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Title: A Scale for Specifying Frequency Levels in Octaves and Semitones
Author: Parry Moon

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'

THE

JOURNAL

OF THE

ACOUSTICAL

SOCIETY

OF AMERICA

VOLUME

25, NUMBER

3

MAY,

1953

A Scale for Specifying Frequency Levels in Octaves and Semitones
PARRY MOON

Massachusetts
Institute of Technology,Cambridge,Massachusetts
(ReceivedSeptember15, 1952)

A newsystem
ofspecifying
frequency
level
isproposed.
It utilizes
•helogarithm
(base
2)oftherelative
frequency,expressedin duodecimalnotation. In this way the designationof musicalintervals and musical
scalesis related directly to frequency,yet the numbersrepeat in eachoctave and the semitonesof the equitemperedchromaticscaleare expressedexactly as 0.100, 0.200, 0.300, ....
The new notation is applied to the macrodesignation
of semitonesand octaves,and to the microdesignation of true intervals and of Pythagoreanand Ptolemaic scales.A slide rule is developedto allow easy
visualizationof the relationsamongmusicalintervals.

1. INTRODUCTION

Various methodsof designationmay be classified
under these two heads:

N outstanding
•haracteristic
of musical
tonesis

frequency.But thoughthis physicaldesignation, Macrostructural
Specification.--(a)Notes on a staff,
expressedin cyclesper second,is necessaryin many (b) letter namesof notes,(c) somaterminology.
applications--particularlyin musical acoustics--such Microstructural
Specification.--(a)Frequency(cycles
a specificationis not satisfactoryin other cases.Par- per sec), (b) log frequency,(c) semitones,(d) cents,
ticularly in the detailed study of musicalscalesand (e) centitones,(f) savarts.
intervals, the specificationof frequencyis not at all
The designations
listedin the first groupareemployed
what is wanted.

by musiciansbut are not always adequate, even for
Thus two general designationshave arisen' the musical purposes.The obvious specificationin the
designation
of the physicistand the designation
of the secondgroupis frequency.Though usuallysatisfactory
musician. And each scheme has its ramifications. A
to the physicist,this specificationdoesnot provide the
studentof musicencountersvariousclefs,solf[gedesig- desired musical information. A basic psychophysionations,Pythagoreanand Ptolemaicwholetones,large logical fact in music is the instantaneousrecognition
and small steps,large and small semitones,as well as of certainfrequencyratios,no matter wherethey appear
commas,cents,savarts, --.. Yet the wide variety of in the musicalscale.The ratio 2' 1 is recognizedas an
specification
refersto the sameth•ng' It is all concerned octave, the ratio 5'4 is a major third, regardlessof
with the quantitative, physical designationof how absolutefrequency.By replacingthe frequencyscaleby
much "higher" one tone is than another. The present a scaleof logf, we come a step closerto meeting the
complexityof the subjectis not inherent;it is largelya needs of music; since with the latter arrangement,a
man-madeconfusionin terminologyand notation.
given frequencyratio is alwaysrepresentedby a fixed
A singlecomprehensive
method of specification
is distanceon the log-f scale.
proposedin the followingpages.This is a physical
But logf doesnot repeatin the next octave,and there
specification
in termsof the logarithmof the frequency. is no simple relationshipbetweenlogarithmsand the
But the logarithmis to the base2 insteadof 10 and is semitonesof the chromaticscale.At this point, many
expressed
in the duodecimal
numbersysteminsteadof experts abandon the logarithmicnumbersand start
the familiar decimal system.The quantity thus ob- anew by dividing the octave into 12 equally spaced
tainedis calledthe frequency
leveland is designated
by intervals correspondingto the 12 semitonesof the
the letter p. The new systemis closelyassociated
with equally temperedscale.Any musicalinterval can then
bothphysicsandmusicand tendsto meetthe needsof be designatedby a number of semitonesplus a fracthe musicianas well as those of the acousticalexpert.
tional part of a semitone.This method has been extendedby dividingeachsemitoneinto 100 equal parts
2. CRITERIA
(on a logarithmicbasis),eachof whichis called2 a cent.
The specification
of frequencylevelis neededin two Any interval can be expressedin cents.A somewMt
kindsof applicatipns,
one dealingwith the macro- similar proposal divides the octave into 301-t- parts
structure of the octave and the other with its microcorresponding
to log2=0.301.... Each of these divistructure.
When a composer
giveswritten instructions sionsis called3 a sayart.The centitoneis anotherpossiby providinga score,he is interestedin specifying bility.4
frequencyonly to the nearestsemitone.This is an
• A. J. Ellis, Proc. Roy. Soc. (London) 13, 93, 392, 404 (1864);
exampleof macrostructure,
• while the questionof the
23, 3 (1874); J. Soc.Arts 33, 1103 (1885); H. L. F. Helmholtz,
exactfrequencies
playedby the artistsis a questionof Sensations
of Tone(PeterSmith,New York, 1948),AppendixXX.

microstructure.

