User:BudjarnLambeth/Sandbox2: Difference between revisions

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[[User:BudjarnLambeth/Draft related tunings section]]
[[User:BudjarnLambeth/Draft related tunings section]]


= Lab =
== Octave stretch or compression ==
 
15edo
* 52ed11
* 11lim WE (79.770)
* 50ed10
* 47zpi (79.715)
* 54ed12
15edo's [[prime]]s 3, 5, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].
 
18edo
* 42ed5
* 13lim WE (66.291)
* 61zpi (66.228)
* 65ed12
* 7lim WE (66.148)
* 47ed6
18edo's [[prime]]s 3, 5, 7 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].
18edo's [[prime]]s 3, 5, 7 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].


25edo
; 18edo
* 95zpi (48.067)
* Step size: NNN{{c}}, octave size: NNN{{c}}  
* 13lim WE (47.946)
Pure-octaves 18edo approximates all harmonics up to 16 within NNN{{c}}.
* 90ed12
{{Harmonics in equal|18|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18edo}}
* 65ed6
{{Harmonics in equal|18|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edo (continued)}}
* 96zpi (47.642)
25edo's [[prime]] 3 is very sharp, and its sharp and flat mapping of 11 and 13 are about equally bad, it can benefit from [[octave shrinking]].
 
26edo
* 13lim WE (46.249)
* 93ed12
* 100zpi (46.268)
26edo's simple [[prime]]s with the most error - 3, 5 and 13 - are all tuned flat, so it can benefit from [[octave stretching]].
 
29edo
* 46edt
* [[116zpi]] (41.465)
* 13lim WE (41.484)
* 107ed13
* 100ed11
* 96ed10
29edo's [[prime]]s 5, 7, 11 and 13 are all tuned flat and the 3 has relatively little error, so 29edo can benefit from [[octave stretching]].
 
30edo
* 39.918zpi (39.918)
* 13lim WE (39.904)
* 11lim WE (79.770)
* 100ed10
* 108ed12
* 78ed6
30edo's simple [[prime]]s with the most error - 3, 5 and 11 - are all tuned sharp, so it can benefit from [[octave shrinking]].
 
34edo
* 11lim WE (35.284)
* 13lim WE (35.276) (octave identical to 113ed10 within 0.1{{c}})
* 79ed5
* 122ed12
* 88ed6
* 144zpi (35.248)
* 126ed13
* 54edt
34edo's [[prime]]s 3, 5, 11 and 13 are all tuned sharp, and it has two about equally bad mappings of 7, so 34edo can benefit from [[octave shrinking]].
 
35edo
* 11lim WE (35.284)
* 13lim WE (35.276)
* 121ed11
* [[149zpi]] (34.359)
* 116ed10
* 98ed7
* 81ed5
* 125ed12
* 90ed6
35edo's [[prime]]s 3, 5, 7 and 11 are all tuned flat, and it has two about equally bad mappings of 13, so 35edo can benefit from [[octave stretching]].
 
37edo
* 137ed13
* [[161zpi]] (32.408) (octave identical to 123ed10 within 0.1{{c}})
* 86ed5
* 104ed7
* 13lim WE (32.383)
* 11lim WE (32.377)
* 133ed12
* 96ed6
37edo's [[prime]]s 3, 5, 7, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].
 
48edo
* 13lim WE (25.005)
* 226zpi (25.006)
* 166ed11
* 172ed12
* 124ed6 (octave identical to 11lim WE within 0.1{{c}})
* 76edt
* 28edf (octave identical to 159ed10 within 0.1{{c}})
Most of 48edo's simple [[prime]]s have low error, but its 5 is substantially flat, so 48edo can benefit from slight [[octave stretching]].
 
; Medium-low priority


10edo
; [[WE|18et, 13-limit WE tuning]]
* 2.5.7.13 WE (120.358)
* Step size: 66.291{{c}}, octave size: 1193.2{{c}}
* 28ed7
Compressing the octave of 18edo by around 7{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
* 37ed13
{{Harmonics in cet|66.291|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning}}
* 26zpi (119.899)
{{Harmonics in cet|66.291|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning (continued)}}
* 2.3.7.13 WE (119.785)
* 13lim WE (119.776)
* 36ed12
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]].


11edo
; [[zpi|61zpi]]
* 28ed6
* Step size: 66.228{{c}}, octave size: 1192.1{{c}}
* 39ed12
Compressing the octave of 18edo by around 8{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 61zpi does this.
* 2.7.11.13 WE (108.821)
{{Harmonics in cet| 66.228 |intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 61zpi}}
* 30zpi (108.722)
{{Harmonics in cet| 66.228 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 61zpi (continued)}}
* 35ed9
* 31ed7
* 41ed13
* 37ed10
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker.


