FUNDAMENTALS
`op -·
`MUSICAL
`ACOUSTICS
`
`Second, Revised Edition
`
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`· Arthur H, Benade
`
`MZ Audio, Ex. 2004, Page 1 of 9
`
`

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`Copyright© 1976, 1990 by Virginia Benade.
`All rights reserved.
`
`.
`.
`.
`ThisDo~r edi~ion, fir~t published in 1990, is an unabridged,
`extensively comcted republication of the edition originally pub(cid:173)
`lished by Oxford University Press in 1976. A Note to the Dover
`Edition luis been added ,
`
`· Library tf Congrtss Cmaloging-in-l'flbli~ion O.au
`Bena,lci, Arthur H:
`Fundamentals of musical acoustics i Arthur H. Benade.
`.
`..
`.
`p. ·
`.cm.
`Reprint. Originally published: Nev,, York : Oxfqrd University
`Press, 1976.
`·
`·
`·
`·
`· Includes index.
`ISBN-13: 978-0-486-26484-4
`ISBN-10: o:486-26484-X
`. 1. Mus"ic...:._Acoustics and physics.
`-Ml,.3805.B328 1990
`78t2...;...:ac20
`· ·
`
`I. Tide.
`
`9040159
`. OP
`MN
`
`Manufactured in the United States by LSC Communications Book LLC
`.·
`26484X17
`. 2021
`www.doverpublications.com
`
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`MZ Audio, Ex. 2004, Page 2 of 9
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`

`

`le
`le
`es
`,t-
`.ie
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`.a(cid:173)
`ed·
`n.g:
`.ve
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`tly
`JW
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`ni~
`:ed
`ea ..
`rd·,
`R
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`in-
`wo
`the
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`·are
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`be-
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`
`Pitch: The Simplest Musical Implication of Characteristic Oscillations
`
`t
`notice some of the rema~kabJe implica(cid:173)
`tions of the first of these observations.
`
`59
`
`fn =rif1
`One reads thiJ mathematical s~nte11ce thus: "I 1ub-11
`IJ equal to n times I sub-I," meaning that the nth
`frequency is n times a.r large as the first orie in the
`Jet.
`
`5. 5. Sounds Having Whole~Number
`Frequency Ratios
`
`Let us imagine ~hat we have available to
`· us a hypothetical stri11g which, when
`plucked or struck, vibrates ii} a family of
`characteristic damped sinusoidal oscilla(cid:173)
`tions whose frequencies are arranged in an
`exact whole-number relation; that is, Q is
`exactly 2P, R = 3P, S =4P, and so on.
`
`In the language of chapter 2, we can say
`that the repetition rate for any one of our
`idealized string's sinusoidal oscillations is
`a whole .number times the repetition rate
`associated with its lowest frequency os(cid:173)
`cillation. Let us look into what happens
`when acco1.1t1t is taken of the fact that the
`string is actually vibrating wfrh a. whole
`set of integrally :related frequencief
`Suppose that for conceptual simplicity
`we assign an imaginary drummer to each
`char_actei-isck . oscillation of our . string,
`giving him the job of tapping with a rep~
`etition rate equal to that measured for his
`"own" string oscilla_tion. The whole(cid:173)
`ntimber · relation . between
`the
`string
`frequendes then requires that the drum(cid:173)
`mer .assigned • to keep time with the_ sec(cid:173)
`ond characteristic osdllation should beat
`twice as fast a,s drwnmer number 1. Simi-
`larly drummer · 3 taps three drries as fast
`
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`
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`
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`.
`
`.
`
`. Digression on the Numerical Labeling of
`Natural Frequencies.
`· We can exprm this iuhole-number relationship be(cid:173)
`tween oscillation frequmcies very compactly as fol(cid:173)
`lows. If we use the -letter n to stand for any one of
`the. integers--that is, n = 1 , or 2, or 3, etc~-.
`. and if the char~cteriJtk frequendes are given the
`serially numbered names f 1, f 2, / 3 imtead of our
`alphabetical names, then the nth one of these
`frequencks can be referred to as f.,,, The desired int-
`eger relation between the suctessive string freq lien cies
`ca11 be written i11 a mathematically tidy fashion a.s
`._follows:
`
`)-----0,-----1 .,
`
`0-
`
`Fig. 5.2. Pattern Made by Tapping Rates Having a Whole-Number Relation
`
`MZ Audio, Ex. 2004, Page 3 of 9
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`

