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Music by the numbers

By Jim Chen

Kyle Gann, Scale in C-major 

Having recently declared on this page that Color is math, math color — That is all / Ye know on earth, and all ye need to know, I figured I would outline some of the basic relationships between mathematics and music. If only for purposes of personal enrichment, I will summarize the snippets of music theory that can be extracted from Wikipedia's pages on meantone intervals and musical cents and from Kyle Gann's explanation of just intonation.

The following table summarizes the basic mathematical relationships for the C-major scale, beginning with middle C:

Cents in 12edo temperament	Note	Interval	Just intonation ratio	Frequency
0	C	unison	1/1	264
200	D	M2	9/8	297
400	E	M3	5/4	330
500	F	P4	4/3	352
700	G	P5	3/2	396
900	A	M6	5/3	440
1100	B	M7	15/8	495
1200	C	octave	2/1	528

Each cent represents 1200th of the interval within an octave. Since doubling the frequency of a note generates the octave above it, each cent represents 2^1/1200, or 1/17.3 percent.

If you know the frequencies a and b of two notes, the following formula yields the number of cents measuring the interval between them:

n = 1200 log₂ (a/b)

Similarly, if you know the frequency of note b and the number of n cents marking the interval to note a, you can compute the frequency of note a with this formula:

a = b · 2^n/1200

Full post as published by Jurisdynamics on March 03, 2008 (boomark / email).