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= Title2 =
= Title2 =
== Octave stretch ==
== Octave stretch or compression ==
Pianos are frequently tuned with stretched octaves anyway due to the slight inharmonicity inherent in their strings, which makes 19edo a promising option for pianos with split sharps. Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-just. For example, if we are using [[49ed6]] or [[30ed3]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57{{c}}, respectively), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well.
58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[92edt]] or [[150ed6]].


What follows is a comparison of stretched-octave 19edo tunings.
What follows is a comparison of stretched- and compressed-octave 58edo tunings.


; 19edo
; [[zpi|288zpi]]
* Step size: 63.158{{c}}, octave size: 1200.0{{c}}  
* Step size: 20.736{{c}}, octave size: 1202.69{{c}}
Pure-octaves 19edo approximates all harmonics up to 16 within NNN{{c}}.
Stretching the octave of 58edo by around 2.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 288zpi does this.
{{Harmonics in equal|19|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo}}
{{Harmonics in cet|20.736|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 288zpi}}
{{Harmonics in equal|19|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo (continued)}}
{{Harmonics in cet|20.736|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 288zpi (continued)}}


; [[WE|19et, 2.3.5.11 WE tuning]]
; 58edo
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: 20.690{{c}}, octave size: 1200.00{{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 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.
Pure-octaves 58edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in cet|63.192|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 2.3.5.11 WE tuning}}
{{Harmonics in equal|58|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edo}}
{{Harmonics in cet|63.192|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 2.3.5.11 WE tuning (continued)}}
{{Harmonics in equal|58|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edo (continued)}}


; [[WE|19et, 13-limit WE tuning]]  
; [[150ed6]]  
* Step size: 63.291{{c}}, octave size: NNN{{c}}
* Step size: 20.680{{c}}, octave size: 1199.42{{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 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
Compressing the octave of 58edo by around half a cent results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 150ed6 does this.
{{Harmonics in cet|63.291|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning}}
{{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed6}}
{{Harmonics in cet|63.291|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning (continued)}}
{{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed6 (continued)}}


; [[49ed6]]  
; [[92edt]]  
* Step size: NNN{{c}}, octave size: 1202.8{{c}}
* Step size: 20.673{{c}}, octave size: 1199.06{{c}}
_ing the octave of 19edo 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 49ed6 does this.
Compressing the octave of 58edo by around 1{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 92edt does this.
{{Harmonics in equal|49|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6}}
{{Harmonics in equal|92|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 92edt}}
{{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6 (continued)}}
{{Harmonics in equal|92|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 92edt (continued)}}


; [[zpi|65zpi]]  
; [[zpi|289zpi]] / [[WE|58et, 7-limit WE tuning]]  
* Step size: 63.331{{c}}, octave size: 1203.3{{c}}
* Step size: 20.666{{c}}, octave size: 1198.63{{c}}
_ing the octave of 19edo 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 65zpi does this.
Compressing the octave of 58edo by just under 1.5{{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. The tuning 289zpi also does this, its octave differing from 7-limit WE by only 0.06{{c}}.  
{{Harmonics in cet|63.331|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi}}
{{Harmonics in cet|20.666|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 289zpi}}
{{Harmonics in cet|63.331|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi (continued)}}
{{Harmonics in cet|20.666|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 289zpi (continued)}}


; [[30edt]]  
; [[WE|58et, 13-limit WE tuning]]  
* Step size: NNN{{c}}, octave size: 1204.6{{c}}
* Step size: 20.663{{c}}, octave size: 1198.45{{c}}
_ing the octave of 19edo 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 30edt does this.
Compressing the octave of 58edo by just over 1.5{{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.
{{Harmonics in equal|30|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt}}
{{Harmonics in cet|20.663|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning}}
{{Harmonics in equal|30|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt (continued)}}
{{Harmonics in cet|20.663|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58et, 13-limit WE tuning (continued)}}
 
; [[11edf]]
* Step size: NNN{{c}}, octave size: 1212.5{{c}}
_ing the octave of 19edo 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 11edf does this.
{{Harmonics in equal|11|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edf}}
{{Harmonics in equal|11|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edf (continued)}}