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`Fundamentals of Musical Acoustics
`60
`
`Fig. 5.4, Combination of Sinusoids Having
`a· Whole-Number Frequency Relation
`
`fer by a factor of two. If our simplified
`string could be excited by some means
`that sets into motion only the first two of
`its characteristic oscillations, then the os(cid:173)
`cilloscope picture produced from a micro~
`phone in its neighborhood woul<i look
`
`as drummer 1, and so on. The upper fuur
`lines of figure 5. 2 show the timing of the
`successive taps produced by the first four
`of out set of drummers . The bottom line
`of the diagram shows the resulting rhyth(cid:173)
`mic pattern that one would hear. Every
`drummer strikes in unison with the blows
`of drummer 1, giving a strongly marked .
`beat, and drwmners 2, 4, . . . strike at
`the midpoints between
`these accented
`taps, giving a somewhat less accented
`tap. The important thing to notice is that
`the repetition rate of the complete rhyth(cid:173)
`mic pattern produced by the composite
`set of tappings is exactly the same as that
`of the lowest frequency member (see sec.
`2. 3, "Repetition Rates of Rhythmic Pat(cid:173)
`terns"). Musiciims should not find this
`idea hard to understand if they compare
`my explanation above with what they
`would expect from a rhythmic pattern
`written out as in figure 5. 3.
`Let us look now at some examples
`using sinusoidal disturbances instead of
`drum~ats . . · The top . two parts of figure
`SA show sinusoids whose frequencies dif-
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`Fig. 5,3.
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`MZ Audio, Ex. 2004, Page 4 of 9
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`

`

`Pitch: The Simplest Musical Implication of Characteristic Oscillations
`
`61
`
`som~thing like the curve sqown· in the
`third part of the figure. This curve is
`two
`produced by the addition of the
`curves immediately abo~e it. Notice that
`the repetition time of tht somewnat spiky
`composite curve (and hence its repetition
`rate) is exactly that of th~ f1 component at
`the. top of the diagram. The bottom part
`of the figure shows the result of combin(cid:173)
`ing additional sinusoids, so chat ~he curve
`is that belonging to the sum of the first
`six oscillations in our specially chosen set.
`
`shown us something that will prove to be
`very important to our undt1rstanding not
`only of the physical basis of tone color
`but also of the special relationships be(cid:173)
`tween notes which underlie formal music
`all over the world: Let us set down some
`of the properties of the tlass of sounds
`that would be made by our hypothetical
`strings.
`
`1. No matter what the strength of excita(cid:173)
`tion of the various oscillations, the repetition
`race for the whole signal as it reaches a micro(cid:173)
`phone (or our ears) would be exactly that of
`the lowest frequency sinusoidal compon~nt
`that is characteristic of the string.
`.
`;.
`2. Because the net repetition rate of the
`vibration is independent of how cir where the
`string • is struck, one would always · get the
`same perceived pitch sensation for the string
`sound. This means that the pitch is ltnam•
`big11011s.
`
`Fig; 5. 5. This figure is identical with the
`Lipper three sections of figure 5 .4 except that
`the second component has been displaced.
`Note that the repetition time is unaffected by
`this change.
`
`Figur~ 5. 5 shows ·~ slightly modified
`version of the upper three sections of fig(cid:173)
`ure 5.4. This time the f2 component is
`"slid ovet" in time 'so that it no longer
`has every second upward excursion coind(cid:173)
`dent with every upward excursion of the
`f1 component. We . notice that the suin(cid:173)
`tnation · of these two oscillations gives a
`resultant pattern whose shape is different
`from the one obtained before, but once
`again we see that the repetition time is
`equal to that of the · lowest frequency (f1)
`oscillation.
`Adding components whose frequencies
`.are
`in whole-number relationships has
`
`Digression:
`Sounds with Only . Even
`.
`.. . , ·.
`H;rmonics.
`In the strictest of logic, one might ask about a po1.,
`sible inadequacy of iiem 1 above. Imagine an
`ingenioui excitation method that fails to excite the
`odd-numbered oscillatiom, so that only /2, f,i, f e,
`. , . are pre,ent. The.re may be written out as
`follws:
`f2 = 2f1 = 1 X (2f1)
`f4 = 4f1 == 2 X (2ft)
`fa == 6f1 = 3 X (2ft) . .
`etc.
`
`This 1hows that our new set of frequency compo(cid:173)
`nents is itself constructed out of integer multiples of
`a new basic frequenry whose value i.r (2/1). The
`repetition rate is thmfore doubled, . and the whole
`game begins again; We would perceive thts altered ·
`sound as having a pitch one octa11(! higher ihan the
`normally excited one. ·
`. As a practical matter, it is not particularly dif(cid:173)
`Jicult to arrange peculiar exdtation.r of the iort de(cid:173)
`scribed in the precedingpariigraph, and if one were
`
`MZ Audio, Ex. 2004, Page 5 of 9
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`

