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


== Octave stretch or compression ==
== Lab ==
18edo's [[prime]]s 3, 5, 7 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].
=== Compressed 49edo ===
{{Harmonics in equal|49|2|1|intervals=integer|columns=12}}
{{Harmonics in cet|24.419|intervals=integer|columns=12}}
{{Harmonics in equal|176|12|1|intervals=integer|columns=12}}
{{Harmonics in equal|170|11|1|intervals=integer|columns=12}}
{{Harmonics in equal|163|10|1|intervals=integer|columns=12}}
{{Harmonics in equal|138|7|1|intervals=integer|columns=12}}
{{Harmonics in equal|127|6|1|intervals=integer|columns=12}}
{{Harmonics in equal|114|5|1|intervals=integer|columns=12}}
{{Harmonics in equal|78|3|1|intervals=integer|columns=12}}


; 18edo
=== Stretched 50edo ===
* Step size: 66.667{{c}}, octave size: 1200.0{{c}}
{{Harmonics in equal|50|2|1|intervals=integer|columns=12}}
Pure-octaves 18edo approximates all harmonics up to 15 within 31.4{{c}}.
{{Harmonics in cet|24.030|intervals=integer|columns=12}}
{{Harmonics in equal|18|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18edo}}
{{Harmonics in equal|185|13|1|intervals=integer|columns=12}}
{{Harmonics in equal|18|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edo (continued)}}
{{Harmonics in equal|179|12|1|intervals=integer|columns=12}}
 
{{Harmonics in equal|166|10|1|intervals=integer|columns=12}}
; [[WE|18et, 13-limit WE tuning]]
{{Harmonics in equal|140|7|1|intervals=integer|columns=12}}
* Step size: 66.291{{c}}, octave size: 1193.2{{c}}
{{Harmonics in equal|129|6|1|intervals=integer|columns=12}}
Compressing the octave of 18edo by around 7{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 15 within 25.3{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in equal|116|5|1|intervals=integer|columns=12}}
{{Harmonics in cet|66.291|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning}}
{{Harmonics in equal|79|3|1|intervals=integer|columns=12}}
{{Harmonics in cet|66.291|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning (continued)}}
 
; [[zpi|61zpi]]
* Step size: 66.228{{c}}, octave size: 1192.1{{c}}
Compressing the octave of 18edo by around 8{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 15 within 28.9{{c}}. The tuning 61zpi does this.
{{Harmonics in cet| 66.228 |intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 61zpi}}
{{Harmonics in cet| 66.228 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 61zpi (continued)}}
 
; [[65ed12]]
* Octave size: 1191.3{{c}}
Compressing the octave of 18edo by around 9{{c}} results in improved primes 3, 5, 7 and 13, but a worse primes 2. This approximates all harmonics up to 15 within 31.4{{c}}. The tuning 65ed12 does this.
{{Harmonics in equal|65|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65ed12}}
{{Harmonics in equal|65|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65ed12 (continued)}}
 
; [[47ed6]]
* Step size: NNN{{c}}, octave size: 1188.0{{c}}
Compressing the octave of 18edo by around 12{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 15 within 29.9{{c}}. The tuning 47ed6 does this.
{{Harmonics in equal|47|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 47ed6}}
{{Harmonics in equal|47|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 47ed6 (continued)}}

Revision as of 06:00, 17 September 2025

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User:BudjarnLambeth/Draft related tunings section

