UD

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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 is rational, while the p for ELD is 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

example: 4-UDO = 4th undertone mode (arranged so that the pitches are in ascending order and still begin on 1/1)
quantity (0) 1 2 3 4
frequency (f) (1/1) 8/7 4/3 8/5 2/1
pitch (log₂f) (0) 0.19 0.42 0.68 1.00
length (1/f) (8/8) 7/8 6/8 5/8 4/8
example: 4-UDO = 4th undertone mode (descending pitches)
quantity (0) 1 2 3 4
frequency (f) (1/1) 4/5 2/3 4/7 1/2
pitch (log₂f) (0) -0.32 -0.58 -0.81 -1.00
length (1/f) (4/4) 5/4 6/4 7/4 8/4