String Inharmonicity and Piano Tuning

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String Inharmonicity and Piano TuningAuthor(s): Rudolf A. Rasch and Vincent HeetveltReviewed work(s):Source: Music Perception: An Interdisciplinary Journal, Vol. 3, No. 2 (Winter, 1985), pp. 171-189Published by: University of California Press

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Music

Perception

© 1985 by the regents

of the

Winter

1985,

Vol.

3,

No.

2,

171-190

university

of

California

String

nharmonicity

nd

Piano

Tuning1

RUDOLF

A. RASCH

&

VINCENT HEETVELT

University f

Utrecht,

The

Netherlands

Inharmonicity

is

a well-known

property

of stiff

strings

such as those used

in

the

modern

piano.

Effects on

piano

tuning

(e.g.,

the stretched

octave)

have been suggested but have never been fully investigated. We have

measured

the inharmonicities

of

the

strings

of a

medium-sized

grand

piano.

The measured

inharmonicities are

in

excellent

correspondence

with

the

predictions

by

formula from the

physical properties

of

the

strings.

Strings

with

higher frequencies usually

have

higher

inharmonicity

than

strings

with lower

frequencies;

this

cancels out

part

of the effect

of

inharmonicity

on beat

frequency.

Introduction

Classical exts

on

piano tuning,

such

as W. B.

White's

Piano

tuning

and

allied

arts

(first

edition

1917,

fifth edition

1946,

twenty-secondprinting

1976),

typically

use

the methodof

beats

or

setting

up

precisepiano

temper-

aments.

Forthis

purpose,

hesebooks contain

ablesof beat

frequencies

or

the

most

important

consonant

intervalsthat are used when

tuning:

the

fifth,

he

fourth,

andthe

major

hird.The

octavesshouldbe tuned

beat-free.

The beat

frequencies

have

been based on the

assumption

of

strictly

har-

monic

partials

of the

tones of the

piano strings.

Under

his

assumption

he

principal

beat

frequency

f an interval

s

easily

calculated

by:

U= fqT, (la)

in which

fz

s the beat

frequency

z

from

zweving,

he

Dutchword for

beat;

see

Rasch, 1984),

f

the

fundamental

requency

f the lower tone

of the

in-

terval,

q

the

higher

numberof the ratio

p:q

that

would

describe

he

untem-

pered

pure,

or

just

ntonation)

nterval,

and

T

the natural

ogarithm

of the

tempering,

hat

s,

the

ratio

between he

tempered

nd

the

untempered

atio

(Rasch,

1984).

Throughout

his

article,

we

will

only

consider he beat

fre-

Requests

for

reprints

may

be

sent

to

Rudolf A.

Rasch,

Institute of

Musicology,

University

of Utrecht,Drift 21, 3512 BR Utrecht, The Netherlands.

1. Some

of

the

data in this

paper

were

presented

in a

paper

given

at the

Fall

1984

meeting

of

the Acoustical

Society

of America in

Minneapolis,

October

9,

1984.

171







1

72 Rudolf A.

Rasch

& Vincent

Heetvelt

Table 1

IntervalSizes

and Relative

Beat

Frequencies

f the Consonant

ntervals

n

EqualTemperament,AssumingHarmonicPartials

Interval

ize Relative

Beat

Frequency

Interval

Ratio Cents

Exact

Approximate

Fifth

1.498307

700 -0.003386

-3/880

Fourth

1.334840 500 +0.004520

1/220

Major

hird

1.259921 400

+0.039684

7/176

Minor hird

1.189207 300 -0.053964

-3/55

Major

ixth

1.681793 900

+0.045378

1/22

Minor ixth

1.587401 800

-0.062995

-7/110

Note. For the

methodused to drive he

approximate

aluesof the

relative

beat

frequencies,

ee

Rasch 1984).

quency

of the first

pair

of

just-noncoinciding

armonics,

which

is

by

far

predominant

n

perception

Vos, 1984).

For

comparison,

we will

give

firstthe harmonicmodel

of intervals

and

beat

frequencies

f

piano

tones.

In

case of

equaltemperament,

hen

an

in-

tervalof

m

semitones

has the

ratio

of

1

:

=

1

:2w/12,

eat

frequency

s

given

by:

fz= f[pi-q], db)

The factorbetween he

square

brackets s the relativebeat

frequency.

nter-

val

sizes

and

relativebeat

frequencies

f

equal temperament

re

given

in

Table

1.

For

severaldecades t has

been known

that the

partials

of

piano

string

tones

do

not

obey

simple

harmonic

requency

elationships.

nstead,

hey

deviate rom a

truly

harmonic

erieswith a

difference

hat

becomes

arger

the

higher

he

partial

numbersare.

Since he

frequencies

f the

piano-string

partials

deviate roma

harmonic

eries,

t is

better

o use

the term

"partial"

than

the term

"harmonic" o

indicate

a

component

of the

complex

tone

generated

n the

string.

The

theory

of

piano-string

nharmonicity

asbeen

well

developed

n

such

papers

as those

by

Shankland nd

Coltman

1939),

Schuckand

Young

(1943),

Young

(1952),

and

Fletcher

1964).

In

general,

the

frequency

f

a

(plain)

piano-string artial

s described

by:

U

=

nfo(l

+

n2B)°-s,

(2)

in

which

n

(

=

1,

2, 3,

. .

