User:BudjarnLambeth/Sandbox2: Difference between revisions
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= Title1 = | = Title1 = | ||
== Octave stretch or compression == | == Octave stretch or compression == | ||
{{main|23edo and octave stretching}} | |||
; [[zpi| | 23edo is not typically taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention. | ||
* Step size: | |||
However, when using a slightly [[stretched tuning|stretched octave]] of around 1216 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well. | |||
Stretched 23edo is one of the best tunings to use for exploring the [[antidiatonic]] scale (and its 9-note extension, the [[superantidiatonic]] scale), since its fifth is more [[consonant]] and less "[[Wolf interval|wolfish]]" than fifths in other [[pelogic family]] temperaments. | |||
What follows is a comparison of stretched- and compressed-octave 23edo tunings. | |||
; [[zpi|86zpi]] | |||
* Step size: 51.653{{c}}, octave size: NNN{{c}} | |||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this. | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this. | ||
{{Harmonics in cet| | {{Harmonics in cet|51.653|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} | ||
{{Harmonics in cet| | {{Harmonics in cet|51.653|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} | ||
; [[ | ; [[60ed6]] | ||
* Step size: NNN{{c}}, octave size: NNN{{c}} | * Step size: NNN{{c}}, octave size: NNN{{c}} | ||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. | ||
{{Harmonics in equal| | {{Harmonics in equal|60|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | ||
{{Harmonics in equal| | {{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} | ||
; [[ | ; [[zpi|85zpi]] | ||
* Step size: | * Step size: 52.114{{c}}, octave size: NNN{{c}} | ||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this. | ||
{{Harmonics in cet| | {{Harmonics in cet|52.114|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} | ||
{{Harmonics in cet| | {{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} | ||
; | ; 23edo | ||
* Step size: NNN{{c}}, octave size: NNN{{c}} | * Step size: NNN{{c}}, octave size: NNN{{c}} | ||
Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}. | Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}. | ||
{{Harmonics in equal| | {{Harmonics in equal|23|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}} | ||
{{Harmonics in equal| | {{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}} | ||
; [[WE|ETNAME, SUBGROUP WE tuning]] | ; [[WE|23et, 13-limit WE tuning]] | ||
* Step size: | * Step size: 53.237{{c}}, octave size: NNN{{c}} | ||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. | |||
{{Harmonics in cet|53.237|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}} | |||
{{Harmonics in cet|53.237|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}} | |||
; [[WE|23et, 2.3.5.13 WE tuning]] | |||
* Step size: 53.447{{c}}, octave size: NNN{{c}} | |||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. | ||
{{Harmonics in cet| | {{Harmonics in cet|53.447|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}} | ||
{{Harmonics in cet| | {{Harmonics in cet|53.447|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}} | ||
; [[ | ; [[59ed6]] | ||
* Step size: NNN{{c}}, octave size: NNN{{c}} | * Step size: NNN{{c}}, octave size: NNN{{c}} | ||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. | ||
{{Harmonics in equal| | {{Harmonics in equal|59|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | ||
{{Harmonics in equal| | {{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} | ||
; [[zpi|ZPINAME]] | ; [[zpi|84zpi]] | ||
* Step size: 52.615{{c}}, octave size: NNN{{c}} | |||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this. | |||
{{Harmonics in cet|52.615|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} | |||
{{Harmonics in cet|52.615|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} | |||
; [[36edt]] | |||
* Step size: NNN{{c}}, octave size: NNN{{c}} | * Step size: NNN{{c}}, octave size: NNN{{c}} | ||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. | ||
{{Harmonics in | {{Harmonics in equal|36|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | ||
{{Harmonics in | {{Harmonics in equal|36|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} | ||
= Title2 = | = Title2 = | ||
Revision as of 03:09, 28 August 2025
Quick link
User:BudjarnLambeth/Draft related tunings section
Title1
Octave stretch or compression
23edo is not typically taken seriously as a tuning except by those interested in extreme xenharmony. Its fifths are significantly flat, and is neighbors 22edo and 24edo generally get more attention.
However, when using a slightly stretched octave of around 1216 cents, 23edo looks much better, and it approximates the perfect fifth (and various other intervals involving the 5th, 7th, 11th, and 13th harmonics) to within 18 cents or so. If we can tolerate errors around this size in 12edo, we can probably tolerate them in stretched-23 as well.