24edo
; [[65ed12]]
((13lim WE's octave is only 1/10th of a cent different from 24edo))
* Step size: NNN{{c}}, octave size: 1191.3{{c}}
* 56ed5
Compressing the octave of 18edo by around 9{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 65ed12 does this.
* 80ed10
{{Harmonics in equal|65|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65ed12}}
* 89ed13
{{Harmonics in equal|65|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65ed12 (continued)}}
* 2.3.5.11.13 WE (49.942)
* 90zpi (49.988)
* 11lim WE (50.017)
* 83ed11
* 86ed12
* 62ed6
* 38edt
If one wishes to use 24edo as a full 19-or-lower-limit tuning, then it benefits from slight [[octave stretching]], mostly to improve its [[prime]] 7. If one wishes to use 24edo as a no-7s 19-or-lower-limit tuning, then it benefits from slight [[octave shrinking]], mostly to improve its primes 5 and 13.


5edo
; [[WE|18et, 7-limit WE tuning]]
* 14ed7
* Step size: 66.148{{c}}, octave size: 1190.7{{c}}
* 2.3.7 WE (239.426)
Compressing the octave of 18edo by around 9.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
* 18ed12
{{Harmonics in cet| 66.148 |intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18et, 7-limit WE tuning}}
If one wishes to use 5edo as a  2.3.7 [[subgroup]] tuning, then it benefits from slight [[octave shrinking]] to improve its prime 3.
{{Harmonics in cet| 66.148 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18et, 7-limit WE tuning (continued)}}


6edo
; [[47ed6]]
* 19ed9
* Step size: NNN{{c}}, octave size: 1188.0{{c}}
* 2.9.5 WE (199.736)
Compressing the octave of 18edo by around 12{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 47ed6 does this.
* 2.9.5.7 WE (199.329)
{{Harmonics in equal|47|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 47ed6}}
* 20ed10
{{Harmonics in equal|47|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 47ed6 (continued)}}
* 14ed5
* 12zpi (198.843)
* 17ed7
If one wishes to use 6edo as a 2.9.5 or 2.9.5.7 [[subgroup]] tuning, then it benefits from [[octave shrinking]].

Revision as of 07:24, 16 September 2025

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User:BudjarnLambeth/Draft related tunings section

Octave stretch or compression

18edo's primes 3, 5, 7 and 13 are all tuned sharp, so it can benefit from octave shrinking.

18edo
  • Step size: NNN ¢, octave size: NNN ¢

Pure-octaves 18edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 18edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +31.4 +0.0 +13.7 +31.4 +31.2 +0.0 -3.9 +13.7 -18.0 +31.4
Relative (%) +0.0 +47.1 +0.0 +20.5 +47.1 +46.8 +0.0 -5.9 +20.5 -27.0 +47.1
Steps
(reduced) 		18
(0) 		29
(11) 		36
(0) 		42
(6) 		47
(11) 		51
(15) 		54
(0) 		57
(3) 		60
(6) 		62
(8) 		65
(11)
Approximation of harmonics in 18edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +26.1 +31.2 -21.6 +0.0 +28.4 -3.9 -30.8 +13.7 -4.1 -18.0 -28.3 +31.4
Relative (%) +39.2 +46.8 -32.4 +0.0 +42.6 -5.9 -46.3 +20.5 -6.2 -27.0 -42.4 +47.1
Steps
(reduced) 		67
(13) 		69
(15) 		70
(16) 		72
(0) 		74
(2) 		75
(3) 		76
(4) 		78
(6) 		79
(7) 		80
(8) 		81
(9) 		83
(11)
18et, 13-limit WE tuning
  • Step size: 66.291 ¢, octave size: 1193.2 ¢

Compressing the octave of 18edo by around 7 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 18et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -6.8 +20.5 -13.5 -2.1 +13.7 +12.0 -20.3 -25.3 -8.9 +25.0 +7.0
Relative (%) -10.2 +30.9 -20.4 -3.2 +20.7 +18.1 -30.6 -38.2 -13.4 +37.7 +10.5
Step 18 29 36 42 47 51 54 57 60 63 65
Approximation of harmonics in 18et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +1.0 +5.3 +18.4 -27.0 +0.6 -32.1 +6.9 -15.6 +32.5 +18.3 +7.6 +0.2
Relative (%) +1.5 +7.9 +27.7 -40.8 +0.9 -48.4 +10.4 -23.6 +49.0 +27.5 +11.4 +0.3
Step 67 69 71 72 74 75 77 78 80 81 82 83
61zpi
  • Step size: 66.228 ¢, octave size: 1192.1 ¢

Compressing the octave of 18edo by around 8 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 61zpi does this.

