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An '''APS''', or '''arithmetic pitch sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
{{Editable user page}}
An '''APS''', or '''arithmetic pitch sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] [[tuning]]. It can also be called an '''equal multiplication'''.


(n-)APSp: (n pitches of an) arithmetic pitch sequence adding by p (equivalent to rank-1 temperament with generator p)
== Specification ==


An AS is a specific (rational) type of APS.
Its full specification is (''n''-)APS-''p'': (''n'' pitches of an) arithmetic pitch sequence adding by interval ''p''.
 
'''Note''':
* The ''n'' is optional. If not provided, the sequence is open-ended.
* The ''p'' can be dimensionless, in which case it refers to an interval by its [[frequency ratio]]. It can also take a unit proportional to [[octave]]s, in which case it refers to an interval by its pitch relation.
 
== Formula ==
 
The pitch of ''k'' steps of APS-''p'' is quite simply ''k''⋅''p'' for a pitch (log-frequency) quantity ''p''.
 
== Relationship to other tunings ==
 
=== Vs. rank-1 temperaments ===
 
By applying a [[mapping]], APS-''p'' becomes an [[equal temperament]] with generator ''p''.
 
=== Vs. EPD ===
 
If the ''n'' is not specified, an APS will be equivalent to an [[EPD|equal pitch division (EPD)]]. Specifically, ''n''-EPD-''p'' = APS(''p''/''n'') for a pitch quantity ''p''. For example, 12-EPD1200¢ = APS(1200¢/12) = APS100¢.
 
=== Vs. AS ===
 
The only difference between an APS and an [[AS|AS (ambitonal sequence)]] is that the ''p'' for an AS must be rational.
 
== Examples ==


{| class="wikitable"
{| class="wikitable"
|+example:
|+Example: APS⁴√2 ≈ APS1.189 = 4-EDO = rank-1 temperament w/ generator 300¢ = equal multiplication of 300¢
|-
|-
! quantity
! Quantity
! (0)
! 1
! 1
! 2
! 2
! 3
! 3
! 4
! 4
! 5
! 6
! 7
! 8
! 9
|-
|-
! frequency
! frequency (''f'', ratio)
|
| (1)
|
| 1.19
|
| 1.41
|
| 1.68
|
| 2
|
|
|
|
|-
|-
! pitch
! length (1/''f'', ratio)
|
| (0/4)
|
| 1/4
|
| 2/4
|
| 3/4
|
| 4/4
|
|
|
|
|-
|-
! length
! Length (1/''f'')
|
| (1)
|
| 0.84
|
| 0.71
|
| 0.59
|
| 0.5
|
|
|
|
|}
|}
== List of notable APSs ==
{{See also| AS #List of ASs }}
* APS35.099¢, tuning of [[Carlos Gamma]]
* APS63.833¢, tuning of [[Carlos Beta]]
* [[1ed69c|APS69¢]]
* APS77.965¢, tuning of [[Carlos Alpha]]
* [[1ed86.4c|APS86.4¢]]
* [[88cET|APS88¢]]
* [[1ed97.5c|APS97.5¢]]
* [[1ed125c|APS125¢]]
For a more complete list, see [[Gallery of arithmetic pitch sequences]]. But do note that the gallery includes many obscure tunings that are of less importance to most xenharmonicists compared to the more curated selection listed above.
[[Category:Equal-step tuning‏‎]]
[[Category:Xenharmonic series]]

Latest revision as of 01:40, 19 May 2025

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An APS, or arithmetic pitch sequence, is a kind of arithmetic and harmonotonic tuning. It can also be called an equal multiplication.

Specification

Its full specification is (n-)APS-p: (n pitches of an) arithmetic pitch sequence adding by interval p.

Note:

  • The n is optional. If not provided, the sequence is open-ended.
  • The p can be dimensionless, in which case it refers to an interval by its frequency ratio. It can also take a unit proportional to octaves, in which case it refers to an interval by its pitch relation.

Formula

The pitch of k steps of APS-p is quite simply k⋅p for a pitch (log-frequency) quantity p.

Relationship to other tunings

Vs. rank-1 temperaments

By applying a mapping, APS-p becomes an equal temperament with generator p.

Vs. EPD

If the n is not specified, an APS will be equivalent to an equal pitch division (EPD). Specifically, n-EPD-p = APS(p/n) for a pitch quantity p. For example, 12-EPD1200¢ = APS(1200¢/12) = APS100¢.

Vs. AS

The only difference between an APS and an AS (ambitonal sequence) is that the p for an AS must be rational.

Examples

Example: APS⁴√2 ≈ APS1.189 = 4-EDO = rank-1 temperament w/ generator 300¢ = equal multiplication of 300¢
Quantity (0) 1 2 3 4
frequency (f, ratio) (1) 1.19 1.41 1.68 2
length (1/f, ratio) (0/4) 1/4 2/4 3/4 4/4
Length (1/f) (1) 0.84 0.71 0.59 0.5

List of notable APSs

See also: AS § List of ASs

For a more complete list, see Gallery of arithmetic pitch sequences. But do note that the gallery includes many obscure tunings that are of less importance to most xenharmonicists compared to the more curated selection listed above.

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