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INW田Ⅳl〕IICITY OF A S口EIN酔Ⅳ

MEL L OAND PIttЮAND lTS EFFECT ON TUNING

June 13, 1984

By: Hideo Suzuki

CBS Technology Center

227 High Ridge Road

Stamford, CT 069o5

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INHAR‖ONICITY OF A STEINWAY HODEL L mND PIAN0

EFFECT ON TUNINC脚 ITS

I. INTRODUCtt10N

The effect of the ■nharmonicity of piano strings on the tun■ ng of the

octave (F3-F4) was discussed in the lnterim RepOrt, ・・REVIEW OF THE TUNINC

PROCEDURE TAKINC INTO ACCOUNT ttHE INHARMONICITY OF A PIANO SttRING'1。 In that

report, the inharmonicity indexes (B) of the strings were assuned to be equal

to O。 0004 、hroughOut the range F3-F4. This was done to roughly approximate

the inharmonicity characteristic in the F3-F4 octave of an upright piano which

was used by Fletcher in his study (3ASA, 36, 203-214, 1964). It was shown

that the theoretically calculated beat rates can never be achieved when

applied to a piano with the above mentioned inharmon■ c■ ty character■ stic. The

■nharmonicity character■ stics of Ste■ nway pianos have only been reported by

Young (3ASA, 24, 267-273, 1952). He also published a paper discussing the

effect of inharmonicity on tuning (3ASA, 5, 1-11, 1943).

For this reason, the inharmonicity characteristic of the Model "L" piano

at CTC was measured, and its effect on tuning was discussed. This report w■ 11

descr■be these results and also discuss a new method to calculate beat rates

of a smoothly stretched octave in a specific piano with a known inharmonic■ ty

characteristic.

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Ⅱ . Ⅲ臥Ⅲ ONICI「Y‖臥StRENT

A theoretical (and simplified) formula tO calculate the resona"℃ e

frequencies of the partials of a stiff string is given by

fn = F ・ n(1 + 1/2Bn2), n = 1, 2 . . . (1)

and

B = (π2ESR2ノ丁22) (2)

where, F is a constant, fn iS the resonance frequency of the nth mOde, and B

is the inharmonicity index deternined by the Young's modulus E, cross

sectional area S, radius of gyration R (R = dィ ,'With String diameter d),tension T, and length 2。 To deterlnine the inharmonicity of a string, at least

two frequencies for different n must be measured since there are twO unknowns

in Eq. (1) (F and B). If fm and fn are the measured resonance frequencies

of the mth and nth modes, respectively, it follows from Eq。 (1)that

F2 = {[(m/n)fn]2 _ [(nノ m)fm]21, (3)

B = [(fn/fm)2(m/n)2_1]/[n2_(fn/fm)2(m/n)2m2]

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(4)

To measure the resonance frequencies of each note, an acceleroneter is

attached to the bridge as closely as possible to the strings of the noteo The

Structural Dynamics Analyzer HP5423A starts to analyze the signal frOm the

accelerometer with some time delay after the key is struck. The time delay

was set approximately O.5 sec or larger for low‐ frequency notes, and O.2 sec

for highrfrequency noteso using this lllethod, the resonance frequencies of the

string should be accurately measured even though the magnitudes of the

partials are affected by the vibration characteristics of the soundboard.

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In general, hOwever, measuring the resonance frequencies with g00d

precislon is not an easy task. 丁he possible reasons are:

(1) The duratiOns (in other wOrds, decay rates)

for different mode numbers.

(2) The frequency of each partial seems tO vary

time fron the initial attack.

of the partials are different

slightly as a functiOn Of the

(3) Some partials shOw a blunt peak due either tO

capability Of the analyzer or due to the overlap

soundboard resonances.

the limited analysis

of the string and the

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(4) ExtranQous reSOnances exist, probably, due to the resonances Of the

adjacent strings even though they are damped by a felt strip and f00t_

pedal dampers. Another possible cause of the extraneous resOnances ■s

the longitudinal vibratiOn of the string。 (Baldwin Piano Company has apatent of a wOund string which is suppOsedly free from this prOblem,

Synchro ttone Strings, U.s. Pat. No. 3, 523, 480。)

(5) 丁he resonance frequencies of partials may not Obey Eq. (1) since the

boundary condition at the bridge is nOt ideal.