.)

is

the

partial

number

and B the

inharmonicity

coefficient,

fo

is a

kind

of

virtual,

imaginary

undamental

component,

which

s

not a

part

of

the

sounding

one,

but

only

a base

for the

computa-

tion

of the

frequencies

f

the

soundingpartials.

We will

use the

inharmonic-

ity coefficientB as thebasicquantitative ariable o indicate heamountof

inharmonicity.

ts

calculation s shown in

Eqns.

3a

and 3b.







String Inharmonicity

and

Piano

Tuning

173

Several

apers

are available hat have

quantitative

ata aboutthe

inhar-

monicity

of

piano

strings

(Young,

1952; Fletcher,

Blackham,

&

Stratton,

1962;Fletcher,1964;Wolf& Müller,1968;Müller,1968;Lieber,1975).

In

general,

nharmonicity

s

larger

for smaller

pianos (especially

pinets)

andsmaller

or

the

larger

ones

(especially rands).

Usually,

he lower tones

of

a

piano

have "wound"

strings,

he middle

and

the

higher

ones

"plain"

strings.

The

inharmonicity

f the wound

strings

s,

on the

average,

ower

than

for

the

plain

strings.

But

also

within

the

wound and the

plain

sections,

inharmonicity

s

not

constant. For the wound

section,

inharmonicity

n-

creases

when

one

goes

down

the

keyboard.

In

the

plain

section,

inhar-

monicity

s

minimal

n

the octave

below

C4

(middle

C);

it increases

n

the

upward

direction.Because

of the

scaling

actorsused

n

piano

construction,

inharmonicityeemsto be related o fundamentalrequencyby way of a

power

function,

ike

B

=

mfk

(m

and k

being

constants).

This

means,

for

example,

hat

for each octave

frequency

iseor

fall,

inharmonicity

hanges

by

a certain

factor.

Roughly,

one can assume that

inharmonicity riples

with

every

ascending

ctavefromaboutmiddle

C on

(k

=

1.6),

that

it

dou-

bles

with

every

descending

octave from about C3 down

(k

=

-1;

apart

from the

jump

at

the

transition rom

plain

to wound

strings),

and that

it

may

be

more

or less constant

n

the octavebetweenC3

and

C4

(k

=

0).

But

it must

be stated

hat

theserates

are,

n

the first

place,

estimates

of the

order

of

magnitude.

n

practical

ases,

a

lot

of variation

may

occur.

Some

sample

valuesof inharmonicities regiven nTable2. Since he variationn inhar-

monicity

over

the instrument ollows

a

more

or less standard

pattern,

he

inharmonicity

alue

(coefficientB)

of

the

middleC

strings

ouldbe

taken

as

Table 2

Some

Sample

Values of

Inharmonicity

(B)

of

Piano

Strings

Inharmonicity

B)

Nominal

Key FrequencyHz) (1) (2) (3)

A0

27.5 0.000352 0.000397 0.000530

Al

55 0.000135 0.000217

0.000280

A2

110

0.000114 0.000144

0.000150

A3

220 0.000401 0.000361

0.000050

A4

440

0.000526

0.000722 0.000400

A5

880 0.001863 0.002165

0.000200

A6

1760 0.003800

0.005413 0.000200

A7

3520 0.014895 0.015156

-

Note.

(1)

Hamilton

upright piano; computed

after data

in

Fletcher

(1964).

(2)

Upright

piano

of

average

size,

after Lieber

(1975). (3)

Baldwin

grand

piano,

after Fletcher, Blackham, & Stratton (1962); these values refer to the G keys,

except

in

the

lowest octave.







1

74

Rudolf A. Rasch

&

Vincent

Heetvelt

a

Standard

value,

roughly

indicative of

the

inharmonicity

of

the

whole

in-

strument.

It is evident that inharmonic partial tones change the beat frequenciesof

intervals,

and

provide

other

beat

frequencies

than

the same

intervals

would

have with

harmonic

partials.

That means that

when the

"harmonic"

beat

frequencies

are

used when

tuning

a

piano

with

non-negligible

string

inhar-

monicity,

this

may

lead to

erroneous

results. From 1960

on,

this

phenome-

non

has been

noted

several times

by

piano

tuners and technicians

publish-

ing

in

such

journals

as Das

Musikinstrument

(Fenner,

1960;

Jung,

1961,

1964; Fuchs, 1964;

Rakowski

&

Bobilewicz, 1971;

Engelhardt,

1973:

Lieber,

1977;

Leipp,

1977)

and

the Piano

Technician's

Journal

(Coleman,

1961, 1964; Howell, 1961;

Fenner, 1961; Lockhart, 1961a, b;

McMahan,

1962; Kent, 1972).

The

supposition

that

string-partial

nharmonicity may

be the

cause

of the

so-called stretched octave of

piano

tuning

was first stated

by

Schuck

and

Young

(1943)

and

repeated

many

times

by

other authors. There

seems

to

be

no

reason

to doubt this

thesis. Cohen

(1984)

found

in

psychoacoustical

ex-

periments

that,

when

tones are

slightly

inharmonic,

intervals

are tuned

in

such a

way

that the relevant

partials

match

in

frequency.

Schuck

and

Young

(1943)

already

mentioned

the fact that

using

normal beat

frequencies

for

tuning

would

lead to

deviating

sizes

of

the fifths.