Stretched 23edo is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the superantidiatonic scale), since its fifth is more consonant and less "wolfish" than fifths in other pelogic family temperaments.
What follows is a comparison of stretched- and compressed-octave 23edo tunings.
- Step size: 51.653 ¢, octave size: NNN ¢
_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning ZPINAME does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -12.0 | +9.2 | -24.0 | +2.9 | -2.8 | -11.4 | +15.7 | +18.4 | -9.0 | -19.1 | -14.8 |
| Relative (%) | -23.2 | +17.8 | -46.4 | +5.7 | -5.4 | -22.0 | +30.4 | +35.6 | -17.5 | -36.9 | -28.6 | |
| Step | 23 | 37 | 46 | 54 | 60 | 65 | 70 | 74 | 77 | 80 | 83 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.6 | -23.4 | +12.2 | +3.7 | +2.1 | +6.4 | +16.1 | -21.0 | -2.2 | +20.6 | -4.7 | +24.9 |
| Relative (%) | +3.2 | -45.2 | +23.5 | +7.2 | +4.0 | +12.5 | +31.2 | -40.7 | -4.2 | +39.9 | -9.1 | +48.2 | |
| Step | 86 | 88 | 91 | 93 | 95 | 97 | 99 | 100 | 102 | 104 | 105 | 107 | |
- Step size: NNN ¢, octave size: NNN ¢
_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning EDONOI does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -10.9 | +10.9 | -21.8 | +5.4 | +0.0 | -8.4 | +18.9 | +21.8 | -5.5 | -15.4 | -10.9 |
| Relative (%) | -21.1 | +21.1 | -42.2 | +10.5 | +0.0 | -16.2 | +36.6 | +42.2 | -10.6 | -29.7 | -21.1 | |
| Steps (reduced) |
23 (23) |
37 (37) |
46 (46) |
54 (54) |
60 (0) |
65 (5) |
70 (10) |
74 (14) |
77 (17) |
80 (20) |
83 (23) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.6 | -19.3 | +16.4 | +8.0 | +6.5 | +10.9 | +20.7 | -16.4 | +2.5 | +25.4 | +0.1 | -21.8 |
| Relative (%) | +10.8 | -37.3 | +31.7 | +15.5 | +12.5 | +21.1 | +40.1 | -31.7 | +4.9 | +49.1 | +0.3 | -42.2 | |
| Steps (reduced) |
86 (26) |
88 (28) |
91 (31) |
93 (33) |
95 (35) |
97 (37) |
99 (39) |
100 (40) |
102 (42) |
104 (44) |
105 (45) |
106 (46) | |
- Step size: 52.114 ¢, octave size: NNN ¢
_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning ZPINAME does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.4 | -25.9 | -2.8 | -24.3 | +24.9 | +18.6 | -4.1 | +0.4 | -25.6 | +17.8 | +23.5 |
| Relative (%) | -2.6 | -49.6 | -5.3 | -46.6 | +47.8 | +35.7 | -7.9 | +0.8 | -49.2 | +34.2 | +45.1 | |
| Step | 23 | 36 | 46 | 53 | 60 | 65 | 69 | 73 | 76 | 80 | 83 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -10.8 | +17.2 | +2.0 | -5.5 | -6.2 | -1.0 | +9.7 | +25.1 | -7.3 | +16.4 | -8.4 | +22.1 |
| Relative (%) | -20.8 | +33.0 | +3.8 | -10.6 | -12.0 | -1.9 | +18.5 | +48.1 | -13.9 | +31.5 | -16.2 | +42.5 | |
| Step | 85 | 88 | 90 | 92 | 94 | 96 | 98 | 100 | 101 | 103 | 104 | 106 | |
- 23edo
- Step size: NNN ¢, octave size: NNN ¢
Pure-octaves EDONAME approximates all harmonics up to 16 within NNN ¢.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -23.7 | +0.0 | -21.1 | -23.7 | +22.5 | +0.0 | +4.8 | -21.1 | +22.6 | -23.7 |
| Relative (%) | +0.0 | -45.4 | +0.0 | -40.4 | -45.4 | +43.1 | +0.0 | +9.2 | -40.4 | +43.3 | -45.4 | |
