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A '''UD''', or '''utonal division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
A '''UD''', or '''utonal division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.


A UD is a specific (rational) type of ELD.
== Specification ==


undertone mode, or under-n scale (equivalent to n-UDO)
Its full specification is n-UDp: n utonal divisions of rational interval p.


n-UDp: n utonal divisions of 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 <span><math>p</math></span> (or <span><math>\frac p1</math></span>), if you are going to move arithmetically (by repeated addition) from <span><math>1</math></span> to <span><math>p</math></span>, then the difference in string length that you need to cover is not actually <span><math>p</math></span>, but only <span><math>p - 1</math></span>. And because you are dividing it into <span><math>n</math></span> parts, each step will have a size of <span><math>\frac{p-1}{n}</math></span>. So, the formula for the length of step <span><math>k</math></span> of an n-UDp is:
 
<math>
L(k) = 1 + (\frac kn)(p-1)
</math>
 
This way, when <span><math>k</math></span> is <span><math>0</math></span>, <span><math>L(k)</math></span> is simply <span><math>1</math></span>. And when <span><math>k</math></span> is <span><math>n</math></span>, <span><math>L(k)</math></span> is simply <span><math>1 + (p-1) = p</math></span>.
 
== 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 [[Overtone_scale#Next_Steps|undertone mode, or under-n scale]].
 
=== Vs ELD ===
 
The only difference between n-UDp and [[ELD|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|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 [[EDL|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).
 
== Examples ==


{| class="wikitable"
{| class="wikitable"
|+example: 4-UDO = 4th undertone mode
|+example: 4-UDO = 4th undertone mode ''(arranged so that the pitches are in ascending order and still begin on 1/1)''
|-
|-
! quantity
! quantity
! (0)
! 1
! 1
! 2
! 2
! 3
! 3
! 4
! 4
! 5
! 6
! 7
! 8
! 9
|-
|-
! frequency
! frequency (''f'', ratio)
|(1/1)
|8/7
|4/3
|8/5
|2/1
|-
! pitch (log₂''f'', octaves)
|(0)
|0.19
|0.42
|0.68
|1.00
|1.00
|1.15
|1.31
|1.46
|1.62
|1.87
|2.12
|2.37
|2.62
|-
|-
! pitch
! length (1/''f'', ratio)
|0.00
|(8/8)
|0.21
|7/8
|0.39
|6/8
|0.55
|5/8
|0.69
|4/8
|0.90
|}
|1.08
 
|1.24
{| class="wikitable"
|1.39
|+example: 4-UDO = 4th undertone mode ''(descending pitches)''
|-
|-
! length
! quantity
|1.00
! (0)
|0.87
! 1
|0.76
! 2
|0.68
! 3
|0.62
! 4
|0.54
|-
|0.47
! frequency (''f'', ratio)
|0.42
|(1/1)
|0.38
|4/5
|2/3
|4/7
|1/2
|-
! pitch (log₂''f'', octaves)
|(0)
| -0.32
| -0.58
| -0.81
| -1.00
|-
! length (1/''f'', ratio)
|(4/4)
|5/4
|6/4
|7/4
|8/4
|}
|}
[[Category:Utonality]]
[[Category:Subharmonic]]
[[Category:Subharmonic series‏‎]]
Retrieved from "https://en.xen.wiki/w/UD"