UD
A UD, or utonal division, is a kind of arithmetic and harmonotonic tuning.
Specification
Its full specification is n-UDp: n utonal divisions of rational interval p.
Formula
To find the steps for an n-UDp, begin by recognizing that while the ratio between your root pitch's string length and the length you would pluck to get the lowest pitch is [math]p[/math] (or [math]\frac p1[/math]), if you are going to move arithmetically (by repeated addition) from [math]1[/math] to [math]p[/math], then the difference in string length that you need to cover is not actually [math]p[/math], but only [math]p - 1[/math]. And because you are dividing it into [math]n[/math] parts, each step will have a size of [math]\frac{p-1}{n}[/math]. So, the formula for the length of step [math]k[/math] of an n-UDp is:
[math] L(k) = 1 + (\frac kn)(p-1) [/math]
This way, when [math]k[/math] is [math]0[/math], [math]L(k)[/math] is simply [math]1[/math]. And when [math]k[/math] is [math]n[/math], [math]L(k)[/math] is simply [math]1 + (p-1) = 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
Vs ED
It is possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by length. You will have 12-ELDO.
Vs under-n series
An n-UDO is equivalent to the nth undertone mode, or under-n scale.
Vs ELD
The only difference between n-UDp and n-ELDp (equal length division) is that the p for UD must be rational, while the p for an ELD is probably irrational.
Vs US
A UD will be equivalent to some US (utonal sequence). E.g. 8-UD7 = 8-US3/4, because to get from 1 to 7 you cover 6 undertones, and 6 divided by 8 is 3/4.
Vs EDL
An n-EDL is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).
Examples
| quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| frequency (f, ratio) | (1/1) | 8/7 | 4/3 | 8/5 | 2/1 |
| pitch (log₂f, octaves) | (0) | 0.19 | 0.42 | 0.68 | 1.00 |
| length (1/f, ratio) | (8/8) | 7/8 | 6/8 | 5/8 | 4/8 |
| quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| frequency (f, ratio) | (1/1) | 4/5 | 2/3 | 4/7 | 1/2 |
| pitch (log₂f, octaves) | (0) | -0.32 | -0.58 | -0.81 | -1.00 |
| length (1/f, ratio) | (4/4) | 5/4 | 6/4 | 7/4 | 8/4 |