| Steps (reduced) |
23 (0) |
36 (13) |
46 (0) |
53 (7) |
59 (13) |
65 (19) |
69 (0) |
73 (4) |
76 (7) |
80 (11) |
82 (13) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.7 | +22.5 | +7.4 | +0.0 | -0.6 | +4.8 | +15.5 | -21.1 | -1.2 | +22.6 | -2.2 | -23.7 |
| Relative (%) | -11.0 | +43.1 | +14.2 | +0.0 | -1.2 | +9.2 | +29.8 | -40.4 | -2.3 | +43.3 | -4.2 | -45.4 | |
| Steps (reduced) |
85 (16) |
88 (19) |
90 (21) |
92 (0) |
94 (2) |
96 (4) |
98 (6) |
99 (7) |
101 (9) |
103 (11) |
104 (12) |
105 (13) | |
- Step size: 53.237 ¢, octave size: NNN ¢
_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its SUBGROUP WE tuning and SUBGROUP TE tuning both do this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +24.5 | +14.6 | -4.3 | -18.0 | -14.2 | -14.9 | +20.1 | -24.1 | +6.5 | +1.2 | +10.2 |
| Relative (%) | +45.9 | +27.4 | -8.1 | -33.8 | -26.7 | -28.0 | +37.8 | -45.2 | +12.1 | +2.2 | +19.2 | |
| Step | 23 | 36 | 45 | 52 | 58 | 63 | 68 | 71 | 75 | 78 | 81 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -21.9 | +9.6 | -3.4 | -8.7 | -7.2 | +0.4 | +13.2 | -22.3 | -0.3 | +25.6 | +1.9 | -18.5 |
| Relative (%) | -41.1 | +18.0 | -6.4 | -16.3 | -13.4 | +0.7 | +24.9 | -41.9 | -0.6 | +48.1 | +3.6 | -34.8 | |
| Step | 83 | 86 | 88 | 90 | 92 | 94 | 96 | 97 | 99 | 101 | 102 | 103 | |
- Step size: 53.447 ¢, octave size: NNN ¢
_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its SUBGROUP WE tuning and SUBGROUP TE tuning both do this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -24.2 | +22.1 | +5.1 | -7.1 | -2.0 | -1.7 | -19.1 | -9.2 | +22.2 | +17.5 | -26.2 |
| Relative (%) | -45.2 | +41.4 | +9.6 | -13.2 | -3.8 | -3.1 | -35.6 | -17.2 | +41.6 | +32.8 | -49.0 | |
| Step | 22 | 36 | 45 | 52 | 58 | 63 | 67 | 71 | 75 | 78 | 80 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.4 | -25.8 | +15.1 | +10.2 | +12.2 | +20.1 | -20.0 | -2.0 | +20.5 | -6.6 | +23.3 | +3.1 |
| Relative (%) | -8.3 | -48.3 | +28.2 | +19.1 | +22.8 | +37.6 | -37.5 | -3.7 | +38.3 | -12.4 | +43.6 | +5.8 | |
| Step | 83 | 85 | 88 | 90 | 92 | 94 | 95 | 97 | 99 | 100 | 102 | 103 | |
- Step size: NNN ¢, octave size: NNN ¢
_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning EDONOI does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +9.2 | -9.2 | +18.5 | +0.2 | +0.0 | -4.0 | -24.9 | -18.5 | +9.4 | +2.1 | +9.2 |
| Relative (%) | +17.6 | -17.6 | +35.1 | +0.4 | +0.0 | -7.6 | -47.3 | -35.1 | +17.9 | +4.1 | +17.6 | |
| Steps (reduced) |
23 (23) |
36 (36) |
46 (46) |
53 (53) |
59 (0) |
64 (5) |
68 (9) |
72 (13) |
76 (17) |
79 (20) |
82 (23) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -24.2 | +5.2 | -9.0 | -15.6 | -15.4 | -9.2 | +2.3 | +18.7 | -13.2 | +11.4 | -13.0 | +18.5 |
| Relative (%) | -46.0 | +10.0 | -17.2 | -29.7 | -29.4 | -17.6 | +4.4 | +35.5 | -25.2 | +21.7 | -24.7 | +35.1 | |
| Steps (reduced) |
84 (25) |
87 (28) |
89 (30) |
91 (32) |
93 (34) |
95 (36) |
97 (38) |
99 (40) |
100 (41) |
102 (43) |