`

`to meet such a situation it could ea.rHy be recog- _
`nized as such with· the· help of simple auxiliary ex(cid:173)
`periments. One would need only to pluck or strike
`the string at random spots once or twice in ortkr to
`.
`find out the true nature. of the string.
`
`There is something intellect1.U1.Ily very
`attractive about the apparent simplicity of
`sounds made up of components having
`integer frequency ratios, and it ·is easy to
`devi~
`lengthy · numerological· games
`based • on their presumed properties, Be(cid:173)
`fore we· fall into this trap, however, _it
`would be advisable to find out whether
`such sounds can •in fact be generated. If
`such sounds can be · generated, we then
`-must ask whether om- ears and nervous
`system deal with them· in a. way that cor(cid:173)
`re~ponds at all with experiencing the
`,sounds from real sttfr1gs. The first ques(cid:173)
`. tion can be answe·red affirmatively in two
`.ways:
`i. A truly uniform slender string of suit(cid:173)
`able material, stretched tightly enough be(cid:173)
`tween sufficiently rigid supports, will.produce.
`sounds whose components have frequencies
`that are in very nearly perfect integer. relation.
`The sounds · from such a string differ only
`. subtly from those produced l:>y. a string vibrat(cid:173)
`ing under less formalistic conditions. That is,
`nothing. drastic happens to the perceived
`sound as long as the string has. nearly integer
`frequency relations.
`·
`2. we· 4od that th~re is a large class of fa(cid:173)
`miliar sound• sources that Oormally produce
`sounds whbse frequenc:y components are found
`to be related in the precisely Whqle-number
`manner that we pos~ulated for our hypotheti(cid:173)
`cal iitrings. E.xamples of sources of this kind
`are very .· common. The human voice is the
`most familiar one, while the woodwind and
`brass instruments join with the violin family
`to provide · orchestral examples. Th~se · diverse
`sound sources have one common demerit in
`their na.ture that sets them apart from the
`bells, chimes, and strings we have considered
`
`Fundamentals of Musical. Acoustics
`62
`so far. Instead of simply ringing (and decay~
`-.ing away) in response to an impulsive stimu(cid:173)
`lus, all of these instruments are capable of
`producing sustained sounds. They are devices
`that are capable of converting the steady Bow
`of air from a man's lungs, or the steady rµo(cid:173)
`tion of the bow in his hand, into the osciUa(cid:173)
`tory · vibrations which give rise to the sound
`we hear. We shall see in a later chapter tpat
`only under vety special circumstances can
`such devices be persuaded to maintain steady
`oscillations whose frequency components are
`not in an ei(act whole-number relation to the
`basic repetitio~ rate.
`
`It turns out that the vast majority of
`our musical listening experiences are with
`sounds whose frequency components are .
`in exact whole-number re.lation, or very
`nearly so. It i,s not surprising, then, that
`the formal structure of music (wherever it
`has developed over the world) is, strongly
`influen~ed by the properties _ of sounds
`each ofwhich has whole-number relations
`among its components. We also find that
`many subtleties in music arise through
`the , slight
`inharmonicities which a~
`present in the tones of some. instruments. ·
`This book has opened with an inves(cid:173)
`tigatio~ of iCQpulsive and heterogeneous
`sounds from struck objects, not only ne(cid:173)
`caus~ of' the 'simplicity of initial expo~i(cid:173)
`tiori •but also' as a means for underlining
`the special nature of the sound~producers
`that man· has selected for his musical ac~
`dvities. ,It is time therefore to return to
`the ;;ounds of bells and chimes in o~der to
`coi.n'pare them with the sounds of plucked
`or •~}ruck strings.
`·
`·
`
`5. 6. Th~ Pitch of Chimes and Bells:
`Hints .of Pattern Recognition
`
`that the characteristic
`We have founc:I
`frequencies that make up any one sound
`
`MZ Audio, Ex. 2004, Page 6 of 9
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`

