Frequency temperament

Arithmetic temperaments are the arithmetic counterpart to regular temperaments. Whereas regular temperaments are created by reducing integer powers of a generator, an arithmetic temperament is created by reducing integer multiples of a generator. The n-th interval in an arithmetic temperament prior to octave-reduction is given by n*g + 1, where g is the generator.

For example, this is the interval chain of an arithmetic temperament with a generator of 0.29 and period 2/1:

1+0.29 = 1.29
1+2*0.29 = 1.58
1+3*0.29 = 1.87
1+4*0.29 = 2.16 -> 1.08
1+5*0.29 = 2.45 -> 1.225
1+6*0.29 = 2.74 -> 1.37
...

Arithmetic temperaments also temper out commas, but these commas represent differences between intervals rather than ratios between them. For example, equating 9/7 with 13/10 in an arithmetic temperament tempers out the "arithmetic comma" (13/10)-(9/7) = 1/70.

List of arithmetic temperaments

• Sqrttwo