Approximation of harmonics in 61zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -7.9 +18.7 -15.8 -4.7 +10.8 +8.8 -23.7 -28.9 -12.6 +21.0 +2.9
Relative (%) -11.9 +28.2 -23.8 -7.2 +16.2 +13.3 -35.8 -43.7 -19.1 +31.8 +4.3
Step 18 29 36 42 47 51 54 57 60 63 65
Approximation of harmonics in 61zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.3 +0.9 +13.9 -31.6 -4.1 +29.4 +2.0 -20.5 +27.5 +13.2 +2.4 -5.0
Relative (%) -4.9 +1.4 +21.0 -47.7 -6.2 +44.4 +3.1 -31.0 +41.5 +19.9 +3.7 -7.6
Step 67 69 71 72 74 76 77 78 80 81 82 83
65ed12
  • Step size: NNN ¢, octave size: 1191.3 ¢

Compressing the octave of 18edo by around 9 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 65ed12 does this.

Approximation of harmonics in 65ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -8.7 +17.4 -17.4 -6.6 +8.7 +6.6 -26.1 -31.4 -15.3 +18.3 +0.0
Relative (%) -13.1 +26.3 -26.3 -10.0 +13.1 +9.9 -39.4 -47.5 -23.1 +27.6 +0.0
Steps
(reduced) 		18
(18) 		29
(29) 		36
(36) 		42
(42) 		47
(47) 		51
(51) 		54
(54) 		57
(57) 		60
(60) 		63
(63) 		65
(0)
Approximation of harmonics in 65ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -6.2 -2.1 +10.8 +31.4 -7.3 +26.1 -1.4 -24.0 +23.9 +9.6 -1.2 -8.7
Relative (%) -9.4 -3.2 +16.3 +47.5 -11.1 +39.4 -2.0 -36.2 +36.2 +14.5 -1.8 -13.1
Steps
(reduced) 		67
(2) 		69
(4) 		71
(6) 		73
(8) 		74
(9) 		76
(11) 		77
(12) 		78
(13) 		80
(15) 		81
(16) 		82
(17) 		83
(18)
18et, 7-limit WE tuning
  • Step size: 66.148 ¢, octave size: 1190.7 ¢

Compressing the octave of 18edo by around 9.5 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this.

Approximation of harmonics in 18et, 7-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -9.3 +16.3 -18.7 -8.1 +7.0 +4.7 -28.0 +32.7 -17.4 +16.0 -2.3
Relative (%) -14.1 +24.7 -28.2 -12.2 +10.6 +7.1 -42.3 +49.4 -26.4 +24.2 -3.5
Step 18 29 36 42 47 51 54 58 60 63 65
Approximation of harmonics in 18et, 7-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -8.6 -4.6 +8.2 +28.8 -10.0 +23.3 -4.1 -26.8 +21.1 +6.7 -4.1 -11.7
Relative (%) -13.0 -7.0 +12.5 +43.5 -15.1 +35.3 -6.2 -40.5 +31.8 +10.1 -6.3 -17.6
Step 67 69 71 73 74 76 77 78 80 81 82 83
47ed6
  • Step size: NNN ¢, octave size: 1188.0 ¢

Compressing the octave of 18edo by around 12 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 47ed6 does this.

Approximation of harmonics in 47ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -12.0 +12.0 -24.0 -14.4 +0.0 -2.9 +29.9 +24.0 -26.4 +6.6 -12.0
Relative (%) -18.2 +18.2 -36.4 -21.7 +0.0 -4.4 +45.4 +36.4 -40.0 +10.0 -18.2
Steps
(reduced) 		18
(18) 		29
(29) 		36
(36) 		42
(42) 		47
(0) 		51
(4) 		55
(8) 		58
(11) 		60
(13) 		63
(16) 		65
(18)
Approximation of harmonics in 47ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -18.6 -14.9 -2.3 +17.9 -21.0 +12.0 -15.6 +27.6 +9.1 -5.4 -16.4 -24.0
Relative (%) -28.2 -22.6 -3.5 +27.2 -31.9 +18.2 -23.6 +41.8 +13.9 -8.2 -24.8 -36.4
Steps
(reduced) 		67
(20) 		69
(22) 		71
(24) 		73
(26) 		74
(27) 		76
(29) 		77
(30) 		79
(32) 		80
(33) 		81
(34) 		82
(35) 		83
(36)
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