A careful measurement, with attentlon to the above mentiOned problems, is

necessary to obta■ n a falrly consistent value of B fOr each nOte. For low― tOmid― frequency notes, it is possible tO measure fn fOr n up to 15 or 20。 Forhigh― frequency nOtes, this is linited to n

≦ 10 beCause Of 10w signal― tO― noiseratios. when the frequencies of n partials are measured, there are n(n-1)/2

possible pairs. Each pair gives the values of F and B using Eqs. (2) and

(3). 丁he averaged F and B can be used as an estinate Of F and B. If there

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are any measured partial frequencies which do not seem rellable, these

frequencies should be excluded from the averagingo Since the lower order

partial frequencies deviate very slightly from the harmon■ c relationship, a

same frequency―measurement error causes a larger error in the estimation of′ B

when the modes of lower orders are used espec■ ally for smal1 3. If this is

the case, it is recommended not to use severa1 lowest modes for the estimation

of B.

Another convenient formula derived froln Eq。 (1) is given by

fn = nfl[1 + 1/2 3(n2_1)] (5)

This formula describes the relationship between the fundamental frequency fl

and the higher partial frequencies fn・

III. …

ONICIttY OF A HODEL L ORAND PIAN0

丁he measured inharmon■ c■ty index B of the Model L Crand Piano at CTC is

shown by circles in Fig. 1. 丁he keyboard is used as the horizontal coordinate

showing the notes of the inharmonicity measurement. Figure l also shows data

of the upright piano in the Fletcheris paper. The transition from wound

strings to solid strings occurs between key number 26(A#3) and 27(B3) for the

Model L Grand Piano. 丁he same thing seems to happen around key number 31(D#3)

for the upright piano. Below key number 10, the grand piano has lower B than

the upright except for a couple of the lowest keys. Between key numbers 10

and 30, both the grand and the upright pianos have approximately the same

inharmon■ c■ tyo The upright piano has a big jump when the str■ ngs change from

a wound to a solid type。 On the other hand, the grand piano gradually

increases the ■nharmonic■ ty around the trans■ tion from wound to solid

stringse Around key number 42, both pianos have approximately the same

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■nharmonicity. It is surprising that the grand piano has a larger

inharmonicity than the upright piano above key number 42.・

丁he inharmonicity of note C4 is approximately equal to the value

calculated from Eq. (2). Especially for the solld strings, Eq。 (2) seems

valid and can be used to estimate inharmonicities instead of actually

measuring the partial frequencies and calculating B using Eqse (3) and (4).

The reason for the larger inharmon■cities of the mid to treble notes of the

grand piano may be that it uses a shorter or a thicker string than the upright

piano for the same note.

The inharmonicity character■ stic in the F3-F4 octave is shown in deta■ l

■n Fig. 2. In this range, the inharmon■ c■ty characteristic seems to be well

approximated by a straight line (shown by the dotted line in Fig. 2).

If the integer numbers from l to 13 are assigned for each note from F3 to

F4, the inharmonicity index B(N) is given by

Ⅸ鴫 =助・ 筒Ψ b山 ノ助 )

(6)

where Bl and B13 are the assumed inharmonicity idexes of F3 and F4,

respectively. 丁he values Bl=0.18x10~3 and B13=006x10~3 。。rrespond

to a straight line in Fig。 2。 丁he approximated inharmon■ city characteristic

of Eq。 (5) will be used to discuss the effect of inharmonicity on tuning in

the next section.

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IV. EFFECT OF コNHAR‖ONICIttY ON TUNINC

The difficulty with aural tuning is that a tuner must tune one note to

another already tuned note using a beat rate assigned to this specific tuning

intervale lf the beat rate were always zero, the tuning night be easier since

he 口ould not have to count the beat ratee For aural tuning, however, this is

only the case when octave intervals are tuned. (Even in this case, a perfect

zero― beat tuning is impossible because of the inharmonicity.) Another

difficulty with aural tuning is that there is more than one beat when two

notes of a tuning interval are played at the same tineo Electronic tuning, on

the other hand, ls free from both of these problemse Thus, lf the plano tone

does not have the inharmonicity property, the electronic tuning may work

perfectly well. The problem with the electronic tuning is probably that the

octave stretching is necessary to properly tune an actual pianoc Present―day

electronic tuning devices are not designed to completely accomodate the

stretched octave, even though they have an adJuStment knob to slightly sharpen

or flatten the pre― set frequencies. Here, a method of determining the

frequenc■ es of notes within the F3-F4 octave wlll be described which are

gradually shifted to match the stretched F3-F4 octave. The beat rates

calculated from these frequencies can be used for the aural tuning as well as

the electronic tuning.