Although

several

authors

(Coleman, 1961; Kent,

1965; Suzuki,

1984)

reported

about the

deviating

beat frequencies of pianos with significant string-partialinharmonicity, a

theoretical

foundation

for the

calculation of

beat

frequencies

of such

inter-

vals

has not

been

constructed

yet.

In an

earlier article

(Rasch,

1984),

we

set

up

a

framework

for the calcula-

tion of

beat

frequencies

of

tempered

intervals

in

musical

tunings

assuming

strictly

harmonic

partials.

In

this

article,

we

will

extend this

framework

to

include

equations

for

the calculation

of

beat

frequencies

of intervals

of

tones with inharmonic

partials.

Whereas

in

the case of

harmonic

partials

both the

temperings

and

the beat

frequencies

of the various

consonant

in-

tervals

in a

particulartuning

or

temperament system

are

systematically

in-

terrelated

(Rasch,

1983, 1984),

this is no

longer

the case

with

inharmonic

partials.

We

will

limit our

attention

to

equal

temperament,

as far as

tuning

and

temperament systems

are

concerned. Given an

octave

of a certain

size,

we

will

always

divide this octave

into

12

equal-ratio

semitones.

Interval

ra-

tios will

always

refer

to

the

ratios of

the first

(lowest

sounding) partials.

Numerical values of

the

inharmonicity

coefficient

may vary

in

practical

cases from about 0.00001

to

about

0.015.

In

general,

values well

under

Ò.0001

can

be

neglected.

Values

between

0.0001 and 0.001 have

small

but

discernable

effects.

Values

greater

than

0.001

significantly

change partial

and beat frequenciesand influencetuning practice.We will use the value of

B

=

0.001

as the

probe

value

in

our

calculated

examples.

By using

this







String

Inharmonicity

and Piano

Tuning

1

75

value,

the

examples

demonstrate

he effectsof a small

but

certainly

not to

be

neglected

amount

of

inharmonicity

n

string

ones.

The

tones of

harpsi-

chordstrings,being ongerand thinner hanpianostrings,have,as a rule,

inharmonicities

hat

stay

under

0.0005

and

can be

neglected

n

the

tuning

process

Rasch,

1980).

The remainder

f

this article alls into

three

parts.

The

first

part

ncludes

some

algebraic

work about

nharmonicity

nd beat

frequencies

f intervals

of

inharmonic

tring

tones.

The

second

part

is a

report

of

a

seriesof

mea-

surements

nd calculations

of

inharmonicities

f

the tones

of an

August

Förster

medium-sized

randpiano.

The third

part

contains

comments

on

the effects

of

inharmonicity

n

the

beat

frequencies

f intervals

of

inhar-

monic ones

as shown

in

our

measurements.

Inharmonicity

and

Beats

Single

Tones

If the

partials

of

the tones

of a

struck

tring

areaffected

by

non-negligible

stiffness

of the

string,

their

frequencies

will

deviatefrom the

strictly

har-

monic

series

described

y fn

=

nf\.

In

fact,

the

frequencies

f the

partials

are

bestdescribedby

fn

=

nfo(l

+

n2B)°-5

nfo(l

+

0.5rc2B),

(2)

in

which

o

s the

frequency

f an

imaginary

undamental

omponent,

and

B

is the

inharmonicity oefficient,

qual

to

B

=

iT2QSK2/TL\

(3a)

in

which

Q

is

Young's

modulus

n

g/cm-sec2,

S the cross

section

n

cm2,

K

the

gyration

adius

n

cm,

Tthe

tension

n

g-

cm/sec2,

nd

L

the

length

n

cm.

By

using

relations

ike S

=

irD2

D

is

diameter),

K

=

D/4,/"

=

T°-5/2Lp°-5or

T

=

4L2f2p

(f

is

fundamental

requency,

p

is

linear

density

n

g/cm)

and

p

=

aS

(a

is

density

n

g/cm3),

he formulacan be

rewritten

s

which

s the

easiest

or

practical omputations.

The first actor

s

a constant

(equal

o

0.154213),

the second

actor

depends

on the material

f the

string

(for

steel

strings

Q

=

1.9

x

1012g/cm-sec2

nd

a

=

7.8

g/cm3,

so

that

Q/v

=

0.243

x

1012cm2/sec2),

hilethe third actor

depends

on the

dimen-

sions

and the

fundamental

requency

of the

string.

Some

authors

(e.g.,

Schelleng,

973; Fletcher,

1977)

haveuseddefinitions f the

inharmonicity

coefficienthatresult n half valuescomparedo "our"definition,norder

to

avoid

this

factor

n

Eq.

2 and

many

other

equations.







176 Rudolf

A.

Rasch & Vincent

Heetvelt

The

partials

of

inharmonic

ones have

the

followingfrequencies

nd

in-

terrelations:

fH

=

nfo[\ + 0.5«2B], (4a)

/•w «A[l

+

0.5(«2-l)BL

(4b)

f°-n

+

0.5n>B

(4c)

n

fn

n

n +

0.5(n2-l)B.