Parry Moon, J. Franklin Inst. 253, 125 (1952).

aj. Sauveur,M6m. Acad. Roy. Sci.Belgique,403 (1701).
4j. Yasser,A Theoryof EvolvingTonality (AmericanLibrary of
Musicology,New York, 1932).

5O6

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SPECIFICATION

OF

TABLEI. Octavedesignation.Basedon equitemperedscalewith
f--440.0 for violin A. To obtain absolutefrequencyof any note,
find (fife) and multiply by oneof the followingvaluesof ft.
Octave
no.

f•
(hertz)

Old designation

0.

fContra-octave,

C, to B,

32.70320

1.

Greatoctave,

C to B

65.40639

fSmalloctave,

c to b

•,32-ft octave,

16-ft octave,

2.
3.

ß
1,8-ftoctave,

130.81278

MUSICAL

PITCH

507

to internationalpitch of 440 for violin A). Then frequencylevel is definedin the new systemas

p=log2(f/fo),

(1)

where the logarithm is expressedin the duodecimal
number system.
The characteristic of the logarithm specifiesthe
octave.This integer is ordinarily lessthan ten, so it is
identical in the duodecimalnumber systemand in the
ordinary system.The numberingof octavesis given in

Table I. The newdesignation
is simpleand mightwell'
261.62557

One-lined
octave, c' to b'
4-ft octave,

4.

fTwo-linedoctave, c" to b"

523.25113

5.

fThreelinedoctavec"' to b'"

1046.50226

1.2-ftoctave,
1,1-ftoctave,

6.

Four-lined octave, c'"' to b""

2093.00452

7.

Five-lined octave, c'"" to b""'

4186.00904

replace the clumsy methods of octave specification
usedat present.
The first integerof the mantissa,Eq. (1), designates
the semitonein the octave(Fig. 1). In the equallytempered scale,this completesthe specification,all further
integers being zero) For instance, middle C of the
equally temperedscaleis specifiedin the new system
as 3.0000, the C• above middle C is 3.1000, the D is
3.2000, etc. Moreover, if we are not interestedin microstructure,we may specifyfrequencylevel in any scale

by givingonly two figures,as 3.0, 3.f, 3.2. This desigEvidently none of the above methodscoversboth
macrostructureand microstructure.The multiplicity of
schemesand the difficulty of interrelating them is a
potent sourceof confusion.The requirementsfor an
ideal specificationare as follows:
1. A singlesystemof specificationshouldapply to both macrostructure

and microstructure.

2. The quantity p should be directly proportional to Iogf so
that a givenmusicalinterval appearsas a fixeddistancealongthe
logf axis, irrespectiveof absolutefrequency.
3. The designationmust repeatfor eachoctave,sothat a given
melody will be instantly recognizedfrom its designationwhen
written an octavehigher or an octave lower.
4. The designationmust be suchthat the octaveis divided into
semitones,not into 10 parts or 301 parts.

nates the octave and the nearest semitone,which is all
that is neededin ordinary music.
Such a macroscopicspecificationof the chromatic
scale is exhibited

in Table

4. DUODECIMAL
3. THE

PROPOSED

SYSTEM

The new method is basedon a scale that is proportional to logf. Use of base 2 instead of 10 makes the
mantissa repeat for each octave, thus satisfying re-

II.

The semitones are num-

bered upward from C, which is taken as the beginning
of the octave. The only peculiarity is that, since the
duodecimalsystemis used,new symbolsmust be introducedfor 10 and 11, which are digits in the new system.
In place of 10, employthe symbolV (called "dec");
and for 11, use • (called "elf"). The simplenumerical
designationgiven in Tables I and II is capable of
replacingthe old macroscopicdesignationswhich employ lettersof the alphabetand solf[genames2
ARITHMETIC

Now turn to the microstructuralspecification,which
requiresa slightknowledgeof duodecimalarithmetic.5
In the customarydecimalnotation(radix10) a symbol
such as 2.143 constitutes

a shorthand

notation

for

quirements2 and 3. The octaveis designatedby the
2.143= 2+ 1/10+4/100+3/1000.
numeral to the left of the decimalpoint, while position
within the octave is specifiedby the numeralsto the Similarly, in the duodecimalsystem (radix 12), the
right of the decimalpoint (Fig. 1).
symbol 2.143 means
Requirement4 is still not satisfied,sincethe octave
2+ 1/12+4/144+3/1728.
is divided decimally.By expressing
the mantissain the
duodecimalnumber system, however, we satisfy all
The arithmetical operations proceed in the same
four criteria. The numeral to the ]eft of the dot still
mannerin both systems,the only differencebeing that
specifiesthe octave, the numeral immediately to the
in the duodecimalsystema digit increasesto a dozen
right of the dot specifiestwelfths of an octave and is
before affecting the next digit. A few numerical exthus identical with the number of equitemperedsemiamplesmay be helpful.
tones. Further numerals to the right designatemicrostructure.

