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A '''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.


The full specification of a US is (n-)USp: (n pitches of a) utonal sequence adding by p.
== Specification ==


A US is a specific (rational) type of [[ALS|ALS, or arithmetic length sequence]]. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and varying it to an irrational number you can produce an 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.
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: 9-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
! (0)
! 1
! 1
! 2
! 2
Line 17: Line 47:
! 7
! 7
! 8
! 8
! (9)
|-
|-
! frequency
! frequency (''f'', ratio)
|28/28
|(28/28)
|28/25
|28/25
|28/22
|14/11
|28/19
|28/19
|28/16
|7/4
|28/13
|28/13
|28/10
|14/5
|28/7
|4/1
|(28/4)
|7/1
|-
|-
! pitch
! pitch (log₂''f'', octaves)
|0.00
|(0)
|0.16
|0.16
|0.35
|0.35
Line 39: Line 68:
|1.49
|1.49
|2.00
|2.00
|(2.81)
|2.81
|-
|-
! length
! length (1/''f'', ratio)
|28/28
|(28/28)
|25/28
|25/28
|22/28
|22/28
Line 50: Line 79:
|10/28
|10/28
|7/28
|7/28
|(4/28)
|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]]

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

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 (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
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 (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
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