(4d)

The

nharmonicity

oefficient

B

can be

calculated

rom he

frequencies

f a

pair

of

partials

with

numbers

m

and

n)

in

the

followingway:

B = [2(mfn nfm)}[n% - m%\ (5)

The

frequency

differencebetweena

partial

of

the

inharmonic

one

of

a

stiff

string

and the

corresponding artial

of a harmonic one

of an

ideally

flexible

tring

of

the same

dimensions,

ension,

and material

s

b»

=

0.5n2f0B,

(6a)

while the

interval

frequency

atio)

between hese

two

partials

s

/„

=

1 +

0.5n2B,

(6b)

or,

in

cents:

In

=

865.62n2B

cents).

(6c)

Some authors

(e.g.,

Schuck

&

Young,

1943;

Young,

1952)

have used

an

equivalent f 865.62B as thecoefficient f inharmonicity. hegraphicpre-

sentationof

calculatedor

measured nharmonicities

y

the

magnitude

f

Jw

has remained

popular

since

Schuckand

Young

(1943).

Beats

of

Intervals

of

InharmonicTones

Inharmonic

artials

can

change

he beat

frequencies

f

tempered

r

mis-

tuned ntervalsbuilt

up

with

inharmonic ones rather

drastically ompared

to the

corresponding

armonic

cases.

In

fact,

they

makethe

definition

of

a

pureor untemperedntervalrathercomplicated.With strictlyharmonic

partials,

a

consonant

musical

nterval s

pure

or

untempered

when

some

low

partials

of

the two

tones

coincide

n

frequency,

which is

the case

when

the

fundamental

requencies

avea

ratio hatcan be

expressed

by

two

small

integersp:q,

like 1:2

(for

the

octave),

2:3

(for

the

fifth),

etc.

When he

fundamental

requencies

f two inharmonic ones

have such

a

simple

ratio,

some

partials

will

be

just-noncoinciding,

which

will

cause

beats when

they

sound

simultaneously.

First,

we

will assume

that

inhar-

monicities

f

the two

tones

are

equal.

The

beat

frequency

of

the

first

pair

of

just-noncoincidingarmonics,

owest

in

frequency)

will be

equal

to

fz= OSqUtf-tfE. (7)







String Inharmonicity

and Piano

Tuning

1

77

Whenthe fundamental

requencies

onstitute

a

tempered

or mistuned

n-

terval,

with ratio

p:qt

(or

i

=

qt/p)9

hen

the

corresponding

eat

frequency

fz

=

qfi[(t-D

+

0.5(t{pi-l}

-

{<?-

1})B], (8a)

or

fz

=

fiUpi-q)

+

0.5(/{p3-p}

-

[q>-q})B].

(8b)

In

order

o make

he consonant nterval

without

beats,

he fundamental

re-

quencies

must

deviate

omewhat

rom

the

simple

nteger

atio

p:q.

If we

let

the

qth partial

of

the lower tone

coincidewith the

pth

partial

of

the

higher

tone,

then the ratio of the first

partials

s,

with

verygood approximation:

fi:gi=p:q[l

+

0.5(q2-p2)B].

(9a)

We

will

call such

a

beat-free

ntervalan

inharmonically ure

interval,

o

distinguisht fromtheharmonically ure ntervalwithsimpleratiop:q.An

inharmonically ure

nterval an

approximately

e

expressed

n

cents

n

the

followingway:

/inh=/harm 865.62(^-^)B

(cents),

(9b)

in

which

inh

s the

inharmonically

ure

consonant nterval

and

harm

s

the

normal,

harmonicallypure

consonant.

Numericalvalues

for

inharmoni-

cally pure

consonant

ntervalsare

given

in

Table 3A. The

inharmonically

pure

interval

s

always

larger

than

the

corresponding armonicallypure

one.

The

inharmonically

pure

octave

equals

1200

+

2597B cents.

With

B

=

0.001,

our

typical

value,

this

gives

rise

to

a

stretch

of

such

an octave

of

2.6 cents,which sagainafairly ypicalvalue.Itmaybe notedthat theratio

between

partials

hat have

equal

numberof the two

tones s

the same

for all

of

such

pairs.

In

Eqns.

7-9,

equal

nharmonicity

was

assumed

or the two

tones

of the

interval.

n

practical

pplications,

owever,

hiswill

only

rarely

be the

case.

When he

inharmonicity

f the

two

tones

of

the

interval

are

B\

and

B2,

the

beat

frequency

f a

tempered

nterval

of

these

tones will be

fz

=

qfi[(t

-

1)

+

0.5(t{p2

-

1}B2

{q2

l}Bi)],

(10a)

or

fz

=

fiUpi-q)

+

0.5(i{p>-p}B2-{q*-q}B1)]. (10b)

Eq.

10 has some

unexpected

onsequences.

The effectof

inharmonicity

n

the

beat

frequency

will be

cancelled

out

in

case of

t{p2

1)B2

=

(<72

l)Bi>

or,

with

very

good approximation,

when

B2/Bi

=

(q2 l)/(p2

-

1)

or,

with

less

but

still

sufficiently

ood approximation,

when

B2/Bi

=

q2/p2.

We

now

assume

hat

inharmonicity

ncreaseswith

frequency

ccording

o

a

power

function

or,

exponentially

with

pitch).

The

ratio of the

inharmonicities

f

two

tones

one octave

apart

will be called he octave

nharmonicity

atio.