`

`Pitch: The Simplest M!-15ical Implication of Characteristic . Oscillations
`
`63
`
`from any one of the commoner orchestral
`1nsttuments are arrariged as exact (or ~~ry
`nearly exact) integer multiples of a certain
`basic frequency. It is i:his bask frequJ~cy
`component that determines the repetition
`rate of the sound we hear and· also, as we
`have learned, its musical pi~ch .. Let us use
`this -knowledge tci help ourselves gain
`some understanding of the' way in which
`'we assign pitches to chimes ind bells,
`whosecharacteristic trequehcies do not ar(cid:173)
`range the~selves in whoJe-number rela(cid:173)
`tionships.
`
`Digression on Terminology:
`Some Partials Are Harmonic.
`It will save a great deal of circum/oc11tion if we
`provide ourselves with some terminology carefully
`chosen for the description of the various compon"ents
`making up the sound we are ikq/ing with. First of
`all, in atJy sound mafk up of {intisoidal compo~ ·
`mnts, we will continue to assign identifying letters
`from the latter part of the alpha/Jet, or serial num:
`bet's, a.11igning them according to their order,
`beginning at ihe lowest one. That is, we will call
`· these /requenciei P, Q, R, . . , or f1, f2, fa,
`, . . SometimeJ it will be _useful to refer to these
`components -as the partials of the sound in que!fion.
`When this word is used, tue °will unikrstand th4t
`no particular relationship is to be "a.rsumed between
`the frequencies of theJe partials,· their fmJ.uencies
`may or may not have a whole-number. relationship.
`The1e components will still be referred to hy their
`serial numb~s a.r first partial ( referring to the
`componenOabeled P or f 1 ), second partial (also
`known asQ or /J.), etc.
`__ ·We tum now to the special ca.re of sounds in
`which the frequenciei of the t1arious sin11Joidal par(cid:173)
`tials are whole-n_11mber multiples . of some ba1ic .repe(cid:173)
`rate. Tin sinusoidal component . "whose
`tition
`frequenry matchei that of the ;repetition rate will be
`referred to _as the funda.piental component, and its
`frequency as the fundamental frequency : It is
`-0/ten referred to also as the first harmonic. The
`Pani;./ ,w~ie·fre,quency is exactly double that of
`the fun<k,mental will . be said to have a frequency
`whic/, is the second harmonic of the fundamental
`.
`. .
`
`'
`
`frequency. Similarly we will say that sim1Joidal
`oscitlaiions running at three times the fundamental
`frequency are vibrating at the third harmonic of
`the fundamental frequency . -
`We will have_ to be t1ery-,trict i_n o~r terminoiogy
`or endless . confusion can result. The word har(cid:173)
`monic is to be used only .when ,;;e m~an to imply
`an exact whole-number frequency relationihip. To
`help make things clear, we may notice that the par(cid:173)
`tials pf a guitar string have frequencies which are
`very nearly, but not exactly, harmonic1 of the
`.
`frequenry of the fint (lowest) partial.
`.
`
`We learned earlier in this chapter that
`musically experienced people won't neces(cid:173)
`sarily ag)'.'ee on wh.at pitch to assign co the
`sound of a grandfather clock chime . Iri'
`the context of out present understanding '
`of musical soun_d~; we may wonder
`whether the frequencies of the chimd'
`partials can be recognized by our nervous
`system as · belonging to two differently
`organized sequences. of harmonics. _That
`is, can we find hints of a series of har(cid:173)
`monics whose fundamental . corresponds to
`the approximate F3 that some listeners
`hear? Similarly, can we detect signs of a
`har_monic series .· whose fundamental im(cid:173)
`plies the pitch just ·above the (: 5 perceived
`by others? In our earlier investigation of
`this sound we recognized that the .second
`partial has a frequency consistent · with
`one of these pitch assignments while the
`two dosely spaced . partials (which were
`labeled Ra and Rb) are associated with the
`othe.r oi;:ie. _ Our ·earlier difficulty stemmed
`from -our inability to dispose of all the
`other partials making . up the tone; co_uld
`these be members of harmonic se.des
`based on the assigned pitches?
`:Figure 5 .6 shows the frequencies of aU
`the partials up through £4 (S) laid out as
`dots along a frequency scale . Above the
`frequency, axis of this diagram . we see · a
`pa,ir of arrows located at frequencies corre-
`
`MZ Audio, Ex. 2004, Page 7 of 9
`
`