Latest revision as of 23:42, 26 August 2025

Title1

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.1 -8.5 -8.2 +4.1 -12.6 +19.5 -12.3 -16.9 +0.0 +34.3 -16.7
Relative (%) -4.1 -8.5 -8.2 +4.1 -12.6 +19.6 -12.4 -17.0 +0.0 +34.4 -16.7
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(28) 		31
(31) 		34
(34) 		36
(36) 		38
(38) 		40
(0) 		42
(2) 		43
(3)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.4 +3.4 +6.7 +21.5 +6.7 +40.7 +10.1 +6.7 +24.9 -39.9 +10.1
Relative (%) +3.3 +3.3 +6.7 +21.4 +6.7 +40.6 +10.0 +6.7 +24.8 -39.8 +10.0
Steps
(reduced) 		12
(5) 		19
(5) 		24
(3) 		28
(0) 		31
(3) 		34
(6) 		36
(1) 		38
(3) 		40
(5) 		41
(6) 		43
(1)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.2 +0.0 +2.5 +16.6 +1.2 +34.7 +3.7 +0.0 +17.8 -47.1 +2.5
Relative (%) +1.2 +0.0 +2.5 +16.6 +1.2 +34.6 +3.7 +0.0 +17.8 -47.1 +2.5
Steps
(reduced) 		12
(12) 		19
(0) 		24
(5) 		28
(9) 		31
(12) 		34
(15) 		36
(17) 		38
(0) 		40
(2) 		41
(3) 		43
(5)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.8 -0.8 +1.5 +15.5 +0.0 +33.3 +2.3 -1.5 +16.2 -48.7 +0.8
Relative (%) +0.8 -0.8 +1.5 +15.4 +0.0 +33.3 +2.3 -1.5 +16.2 -48.7 +0.8
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(28) 		31
(0) 		34
(3) 		36
(5) 		38
(7) 		40
(9) 		41
(10) 		43
(12)

Title2

Octave stretch or compression

58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as 92edt or 150ed6.

What follows is a comparison of stretched- and compressed-octave 58edo tunings.

288zpi
  • Step size: 20.736 ¢, octave size: 1202.69 ¢

Stretching the octave of 58edo by around 2.5 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 288zpi does this.

Approximation of harmonics in 288zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.69 +5.76 +5.38 -7.69 +8.44 -9.59 +8.06 -9.22 -5.00 -4.12 -9.60
Relative (%) +13.0 +27.8 +25.9 -37.1 +40.7 -46.3 +38.9 -44.5 -24.1 -19.9 -46.3
Step 58 92 116 134 150 162 174 183 192 200 207
Approximation of harmonics in 288zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.02 -6.91 -1.93 -9.98 +9.48 -6.53 +3.54 -2.31 -3.84 -1.43 +4.56 -6.92
Relative (%) -14.6 -33.3 -9.3 -48.1 +45.7 -31.5 +17.1 -11.2 -18.5 -6.9 +22.0 -33.3
Step 214 220 226 231 237 241 246 250 254 258 262 265
58edo
  • Step size: 20.690 ¢, octave size: 1200.00 ¢

Pure-octaves 58edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 58edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.00 +1.49 +0.00 +6.79 +1.49 +3.59 +0.00 +2.99 +6.79 +7.30 +1.49
Relative (%) +0.0 +7.2 +0.0 +32.8 +7.2 +17.3 +0.0 +14.4 +32.8 +35.3 +7.2
Steps
(reduced) 		58
(0) 		92
(34) 		116
(0) 		135
(19) 		150
(34) 		163
(47) 		174
(0) 		184
(10) 		193
(19) 		201
(27) 		208
(34)
Approximation of harmonics in 58edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +7.75 +3.59 +8.28 +0.00 -1.51 +2.99 -7.86 +6.79 +5.08 +7.30 -7.58 +1.49
Relative (%) +37.4 +17.3 +40.0 +0.0 -7.3 +14.4 -38.0 +32.8 +24.6 +35.3 -36.7 +7.2
Steps
(reduced) 		215
(41) 		221
(47) 		227
(53) 		232
(0) 		237
(5) 		242
(10) 		246
(14) 		251
(19) 		255
(23) 		259
(27) 		262
(30) 		266
(34)
150ed6
  • Step size: 20.680 ¢, octave size: 1199.42 ¢

Compressing the octave of 58edo by around half a cent results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 150ed6 does this.