An

octave

nharmonicity

atio that

will

lead

to no

effect

on the

beat

frequency

of a

(tempered)

onsonant

nterval

with

basic ratio

p:q)

is

approximately

equalto l:(^2/p2)1/loS2^),hich is equalto 1:4 independentof q/p. Since

inharmonicity

ithinthe

upper

half of

the

piano keyboard

s

usually

higher







178

RudolfA. Rasch

&

VincentHeetvelt

Table 3a

Inharmonically

ureIntervals

Ratio Size (cents)

Interval

Harmonie

+

Inharmonic

Part Harmonic

+ Inharm.

Part

Octave

2.000000

+

3B

1200

+ 2597B

Fifth

1.500000

+

3.75B

701.955

+ 7328B

Fourth

1.333333

+

4.667B

498.045

+

6059B

Major

hird

1.250000+

5.625B

386.314+

7791B

Minor hird

1.200000+

6.6B

315.641

+

9522B

Major

ixth

1.666667

+

13.333B

884.359

+ 13850B

Minorsixth

1.625000

+31.

IB

813.686

+33759B

Note.

The

ratio

numbers

must

be readas

i in

the

ratio

1

: .

The

nhar-

monicrationumbersthe sumof aharmonic ndaninharmonicart,

he

valueof the latter

part

determined

y

the

inharmonicity

oefficientB.

When

he sizes

of the intervals

re

expressed

n

cents,

he

same

applies.

Table 3

b

Sample

Values

of

Inharmonically

ure

ntervals

Octave

Ratio of

Inharmonicity

05

Ï

3

Interval

Ratio

Cents

Ratio

Cents

Ratio

Cents

Octave

2.00300

1202.60 2.00300

1202.600 2.00300

1202.600

Fifth

1.50450 707.141

1.50375

706.278 1.50172

703.941

Fourth

1.33933

505.818 1.33800 504.094

1.33492

500.103

Major

hird

1.25750 396.670

1.25563 394.087

1.25165

388.594

Minor

hird

1.20900

328.577

1.20660

325.137

1.20178

318.201

Major

ixth

1.68267

900.899 1.68000 898.153

1.67169

889.564

Minorsixth

1.63840

854.745

1.63120

847.120

1.60996

824.429

Note. All valuescalculated

with

Bi

=

0.001,

but

with

differing

ctave

ratiosof

inhar-

monicities,

amely

0.5,

1,

and

3.

for

tones

with

higher

requencies,

here

will often be

at

least

a

partial

com-

pensation

of the deviations

n

beat

frequency

due to

inharmonicity,

f not

a

complete

anceling.

f

the octave

nharmonicity

atio

n

a

particular

nstru-

mentwere

4

or

nearly

o,

there

would be

no or

only

littleeffect

of the

inhar-

monicity

on

the beat

frequencies.

n

practical

ases,

the

octave

nharmonic-

ity

is

usually

lower

than

4,

say,

about

3,

which

implies

a

partial

compensation

f the

inharmonicity

ffects.

An

inharmonically

ure

ntervalof

two tones

with

different

nharmoni-

citieshastheratio of

fi:gi

=

p:q[l

+

0.5(t{p2-l}B2-{tf-l}B1)].

(11)







String

Inharmonicity

and Piano

Tuning

1

79

For the

octave,

the

inharmonicity

of the

higher

tone does not count

in

the

calculation

of the

inharmonically

pure

interval.

Sample

values

of

inhar-

monically pure intervals for various octave inharmonicity ratios are given

in

Table 3B.

Often,

we will

work with the

ratio

of

the beat

frequency

of

the

inhar-

monic interval

divided

by

that

of the harmonic

interval,

to see the

effects

of

the

inharmonicities

on

beat

frequencies.

The deviations

of

the relative beat

frequencies

caused

by

inharmonicity

are not

equally

large

for

all

the

various consonant

intervals.

Inharmonically

pure

intervals

are

always larger

than the

corresponding

harmonically pure

intervals

(see

Table

3b).

For the

intervals that

are

positively

tempered

in

equal

temperament

(enlarged

compared

to

their

just-intonation

size),

namely, the fourth, the major third, and the major sixth, the equal-

tempered

sizes are nearer

to

the

inharmonically pure

sizes

than

they

are to

the

harmonically

pure

sizes. This means that

inharmonicity

usually

lowers

the

beat

frequencies.

It

may

even become zero or

change sign.

The

other

consonant

intervals

(the

fifth,

the minor

third,

and the

minor

sixth)

are

smaller

than

pure

in

equal temperament.

The difference

between

the

equal-

tempered

and the

inharmonically pure

size is

larger

than

between

the

equal-

tempered

and the

harmonically pure

sizes. This means

that beat

frequencies

are raised

by inharmonicity.

Beat

frequencies

of

major

sixths,

major

thirds,

and minor

thirds

are,

in

general,

relatively

little affected

by inharmonicity

when the octave inhar-

monicity

ratio is

larger

than 2.

Beat

frequencies

of the fifths

are

more seri-

ously

changed

in

this condition

(they

increase),

while

those of

the

minor

sixths

are altered

by

an

even

larger

increment.

Beat

frequencies

of the

fourths

can

be

reduced,

even to such an extent that

they change

sign

(which

means that

the interval becomes smaller than its

pure

ratio)

and

get higher,

but

"negative"

values.

The

large

changes

in

beat

frequencies

of

the

minor

sixths need

not

worry

the

piano

tuner.

