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An '''AFS''', or '''arithmetic frequency sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
An '''AFS''', or '''arithmetic frequency sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.


An OS is a specific (rational) type of AFS.
== Specification ==


(n-)AFSp: (n pitches of an) arithmetic frequency sequence adding by p
Its full specification is (n-)AFSp: (n pitches of an) arithmetic frequency sequence adding by (irrational) interval p. The n is optional. If not provided, the sequence is open-ended.


shifted overtone series (± frequency) (equivalent to AFS)
== Formula ==


AFS(1/7π,7) is saying start at 1/7π and then just move by 1's, so the next step is 1+1/7π which equals (7π+1)/7π, and the next step would be 2+1/7π = (14π+1)/7π, and you'd keep going until you had 7 pitches, so the last one would be (42π+1)/7π. Though that's just the first step, because as I mentioned in my previous comment here, you want the first pitch to be 1/1, so you multiply everything by 7π, so in the end the scale is 1, 7π+1, 14π+1, 21π+1 ... 42π+1. A good way to read AFS(1/p,n) is "start on 1, then add p each step, and go until you have n pitches."
The formula for step <span><math>k</math></span> of an AFSp is:


If the new step size is irrational, the tuning is no longer JI, so we use a different acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: <span><math>1, 1+φ, 1+2φ, 1+3φ...</math></span> etc. we could have the AFSφ.
<math>
f(k) = 1 + k⋅p
</math>


OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[Monotonic tunings#Derivation of OS|derivation of OS]].
== Relationship to other tunings ==
 
=== Vs. OS ===
 
The only difference between an [[OS|OS (overtone sequence)]] and AFS is that for OS the p must be rational.
 
=== As shifted overtone series ===
 
An AFS could also be described as a shifted [[overtone series]] (± frequency). Both AFS and OS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see [[OS#Derivation|derivation of OS]].
 
=== Vs. EFD ===
 
By specifying n, your sequence will be equivalent to one period of some [[EFD|EFD (equal frequency division)]]. Specifically, n-EFDp = n-AFS((p-1)/n).
 
=== Vs. ALS ===
 
The analogous utonal equivalent of an AFS is an [[ALS|ALS (arithmetic length sequence)]].
 
== Examples ==
 
If we wanted to move by steps of φ, like this: <span><math>1, 1+φ, 1+2φ, 1+3φ...</math></span> etc. we could have the AFSφ.
 
Here's another example:


{| class="wikitable"
{| class="wikitable"
|+example: (1/⁴√2)-shifted overtone series segment = 9-AFS(1/⁴√2)
|+example: (1/⁴√2)-shifted overtone series segment = 8-AFS(1/⁴√2) ≈ 8-AFS0.841
|-
|-
! quantity !! 1 !! 2
! quantity !! (0) !! 1
!2
!3
!3
!4
!4
Line 23: Line 48:
!7
!7
!8
!8
!9
|-
|-
! frequency  
! frequency (''f'', ratio)
| 1.00 || 1.84
|(1 + 0/⁴√2)
|2.68
|1 + 1/⁴√2
|3.52
|1 + 2/⁴√2
|4.36
|1 + 3/⁴√2
|5.20
|1 + 4/⁴√2
|6.05
|1 + 5/⁴√2
|6.89
|1 + 6/⁴√2
|7.73
|1 + 7/⁴√2
|1 + 8/⁴√2
|-
|-
! pitch  
! pitch (log₂''f'', octaves)
| 0.00 || 0.88
| (0) || 0.88
|1.42
|1.42
|1.82
|1.82
Line 45: Line 70:
|2.95
|2.95
|-
|-
! length  
! length (1/''f'', ratio)
| 1.00 || 0.54
| (1) || 0.54
|0.37
|0.37
|0.28
|0.28
Line 55: Line 80:
|0.13
|0.13
|}
|}
[[Category:Otonality]]
[[Category:Harmonic]]
[[Category:Harmonic series‏‎]]
[[Category:Xenharmonic series]]

Latest revision as of 20:36, 19 October 2023

An AFS, or arithmetic frequency sequence, is a kind of arithmetic and harmonotonic tuning.

Specification

Its full specification is (n-)AFSp: (n pitches of an) arithmetic frequency sequence adding by (irrational) interval p. The n is optional. If not provided, the sequence is open-ended.

Formula

The formula for step [math]\displaystyle{ k }[/math] of an AFSp is:

[math]\displaystyle{ f(k) = 1 + k⋅p }[/math]

Relationship to other tunings

Vs. OS

The only difference between an OS (overtone sequence) and AFS is that for OS the p must be rational.

As shifted overtone series

An AFS could also be described as a shifted overtone series (± frequency). Both AFS and OS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see derivation of OS.

Vs. EFD

By specifying n, your sequence will be equivalent to one period of some EFD (equal frequency division). Specifically, n-EFDp = n-AFS((p-1)/n).

Vs. ALS

The analogous utonal equivalent of an AFS is an ALS (arithmetic length sequence).

Examples

If we wanted to move by steps of φ, like this: [math]\displaystyle{ 1, 1+φ, 1+2φ, 1+3φ... }[/math] etc. we could have the AFSφ.

Here's another example:

example: (1/⁴√2)-shifted overtone series segment = 8-AFS(1/⁴√2) ≈ 8-AFS0.841
quantity (0) 1 2 3 4 5 6 7 8
frequency (f, ratio) (1 + 0/⁴√2) 1 + 1/⁴√2 1 + 2/⁴√2 1 + 3/⁴√2 1 + 4/⁴√2 1 + 5/⁴√2 1 + 6/⁴√2 1 + 7/⁴√2 1 + 8/⁴√2
pitch (log₂f, octaves) (0) 0.88 1.42 1.82 2.13 2.38 2.60 2.78 2.95
length (1/f, ratio) (1) 0.54 0.37 0.28 0.23 0.19 0.17 0.15 0.13
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