US
A US, or utonal sequence, is a kind of arithmetic and harmonotonic tuning.
The full specification of a US is (n-)USp: (n pitches of a) utonal sequence adding by p.
The n is optional. If not provided, the sequence is open-ended. By specifying n, your sequence will be equivalent to some UD (utonal division). E.g. 8-US3/4 = 8-UD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.
A US is a specific (rational) type of ALS, or arithmetic length sequence. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and varying it to an irrational number you can produce an ALS. In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.
| quantity | (0) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| frequency (f) | (28/28) | 28/25 | 14/11 | 28/19 | 7/4 | 28/13 | 14/5 | 4/1 | 7/1 |
| pitch (log₂f) | (0) | 0.16 | 0.35 | 0.56 | 0.81 | 1.11 | 1.49 | 2.00 | 2.81 |
| length (1/f) | (28/28) | 25/28 | 22/28 | 19/28 | 16/28 | 13/28 | 10/28 | 7/28 | 4/28 |
| quantity | (0) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| frequency (f) | (1/1) | 4/7 | 2/5 | 4/13 | 1/4 | 4/19 | 2/11 | 4/25 | 1/7 |
| pitch (log₂f) | (0) | -0.81 | -1.32 | -1.70 | -2.00 | -2.25 | -2.46 | -2.64 | -2.81 |
| length (1/f) | (4/4) | 7/4 | 10/4 | 13/4 | 16/4 | 19/4 | 22/4 | 25/4 | 28/4 |