Lab

Compressed 49edo

Approximation of harmonics in 49edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +8.2 +0.0 +5.5 +8.2 +10.8 +0.0 -8.0 +5.5 +11.9 +8.2 -7.9
Relative (%) +0.0 +33.7 +0.0 +22.6 +33.7 +44.0 +0.0 -32.6 +22.6 +48.8 +33.7 -32.2
Steps
(reduced) 		49
(0) 		78
(29) 		98
(0) 		114
(16) 		127
(29) 		138
(40) 		147
(0) 		155
(8) 		163
(16) 		170
(23) 		176
(29) 		181
(34)
Approximation of harmonics in 1ed24.419c
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) -3.5 +2.7 -6.9 -2.5 -0.7 +1.0 -10.4 +5.5 -6.0 -0.1 -4.2 +3.7
Relative (%) -14.2 +11.2 -28.4 -10.4 -3.0 +4.1 -42.6 +22.3 -24.6 -0.4 -17.2 +15.3
Step 49 78 98 114 127 138 147 156 163 170 176 182
Approximation of harmonics in 176ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) -2.3 +4.6 -4.6 +0.2 +2.3 +4.3 -6.9 +9.2 -2.1 +4.0 +0.0 +8.1
Relative (%) -9.4 +18.8 -18.8 +0.7 +9.4 +17.6 -28.2 +37.6 -8.7 +16.3 +0.0 +33.1
Steps
(reduced) 		49
(49) 		78
(78) 		98
(98) 		114
(114) 		127
(127) 		138
(138) 		147
(147) 		156
(156) 		163
(163) 		170
(170) 		176
(0) 		182
(6)
Approximation of harmonics in 170ed11
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) -3.4 +2.8 -6.9 -2.5 -0.7 +1.1 -10.3 +5.5 -5.9 +0.0 -4.1 +3.8
Relative (%) -14.1 +11.3 -28.2 -10.2 -2.8 +4.4 -42.3 +22.7 -24.3 +0.0 -16.9 +15.7
Steps
(reduced) 		49
(49) 		78
(78) 		98
(98) 		114
(114) 		127
(127) 		138
(138) 		147
(147) 		156
(156) 		163
(163) 		170
(0) 		176
(6) 		182
(12)
Approximation of harmonics in 163ed10
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) -1.7 +5.6 -3.3 +1.7 +3.9 +6.1 -5.0 +11.2 +0.0 +6.2 +2.3 +10.4
Relative (%) -6.8 +22.9 -13.6 +6.8 +16.1 +24.9 -20.4 +45.8 +0.0 +25.3 +9.3 +42.7
Steps
(reduced) 		49
(49) 		78
(78) 		98
(98) 		114
(114) 		127
(127) 		138
(138) 		147
(147) 		156
(156) 		163
(0) 		170
(7) 		176
(13) 		182
(19)
Approximation of harmonics in 138ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) -3.8 +2.2 -7.6 -3.4 -1.7 +0.0 -11.5 +4.3 -7.2 -1.3 -5.5 +2.4
Relative (%) -15.7 +8.9 -31.3 -13.8 -6.8 +0.0 -47.0 +17.7 -29.5 -5.4 -22.5 +9.9
Steps
(reduced) 		49
(49) 		78
(78) 		98
(98) 		114
(114) 		127
(127) 		138
(0) 		147
(9) 		156
(18) 		163
(25) 		170
(32) 		176
(38) 		182
(44)
Approximation of harmonics in 127ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) -3.2 +3.2 -6.4 -1.9 +0.0 +1.8 -9.5 +6.4 -5.1 +0.9 -3.2 +4.8
Relative (%) -13.0 +13.0 -26.1 -7.7 +0.0 +7.4 -39.1 +26.1 -20.7 +3.7 -13.0 +19.6
Steps
(reduced) 		49
(49) 		78
(78) 		98
(98) 		114
(114) 		127
(0) 		138
(11) 		147
(20) 		156
(29) 		163
(36) 		170
(43) 		176
(49) 		182
(55)
Approximation of harmonics in 114ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) -2.4 +4.5 -4.7 +0.0 +2.1 +4.1 -7.1 +8.9 -2.4 +3.7 -0.3 +7.8
Relative (%) -9.7 +18.3 -19.4 +0.0 +8.6 +16.7 -29.1 +36.6 -9.7 +15.2 -1.1 +31.9
Steps
(reduced) 		49
(49) 		78
(78) 		98
(98) 		114
(0) 		127
(13) 		138
(24) 		147
(33) 		156
(42) 		163
(49) 		170
(56) 		176
(62) 		182
(68)
Approximation of harmonics in 78edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) -5.2 +0.0 -10.4 -6.5 -5.2 -3.8 +8.8 +0.0 -11.7 -6.0 -10.4 -2.6
Relative (%) -21.3 +0.0 -42.5 -26.8 -21.3 -15.7 +36.2 +0.0 -48.0 -24.7 -42.5 -10.8
Steps
(reduced) 		49
(49) 		78
(0) 		98
(20) 		114
(36) 		127
(49) 		138
(60) 		148
(70) 		156
(0) 		163
(7) 		170
(14) 		176
(20) 		182
(26)

