# US

A US, or utonal sequence, is a kind of arithmetic and harmonotonic tuning.

## Specification

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.

## Formula

The formula for length [math]k[/math] of a USp is:

[math] L(k) = 1 + k⋅p [/math]

## Tip about tunings based on length

Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.

## Relationship to other tunings

### As shifted undertone series

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.

### Vs. UD

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.

### Vs. ALS

A US is the rational version of ALS, or arithmetic length sequence.

## Examples

example: 8-US(3/4) (arranged so that the pitches are in ascending order and still begin on 1/1)
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
example: 8-US(3/4) (descending pitches)
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