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.