• F. E. Andrews, New Numbers(Harcourt, Brace & Company,

Let f= fundamentalfrequencyin hertz (cyclesper Inc. New York, 1935); G. S. Terry, Duodecimal Arithmetic
(Longmans,Green& Company,London,1938); G. H. Hardy and
sec)of the givenmusicaltone,f0= basefrequency,arbi- E.
M. Wright, Theory of Numbers (Oxford University Press,
trarily chosento be C,=32.7032 hertz (correspondingLondon,1945),p. 110.

Downloaded 06 Oct 2013 to 129.173.72.87. Redistribution subject to ASA license or copyright; see http://asadl.org/terms

508

PARRY

MOON

Addition and Subtraction

TABLEII. Macroscopic
specification
of the chromaticscale.

What is the musical interval between two notes
Old specification

specified
by p= 2.143and p= 1.0867
The difference is

New specification

C'
C4•
D
D•
E
F
Fg
G

2.143
-- 1.086

1.079= octaveq-0 semitone

7/12 semitone+9/144semitone.
The 6 cannotbe subtractedfrom 3, so 1 is borrowed
from the preceding
4, giving(12+3)-6=9 as the last

Name

0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7

Gg

0.8

A
Ag
B

0.9
0.V
0. e

C

1.0

"Point zero"
"Point one"
"Point two"
"Point three"
"Point four"
"Point five"
"Point six"
"Point seven"

"Pointeight"

"Point nine"
"Point dec"
"Point elf"

figure.Similarly,the 8 cannotbe subtracted
from3,
but (12+3)-8=7. Thusthe difference
in pitch beDivision
tweenthetwonotesis 1.079,whichisevidently
slightly
morethan oneoctaveplus7/12 of an equitempered An interval of 2.68425 is to be divided into eleven
semitone.
equalparts.Whatis thedifference
of p for eachpart?
We first expressthe divisor in duodecimalform
(eleven=e). The divisionis thencarriedout in the usual
A truefifth is specified
by thefrequency-level
differ- mannerkeepingin mindat eachstepthat theradixis 12'
Multiplication

ence0.702985.
Whatistheintervalbetween
theopenC
of the violaandthe openE of theviolin,the strings

•)2.68425

beingtuned in perfectfifths?

0.295V0q- '

Betweenthe specified
C andE thereare 4 fifths,so

the total interval is

Evidently the number to the left of the dot in the

quotientis zero. Then [-2(12)+6-]/11=2,with remainderof 8. The nextdivision
is [-8(12)+8-]/11=9,

0.702985
4

with remainderof 5. The result is an interval of 0.295V

2.40•298

= 2 equitempered
semitones
+9/12 semitone
q-10/1728
semitone,
elevenof whichaddup to 2.68425.

Multiplyingin the usualmanner,we get 5)<4= 20= 1
Notethatin theforegoing
examples,
eachelementary
dozen+8.Then8)<4=32,32+1=33=2 dozen+9,etc. operation
is performed
in theordinarynumbersystem,
Thusthepitchdifference
between
thetwoopenstrings but theresultismentallytranslatedintotheduodecimal
is 2.40e298or 2 octaves+4equitempered
semitonessystembeforebeingwritten down.A moreelegant
q-11/144 semitone+2/1728semitone+....
procedure
is to workentirelyin duodecimals,
but this
requiresa duodecimal
multiplicationtable.5 Sliderules
and computingmachinescan alsobe madefor radix 12.

For ourpurpose,
however,
theelementary
methodused
OLD NOTATION

(;

C"

O

D•

E

F

Fa

G

G•

A

A•

8

C

hereis probablybest.
Decimal

NEW NOTATION

.0

.I

.2

.:3

.4

.5

.6

.7

.8

.9

.q•

.•

1.0

to Duodecimal

A wholenumberlessthanten is exactlythe samein

eithersystem.A decimalfraction,however,
requires
somemanipulationto change'
it to a duodecimal.
Per-

hapsthe simplest
procedure
is that of R. M. Pierce,
6
whichconsists
in successive
multiplications
by twelve.
Thesemultiplications
are performed
very rapidlyand
easilyon an ordinarycomputing
machine.

For instance,
a frequency
ratio of 1.4 hasa loga-

rithm of

½.-"--•.' c[B ' '
OLD NOTATION
NEW NOTATION

log•.(1.440)
=0.52607.

•-•,•,0 •:J.e

I.

•.

•.

4.

5.

Fig. 1. Comparisonof old and newnotationsfor macrostructure.

Octaves
aredesignated
by numbers
to theleft of thedot,semi-

tonesaredesignated
by numbersto the rightof the dot.

Changethis decimalto a duodecimal.
6R. M. Pierce,Problems
of Numberand Measure(Chicago,

1898); seealso reference5.

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SPECIFICATION

OF

MUSICAL

5O9

PITCH

TABLEI!I. Relationbetweenfrequencylevel and relativefrequency.
Rel.

