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A '''US''', or '''utonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | A '''US''', or '''utonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | ||
== Specification == | |||
The n is optional. If not provided, the sequence is open-ended | 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 <span><math>k</math></span> of a USp is: | The formula for length <span><math>k</math></span> of a USp is: | ||
| Line 14: | Line 12: | ||
L(k) = 1 + k⋅p | L(k) = 1 + k⋅p | ||
</math> | </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 number other than 1, if that number is irrational you can produce an ALS, and if rational you can produce a US (which you could also call an ALS if you really wanted to). 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|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|ALS, or arithmetic length sequence]]. | |||
== Examples == | |||
{| class="wikitable" | {| class="wikitable" | ||
| Line 30: | Line 48: | ||
! 8 | ! 8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(28/28) | |(28/28) | ||
|28/25 | |28/25 | ||
| Line 41: | Line 59: | ||
|7/1 | |7/1 | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
|(0) | |(0) | ||
|0.16 | |0.16 | ||
| Line 52: | Line 70: | ||
|2.81 | |2.81 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
|(28/28) | |(28/28) | ||
|25/28 | |25/28 | ||
| Line 79: | Line 97: | ||
! 8 | ! 8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(1/1) | |(1/1) | ||
|4/7 | |4/7 | ||
| Line 90: | Line 108: | ||
|1/7 | |1/7 | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
|(0) | |(0) | ||
| -0.81 | | -0.81 | ||
| Line 101: | Line 119: | ||
| -2.81 | | -2.81 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
|(4/4) | |(4/4) | ||
|7/4 | |7/4 | ||
| Line 113: | Line 131: | ||
|} | |} | ||
[[Category:Utonality]] | [[Category:Utonality]] | ||
[[Category:Subharmonic]] | [[Category:Subharmonic]] | ||
[[Category:Subharmonic series]] | [[Category:Subharmonic series]] | ||
[[Category:Xenharmonic series]] | |||
Latest revision as of 20:37, 19 October 2023
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]\displaystyle{ k }[/math] of a USp is:
[math]\displaystyle{ 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 number other than 1, if that number is irrational you can produce an ALS, and if rational you can produce a US (which you could also call an ALS if you really wanted to). 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, ratio) | (28/28) | 28/25 | 14/11 | 28/19 | 7/4 | 28/13 | 14/5 | 4/1 | 7/1 |
| pitch (log₂f, octaves) | (0) | 0.16 | 0.35 | 0.56 | 0.81 | 1.11 | 1.49 | 2.00 | 2.81 |
| length (1/f, ratio) | (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, ratio) | (1/1) | 4/7 | 2/5 | 4/13 | 1/4 | 4/19 | 2/11 | 4/25 | 1/7 |
| pitch (log₂f, octaves) | (0) | -0.81 | -1.32 | -1.70 | -2.00 | -2.25 | -2.46 | -2.64 | -2.81 |
| length (1/f, ratio) | (4/4) | 7/4 | 10/4 | 13/4 | 16/4 | 19/4 | 22/4 | 25/4 | 28/4 |