Because of

the

relatively

high

num-

bers of the

pure

ratio

(5:8)

beats are both with rather

high

frequencies

and

ratherhard to perceive, so that this interval is nearlynever used when tun-

ing.

Since

the fifths are

the most

important

intervals

for

tuning,

the

changes

of the

beat

frequencies

of these

intervals

are

potentially

the

most

disturbing

ones

in the

tuning process.

In

Figure

1,

we have summarized the effects of

inharmonicity

on

beat

fre-

quency

as

a

function of

the

octave

inharmonicity

ratio.

Now,

this

ratio

is

varied

from

0.25 to 4. The left

diagram

of

Figure

1 is based on

a

harmoni-

cally pure

octave

(1:2),

the

right

one on an

inharmonically pure

octave

(1:[2

+

3B]).

Inharmonicity

is set to

J3i

=

0.001.

When we

compare

the

predicted

beat

frequencies

with

a

strictly

harmonic

octave to

those

with

an

inharmonically pure octave, the latter ones show somewhat smaller devia-

tions

from the harmonic case than the

former one.

But the

differences







180 Rudolf A. Rasch

&

Vincent

Heetvelt

Fig.

1. Ratios

between beat

frequencies

under

inharmonic and

harmonic

conditions,

for the

various consonant

intervals,

with a

harmonically

pure

octave

(1:2;

left

diagram)

or

an inhar-

monically pure

octave

(1:[2

+

3B];

right

diagram).

caused

by

the

choice

of the

octave are much

smaller

than those

caused

by

the octave inharmonicity ratio. When this ratio is about 4, the effects of the

inharmonicity

on the beat

frequencies

are

only

slight.

But when

the

octave

inharmonicity

ratio

decreases,

deviations

of

beat

frequencies

become

larger

and

larger.

Beat

frequencies

of

minor

thirds,

fifths,

and minor sixths

are

raised with

amounts that increase

in

the

order of mention.

Evidently, taking

into account

unequal

inharmonicities

(preferably

with

an

octave

inharmonicity

not too

different from

4)

and

inharmonically pure

octaves

led to the

prediction

of

beat

frequencies

that deviate

the least

from

the harmonic case.

Both

principles

are known to

appear

in

piano

strings

(inharmonicity increasing with frequency) and piano tuning (inharmoni-

cally

pure

octaves,

leading

to

a

stretched

tuning),

so that

both

can do their

compensating

work

as

regards

beat

frequencies.

Measurements

Recording

and

Analysis

Tape recordings

were made of

the sound of

the tones of

single

strings

from the string choirs in the octave from A3 to A4 of a medium-sizedAu-

gust

Förster

grand piano

(No.

57825).

Some

data

about

these

strings

are

in







String

nharmonicity

nd

Piano

Tuning

181

Table 4

Diameter

and

Length

of

Strings

of the Piano

Used

for the Measurements, with Nominal Frequency,in

the Octave

from A3 to

A4

String

Diameter

Length Frequency

(cm)

(cm)

(Hz)

A3

0.1050

74.9

220.00

Asharp3

0.1050

70.7 233.08

B3

0.1050

67.0

246.94

C4

0.1025

63.8

261.63

Csharp4

0.1025 60.3 277.18

D4

0.1025

57.3

293.66

Dsharp4

0.1025

54.3

311.13

E4 0.1025 51.4 329.63

F4

0.1000 48.9 349.23

Fsharp4

0.1000 46.3 369.99

G4

0.1000

44.1

392.00

Gsharp4

0.1000

42.0 415.30

A4

0.0975

39.8 440.00

Table

5

Measured

Frequencies

(in Hz)

of the First

Eight

Piano-String

Partials,

for

Single Strings

in

the Octave

from A3

to

A4

Partial

A3

Asharp3

B3

C4

Csharp4

D4

0

219.8

232.6

247.0

260.8

276.8

292.7

1

219.7

232.6

247.0 260.8

276.9

292.7

2

439.8

465.5

494.2

522.3

554.4

586.1

3

659.9

698.9

741.9

784.1

832.6

880.1

4

880.9

932.8

990.4

1047.3

1111.3

1174.0

5

1102.4

1167.6

1239.9

1310.9

1391.7

1471.7

6

1324.9

1403.5 1490.6

1576.2

1673.9

1770.2

7

1548.3 1642.1

1743.2

1843.3

1957.6

2070.9

8

1773.5

1879.8 1997.3

2112.6

2243.6

2374.6

Dsharp4

E4

F4

Fsharp4

G4

Gsharp4

A4

310.5

329.5

348.1

369.6

390.0

415.3

439.4

310.6

329.5

348.1

369.7

390.1

415.4

439.7

621.8

659J

696.7

740.6

781.8

831.9

880.5

933.7

990.7

1047.4

1112.5

1174.5

1250.4

1322.9

1247.2

1323.6

1398.3

1486.4

1569.9

1672.1

1768.5

1562.3

1658.8

1752.4

1863.3

1968.5

2096.3

2217.8

1880.5

1995.5

2109.0

2243.5 2370.3

2524.1

2673.8

2200.2

2336.2

2469.7 2627.8

2777.4

2960.3

3133.0

2522.7

2679.7

2834.1

3017.0

3190.5

3398.6

3601.3

Note. The zerothpartials theimaginaryundamentalomponent f thecorresponding

harmonic

one,

calculated

with

Eq.

5 from

he

frequencies

f

the

firstand

he

eighthpartials.