Rel.

freq.

freq.
(I/It)

log2(f / f c)

P

1.00

o

o

(f/fc)

Ap

log:(f/f,)

1.33

0.41142 62457

0.4•2 •4-0 22•

1.34

0.42223 30007

0.509 750 •99

16 810 96V

.020 980 813
1.01

0.01435 52930

0.020 980 813

1.02

0.02856 91522

o.o41 44•

1.03

0.04264 43374

o.o61 833 386

16 650 V66

2o 68v 433

1.04

0.05658 35284

046
20 3v4 340

1.35

0.43295 94073

0.524 1Vl X743

20 lO6 250

1.36

0.44360 66515

0.53V 676 444

1 • v33 v91

1.37

0.45417 58932

0.554 995

1.38

0.46466 82670

0.56V •44 016

1.39

0.47508 48829

0.584 •43 V38

16 494 601

16 31•

o.o81 939 616

706

16 16V 088
1.05

0.07038 93279

o.ovl

771 4v7

1.06

0.08406 42648

O.lOl 31v 672

1.07

0.09761 07966

o. 12o 808 33v

1.08

0.11103

o.13• v41

1• 769 187

15 •

15 V54 V69

1 • 4v9 888

1.09

13124

0.12432 81350

1• 235 542

1.40

0.48542 68272

0.59V 998 8V5

1v •88 124

1.41

0.49569 51626

0.5•4 689 9V9

1v 925 600

1.42

0.50589 09297

0.60V 21V 27 •

1.43

0.51601 51470

0.623 810 •81

1.44

0.52606 88117

0.639 069 07V

880

o.15v vo9 9v4

1.10

0.1375o 35238

o.179 733 3v4

1.11

0.15055 96766

o. 198 200 951

1.12

0.16349 87324

o.1•6

638 723

1.13

o.17632 27726

0.214

828

1v 689 569

15 8•1

1.14

0.18903 38244

15 5•2 902
15 458 0•9
15 304 3 •4

1v 1 •o 466

1.45

0.53605 29002

0.652 371 472

19 e6V ee8

1.46

0.54596 83691

0.667 524 994

19 933 612

1.47

0.55581 61551

0.680 54V 190

1.48

0.56559 71759

0.695 428 14•

1.49

0.57531 23307

0.6VV 181 4•8

0.232 797 e85

1.15

o.2ol 63 38612

0.250 50• 597

1.16

0.21412 48054

0.26V 011 2•0

ß 1.17

0.22650 85298

0.287 4V6

1.18

0.23878 68596

0.2V4 759 8VV

19 701 915

15 173 522
15 025 3•8

14 V99 •7•
14 955 369
14 813 306

19 495 72V
19 272 V90
19 055 602
1.19

0.25096 15735

0.301 7 •3 2 •0
18 V41 350

1.20

0.26303 44058

0.31V 634 640

1.21

0.27500 70475

0.337 266 788

1.22

0.28688 11478

0.353 892 625

1.23

0.29865 83156

0.370 0•8

18 832

148

1.50

0.58496 25007

0.702 994 802

1.51

0.59454 85495

0.717 468 558

1.52

0.60407 13237

0.72• V03 383

1.53

0.61353 16529

0.744 223 848

1.54

0.62293 03509

0.758 510 089

14 693 956

14 556 V27
14 420 485
14 2V8 441

14 176 86•

18 627 V59
18 426 528

•51
18 229 824

1.24

0.31034 o 1206

0.388 326 775
18 035 60V

1.25

0.32192 80949

0.3V4 360 183

1.26

0.33342 37337

0.400

1.27

0.34482 84970

0.417 V44 460

1.28

0.35614 38102

0.433 4•

1.29

0.36737 10656

0.44V 998 15V

1.55

0.63226 82155

0.770 686 938

1.56

0.6415460291

0.784 712 218

1.57

0.65076 45591

0.798 630 68V

1.58

- 0.65992 45584

0.7•0 424 20V

1.59

0.66902 67655

0.804 0•3 274

14 047 4V0

13 •iV 472

13 9•3 740
13 88•

17 V45 963
1V5 •26

17 677 416
17 498 4V4
17 301 63e
1.30

0.37851 16233
0.38956 68118

1.60

0.67807 19051

0.817 85• V39

1.61

0.68706 06883

0.82• 2V8 280

1.62

0.69599 38131

0.842 816 482

1.63

0.70487 19645

0.856 028 4V9

1.64

0.71369 58148

0.86• 324 34•

13 648 443
13 52V 202

13 412 027
13 2•7 V62

0.466 099 799
17 12V 748

1.31

066

13 768 785

17 85V 536

876

104

15 750 492

1v 437 992

•89

V22

13 1V3 857

0.481 208 325
16 •5•

1.32

0.40053 79296

1.65

0.498 167 V23

0.72246 60245

0.880 507

16 994 408

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13 091 57V

510

PARRY
TABLE Ill.--Continued.

MOON
TABLEIV. Frequencyratiosfor the equitempered
scale.

Rel.

New notation

freq.

