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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: | |||
<math> | |||
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|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" | ||
|+example: 8-US(3/4) | |+example: 8-US(3/4) | ||
''(arranged so that the pitches are in ascending order and still begin on 1/1)'' | |||
|- | |- | ||
! quantity | ! quantity | ||
| Line 21: | Line 48: | ||
! 8 | ! 8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(28/28) | |(28/28) | ||
|28/25 | |28/25 | ||
| | |14/11 | ||
|28/19 | |28/19 | ||
| | |7/4 | ||
|28/13 | |28/13 | ||
| | |14/5 | ||
| | |4/1 | ||
| | |7/1 | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
|(0) | |(0) | ||
|0.16 | |0.16 | ||
| Line 43: | Line 70: | ||
|2.81 | |2.81 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
|(28/28) | |(28/28) | ||
|25/28 | |25/28 | ||
| Line 55: | Line 82: | ||
|} | |} | ||
{| class="wikitable" | |||
|+example: 8-US(3/4) | |||
''(descending pitches)'' | |||
|- | |||
! 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 (log₂''f'', 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 | |||
|} | |||
[[Category:Utonality]] | [[Category:Utonality]] | ||
[[Category:Subharmonic]] | [[Category:Subharmonic]] | ||
[[Category:Subharmonic series]] | [[Category:Subharmonic series]] | ||
[[Category:Xenharmonic series]] | |||