103 (44) |
105 (46) | |
- Step size: 52.615 ¢, octave size: NNN ¢
_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning ZPINAME does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.1 | -7.8 | +20.3 | +2.3 | +2.3 | -1.5 | -22.2 | -15.6 | +12.4 | +5.3 | +12.5 |
| Relative (%) | +19.3 | -14.9 | +38.6 | +4.3 | +4.4 | -2.8 | -42.2 | -29.7 | +23.6 | +10.0 | +23.7 | |
| Step | 23 | 36 | 46 | 53 | 59 | 64 | 68 | 72 | 76 | 79 | 82 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -20.9 | +8.7 | -5.5 | -12.0 | -11.8 | -5.5 | +6.1 | +22.6 | -9.3 | +15.4 | -8.9 | +22.6 |
| Relative (%) | -39.7 | +16.5 | -10.5 | -22.9 | -22.4 | -10.4 | +11.7 | +42.9 | -17.6 | +29.3 | -17.0 | +43.0 | |
| Step | 84 | 87 | 89 | 91 | 93 | 95 | 97 | 99 | 100 | 102 | 103 | 105 | |
- Step size: NNN ¢, octave size: NNN ¢
_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning EDONOI does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +15.1 | +0.0 | -22.6 | +13.8 | +15.1 | +12.4 | -7.4 | +0.0 | -23.9 | +22.4 | -22.6 |
| Relative (%) | +28.7 | +0.0 | -42.7 | +26.1 | +28.7 | +23.5 | -14.0 | +0.0 | -45.3 | +42.4 | -42.7 | |
| Steps (reduced) |
23 (23) |
36 (0) |
45 (9) |
53 (17) |
59 (23) |
64 (28) |
68 (32) |
72 (0) |
75 (3) |
79 (7) |
81 (9) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.6 | -25.3 | +13.8 | +7.7 | +8.4 | +15.1 | -25.6 | -8.8 | +12.4 | -15.3 | +13.4 | -7.4 |
| Relative (%) | -5.0 | -47.8 | +26.1 | +14.6 | +16.0 | +28.7 | -48.5 | -16.6 | +23.5 | -28.9 | +25.4 | -14.0 | |
| Steps (reduced) |
84 (12) |
86 (14) |
89 (17) |
91 (19) |
93 (21) |
95 (23) |
96 (24) |
98 (26) |
100 (28) |
101 (29) |
103 (31) |
104 (32) | |
Title2
Possible tunings to be used on each page
You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.
(Used https://x31eq.com/temper-pyscript/net.html, used WE instead of TE cause it kept defaulting to WE and I kept not remembering to switch it)
- High-priority
23edo
- Main: "23edo and octave stretching"
- 36edt
- 84zpi (52.615c)
- 59ed6
- 2.3.5.13 WE (52.447c)
- 13-limit WE (52.237c)
- 85zpi (52.114c)
- 60ed6
- 86zpi (51.653c)
60edo (narrow down edonoi & ZPIs)
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +15.1 | +0.0 | +13.8 | +12.4 | +22.4 | -2.6 | +8.4 | -25.6 | +13.4 | -18.0 | +25.0 |
| Relative (%) | +28.7 | +0.0 | +26.1 | +23.5 | +42.4 | -5.0 | +16.0 | -48.5 | +25.4 | -34.2 | +47.3 | |
| Steps (reduced) |
23 (23) |
36 (0) |
53 (17) |
64 (28) |
79 (7) |
84 (12) |
93 (21) |
96 (24) |
103 (31) |
110 (2) |
113 (5) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.4 | -25.9 | -24.3 | +18.6 | +17.8 | -10.8 | -6.2 | +9.7 | -8.4 | +7.2 | -4.0 |
| Relative (%) | -2.6 | -49.6 | -46.6 | +35.7 | +34.2 | -20.8 | -12.0 | +18.5 | -16.2 | +13.8 | -7.8 | |
| Step | 23 | 36 | 53 | 65 | 80 | 85 | 94 | 98 | 104 | 112 | 114 | |
- 95edt
- 35edf
- 139ed5
- 155ed6
- 208ed11
- 215ed12
- 255ed19
- 272ed23 (great for catnip temperament, maybe there's a similar but simpler tuning w similar benefits?)
