Alpha, beta, and gamma family of equal divisions

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Wendy Carlos invented in the 1980's the Alpha, Beta, and Gamma scales. These are scales that divides 3/2 with steps very near to the successive superparticular complementary pair folding in 3/2, namely 6/5 and 5/4. The happy equal divisions are 9edf, 11edf, and 20edf.

These scales belong to a much vaster family, where also applies the same principle of divisions of a ratio with steps very near to the successive superparticular complementary pair folding into it. Below is a table showing the first members of this family:

The Alpha–Beta–Gamma family
Tuning Intervals Comment
Equal division Type Cents
per steps 		Steps
per octave 		Ratio divided Successive superparticular
complementary pair folding
in the ratio divided 		Approximation
in cents of these
three intervals
3edt Alpha 633.985 1.893 3/1 3/2, 2/1 0, -67.970, 67.970 Too much off to be useful
5edt Beta 380.391 3.155 0, 58.827, -58.827
8edt Gamma 237.744 5.047 0, 11.278, -11.278 Pairs are off but still recognizable
5edo Alpha 240 5 2/1 4/3, 3/2 0, -18.045, 18.045
7edo Beta 171.429 7 0, 16.241, -16.241
12edo Gamma 100 12 0, 1.955, -1.955 Happy divisions musically useful
7ed5/3 Alpha 126.337 9.498 5/3 5/4, 4/3 0, -7.303, 7.303
9ed5/3 Beta 98.262 12.212 0, 6.735, -6.735
16ed5/3 Gamma 55.272 21.711 0, 0.593, -0.593
9edf Alpha 77.995 15.386 3/2 6/5, 5/4 0, -3.661, 3.661
11edf Beta 63.814 18.805 0, 3.429, -3.429
20edf Gamma 35.098 34.190 0, 0.238, -0.238
11ed7/5 Alpha 52.956 22.660 7/5 7/6, 6/5 0, -2.093, 2.093
13ed7/5 Beta 44.809 26.781 0, 1.981, -1.981
24ed7/5 Gamma 24.271 49.441 0, 0.114, -0.114
13ed4/3 Alpha 38.311 31.322 4/3 8/7, 7/6 0, -1.307, 1.307
15ed4/3 Beta 33.203 36.141 0, 1.247, -1.247
28ed4/3 Gamma 17.787 67.464 0, 0.061, -0.061

A pair of small and big successive superparticulars [math]\displaystyle{ S_n=\dfrac{n+1}{n} }[/math] and [math]\displaystyle{ B_n=\dfrac{n}{n-1} }[/math] has product [math]\displaystyle{ \dfrac{n+1}{n}\cdot\dfrac{n}{n-1}=\dfrac{n+1}{n-1} }[/math]. Thus they are complementary in the ratio [math]\displaystyle{ R_n=\dfrac{n+1}{n-1} }[/math].

For each [math]\displaystyle{ n\ge 2 }[/math] consider the three equal divisions of [math]\displaystyle{ R_n }[/math] where low errors appear for [math]\displaystyle{ S_n }[/math] and [math]\displaystyle{ B_n }[/math] as a converging sequence and pattern:

  • Alpha: [math]\displaystyle{ k_\alpha=2n-1 }[/math]
  • Beta: [math]\displaystyle{ k_\beta=2n+1 }[/math]
  • Gamma: [math]\displaystyle{ k_\gamma=4n=k_\alpha+k_\beta }[/math]
Converging sequence and pattern
n Ratio divided SSCP Number of divisions SSCP mappings
Small Big Alpha Beta Gamma Alpha Beta Gamma
2 3/1 3/2 2/1 3 5 8 1, 2 2, 3 3, 5
3 2/1 4/3 3/2 5 7 12 2, 3 3, 4 5, 7
4 5/3 5/4 4/3 7 9 16 3, 4 4, 5 7, 9
5 3/2 6/5 5/4 9 11 20 4, 5 5, 6 9, 11
6 7/5 7/6 6/5 11 13 24 5, 6 6, 7 11, 13
7 4/3 8/7 7/6 13 15 28 6, 7 7, 8 13, 15

Mappings of [math]\displaystyle{ S_n }[/math] and [math]\displaystyle{ B_n }[/math] are always mapped as follow:

  • On Alpha scales, [math]\displaystyle{ S_n }[/math] is mapped on the [math]\displaystyle{ (n - 1)^\text{th} }[/math] degree step, and [math]\displaystyle{ B_n }[/math] is mapped on the [math]\displaystyle{ n^\text{th} }[/math] degree step.
  • On Beta scales, [math]\displaystyle{ S_n }[/math] is mapped on the [math]\displaystyle{ n^\text{th} }[/math] degree step, and [math]\displaystyle{ B_n }[/math] is mapped on the [math]\displaystyle{ (n + 1)^\text{th} }[/math] degree step.
  • On Gamma scales, [math]\displaystyle{ S_n }[/math] is mapped on the [math]\displaystyle{ (2n - 1)^\text{th} }[/math] degree step, and [math]\displaystyle{ B_n }[/math] is mapped on the [math]\displaystyle{ (2n + 1)^\text{th} }[/math] degree step.

