US: Difference between revisions

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-An '''US''', or '''utonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
+A '''US''', or '''utonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-A US is a specific (rational) type of ALS.
+== Specification ==
-(n-)USp: (n pitches of a) utonal sequence adding by p
+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.
-The same principles that were just described for frequency are also possible for length: by varying the undertone series step size to some rational number you can produce a utonal sequence (US), and varying it to an irrational number you can produce an arithmetic length sequence (ALS). 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.
+== 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"
+|+example: 8-US(3/4)
+''(arranged so that the pitches are in ascending order and still begin on 1/1)''
+|-
+! 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 (log₂''f'', 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
+|}
+{| 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:Subharmonic]]
+[[Category:Subharmonic series‏‎]]
+[[Category:Xenharmonic series]]