WORKING WITH CENTS: A SURVEY

Fred Lieberman

 I. II. III. IV. V. VI.

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 (1955:13-14) 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 currently-used 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; 1965:25-26).

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 hertz--other 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 seven-place table is generally correct to the nearest 0.01 cent (see Example 7). Husmann's five-place table gives results correct to the nearest cent; his tables cover the range from 0 to 1009 hertz in 1-hertz 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 0.1-hertz 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 equal-tempered scale based on A440 in the range 32.703-3951.1 hertz in 1-cent 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 general-purpose cents table.

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V. OTHER METHODS

Two other methods remain to be mentioned, Markus Reiner's music rule (1949; Gerson-Kiwi 1953) and Fritz Bose's nomogram (1952). Both are essentially slide-rule-like 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 264-528 hertz, of the nomogram 200-600 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: 205-208.

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. 446-451.

Gerson-Kiwi, Edith

1953 Towards an exact transcription of tone-relations. Acta Musicologica 25:80-87.

Hornbostel, Erich M. von

1921 Eine Tafel zur Logarithmischen Darstellung von Zählenverhaltnissen. Zeitschrift für Physik 6:29-34.

Husmann, Heinrich

1951 Fünf- und Siebenstellige Centstafeln zur Berechnung musikalischer Intervalle. Leiden: E. J. Brill. Ethno-Musicologica, Vol. II.

1952 Cent in MGG 2:965-966.

Kunst, Jaap

1959 Ethnomusicology. 3rd ed. The Hague: Nijhoff.

Kuttner, Fritz A.

1953 Nochmals: die Steinzeit Lithophone von Annam. Die Musikforschung 6:1-8.

Lieberman, Fred, and Diane Larrabee

1970 A table of cents for frequencies from 0 to 4000.9 Hertz. Providence, R.I.: Brown University computer print-out.

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):441-445.

Sachs, Curt

1943 The rise of music in the ancient world. New York: Norton.

1955 Our musical heritage. 2nd ed. Englewood Cliffs, N.J.: Prentice-Hall.

1965 The wellsprings of music. New York: McGraw-Hill.

Young, R. W.

1952 A table relating frequency to cents. Elkhart, Indiana: Conn.

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