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]\displaystyle{ p }[/math] (or [math]\displaystyle{ \frac p1 }[/math]), if you are going to move arithmetically (by repeated addition) from [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ p }[/math], then the difference in string length that you need to cover is not actually [math]\displaystyle{ p }[/math], but only [math]\displaystyle{ p - 1 }[/math]. And because you are dividing it into [math]\displaystyle{ n }[/math] parts, each step will have a size of [math]\displaystyle{ \frac{p-1}{n} }[/math]. So, the formula for the length of step [math]\displaystyle{ k }[/math] of an n-UDp is:

[math]\displaystyle{ L(k) = 1 + (\frac kn)(p-1) }[/math]

This way, when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ 0 }[/math], [math]\displaystyle{ L(k) }[/math] is simply [math]\displaystyle{ 1 }[/math]. And when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ n }[/math], [math]\displaystyle{ L(k) }[/math] is simply [math]\displaystyle{ 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

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, 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

example: 4-UDO = 4th undertone mode (descending pitches)

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