`

`·:.' 1,i!l:i:l~Ji..f1 -
`
`i1i!t
`]l~;
`
`200
`
`~
`
`Fundamentals of Musical Acoustics
`
`64
`
`i
`
`APPROX .
`
`. c5
`
`f
`
`f~
`1000
`
`1ioo
`
`f
`
`f
`
`600
`400
`FREQUENCY
`
`800
`
`f ± ±
`
`SHARP
`
`,=:
`3
`Fig. 5.6: Assignment of Pitches to Sound from a Cock Cbime
`
`sponding to a fundamental, belongirtg to
`the note C 5, and its second harmonic.
`The fundamental arrow is pointing at the
`pair of Rs, .while the arrow for the second
`
`harmonic points almost exactly at the
`
`me~ured S. It seems possible, then, that
`o,ur. ears can ,seize on the reiacionship of
`these two strong components a:nd accept
`,tl)em jointly as the two lowest members
`· (fundamental and second harmonic) of a
`. set of partials beJonging:to a sound whose
`pitch is near C5 •
`.
`.· Below the frequency axis we fin~ in
`similar fashion a sec of arrows indicating
`the frequencies making up .the set (fun~
`damental and its harmonics) belonging to
`the sharp-pitched F 3 which we associated
`with t.he measured f2 (which was labeled
`Q earlier). This time we find that the
`arrows corresponding to the fundamental,
`the third,· !lnd the sixth harmonics point
`very nearly ·at the <dots indicating the
`measured components Q; R; and S.
`Our search for integer relations among
`the frequency components of a struck
`chime rod has been reasonably successful, ·
`in . that it gives results that seem consis(cid:173)
`tent with the hypothesis that our ears as(cid:173)
`sign pitch (when possible) on the basis of
`
`they can .
`
`any · whole-number sequences
`.find.
`.
`.
`We turn our attention next to the bell
`sounds, to see whether they give any sup(cid:173)
`port to our hypothesis . that pitch is as~
`the- basis of approximately
`signed on
`w hole-nuinber frequency
`relationships.
`The individual lines of · figure 5. 7 show .
`the frequencies of the .first. five partials for
`the first five bells in the Terling Peal, laid
`out by means of dots on a frequency axis
`in exactly the same way as was done for
`the chime rod. The dashed vertical lines
`appeJing on the diagram . indicate the
`fundamental repetition frequency aqd its
`h~monics belonging to a reference sound
`whose pitch matches that of the bells as
`made; uniform. by a variable-speed record(cid:173)
`ing device_.
`•
`Inspection of the line corresponding to
`bell 1 shows that the first partial (marked
`P) has a'frequency quite close to that as(cid:173)
`swped for the fundamental. Furthermore;
`we,-, ~ee t~at partials 4 and 5 agree ex(cid:173)
`treri:iely well with harmonics 3 and 4 of
`the pitch reference tone. We note that
`partials · 2 and 3 do not seem to agree
`with any ,member of the reference har(cid:173)
`monic series.
`
`~:/t;(t
`··.){:;\
`
`,!,_,,.
`ItP
`' /],:)iii,
`
`MZ Audio, Ex. 2004, Page 8 of 9
`
`