First of all,'the note C4 1s tuned to a tuning fork (or an electronic

tunig device) with a standard frequency of 523.252Hz. Because of the

inharmonicity of C4, the fundamental resonance frequency of this note is

sllghtly lower than 523。 252/2 = 261.626Hz. 工n the present case, this is given

as 261。 485Hz (assuming that the inharmonicity index of C4, 88, lS equal to

O.00036 from Fig。 2 or Eq。 (6))臀hich is O。 93 cents lower than 523.252Hz.

The next step is to determine the note frequencies in the range F3 to F4

with the reference frequency C4 unchangedo The frequency ratios of F4/F3 and

F5/F4 are given by (using Eq. 5))

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Xl = 2(1+1.5Bl) ・

and X2 = 2(1+1.5B13)

respectively. If B13-iS larger than Bl (mostly this is true), the F4-F5

interval is stretched more than the F3-F4 intervalo lt seems appropriate to

increase the frequency ratio of adiaCent notes gradually to match the

stretched intervals of F3-F4 and F4-F5。 The conditlon that the frequency

ratio of adiaCent notes increases by a constant frorn F3 to F5 is enough to

determine the frequencies from F3 to F4。 丁he frequencies of note F#4 and

higher need not be determined at this stage, but this condition may ensure a

smooth trans■ tion of the frequency ratio of adjacent notes from E4-F4 to

F4-FfF4。 Figure 3 shows the frequency ratios of adiacent nOtes that satisfy

the above conditione tthe in■ tial frequency ratio of F#3/F3 is b. The next

frequency ratio (G3/F#3) increases by a factor of a. 丁hus, it is equal to

ab. 丁he fo1lowing frequency ratio ls a2b, and so on. From these frequency

ratios of adJacent nOtes, the frequency ratios of F4/F3 and F5/F4 are given by

a66b12 and a210b12, respectively.

relationship:

a66b12 = 2(1 + 1。 5Bl)

and a21 0b12 = 2(1 + 1。 5B13)

Dividing Eq. (8)by Eq. (7), we have

a144 = 1 + 1.5(B13 - Bl)

where B13, 31 〈く 1 lS assumed.

approx■ mately given by

丁herefore, we have the

(7)

(8)

Since (B13 - Bl) iS very small, a is

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a = 1 + 0.010,17(B13 - 31)

Substituting Eq.

b12=2(1-

(9) into Eq. (7)we have

0。 6875 B13 + 2。 1875 Bl)

Again, since B13 and Bl are very srnall, b is approximately given by

b = 21/2(1 _ 0。 057292 B13 + 0・ 182292 Bl)

(9)

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(10)

Using these two constants and starting from the reference frequency of

C4, a1l other frequencies from F3 to F4 are determinedo For example, the

frequency of B4 is given by fc/(a6b), where fc is the reference

frequency of C4. 丁he frequency of A#4 is given by [fc/(a6b)]/(a5b), and

so on. 丁he frequency of CrF4 is given by fc a7b.

Once the frequencies of the 13 notes from F3 to F4 are determined, the

beat rates which can be used for the aural tuning are obta■ ned. 丁he result is

shown in Table l for the case when the ■nharmonicity characteristic is given

by the dotted line of Fig. 2. For example, the beat rate between F3 and C4 is

O.64/seco tthe negative sign indicates that the high note (C4 in this case)

has a partial frequency that is lower than the corresponding partial frequency

of the low note (F3 in this case). In Table II, the beat rates of the ideally

harmonic case are shown for comparison. 丁he beat rates of the 5th intervals

(F3-C4, Ff13-C#4, 。..) of the present case is slightly higher than the ideal

case. 丁he beat rates of 4th intervals (F3-A#4, F#3-B3, 。。.) in the present

case is slightly lower than the ideal case. 丁he beat rates of 3rd and 6th

intervals in the present case increase smoothly within like intervals. 丁his

property is almost always used as a check for proper tun■ ng. Cenerally

spёaking, it seems there is not much difference between ttable l and ttable II.