(f/f c)

log•(f/fc)

p

1.66

0.73118 32416

0.893 599 564

Ap

0.00

C

1.00000 00000

0.10

Ct•

1.05946 30944

0.20

D

1.12246 20482

0.30

D:•

1.18920 71150

12 V72 804

0.40
0.50

E
F

1.25992 10500
1.33483 98542

12 966 051

0.60
0.70
0.80

F•
G
G:•

1.41421 35624
1.49830 70769
1.58740 10519

0.90

A

1.68179 28309

0.170

A•

1.78179 74362

0.•0
1.00

B
C

1.88774 86253'
2.00000 00000

'
12 e81 178

1.67

0.73984 81027

0.8V6 55V 720

1.68

0.74846 12330

0.8e9 411 324

1.69

0.75702 32465

0.910 177 375

1.70

0.76553 47464

0.922 1716 607

12 85•

252

12 756 180

1.71

0.77399 63251

0.935 570 787

1.72

0.78240 85649

0.948 003 558

1.73

0.79077 20379

0.9517 554 7170

1.74

0.79908 73061

0.970 917517•2

Old notation

12 652 991
12 551 244
12 451 312

12 352 •76

1.75

0.80735 49221

0.983 138 1768

1.76

0.81557 54289

0.995 393 222

1.77

0.82374 93603

0.9177 532 4e0

1.78

0.83187 72412

0.9e9 598 170

12 256 376
12 15e 2817
12 065 880

11 e71 910

1.79

0.83995 95875.

0.170e549 1780
11 177e 381

1.80

0.84799 69066

Multiply as follows'
0.52607X12= (6). 31284
0.31284X12= (3). 75408
0.75408X 12: (9). 04896
0.04896X12: (0). 58752
0.58752X12= (7). 05024.

The resultingduodecimal
consists
of thedigitsin parentheses,so the differencein p corresponding
to a frequencyratio of 1.440 is
Ap = 0.63007.

0.1721 409 241
11 98V 419

1.81

0.85598 96973

0.1733 197 6517

1.82

0.86393 84504

0.1744 1776 482

1.83

0.87184 36485

0.1756 667 228

1.84

0.87970 57663

0.I768 16• 414

11 8917 1724

11 7e0 966

11 704 f178

0.88752 52707

0.1779 788 312

1.86

0.89530 26213

0.178e 0ee 376

We cannow considerthe conversion
of frequencyto
frequencylevel. For example,what is the value of p
corresponding
to a frequencyof 1000.0hertz?
ß

11 618 17•17

1.85

Frequency to Frequency Level

f-- 1000.0
f0=32.7032 from Table I.

11 533 O64
11 4617 635

1.87

0.90303 82701

0.17V0 549 917•

1.88

0.91073 26619

0.17el 8e5 1177

1.89

0.91838 62344

0. e02 e717 728

1.90

0.92599 94186

0. el4 163 546

11 367 3e8
11 285 541

11 1174 17117

From an ordinary logarithmtable,

1ogi0
\3-•.7--•/----1Ogl0(30.$7805)---1.48541.
The logarithmto the base2 is found by dividingthe
above by 0.30103'

11 105 6117

1.91

0.93357 26383

0. e25 268 e64

1.92

0.94110 63109

0. e36 294 478

11 027 514
10 e417 697

1.93

0.9486008475

0. e47 222 e53

1.94

0.95605 66524

0.•58 095 17417

1.95

0.96347 41240

0. e68 1772 39e

10 1772 17e7
10 998 551

10 903

1.96

0.97085 36543

0. e79 775 5817

1.97

0.97819 56297

0. e81731745½e

1.98

0.98550 04303

0. e917 e40 625

117e

10 82e 031

10 758 026

1091o(f/fo)
10g2(f/fo)-= 4.93442
O.3O1O3

and the decimalis changedto a duodecimalby successivemultiplicationsby twelve'
0.93442X 12= (s). 21304
0.21304X 12= (2). 55648
0.55648X12= (6). 67776
0.67776X12= (8). 13312.

The completespecification
for f--1000 is therefore

10 686 172

1.99

0.99276 84308

p = 4. t268

0. e17e606 797
10 5 e5 425

2.00

1.00000 00000

1.000 000 000

=4 octaves above C,-+-elevenequitemperedsemitones+2/12 semitone+8/1728 semitone.

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SPECIFICATION

OF

MUSICAL

PITCH

511

TABLEV. Frequencies
of the equitempered
scaleß
(International
pitch,a•=440.0hertz.)
New
notation

0.00

Old
notation

2

3

5

6

65.40639

1

130.81278

261.62557

523.25114

1046.50226

2093.00452

69.29566
73.41619
77.78175
82.40689
87.30706
92.49861
97.99886
103.82617
110.00000
116.54094
123.47083
130.81278

138.59132
146.83238
155.56349
164.81378
174.61412
184.99721
195.99772
207.65235
220.00000
233.08188
246.94165
261.62557

277ß18263
293.66477
311.12698
329.62756
349.22823
369.99442
391.99544
415.30470
440.00000
466.16376
493.88330
523.25114

554.36526
587.32953
622.25397
659.25511
698.45646
739.98884
783.99087
830.60939
880.00000
932.32752
987.76660
1046.50226

