'1 +Pa+ Pi.+ P4 -t P5
`
`• 'II" "i ... ,, ... "•
`
`~ 1 i P1
`
`I
`
`/l
`
`~
`
`·1 Q
`;I s ,.
`~
`
`I
`
`2 •
`
`I P1
`~
`
`• ' •
`5
`
`4
`
`IP•
`5
`
`8
`
`1Pa · 1,,
`,..,
`l
`4
`f" + (,
`f".+ f.
`~'10. 6.7. A. The fundamental, second, third, fourth, and fifth. components of 1
`wave, and the resultant wave consisting of- the combination of these component, of
`the sine series. The spectrum depicts the relative amplitudes of the component, r:l
`the resultant wave as a function of ·the frequency. Ji .., frequency of the fund.
`mental, !N - frequency of the Nth he.rtiionic. B. The funds.mental, third, e.nd fift.11
`. harmonic components of a wave, and the resultant wave consisting of theae oom(cid:173)
`ponents of the odd sine series, The spectrum depicts the relative amplitudes of t,ha
`components of the resultant wave as a function of the frequency, · /1 • frequency cf
`the fundament&l; !N • frequency of the Nth h&rmonic.
`·
`
`The output of musical instruments produces exceedingly complex
`waveforms. A few illustrations of. the simpler complex waves will now
`be given;
`The waveform of the series
`PB = cos wt + 0.7 cos 2wt + 0.5 cos 3wt
`(6.13)
`is shown · in · Fig. 6.8A. The spectrum of. this wave is also · shown in Fi&.
`6.8A.
`The waveform of the series
`Pa = sin wt- 0.5 sin 3wt + 0.33 sin 5wt - 0.25 sin 7wt
`(6.H)
`is shown in Fig. 6.8B. The spectrum of this wave is also shown jn Fic-
`6.SR
`
`11 I I
`
`II I I
`
`I
`
`1 2 3 4 5 6 7 1 2 3 4 5· 1 1
`fN+ fl
`fN-i- f1
`~•10. 6.8. A. A wave consisting of the first three components of a cosine•series. The
`~trum depicts the relative amplitudes of the components of the wave as a. function
`of the frequency. Ji •· frequency of the fundamental, /N = frequency of the Nth
`harmonic. · B. A wave consisting of the first four components of an odd sine series
`with alternate signs. The f!pectrum depicts the relative amplitudes of the components
`of the resultant wave as a function of the frequency. Ii = frequency of the funda(cid:173)
`mental,IN - frequency of ~he }.Ith harmonic.
`
`The resultant waveforms of Figs. 6.7 and 6.8 illustrate how different
`wave shapes are developed from various complexions of the overtone
`structure.
`The phase of the components plays an important part in determining
`the shape of the resultant wave.
`In Fig. 6.9A the equation of the wave
`is given by the series
`P11 = sin wt + ½ sin 3wt + ½ sin 5wt
`In Fig. 6.9B the equation of the wave is given by the series
`PB = sin wt --- ¼ sin wt + ¼ sin 5 wt
`(6.16)
`The phase of the third harmonic of Eq. (6.15) differs in phase by 180
`degrees from the third harmonic of Eq. (6.16). It will be seen that the
`shapes of the two waves are entirely different. However, the spectrums
`are the same; . In general, except for very large phase shifts or very
`intense sounds, phase plays a relatively small role.
`The wave shape and the equation of a saw-tooth wave are shown in
`
`(6.15)
`
`MZ Audio, Ex. 2002, Page 11 of 13
`
`

`

`254
`
`MUSIC, PHYSICS, AND ENGIHEERIHO
`
`UOPUTIES Of MUSIC
`
`255
`
`detect as a function of frequency for various sensation levels' is shoWL in
`Fig. 7.11. These characteristics show that the ear is most sensitive to
`intensity-level changes at the higher sensation levels.
`II
`'\
`\
`'
`.....
`"' w .J
`z"' ~•H
`u I!!
`,.,!9
`ii .J
`
`'
`' ~ '-.
`' '~
`S§,
`..........
`"" ~
`r~, '~
`~ .... r,..., ......
`~
`,_ ......
`~"
`i1I I~
`.... ~
`~ "--
`"'~r---___ :\0
`~ .ao
`50 •
`
`i-,...._
`
`! ~
`~
`
`-:it z-
`i~
`I
`
`t,.,
`
`---
`
`t--...
`
`i,..~
`
`~
`
`ii
`_,/v
`
`~
`
`~
`
`~,II
`... ~
`
`,_ ...
`
`I 0000
`
`~
`
`100
`
`II 1111
`f'lll!:QU[NCY
`IN GYGLts · t"ER SECOND
`FIG. 7.11. The minimum perceptible change in intensity level of pure tones ae a fun~
`tion of the frequency. Numbers on the curves indicate level above threshold. · (Afrer
`Flel.cAer, Speuh. and Hearing in Cinnmu.nication, D. Van Nostrand Compa,.y, foe,,
`Princeton, 1969,)
`C. Timbre (Quality)
`Timbre is the most important fundamental attribute of all music.
`Timbre is that characteristic of a tone which depends on its harmonic
`structure. The timbre of a tone is expressed in the number, fatensity,
`distributiou, and p