WORKING WITH CENTS: A SURVEY
Fred Lieberman
TABLE OF CONTENTS
I.
INTRODUCTION
The cent, defined by Alexander Ellis as 0.01 of an equal
tempered semitone, has proven to be a useful tool for
comparing musical intervals. Information on the mathematics
and history of logarithmic representations of frequency
ratios, and on applications of the cents system, is readily
available and need not be reviewed here (Apel 1969; Husmann
1952).
The purpose of the present article is to compare the most
common methods for the calculation of cents, and determine
their relative accuracy and convenience. The necessity for
this became apparent when, in the course of preparing a new
computer generated cents table (Lieberman and Larrabee
1970), I noticed errors of significant magnitude in
influential and widely disseminated ethnomusicological
sources.
For most musical purposes it is sufficient to state
figures to the nearest whole cent, rounding off (not simply
dropping) fractions. It should be noted that in calculations
involving a sequence of operations (for instance in
calculating a scale or cyclic tuning system) what begin as
small errors can easily cumulate into much larger ones.
Fritz A. Kuttner found that inconsistencies on the order of
3 cents ". . . are much too great to furnish useful
foundations for the interpretation of tonal systems."
(1953:3)
Therefore it is not only sufficient but also necessary to
state cents accurately to the nearest whole cent, and only
those methods which produce this degree of accuracy should
be employed thus insuring that published results can be
confidently used by future researchers. The following survey
evaluates accuracy and convenience for various cents
calculation methods; the reader will then be able to choose
the one most suited to his particular need. As an example
for comparison a single arbitrarily chosen frequency ratio
will be converted into cents. The ratio x:y will be assumed
to be 756:546; the correct figure for this ratio, to the
nearest 0.01 cent, is 563.38 C.
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II.
ARITHMETIC APPROXIMATION
The arithmetic approximation was described by Ellis in
some detail (1954:447) and later reproduced several times
(Kunst 1959; Sachs 1943). If the two frequencies are
represented by the symbols x and y (and are less than an
octave apart) the Ellis method may be stated in the
following formulas:
a) if the intenal is less than a fourth (3x < 4y),
C = {[3477 X (x  y)]
/ (x + y); (1)
b) if the intenal is greater than a fourth but less than
a fifth (3x > 4y and 2x < 3y),
C = 4 [3477 X (3x 
4y)] (3x + 4y)} + 498; (2)
c) if the internal is greater than a fifth (2x > 3y),
C= {[3477 X (2x 
3y)] / (2x + 3y)} + 702. (3)
Example 1.
Step 1. Test interval size:
x:y = 756:546
2x:3y = 1512:1638
3x:4y = 2268:2184
Step 2. Since 3x > 4y, use formula 2:
C = {[3477 X (2268  2184)] / (2268 + 2184)} + 498
= [(3477 X 84) / 4452] + 498
= (292068 / 4452) + 498
= 65.60 + 498
= 563.60, ans.
The result of arithmetic approximation as shown in
Example 1, though deviating by only 0.22 cents from the
correct figure, will round off incorrectly to 564 cents.
Sachs (1955:1314) gives a ''simplified'' version, which
merely employs formula l above, disregarding corrections for
intenals larger than a fourth. Following is the same problem
worked according to the "simplified" method.
Example 2.
C = [3477 X (756  546)] / (756 + 546)
= 560.81, ans.
This result deviates by 2.57 cents from the correct
value, but will round to a difference of 2 cents. Sachs'
simplification can be disqualified on grounds of inaccuracy;
it was, nevertheless, recommended by Kunst (1959:9).
Arithmetic approximation, when used in its complete form,
will no doubt give results +/ 1 cent as stated by Ellis.
However it is much too cumbersome to bother with, demanding
two preliminary test multiplications followed by a series of
additions, subtractions, multiplications, and divisions.
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III. LOG
TABLE CONVERSIONS
The logarithm of a frequency ratio x:y can be expressed
as the difference of the logs of the two frequencies:
log i =
log x  log y (4)
Log i can be converted into cents either with special
tables or by multiplying by a scaling factor k:
C = k
(log i) (5)
K may be determined by starting from the assumption that
the octave 2:1 has exactly 1200 cents (this is, of course,
the definition of a cent). From formula 4,
log i =
log 2  log 1;
since, by the definition of logarithms log 1 is always
equal to zero,
log i = log 2  0
= log 2
Substituting the known values into formula 5,
1200 = k log 2.
Solving for k:
k = 1200 / log 2.
Applying formula 6 in the three currentlyused systems of
logs, we find that for common logarithms (base 10), k =
3986.3; for natural logs, k = 1731.2; for base 2 logs, k =
1200.
Example 3 shows how the problem is figured using a five
place table of common logs (Nielsen 1943) and the scaling
factor k = 3986.3.
Example 3.
log i = log x  log y = log 756  log 546
= 2.87852  2.73719
= .14133
C = 3986.3 X .14133
= 563.38. . ., ans.
This method is precise and relatively easier to figure
than arithmetic approximation, but demands multiplication by
the scaling factor. To avoid this multiplication, several
tables have been published to convert log i into cents
directly. Example 4 shows conversion using the table
published by Sachs (1955:13; 1965:2526).
Example 4.
log i =