1108.73052
1174.65907
1244.50793
1318.51023
1396.91292
1479.97769
1567.98174
1661.21879
1760.00000
1864.65504
1975.53320
2093.00452

2217.46105
2349.31814
2489.01586
2637.02046
2793.82585
2959.95538
3135.96349
3322.43758
3520.00000
3729.31009
3951.06641
4186.00904

0 octave

C

32.70320

0.10

C•

34.64783

0.20

D

36.70810

0.30

D•

38.89087

0.40
0.50

E
F

41.20344
43.65353

0.60

FO

46.24930

0.70

G

48.99943

0.80

G•

51.91309

0.90

A

55.00000

0.V0

A•

58.27047

0.•0
1.00

B
C

61.73541
65.40639

4

Thus the noteisa slightlysharpB, two spacesabovethe
treble clef. This is for internationalpitch (a'=440.0).

Therefore,

If any otherstandardis employed,f will be different
and the specificationwill be altered in a definite

and

manner which is calculatedas in the above example.

log2(f/ f o)= 5.55329

log•o(f/fo)- 0.30103log2(f/fo)= 1.67171.
From an ordinary log table,

Frequency Level to Frequency

fifo-46.958

What is the frequencycorresponding
to p-5.67817
The duodecimal is 0.6781=6/12+7/144+8/1728
q- 1/20736 so
6/12=0.50000,
8/! 728= 0.00463,
1/20736=0.00005
0.55329.
TABLE VI.

Freq. ratio

2:1
3: 2

Perfect fourth

Major third
Major sixth
ß..
ß..

Perfect seventh
ß..

Minor sixth

Largestep(majorwholetone)

5. APPLICATIO

Musical

Octave
Perfect fifth

Minor third

[=46.958(32.7032)= 1535.7 hertz.
NS

The precedingsectionhas sketcheda few applications of the new specification.The duodecimalarithmetic usedin microstructuraldesignationis not unduely
complicatedand can be masteredvery easily.Nevertheless,there is an advantagein reducingthe arith-

7/144=0.04861,

Name

or

intervals.
log•.(ft/j

)

p

4:3

1.00000 00000
0.58496 25007
0.41503 74993

1.000 000 000
0.702 994 802
0.4•9 227 3•V

5:4
5:3

0.3219280949
0.7369655942

0.3V4360 183
0.8V1587 580

6:5
7:6
7:5

0.26303 44058
0.22239 24213
0.48542 68272

0.31V 634 640
0.280 364 265
0.59V 998 8V5

0.983 138 V68

7:4

0.80735 49221

8:7

0.19264 50779

0.238 V83 154

8:5

0.67807 19051

0.817 85• V39

9' 8

0.1699250014

9:7
9:5

Smallstep(minorwholetone)

10:9

Ptolemaic diatonic semitone

16:50

0.09310 94044

0.114 V87 237

Pythagorean
seventh
Pythagorean
majorsixth
Pythagorean
minorthird
Septimalcomma
Commaof Didymus
Pythagoreanmajor third

16:9
27:16
32:27
64:63
81:80
81:64

0.8300749986
0.7548875022
0.2451124978
0.0227200765
0.0179219080
0.3398500029

0.9e6452 7e8
0.908 542 006
0.2e3 649 ee6
0.033 315 950
0.026 e76 645
0.40e 316 808

243:128

0.92481 25036

0.•12

256:243

0.0751874964
0.01629 38073
0.02467 79737
0.00162 81007

0.0V9 •10 7•2

0.024 1V5 06•
0.036 788 050
0.002 991 596

531441:524288

0.0195500087

0.029 948 01•

ß..
ß-.

Great diesis

ß- ß

Pythagorean
diatonicsemitone

Diaskisma
Small diesis
Skisma

Pythagorean
comma

10:7
15:8

128:125
2048:2025
3125:3072
32805:32768

0.36257 00794
0.84799 69066

0.1520030934
0.51457 31728
0.90689 05956

0.03421 57153

'

0.205 769 404

ß-.
ß-.

0.442 630 558
0.V21 409 241

0.19V7•2 97•
0.621 223 317
0.VV7 134 985

0.04e 15e 6e4
0V• 40V

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512

PARRY

metic to a minimumby tabulatingthe data that are
most frequentlyneeded.
One of the greatestneedsof a physicistworkingin
music is the conversion between musical notation and

MOON
TABLEVII. Pythagoreanscale,obtainedby
progressing
upward in fifths.
No.
fifths

p (micro)

p (macro)