1

82

Rudolf

A.

Rasch

&

Vincent Heetvelt

Table

4.

Total

length

of

this

grand

piano

was 187

cm;

the

length

of the

long-

est

string

was

136.6

cm.

The use of

single

strings

was,

of

course,

dictated

by

the wish to avoid interaction effects of the three strings belonging to the

same

string

choir.

Recordings

were also

made

of

the

simultaneously

sound-

ing

fifths and

major

thirds of the same

strings

within the

A3-A4

octave

range.

A

time

span

of

2.5

sec of the recorded

tones

after the

attack

transient

was

digitized by

a 10-bit

AD

converter with

a

16-kHz

sample

rate.

This

digital

signal

was Fourier

analyzed.

The results

of this

analyses,

as

far as the

first

eight

partials

of

the

tones are

concerned,

are

given

in Table

5.

The recorded intervals were

also

digitally

converted.

Temporal

ampli-

tude

envelopes

were

plotted

of

the unfiltered

signal

as well

of

the

signal

band-passfiltered around the frequencyof the expected firstpairof beating

partials

of

the two tones. Some

temporal

amplitude

envelopes,

both

of

un-

filtered

and

filtered

tones,

can be found

in

Fig.

2.

For these

recordings

and

analyses

we could use the

audio and

computer

equipment

of the

Institute of

Sonology

in

Utrecht,

as

well

as

several

of

their

standard

computer

programs.

Fig.2a. Temporal amplitude envelope of the major third A sharp-D. Amplitude scale is ar-

bitrary.







String nharmonicity

nd

Piano

Tuning

183

Fig.

2b.

Temporal

amplitude

envelope

of

the

major

third A

sharp-D,

but filtered

around

the

pair of beating harmonics (1170±50 Hz).

Inharmonicities

The

inharmonicity

of

a

piano

string

can be established

in

two

ways.

First,

one

can measure

the

frequencies

of

the

partials

and

calculate

the

inhar-

monicity

coefficient

with

help

of

Eq.

5, and, second,

one can use

the

proper-

tiesof the stringandcalculate the inharmonicity with help of Eq.3b. We did

both.

We used

the measured

frequencies

of

the first

and the

eighth

partials

of

the tones

for the calculation of

the

inharmonicity

coefficients.

Some

checks

made clear

that

calculations from other

pairs

of

partials

did not

result

in

significantly

different values.

In

the second

place,

we used

Eq.

3b,

with 0.154

213

and 0.243

x

1012

as constant

factors,

and values

of diame-

ters

and

lengths

from Table 4.

We used the first

partials

from

Table

5

as

fundamental

frequencies.

The measured

and

calculated

inharmonicity

coefficients

are

listed

in

Table

6;

it can

easily

be seen that

they

are

in

good

agreement.

In

fact,

the

agreement

makes it

possible

in

general

to

rely

on

inharmonicity values calculated with the help of Eq. 3b. Figure3 gives the

measured

inharmonicity

for each

partial

tone,

using Eq.

6c.







1

84

Rudolf

A.

Rasch

&

Vincent

Heetvelt

TABLE 6

Inharmonicity

Coefficients of

the

Strings

of

the A3-

A4 Octave of the Piano Used

String

Inharmonicity

Inharmonicity

(measured)

(calculated)

A3

0.000 285

0.000 273

Asharp3

0.000

324 0.000

306

B3

0.000

340 0.000

337

C4

0.000 393

0.000 350

Csharp3

0.000

413 0.000

390

D4

0.000

443 0.000

427

Dsharp4

0.000 484 0.000

471

E4

0.000 522

0.000

521

F4 0.000 559 0.000 542

Fsharp4

0.000 640

0.000 598

G4

0.000

706

0.000 653

Gsharp4

0.000

720 0.000

700

A4

0.000

756 0.000

737

Note. The first column of coefficients is calculated from the

frequencies

of

the

first and

the

eighth

partials

of the

tones

(see

Table

5)

and is therefore called

"measured";

the second

column

is calculated

with

help

of

string

properties

and

the

formula for

the

inharmonicity

coefficient.

Fig. 3. Inharmonicityin cents (ordifference with harmonicpartials)of the firsteight partials

of

the

tones of

strings

in the

A3-

A4

octave.







String nharmonicity

nd Piano

Tuning

185

Beats

Sincenone of the consonant

ntervals

except

the

octave)

of a

piano

is

tuned o its exactsmall-number

requency

atio,

the

temperament

n

piano

tuning

s

the

major

cause of

the beats

in

the

sound of the

consonant

nter-

vals.

We calculated he

beat

frequencies

rom the

partial

requencies

f

all

the

fifths

and

major

thirds

within

the

A3-A4 octave

(Table

5)

and

com-

pared

hesevalues o

beat

frequencies

measured romthe

plotted

envelopes

of the

beating

harmonicswhere

possible.

The

resulting

valuesare

in

Table

7;

it can

easily

be seen that the values

are

n

good

agreement.

For

compari-

son,

we

have added the

beat

frequency

alues of the "theoretical"

ase

of

piano

tuning,

with

strictly

harmonic

partials

and

fundamental

requencies

Table

7

Beat

Frequencies

of the

Fifths and

Major

Thirds

in

the

A3-A4 Octave of

the Piano Used

Interval

Beat

Frequency

Beat

Frequency

Beat

Frequency

Ratio

(nominal)

(calculated) (measured)

Calc./Nom.