- 13-limit WE (20.013c)
- 299zpi (20.128c)
- 300zpi (20.093c)
- 301zpi (20.027c)
- 302zpi (19.962c)
- 303zpi (19.913c)
- 304zpi (19.869c)
- Medium priority
13edo
- Main: "13edo and optimal octave stretching"
- 2.5.11.13 WE (92.483c)
- 2.5.7.13 WE (92.804c)
- 2.3 WE (91.405c) (good for opposite 7 mapping)
- 38zpi (92.531c)
32edo (narrow down ZPIs)
- 90ed7
- 51edt
- 75ed5
- 1ed46/45
- 11-limit WE (37.453c)
- 13-limit WE (37.481c)
- 131zpi (37.862c)
- 132zpi (37.662c)
- 133zpi (37.418c)
- 134zpi (37.176c)
33edo (narrow down edonoi)
- 76ed5
- 92ed7
- 52edt
- 1ed47/46
- 114ed11
- 122ed13
- 93ed7
- 23edPhi
- 77ed5
- 123ed13
- 115ed11
- 11-limit WE (36.349c)
- 13-limit WE (36.357c)
- 137zpi (36.628c)
- 138zpi (36.394c)
- 139zpi (36.179c)
39edo (narrow down slightly)
- 62edt
- 101ed6
- 18ed11/8
- 2.3.5.11 WE (30.703c)
- 2.3.7.11.13 WE (30.787c)
- 13-limit WE (30.757c)
- 171zpi (30.973c)
- 172zpi (30.836c)
- 173zpi (30.672c)
42edo (narrow down slightly)
- 42ed257/128 (replace w something similar but simpler)
- AS123/121 (1ed123/121)
- 11ed6/5
- 34ed7/4
- 7-limit WE (28.484c)
- 13-limit WE (28.534c)
- 189zpi (28.689c)
- 190zpi (28.572c)
- 191zpi (28.444c)
45edo
- 126ed7
- 13ed11/9
- 7-limit WE (26.745c)
- 13-limit WE (26.695c)
- 207zpi (26.762)
- 208zpi (26.646)
- 209zpi (26.550)
54edo (narrow down slightly)
- 86edt
- 126ed5
- 152ed7
- 38ed5/3
- 40ed5/3
- 2.3.7.11.13 WE (22.180c)
- 13-limit WE (22.198c)
- 262zpi (22.313c)
- 263zpi (22.243c)
- 264zpi (22.175c)
59edo (narrow down ZPIs)
- 93edt
- 166ed7
- 203ed11
- 7-limit WE (20.301c)
- 11-limit WE (20.310c)
- 13-limit WE (20.320c)
- 293zpi (20.454c)
- 294zpi (20.399c)
- 295zpi (20.342c)
- 296zpi (20.282c)
- 297zpi (20.229c)
64edo (narrow down ZPIs)
- 149ed5
- 180ed7
- 222ed11
- 47ed5/3
- 11-limit WE (18.755c)
- 13-limit WE (18.752c)
- 325zpi (18.868c)
- 326zpi (18.816c)
- 327zpi (18.767c)
- 328zpi (18.721c)
- 329zpi (18.672c)
- 330zpi (18.630c)
103edo (narrow down edonoi, choose ZPIS)
- 163edt
- 239ed5
- 266ed6
- 289ed7
- 356ed11
- 369ed12
- 381ed13
- 421ed17
- 466ed23
- 13-limit WE (11.658c)
- Best nearby ZPI(s)
111edo (choose ZPIS)
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
118edo (choose ZPIS)
- 187edt
- 69edf
- 13-limit WE (10.171c)
- Best nearby ZPI(s)
- Low priority
104edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
125edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
145edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
152edo
- 241edt
- 13-limit WE (7.894c)
- Best nearby ZPI(s)
159edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
166edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
182edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
198edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
212edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
243edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
247edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
- Optional
25edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
26edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
29edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
30edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
34edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
35edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
36edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
37edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
5edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
6edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
9edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
10edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
11edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
15edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
18edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
48edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
20edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
24edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
28edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)