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If the bёat rate of Table II is used to tune a piano with the inharmonic■ ty

characteristic of the present case, however, an unevenness in the beat rates

results. 丁able III shows the result. 丁he double boxed numbers are the beat

rates used for tuning. 丁he 4th intervals F#3-B and C4-F4 that are not used

for tuning have much smaller beat rates than the other 4th intervals. The 3rd

and 6th intervals have uneven changes of beat rates within the like

intervals. This is the problem with the theoretical beat rates, calculated

frorn the ■deally harmonic case, when they are used to tune a real piano. 丁he

numbers shown by the right― side columns of Tables l and III are the frequency

shifts (in cents) of nOtes froln the ideal caseo The column of ttable l shows

that the interval of the adjacent notes increases smoothly from low notes to

high notes as originally intended. On the other hand, the frequency shifts of

丁able III are rather random indicating an improper tuning. 丁he frequency

shifts in cents shown in Table l can be used in the electronic tuning if the

device is able to shift the frequencies in 1/10 cent increments.

When a constant inharmonicity of O。 0004 is assumed throughout the F3-F4

octave (this was the case discussed in the last report), the beat rates

calculated from the suggested procedure are significantly different than those

of the ideal case. Table IV shows the resulte ln this case, the 4th

intervals are almost beatlesso The 5th intervals have higher beat rates than

the ideal casee As Eq. (9) shows, the constant a is equal to unity in this

case since Bl = B13 and, therefore, the frequency ratios of adiacent nOtes

are all equal. Hbwever, this ratio (=b) is slightly larger than 21/2 =

1.059463。 If the theoretical beat rates of ttable II are applied in this case,

the tuner may encounter more difficulty in tuning than in the prevlous case.

丁he result is shown in ttable V. The unevenness of beat rates ■s sign■ ficantly′

large and the frequency shifts of notes from the ■deal case is random and

large。

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As the above discusslon shows, the suggested method to determ■ ne the

frequenc■ es of notes from F3 to F4 and, consequently, to determine the

modified beat rates which can be used to tune pianos with stretched octaves

seems to be appropriate。 Obvlously, the modified beat rates become more

effective when tuning a piano with larger inharmonicity characteristic.

Ve COMLtS10N

丁he inharmon■ c■ty characteristic of a Steinway Model

measuredo lt was surprising that the 6… foot grand

inharmonicities in the treble region than the upright piano

the Fletcheris study. The reason for this is probably due

thicker string of the Model L grand piano.

L grand piano was

piano has larger

which was used in

to the shorter or

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A new method was presented to determ■ne the frequencies of notes within

the F3-F4 octave when the inharmonicity characteristic in the same region is

known. This method takes care of the stretched octave F3-F4 by increas■ ng the

frequency ratio of two adiacent nOtes from 21/2。 Furthermore, this method

makes ■t poss■ ble to gradually increase the frequency ratio of adiacent nOtes

for a smooth transition of frequency ratios around F4。 This smooth trans■ tion

seems important since the F4-F5 interval is mostly w■der than the F3-F4

interval. The modified beat rates obtained by this method seems to comply

with the commonly used tests in the conventional tuning technique. 丁he

validity of applying the modified beat rates to the tun■ ng of a real piano

relies on whether each partial frequency of note obeys Eqc (1)or nOt. On an

average basis, however, the modified beat rates may make the tun■ ng eas■ er

since it has already taken into account thb effect of the inharmonicity.

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Fig。 1 ■nharmonicities of aand an upright pianO

l10 50 60

KEY NUMBERS

Steinway Model(dotted ■ines,

L Crand Piano (circles)after Fletcher)

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Fiε 。 2 Measured (。 )and apprOxinated (dctted line)inharrrlonicities ora Ste■ nway Model L E3rand piano ■n the tcmperament octave F3-F4.

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-13-

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