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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" | ||
| Line 22: | Line 48: | ||
! 8 | ! 8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(28/28) | |(28/28) | ||
|28/25 | |28/25 | ||
| Line 33: | Line 59: | ||
|7/1 | |7/1 | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
|(0) | |(0) | ||
|0.16 | |0.16 | ||
| Line 44: | Line 70: | ||
|2.81 | |2.81 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
|(28/28) | |(28/28) | ||
|25/28 | |25/28 | ||
| Line 71: | Line 97: | ||
! 8 | ! 8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(1/1) | |(1/1) | ||
|4/7 | |4/7 | ||
| Line 82: | Line 108: | ||
|1/7 | |1/7 | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
|(0) | |(0) | ||
| -0.81 | | -0.81 | ||
| Line 93: | Line 119: | ||
| -2.81 | | -2.81 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
|(4/4) | |(4/4) | ||
|7/4 | |7/4 | ||
| Line 105: | Line 131: | ||
|} | |} | ||
[[Category:Utonality]] | [[Category:Utonality]] | ||
[[Category:Subharmonic]] | [[Category:Subharmonic]] | ||
[[Category:Subharmonic series]] | [[Category:Subharmonic series]] | ||
[[Category:Xenharmonic series]] | |||