For tonic-C
old notation

frequency.Sucha conversion
may be employedalsoby
the musicianor the organ builder who usesbeat frequenciesin tuning.Table III givesthe relationbetween
p and relativefrequencyf/ft. Here f is the absolute
frequencyof the note, and fc is the frequencyof the
nearestC belowthe given note.
A columnof first differences
Ap is alsoincludedfor
aid in interpolation.Ordinary linear interpolationis
generallycorrectto-5 figures;if 8-figureaccuracyis
necessary,secondand third differencesmust be used.
For all ordinary musicalpurposes,5 figuresare sufficient; but for refinedphysicalmeasurements
and in the
computationof other tablesby interpolation,the full
potentialitiesof Table III may be needed.A column
of log•.(f/fc) is includedfor thosewho do not care to
useduodecimals.
It givesall the advantagesof the new
specificationexcept that it doesnot divide the octave
into twelve parts.
Table III is used in conjunctionwith Table I to
obtain absolutefrequencies.For example,what is the
value of p correspondingto f=368.89 hertz? From
Table I, the C belowthe givennotehasa frequencyof

--1 1.4e9
227
3eV--0.5
F[
0

0.000 000 000

0.0

C

1
2

0.702
994802
1.205 769 404

0.7
1.2

D

G Diatonic

3 1.908
542
0061.9AI scale
4
5

2.40e 316 808
2.e12 0re 40V

2.4
2.e

E
B

6 3.614
V84
010 3.6F•l
9 5.321
406
016 5.3 D•
I additio
7
8

4.117 858 812
4.81V 631 414

4.1
4.8

C•
G•

10
11

5.V24 19V 818
6.526 •73 420

5.V
6.5

A•
E•

Chromatic

The difference
is, therefore,somewhat
morethan2/12 of
a temperedsemitone;and referenceto Table VI shows
that the differenceis exactlya commaof Didymus.
It is known that 12 perfectfifths constitutevery
nearlyan integralnumberof octaves.Exactly what is
the discrepancy?
Twelvein the radix-10systemis equal
to 10 in the duodecimalsystem.Thus
0.7O2 994 8O2
X10

f•= 261.626,

7.029 948

O2

so

= 7 octaves+0 semitones+2/12semitone+....

368.89

f/fc=•=

1.4100.

261.626

Referenceto Table VI showsthat the discrepancy
is
exactlya Pythagoreancomma.Thus 12 perfectfifths

From Table III, making useof the fact that the note is
in the third octave,

TABLEVIII. Pythagorean scale
(rearrangementof Table VII).

p = 3.5z469.

Old

The table is usefulalsoin convertingfrom p to f.
Becauseof the wideuseof the equallytemperedscale
in keyboard instruments,this scaleis of considerable
importance.Table IV lists the computedfrequency
ratiosfor eachsemitone,while Table V givesthe absolute frequencies.
As statedpreviously,the valueof p for
notesof the equallytemperedscaleis givenexactlyby
one figure to the right of the dot.
Ever sincethe time of Pythagoras,it hasbeenknown
that harmoniousintervalsare associated
with simple
frequencyratios suchas 3'2, 4' 3, 5' 4. The most useful
frequencyratios in music are collectedin Table VI,
with their differencesexpressedin the new notation.
As an exampleof the useof Table VI, what is the differencein p betweena Ptolemaicsemitoneand a Pythagoreansemitone?From the table,
Ptolemaic

semitone

0.114 V87 237

p

nota-

p

(macro)

tion

(micro)

0.0

C

0.000 000 000

0.1

C•

0.117 858 812

Diatonic

zXp

0.117
858
812
}
0.0V9 ½107½2

0.3

D•

0.321 406 016

0.4

E

0.40e 316 808

0.5

¾

0.469 227 36V

0.6

F•

0.614 V84 010

0.7

G

0.702 994 802

0.8

G•

0.81V 631 414

A

0.908 542 006

0.V

A•

0.V24 19V 818

0. e

B

0. el2 0Ve 40V

1.0

C

1.000 000 000

0.205 769 404
(9' 8 ratio)

0.

0.0V9 •10 7•2
0.0V9 el0 7e2

0.9

intervals

769 404

'8)
0.0V9 el0 7e2

(256:243)

0.117
858
812
}
0.117
858
812
}
0.117
858
812
}

0.205 769 404

0.0V0 d0 7 •2

(9:8)

0.205 769 404

0.0V9 d0 7 •2

0.0V9 el0 7e2

(9:8)

0.205 769 404
(9: 8)

0.0V9 el0 7 e2

Pythagoreansemitone=0.0V9 el0 7•2

0.0V9 el0 7 e0

(256:243)
Difference=0.026

e76 645.

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SPECIFICATION

OF

TABLE IX. Building a Ptolemaic scale.
p

(micro)

p (macro)

MUSICAL

261.6 (Table I)
X0.0025 (Table III)

old notation

0.000 000 000

0.0

C

T+3rd
T+4th
T+5th

0.3V4360 183
0.4•9 227 3•V
0.702 994 802

0.4
0.5
0.7

E
F
G

T+4th+3rd

0.8vl 587 581

0.9

A

513

middle C, this discrepancywould correspondto

For tonic=C

Tonic

PITCH

0.65 hertz or 1.3 beatsper sec.