`

`Pitch: The Simplest Musical Implication of Characteristic Oscillations
`
`n REPETITION FREQUENCY
`t CORRESPO~DING TO P~RCEIVED P!TCH
`
`65
`
`.J
`-~ CL2
`, ' .
`!
`.J .J! ....,_ _____________ .,_~._-.... _ _ _ _ _ -MIII - - - -
`LtJ
`a)
`~4 1-----il..,_ _______ __,...,_ ______ ....+ __ _
`a:
`LtJ m
`~51------------1411~ --~--H..----------
`~
`z
`
`I
`
`.
`
`I
`'· I
`
`I
`
`800
`600
`400"
`200
`1000
`FREQUENCY
`· Fig. 5. 7. Frequency Components of Bells Adjusted to tl:ie' Same Pitch
`
`1200·
`
`Sl<ipping now to bells 3, 4, and 5, we
`find that partials 1, 2; 4, and 5 agree
`quite weil · with the fundamental and hat:
`monies 2, 3, · and 4 belonging to our
`pitch reference. Partial 3 never seems to
`fit in. Bell 2 does not show such a clear(cid:173)
`cut relation, although the frequencies of
`partials 1 and 4 are roughly equal to
`those. of the fundamental and · third har~
`nionic of our reference sound. Interest(cid:173)
`ingly enough, most listeners feel quite
`une~y about assigning pit~h to this bell,
`even though they find · no difficulty with
`the other ones.
`Looking o~er the data we come co real(cid:173)
`ize that for a bell to have a reasonably
`it can be .
`well~defined pitch (so -. that
`matched with a normal sort of tone · hav(cid:173)
`ing harmonic partials), our eats do not
`demand any particular set of component
`
`frequencies frotn it. · That is, our ears . do
`not demand that the same. (only approid.:
`iden(cid:173)
`mately harmonic) partials serve
`tically as the· ''pointers" in _the sounds for
`all the bells .. All ·chat is required is a suf•
`ficient · ~umber of sufficie11dy consisteni:
`clues. The frequencies of the skillet clang
`listed on p. 43 are similar to these bell
`sounds in chat they are not harn,1onically
`related.
`
`5. 7. Another Pitch Assigninept
`Phenomenon: The Effect of Suppressing
`·
`· ·
`Upper or u;wer Partials
`
`In the previous section of this chapter we
`found ourselves thinking about the ways
`i~ which our ears respond to sounds fed
`to them from bells and chimes. We no(cid:173)
`deed that the ace of assigning pitch to
`
`MZ Audio, Ex. 2004, Page 9 of 9
`
`