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Using cents

When using cents to specify scales and intervals, the most common requirements are to convert cents to a ratio, or a ratio to cents, and these can be done using two standard formulae. (The log and antilog functions are ordinary base 10 logarithms.) See the following for an explanation of how these relationships are derived.

Every note used in music has a fundamental frequency that is usually the loudest of a series of harmonics related to it. This frequency can be stated in Hertz (cycles-per-second) when required, but because music is formed by the relationship between notes, rather than by their absolute frequencies, either the ratio between two notes, or between a note and its keynote, is almost always of greater interest than their frequencies. Integer ratios accurately describe simple intervals, but cannot represent more dissonant intervals. For this and other reasons, various methods of describing frequency ratios have been devised over the years. The measure most commonly used today is the cent, and uses logarithms to base two. This will likely seem somewhat daunting to non-mathematicians, but by understanding its basis and following a few simple rules, cents soon become as familiar as any other measurement, even if the precise details remain somewhat obscure.

Table 1   ARITHMETIC & GEOMETRIC INTERVALS
IntervalArithmeticGeometric
Keynote100Hz100Hz
1st100Hz + 10Hz = 110Hz100Hz x 1.1 = 110Hz
2nd110Hz + 10Hz = 120Hz110Hz x 1.1 = 121Hz
3rd120Hz + 10Hz = 130Hz121Hz x 1.1 = 133.1Hz
4th130Hz + 10Hz = 140Hz133.1Hz x 1.1 = 146.4Hz
5th140Hz + 10Hz = 150Hz146.4Hz x 1.1 = 161.1Hz
nth nth = 100 + (n x 10)nth = 100 x 1.1n
10th190Hz + 10Hz = 200Hz235.8Hz x 1.1 = 259.4Hz

Each octave is divided into 1,200 cents on the assumption that there are twelve equal semitones per octave, and one hundred equal cents per semitone. The word "equal" here denotes geometric rather than arithmetic equality - that is, a common ratio rather than a common difference, as Table 1 makes clear. Consider an octave between 100Hz and 200Hz in which there are ten arithmetically equal divisions, each of 10Hz, and compare it with the octave between 200Hz and 400Hz in which twenty such divisions occur. It is obvious that an arithmetic sequence will have an increasing number of divisions in each higher octave.

The case with geometric divisions is different. If we choose a ratio of 1.1, ten such divisions above 100Hz gives a frequency of 259.4Hz, and ten more divisions will produce a frequency of

259.4 x ( 259.4 / 100 ) = 672.7Hz

The ratio that gives twelve equal divisions in an octave is the twelfth root of two:

2-12=1.05946...
 =1 semitone
 =s
Or s x s x s x s x s x s x s x s x s x s x s x s=2

Similarly, if we now divide each s into one hundred geometrically equal divisions, we have:

1.05946-100=1.00057779...
or 2-1200=1.00057779...
 =1 cent

A few moments spent with pencil, paper and calculator will clarify these facts. The notes of the equal-tempered scale are then 100 cents apart, as Table 2 shows reading bottom to top:

Table 2   NOTES OF THE ET SCALE
Note¢ to previous¢ to Root
C'1001200
:::
E100400
Eb100300
D100200
C#100100
C00

Finally, as a useful reference, here are the most common musical intervals expressed as ratios and as cents. The examples given are only approximate on an equal-tempered instrument, especially the tones and semitones, but serve to illustrate the musical effect. The last column (Equal Tempered Cents) give the closest available interval on the equal-tempered scale.

Table 3   MUSICAL INTERVALS
Interval Ratio Cents ExampleET Cents
Unison1:10C - C0
Octave2:11200C - C'1200
Perfect Fifth3:2702C - G700
Perfect Fourth4:3498 C - F500
Major Sixth5:3884C - A900
Major Third5:4386C - E400
Minor Seventh7:4969C - Bb1000
Major Tone9:8204( C - D )200
Minor Tone10:9182( C - D )200
Major Semitone16:15112( C - C# )100
Minor Semitone17:16105( C - C# )100

From this can be seen the usefulness of cents in appreciating the relative size of intervals, something difficult or impossible to ascertain from a ratio. It is also obvious that neither the equal-tempered semitone of 100 cents nor the equal-tempered tone of 200 cents bear any simple relationship to the natural intervals in the Chord of Nature.

- Taken from Appendix K of EarthSong