User:IlL/Tempered scales

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27edo subsets

7-note subsets

  • 0-3-6-11-16-18-22-27
  • 0-3-6-11-16-19-22-27
  • 0-3-6-11-16-20-22-27
  • 0-5-7-14-16-21-25-27
  • 0-4-6-13-16-20-25-27
  • 0-5-10-12-16-21-25-27 (approximates 28:32:36:38:42:48:53:56)

Dipentatonic subsets

See dipentatonic scales.

11 note subset

(Maximal subset without 44.444 cent interval) [1]

18 note subset

[2]

Inthar's 13-note well temperament

The worst intervals in 13edo are 9/8 and 5/4, and it'd be nice if the 17/8 was way better too. We want a well-temperament that just uses 13 notes and doesn't hurt the 11/8 and 13/8 too much either. Taking User:CritDeathX's 19-WT approach we use different generator chains to optimize different intervals.

Generators

The generators we use are:

13&19: POTE gen. 188.7573c, 2.9.5.13 (for a better meantone 9/8).

Generator chain A
13/10 13/9 8/5 9/5 1/1 10/9 5/4 18/13 20/13
444.9708c 633.7281c 822.4854c 1011.2427c 0.0c 187.84c 375.68c 563.52c 751.36c

13&17: POTE gen. 279.3176c, 2.11.13

Here, An means the nth note in generator chain A, etc.

Generator chain B
1/1 13/11 11/8 13/8 21/11 18/17 21/16 17/11 20/11
0.0c 279.3176c 558.6352c 837.9528c 1117.2704c 96.588c 475.9056c 755.2232c 1034.5408c

13&23: POTE gen. 365.4498c, 2.9.13.21.17

Generator chain C
1/1 16/13, 21/17 32/21 32/17 7/6 23/16 16/9 11/10, 23/21
0.0c 365.4498c 730.8996c 1096.3494c 261.7992c 627.249c 992.6988c 158.1486c
Degree Generator Stack
0
1 C-3
2 A1
3 B1
4 A2
5 C-2
6 B2
7 653.5264
8 C2
9 B3
10 B-1
11 A-1
12 19/10

For my 7\13 "minor fifth", I'm choosing the midpoint between my 5\13 and my 9\13 because the Dylathian mode of A-Team[8] works better that way. For the 12\13 major seventh let's choose 19/10 (about 1111 cents), so that it becomes a maximally buzzy 19/5 when widened by an octave.

The scale

Inthar's 13-note well temperament