Stretched 50edo

Approximation of harmonics in 50edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -6.0 +0.0 -2.3 -6.0 -8.8 +0.0 -11.9 -2.3 +0.7 -6.0 -0.5
Relative (%) +0.0 -24.8 +0.0 -9.6 -24.8 -36.8 +0.0 -49.6 -9.6 +2.8 -24.8 -2.2
Steps
(reduced) 		50
(0) 		79
(29) 		100
(0) 		116
(16) 		129
(29) 		140
(40) 		150
(0) 		158
(8) 		166
(16) 		173
(23) 		179
(29) 		185
(35)
Approximation of harmonics in 1ed24.03c
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +1.5 -3.6 +3.0 +1.2 -2.1 -4.6 +4.5 -7.2 +2.7 +5.9 -0.6 +5.0
Relative (%) +6.2 -14.9 +12.5 +4.9 -8.7 -19.3 +18.7 -29.8 +11.1 +24.4 -2.4 +20.9
Step 50 79 100 116 129 140 150 158 166 173 179 185
Approximation of harmonics in 185ed13
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.1 -5.7 +0.3 -2.0 -5.6 -8.4 +0.4 -11.5 -1.8 +1.2 -5.4 +0.0
Relative (%) +0.6 -23.9 +1.2 -8.3 -23.3 -35.1 +1.8 -47.7 -7.7 +4.9 -22.7 +0.0
Steps
(reduced) 		50
(50) 		79
(79) 		100
(100) 		116
(116) 		129
(129) 		140
(140) 		150
(150) 		158
(158) 		166
(166) 		173
(173) 		179
(179) 		185
(0)
Approximation of harmonics in 179ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +1.7 -3.3 +3.3 +1.5 -1.7 -4.2 +5.0 -6.7 +3.2 +6.4 +0.0 +5.6
Relative (%) +6.9 -13.8 +13.8 +6.4 -6.9 -17.3 +20.8 -27.7 +13.4 +26.8 +0.0 +23.4
Steps
(reduced) 		50
(50) 		79
(79) 		100
(100) 		116
(116) 		129
(129) 		140
(140) 		150
(150) 		158
(158) 		166
(166) 		173
(173) 		179
(0) 		185
(6)
Approximation of harmonics in 166ed10
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.7 -4.9 +1.4 -0.7 -4.2 -6.9 +2.1 -9.7 +0.0 +3.1 -3.5 +2.1
Relative (%) +2.9 -20.2 +5.8 -2.9 -17.3 -28.6 +8.7 -40.4 +0.0 +12.9 -14.4 +8.5
Steps
(reduced) 		50
(50) 		79
(79) 		100
(100) 		116
(116) 		129
(129) 		140
(140) 		150
(150) 		158
(158) 		166
(0) 		173
(7) 		179
(13) 		185
(19)
Approximation of harmonics in 140ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +3.2 -1.0 +6.3 +5.0 +2.2 +0.0 +9.5 -1.9 +8.2 +11.6 +5.3 +11.1
Relative (%) +13.1 -4.1 +26.2 +20.8 +9.0 +0.0 +39.3 -8.1 +33.9 +48.2 +22.1 +46.3
Steps
(reduced) 		50
(50) 		79
(79) 		100
(100) 		116
(116) 		129
(129) 		140
(0) 		150
(10) 		158
(18) 		166
(26) 		173
(33) 		179
(39) 		185
(45)
Approximation of harmonics in 129ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +2.3 -2.3 +4.6 +3.0 +0.0 -2.4 +6.9 -4.6 +5.3 +8.7 +2.3 +8.0
Relative (%) +9.6 -9.6 +19.2 +12.6 +0.0 -9.8 +28.8 -19.2 +22.2 +36.0 +9.6 +33.3
Steps
(reduced) 		50
(50) 		79
(79) 		100
(100) 		116
(116) 		129
(0) 		140
(11) 		150
(21) 		158
(29) 		166
(37) 		173
(44) 		179
(50) 		185
(56)
Approximation of harmonics in 116ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +1.0 -4.4 +2.0 +0.0 -3.4 -6.0 +3.0 -8.8 +1.0 +4.1 -2.4 +3.2
Relative (%) +4.2 -18.2 +8.3 +0.0 -14.1 -25.1 +12.5 -36.5 +4.2 +17.2 -9.9 +13.2
Steps
(reduced) 		50
(50) 		79
(79) 		100
(100) 		116
(0) 		129
(13) 		140
(24) 		150
(34) 		158
(42) 		166
(50) 		173
(57) 		179
(63) 		185
(69)
Approximation of harmonics in 79edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +3.8 +0.0 +7.5 +6.4 +3.8 +1.7 +11.3 +0.0 +10.2 -10.4 +7.5 -10.7
Relative (%) +15.7 +0.0 +31.3 +26.7 +15.7 +7.2 +47.0 +0.0 +42.4 -43.0 +31.3 -44.3
Steps
(reduced) 		50
(50) 		79
(0) 		100
(21) 		116
(37) 		129
(50) 		140
(61) 		150
(71) 		158
(0) 		166
(8) 		172
(14) 		179
(21) 		184
(26)
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