The above types of computation may be made
directly in the duodecimalsystem, as shown. Or the
computationcan be performedin the ordinary decimal
T+5th+3rd
0.VV7 134 985
0. e B
system,usinglog2(f/fc) and translatingto p as a final
Tq- 5thq- 5th
1.205 769 404
1.2
D
step.7 In either case, we are using a form of base-2
Tq-3rdq-3rd
0.788 700 $46 0.8
G•
logarithms, which automatically takes care of the
octavesand showsthe positionwithin the octaveby the
Tq-4thq-3rdq-3rd 1.085927744 1.1 C• Chromatic quantity to the right of the decimalpoint.
Tq-5thq-3rdq-3rd 1.28•494•48 1.3 D•
additions
Tq-5thq-3rdq-5th
1.5V9 •09 587 1.6 F•
Incidentally, this exampleillustratesan advantageof
the
new system over the Ellis designationin cents2
Tq-5thq-3rdq-5th+3rd
1.992 269 74V 1.V A•
A perfectfifth in the Ellis notation is 701.955cents,and
therefore12 fifths are equal to
Diatonic
scale

12X 701.955- 8423.46 cents.
exceed7 octavesby a Pythagoreancomma. A still
smallerdiscrepancyoccurswith 53 fifths. Then, chang- Here is the answer,but it givesno immediateindication
ing 53 to a duodecimal,
of wherethe new note lies in the octave.Only after we
have
dividedby 1200and subtractedthe quotientfrom
0.702 994 802
8423.46 do we find that

X45

8423.46 cents= 7 octaves+ 23.46 cents.

2d20V•40V
240•31 6808

6. PYTHAGOREAN

AND

PTOLEMAIC

SCALES

27.005 257 48V.

As a final example,considerthe formation of musical
scales.A Pythagoreanscalea can be formed by piling
Therefore,53 perfectfifths make 31 octavesplusa small fifths on top of each other. The procedureis indicated
discrepancyof approximately 5/144 semitone. At in Table VII, startingwith C, movingdownwardto get
F and upward to get the other notes.The valuesof p
TABLE X. Ptolemaic scale
obtained in this way are reassembledin Table VIII,
(rearrangementof Table IX).
which allowseasy comparisonof the Pythagoreanwith
the equitemperedscale.
Old

p

nota-

p

(macro)

tion

(micro)

0.0

C

0.000

Diatonic

Ap

intervals

dominant,etc. (Table IX). The first sevenvaluesof p,

000 000

0.!

C$

0.085 927 744

0.2

D

0.205

0.085
927
744
}
0.085
927
744
}
0.13eV41 880

769 404

0.3

DS

0.28• 494 e48

0.4

E

0.374

360 183

F

0.6

F$

0.5V9 •09 587

G

0.702 994 802

0.8

G$

0.788 700 346

0.9

A

0.8Vl

587 581

0.V

AS

0.992 269 74V

0.•

B

0.VV7 134 985

1.o

c

237

0.4•9 227 3•V

0.7

1.ooo ooo ooo

0.205 769 404
(9:8 ratio)

0.19V 7•2 97 •

0.144 787 237
0.114V87

0.5

A Ptolemaicor "true" scale
ø may be obtainedby
building major triads on the tonic, subdominant,

(10:9)

0.114V87

237

(16:15)

0.0•0
8V2
189
}
0.085
927
744
}
0.085
927
744
}

0.205 769 404

0.114 V87 237

(9' 8)

(10:9)

0.114V87 237

0.205 769 404
(9:8)

0.114V87

0.114V87

237

(16:15)

?The first proposalto uselogsin musicseemsto have beenmade
by Juan Caramuel de Lobkowitz, Nova Musica (Vienna, 1645).
Euler also employed a few logarithms to the base 2: L. Euler,
TentamenNovae TheoriaeMusicae (Petropolis, 1739). Despite
the antiquity of the suggestion,no extensive table of binary
logarithmsseemsto exist exceptG. H. Pohland, Binary Logarithms
(Chicago,1931), and this table is not very convenientfor musical
use.

0.19V 7•2 97 •

0.114 787 237

obtainedin this way, constitutea diatonic scale.Building major triads on these notes producesthe intermediatenoteswhich are usuallydesignatedby sharps.
A continuationupwardgivesdoublesharps,a continua-

237

8 Pythagoras,6th century B.C. The scaleis still usedby players
of string instruments for melodic passages.P. C. Greene, University of Iowa Studies, IV (1937); J. Acoust. Soc. Am. 9, 43
(1937); J. F. Nickerson,J. Acoust.Soc.Am. 21, 593 (1949).
0 Claudius Ptolemy, 2nd century A.D., Harmony, translated
into Latin by John Wallis (Oxford University Press, London,
1680). Other ancient contributors to the theory of musical scales
are Euclid and Aristoxenus.Euclid, "Theory of intervals," given
in Heiberg and Menge, Euclidisoperaomnia (Leipzig, 1916), Vol.
8; H. S. Macran, The Harmonics of Aristoxenus(Oxford University Press,London, 1902); L. Laloy, Aristox•ne de Tarente et la
musiquede l'antiquit• (Pads, 1904).

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514

PARRY

MOON

e

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