Alpha types flatten the smaller interval and sharpen the larger; Beta types do the reverse; Gamma types also sharpen the smaller and flatten the larger.

The Alpha, Beta, and Gamma types bring their interval pairs increasingly close to just intonation. n also brings the interval pairs increasingly near to pure.

User:Contribution/Successive superparticular complementary pair #The converging Alpha-Beta-Gamma sequence shows a bigger picture of the converging Alpha–Beta–Gamma sequence.

Musically, scales of the Alpha–Beta–Gamma family are perfect for making almost pure chords filling the vertical space. For example, with 7ed5/3, 9ed5/3 and 16ed5/3, it is possible to make big chords stacking a repetition of 5/4 4/3 repeating at 5/3, or a repetition of 4/3 5/4 repeating at 5/3; with 9edf, 11edf and 20edf, it is possible to make big chords stacking a repetition of 6/5 5/4 repeating at 3/2 or 5/4 6/5 repeating at 3/2; with 11ed7/5, 13ed7/5 and 24ed7/5, big chords stacking a repetition of 7/6 6/5 repeating at 7/5 or 6/5 7/6 repeating at 7/5; with 13ed4/3, 15ed4/3 and 28ed4/3, big chords stacking a repetition of 8/7 7/6 repeating at 4/3 or 7/6 8/7 repeating at 4/3; and so on.

Associated commas

Equal temperaments of the Alpha–Beta–Gamma family are characterized by comma association:

  • Alpha-related comma: Cα(n), rational intervals of the form ⁠(n + 1)n(n − 1)n−1/n2n−1⁠.
    Tempering out Cα(n) splits ⁠n + 1/n − 1⁠ into 2n−1 equal parts.
  • Beta-related comma: Cβ(n), rational intervals of the form ⁠n2n+1/(n + 1)n+1(n − 1)n⁠.
    Tempering out Cβ(n) splits ⁠n + 1/n − 1⁠ into 2n+1 equal parts.
  • Gamma-related comma: Cγ(n), rational intervals of the form ⁠n4n/(n + 1)2n+1(n − 1)2n−1⁠.
    Tempering out Cγ(n) splits ⁠n + 1/n − 1⁠ into 4n equal parts.
Table of alpha-related commas
Associated ET Interval relation Ratio
3ed3/1 (3/1)2/(2/1)3
(3/2)3/(3/1) 		9/8
5edo (2/1)3/(3/2)5
(4/3)5/(2/1)2 		256/243
7ed5/3 (5/3)4/(4/3)7
(5/4)7/(5/3)3 		16875/16384
9ed3/2 (3/2)5/(5/4)9
(6/5)9/(3/2)4 		1990656/1953125
11ed7/5 (7/5)6/(6/5)11
(7/6)11/(7/5)5 		367653125/362797056
13ed4/3 (4/3)7/(7/6)13
(8/7)13/(4/3)6 		97844723712/96889010407
Table of beta-related commas
Associated ET Interval relation Ratio
5ed3/1 (2/1)5/(3/1)3
(3/1)2/(3/2)5 		32/27
7edo (3/2)7/(2/1)4
(2/1)3/(4/3)7 		2187/2048
9ed5/3 (4/3)9/(5/3)5
(5/3)4/(5/4)9 		262144/253125
11ed3/2 (5/4)11/(3/2)6
(3/2)5/(6/5)11 		48828125/47775744
13ed7/5 (6/5)13/(7/5)7
(7/5)6/(7/6)13 		13060694016/12867859375
15ed4/3 (7/6)15/(4/3)8
(4/3)7/(8/7)15 		4747561509943/4696546738176
Table of gamma-related commas
Associated ET Interval relation Ratio
8ed3/1 (2/1)8/(3/1)5
(3/1)3/(3/2)8 		256/243
12edo (3/2)12/(2/1)7
(2/1)5/(4/3)12 		531441/524288
16ed5/3 (4/3)16/(5/3)9
(5/3)7/(5/4)16 		4294967296/4271484375
20ed3/2 (5/4)20/(3/2)11
(3/2)9/(6/5)20 		95367431640625/95105071448064
24ed7/5 (6/5)24/(7/5)13
(7/5)11/(7/6)24 		4738381338321616896/4730908711279296875
28ed4/3 (7/6)28/(4/3)15
(4/3)13/(8/7)28 		459986536544739960976801/459532317997325522829312
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