Fifths

A3-E4

-0.7

-0.3

*****

0.34

Asharp3-F4 -0.8 -2.0 -2.0 2.54

B3-Fsharp4

-0.8

-1.3 -1.4

1.50

C4-G4 -0.9

-2.3 -2.2

2.55

Csharp4-Gsharp4

-0.9

-0.7

*****

0.72

D4-A4 -1.0

+0.4

*****

-0.42

Mean

1.20

Major

Thirds

A3-Csharp4

+8.7

+8.9

+9.0

1.02

Asharp3-D4

+9.3

+

7.5 +7.4

0.81

B3-sharp4

+9.8 +7.3 +7.2

0.74

C4-E4

+10.4

+12.8 +12.3 1.23

Csharp4-F4

+11.0

+

6.6 +7.0

0.60

D4-Fsharp4 +11.7 +14.7 +15.0 1.26

Dsharp4-G4

+12.4

+7.6 +8.2

0.62

E4-Gsharp4

+13.1

+13.4 +12.8

1.02

F4-A4 +13.7

+16.2

+16.4

1.17

Mean 0.94

Note.

The firstcolumn

gives

he

"nominal"

eat

frequencies,

he beat

frequencies

n the

caseof

strings

with harmonic

artials

nd

fundamentalsuned

o

the nominal

requencies

f

the

string.

The

second

column

gives

the beat

frequencies

alculated romthe

stringpartial

frequencies

n

Table 5.

The

third column

gives

the beat

frequencies

measured

rom

the

filtered

nvelopes

of the

tones;

asterisks

ndicatethat a measurementwas

not

possible,

which

was

always

he case when the beat

frequency

was below 1 Hz.

The far

right

column

gives

the ratio between

he

"calculated" nd the "nominal"

beat

frequencies.

All beat

fre-

quencies rein Hz. A negativevalue means hat the intervalwas smaller han the untem-

pered

nterval.







String

Inharmonicity

and Piano

Tuning

187

fourths,

and 0.82

for the

major

thirds. This fits

rather well

with

an inhar-

monicity

coefficient

of 0.001 and an

octave

inharmonicity

ratio of

2.9,

which lead to 2.07,

-

0.11, and 0.85 as ratios calculated from theory.

In

his

Figure

2,

Kent

(1965)

compared

observed inharmonic and ex-

pected

harmonic

beat

frequencies,

straightened

out

in

a

model-like

fashion

in his

Figure

6.

There is

quite

a

good agreement

in a

qualitative

sense

be-

tween these

figures

and

the main

trends of our model.

Beat

frequencies

of

the

fifths

and the

minor

thirds are

higher

in

the inharmonic

than

in

the

har-

monic

case,

those

of the

major

thirds

and

the

major

sixths lower.

The beat

frequencies

of the

fourths are

also

lower,

even

reduced

so much

that

they

almost vanish

or

change

into

"negative"

beats.

A few words

about the octave

inharmonicity

ratio

may

be

in

order

here.

It seems that the length ratio of two piano stringsone octave apart is rather

well

approximated

by

the fraction

0.53,

and

the diameter

ratio is

about

0.94.

Using

the

third factor

of

the

right-hand

side

of

Eq.

3b,

this

gives

an

octave

inharmonicity

ratio of

2.8,

which seems to be

typical

for

a

fairly

large

variety

of

instruments

(see

also

Benade, 1976,

p.

343).

Up

to

now,

we have

not

talked about the lowest

region

of the

piano

key-

board,

with

strings

that have

a

winding

around a

solid core.

Inharmonicity

of

these

strings

is lower

than

that of the

plain

strings

(the

reduction

of

inhar-

monicity being

the

background

of

their introduction

in

history),

but the

oc-

tave

inharmonicity

ratio is often less than

one

in

this

part

of the

keyboard.

That means that beat frequencies must deviate rather drastically from the

harmonic

case.

Yet,

although

the

inharmonicity

leads to

an

octave

stretch

also

in this

part

of the

keyboard,

its

consequences

for beat

frequencies

are

quite

modest.

The first reason is that the low beat

frequencies

are

accompa-

nied

by

rather

low inharmonicities.

A

second reason is that

this

part

of the

keyboard

is

tuned

almost

exclusively by

octaves.

Finally,

consonant

inter-

vals

smaller

than

the octave are used less

in

the bass

register

than

in

the

mid-

dle

and descant

registers

in

musical

composition,

so that the

beats of

these

intervals

are

not

so

important

at all.

The most critical deviations as far as the effects of inharmonicityon

the

tuning process

are concerned are those that occur

in

the

octave

used

for

"laying

the

bearings"

of the

temperament,

most often the octave

from

F3

to

F4.

Actually,

if there is

appreciable inharmonicity

in

the

strings'

tones,

it is

to be

hoped

that

this

inharmonicity

is

increasing

with

frequency,

so

that

its

effects

are set

off a little bit

by

this increase.

In

the latter

case,

the

tuning

deviations

from

the standard

scale

can remain of limited

magnitude,

some-

thing

that is

especially

to be wished for when the

piano

is used

together

with

other

instruments,

as

in

concert

and

chamber music.







1

8

8

Rudolf A.

Rasch

&

Vincent

Heetvelt

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