User:Cmloegcmluin/APS
An APS, or arithmetic pitch sequence, is a kind of arithmetic and harmonotonic tuning. It can also be called an equal multiplication.
Specification
Its full specification is (n-)APS-p: (n pitches of an) arithmetic pitch sequence adding by interval p. The n is optional. If not provided, the sequence is open-ended.
Formula
The pitch of the k-th step of an APS-p is quite simply k⋅p for a pitch (log-frequency) quantity p.
Relationship to other tunings
Vs. rank-1 temperaments
By applying a mapping, APS-p becomes an equal temperament with generator p.
Vs. EPD
If the n is not specified, an APS will be equivalent to an equal pitch division (EPD). Specifically, n-EPD-p = APS(p/n) for a log-frequency quantity p. For example, 12-EPD1200¢ = APS(1200¢/12) = APS100¢.
Vs. AS
The only difference between an APS and an AS (ambitonal sequence) is that the p for an AS must be rational.
Examples
| Quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Frequency (f) | (1) | 1.19 | 1.41 | 1.68 | 2 |
| Pitch (log₂f) | (0/4) | 1/4 | 2/4 | 3/4 | 4/4 |
| Length (1/f) | (1) | 0.84 | 0.71 | 0.59 | 0.5 |