.14133


(from Ex. 3)


.12500 = 500

C

(from table)


.01633


(remainder)


.01510 = + 60

C

(from table)


.00123


(remainder)


.00125 = + 5

C

(from table)


(.00002)


(remainder)


565

C

(answer)

Here the rounded inaccuracy is 2 cents, and the one long
multiplication has been replaced by a series of
subtractions. Ellis's table, reproduced by Kunst (1959:6)
gives a result considerably more accurate than Sachs' but
not in any significant way simpler than the method of
Example 3 (see Example 5).
Example 5.
log i =

.14133


(from Ex. 3)


.12543 = 500

C

(from table)


.01590


(remainder)


.01505 = + 60

C

(from table)


.00085


(remainder)


.00075 = + 3

C

(from table)


.00010


(remainder)


.00010 = + .4

C

(from table)


.00000




563.4

C

(answer)

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IV. CENTS
TABLES
Cents tables are essentially log tables in which the log
for each number has already been converted into a cents
figure by the scaling factor k. Thus the two numbers looked
up in the table are (k log x) and (k log y), and a simple
subtraction gives the answer (k log i) directly in cents:
C=k log x  k log y
= k (log x  log y)
= k log i.
Of the four cents tables currently available, the most
widely known is that of Hornbostel (1921; Kunst 1959); it is
also the least accurate, and most restricted in range (340 
809 hertzother frequencies must be brought into this range
by octave displacement). From Example 6 the simplicity of
this cents table is obvious, but the result is in error by
1.38 cents.
Example 6.
C= 1382  820 = 562
In fact 59% of Hornbostel's cents figures are incorrect
to the nearest cent, and 13% deviate from 1.0 to 1.5 cents.
Use of this table, then, for any series of calculations may
lead to significant error.
The tables published by Husmann (1951) are much more
satisfactory; the sevenplace table is generally correct to
the nearest 0.01 cent (see Example 7). Husmann's fiveplace
table gives results correct to the nearest cent; his tables
cover the range from 0 to 1009 hertz in 1hertz increments.
Example 7.
C = 1147469  1091131 = 563.38 (decimal point added)
The table recently prepared at Brown University
(Lieberman and Larrabee 1970) differs from Husmann's
primarily in size. By using a computer to generate the table
from the formula
C = 1200 X log2 n,
it was simple and practical to cover the range of
frequencies from 0.0 to 4000.9 hertz in 0.1hertz
increments, with the assurance of accuracy to any desired
degree. Thus there are 40,010 entries on 81 pages in our
table as compared to 1,010 on 2 pages in Husmann's. Example
7 also illustrates the use of this table.
R. W. Young's is a special purpose table (1952), quite
accurate, arranged so as to give frequencies corresponding
to cents deviations from steps of the equaltempered scale
based on A440 in the range 32.7033951.1 hertz in 1cent
increments (frequencies are figured to five significant
places). This is very convenient for working with data
obtained in stroboconn measurements, which take the form
A+40C,F  27 C, etc.
It can, however, be inconvenient for other kinds of data,
and can most efficiently be used in conjunction with a
generalpurpose cents table.