Approximation of harmonics in 150ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.58 +0.58 -1.15 +5.45 +0.00 +1.97 -1.73 +1.15 +4.87 +5.30 -0.58
Relative (%) -2.8 +2.8 -5.6 +26.3 +0.0 +9.5 -8.4 +5.6 +23.5 +25.6 -2.8
Steps
(reduced) 		58
(58) 		92
(92) 		116
(116) 		135
(135) 		150
(0) 		163
(13) 		174
(24) 		184
(34) 		193
(43) 		201
(51) 		208
(58)
Approximation of harmonics in 150ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.61 +1.39 +6.02 -2.31 -3.87 +0.58 -10.31 +4.29 +2.54 +4.72 -10.19 -1.15
Relative (%) +27.1 +6.7 +29.1 -11.2 -18.7 +2.8 -49.8 +20.7 +12.3 +22.8 -49.3 -5.6
Steps
(reduced) 		215
(65) 		221
(71) 		227
(77) 		232
(82) 		237
(87) 		242
(92) 		246
(96) 		251
(101) 		255
(105) 		259
(109) 		262
(112) 		266
(116)
92edt
  • Step size: 20.673 ¢, octave size: 1199.06 ¢

Compressing the octave of 58edo by around 1 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 92edt does this.

Approximation of harmonics in 92edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.94 +0.00 -1.88 +4.60 -0.94 +0.94 -2.82 +0.00 +3.66 +4.04 -1.88
Relative (%) -4.6 +0.0 -9.1 +22.2 -4.6 +4.6 -13.7 +0.0 +17.7 +19.5 -9.1
Steps
(reduced) 		58
(58) 		92
(0) 		116
(24) 		135
(43) 		150
(58) 		163
(71) 		174
(82) 		184
(0) 		193
(9) 		201
(17) 		208
(24)
Approximation of harmonics in 92edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.26 +0.00 +4.60 -3.77 -5.35 -0.94 +8.82 +2.72 +0.94 +3.10 +8.84 -2.82
Relative (%) +20.6 +0.0 +22.2 -18.2 -25.9 -4.6 +42.7 +13.1 +4.6 +15.0 +42.7 -13.7
Steps
(reduced) 		215
(31) 		221
(37) 		227
(43) 		232
(48) 		237
(53) 		242
(58) 		247
(63) 		251
(67) 		255
(71) 		259
(75) 		263
(79) 		266
(82)
289zpi / 58et, 7-limit WE tuning
  • Step size: 20.666 ¢, octave size: 1198.63 ¢

Compressing the octave of 58edo by just under 1.5 ¢ 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. The tuning 289zpi also does this, its octave differing from 7-limit WE by only 0.06 ¢.

Approximation of harmonics in 289zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.37 -0.68 -2.74 +3.60 -2.06 -0.27 -4.12 -1.37 +2.22 +2.55 -3.43
Relative (%) -6.6 -3.3 -13.3 +17.4 -9.9 -1.3 -19.9 -6.6 +10.8 +12.3 -16.6
Step 58 92 116 135 150 163 174 184 193 201 208
Approximation of harmonics in 289zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.66 -1.64 +2.91 -5.49 -7.11 -2.74 +6.99 +0.85 -0.95 +1.18 +6.88 -4.80
Relative (%) +12.9 -7.9 +14.1 -26.6 -34.4 -13.2 +33.8 +4.1 -4.6 +5.7 +33.3 -23.2
Step 215 221 227 232 237 242 247 251 255 259 263 266
58et, 13-limit WE tuning
  • Step size: 20.663 ¢, octave size: 1198.45 ¢

Compressing the octave of 58edo by just over 1.5 ¢ 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 58et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.55 -0.96 -3.09 +3.19 -2.51 -0.76 -4.64 -1.92 +1.65 +1.95 -4.05
Relative (%) -7.5 -4.6 -15.0 +15.4 -12.1 -3.7 -22.4 -9.3 +8.0 +9.4 -19.6
Step 58 92 116 135 150 163 174 184 193 201 208
Approximation of harmonics in 58et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.02 -2.30 +2.23 -6.18 -7.82 -3.46 +6.25 +0.10 -1.72 +0.40 +6.09 -5.60
Relative (%) +9.8 -11.1 +10.8 -29.9 -37.9 -16.8 +30.2 +0.5 -8.3 +1.9 +29.5 -27.1
Step 215 221 227 232 237 242 247 251 255 259 263 266
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