# 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

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 |