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V. OTHER
METHODS
Two other methods remain to be mentioned, Markus Reiner's
music rule (1949; GersonKiwi 1953) and Fritz Bose's
nomogram (1952). Both are essentially sliderulelike
devices in which a linear scale (cents) is juxtaposed with
an exponential scale (frequency); cents can then be read off
directly for any pair of frequencies. These methods are
handy for quick approximate figuring (for example, in
comparing a large number of scales), but they cannot
approach 1 cent accuracy, and even 2 cent accuracy might be
difficult to attain. The direct frequency range of the music
rule is 264528 hertz, of the nomogram 200600 hertz.
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VI.
CONCLUSIONS
Of the various methods for doing problems involving
cents, Husmann's tables can be recommended as the best
combination of accuracy and simplicity. Lieberman and
Larrabee's table is available for exceptionally precise and
detailed work, though probably unwieldy for everyday use.
Particularly to be avoided are Hornbostel's cents table
and Sachs' log conversion tables. Note that two scholars
correctly figuring the sample problem using these two
methods will differ in their results by a full 3 cents (Ex.
4, 565 C; Ex. 6, 562 C).
Though the deviations from correct figures caused by
inadequate methods may seem insignificant to some readers,
accuracy can be attained with very little effort, since
reliable and simple tools are readily at hand.
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REFERENCES
CITED
Apel, Willi, ed.
1969 Harvard dictionary of music. 2nd ed.
Cambridge: Harvard University Press. p. 420.
Bose, Fritz
1952 Ein Hilfsmittel zur Bestimmung der
Schrittgrösse beliebiger Intervalle. Die Musikforschung
5: 205208.
Ellis, Alexander
1954 On the calculation of cents from
interval ratios. In Hermann Helmholz, On the sensations of
tone. 2nd ed., reprint. New York: Dover. Appendix XX,
Section C, pp. 446451.
GersonKiwi, Edith
1953 Towards an exact transcription of
tonerelations. Acta Musicologica 25:8087.
Hornbostel, Erich M. von
1921 Eine Tafel zur Logarithmischen
Darstellung von Zählenverhaltnissen. Zeitschrift
für Physik 6:2934.
Husmann, Heinrich
1951 Fünf und Siebenstellige
Centstafeln zur Berechnung musikalischer Intervalle. Leiden:
E. J. Brill. EthnoMusicologica, Vol. II.
1952 Cent in MGG 2:965966.
Kunst, Jaap
1959 Ethnomusicology. 3rd ed. The Hague:
Nijhoff.
Kuttner, Fritz A.
1953 Nochmals: die Steinzeit Lithophone von
Annam. Die Musikforschung 6:18.
Lieberman, Fred, and Diane Larrabee
1970 A table of cents for frequencies from 0
to 4000.9 Hertz. Providence, R.I.: Brown University computer
printout.
Nielsen, Kaj L.
1943 Logarithmic and trigonometric tables to
five places. New York: Barnes and Noble. College Outline
Series, #44.
Reiner, Markus
1949 The music rule. Experientia
5(11):441445.
Sachs, Curt
1943 The rise of music in the ancient world.
New York: Norton.
1955 Our musical heritage. 2nd ed. Englewood Cliffs,
N.J.: PrenticeHall.
1965 The wellsprings of music. New York: McGrawHill.
Young, R. W.
1952 A table relating frequency to cents.
Elkhart, Indiana: Conn.
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