User:BudjarnLambeth/Sandbox2: Difference between revisions

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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.
What follows is a comparison of stretched- and compressed-octave 22edo tunings.


What follows is a comparison of stretched-octave 19edo tunings.
; [[zpi|ZPINAME]]
 
* Step size: NNN{{c}}, octave size: NNN{{c}}
; 19edo
_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.
* Step size: 63.158{{c}}, octave size: 1200.0{{c}}  
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}}
Pure-octaves 19edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME (continued)}}
{{Harmonics in equal|19|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo}}
{{Harmonics in equal|19|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo (continued)}}


; [[WE|19et, 2.3.5.11 WE tuning]]  
; [[EDONOI]]  
* 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}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[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}}. The tuning EDONOI does this.
{{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|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}}
{{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|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}


; [[WE|19et, 13-limit WE tuning]]
; 22edo
* Step size: 63.291{{c}}, octave size: NNN{{c}}
* Step size: 54.545{{c}}, octave size: 1200.0{{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.
Pure-octaves 22edo approximates all harmonics up to 16 within NNN{{c}}. The optimal 13-limit [[WE]] tuning has octaves only 0.01{{c}} different from pure-octaves 22edo, and the 13-limit [[TE]] tuning has octaves only 0.08{{c}} different, so by those metrics pure-octaves 22edo might be considered already optimal.
{{Harmonics in cet|63.291|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning}}
{{Harmonics in equal|22|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo}}
{{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|22|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 22edo (continued)}}


; [[49ed6]]  
; [[WE|22et, 11-limit WE tuning]]  
* Step size: NNN{{c}}, octave size: 1202.8{{c}}
* Step size: 54.494{{c}}, octave size: NNN{{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 22edo by around half a cent results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
{{Harmonics in equal|49|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6}}
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6 (continued)}}
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}


; [[zpi|65zpi]]  
; [[zpi|80zpi]]  
* Step size: 63.331{{c}}, octave size: 1203.3{{c}}
* Step size: 54.483{{c}}, octave size: NNN{{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 22edo 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 80zpi does this.
{{Harmonics in cet|63.331|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi}}
{{Harmonics in cet|54.483|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi}}
{{Harmonics in cet|63.331|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi (continued)}}
{{Harmonics in cet|54.483|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80zpi (continued)}}


; [[30edt]]  
; [[13edf]]  
* Step size: NNN{{c}}, octave size: 1204.6{{c}}
* Step size: NNN{{c}}, octave size: NNN{{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.
_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|30|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt}}
{{Harmonics in equal|13|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|30|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt (continued)}}
{{Harmonics in equal|13|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}}


; [[11edf]]  
; [[35edt]]  
* Step size: NNN{{c}}, octave size: 1212.5{{c}}
* Step size: NNN{{c}}, octave size: NNN{{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.
_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|11|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edf}}
{{Harmonics in equal|35|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|11|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edf (continued)}}
{{Harmonics in equal|35|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}}

Revision as of 01:08, 24 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

What follows is a comparison of stretched- and compressed-octave 22edo tunings.

ZPINAME
  • 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 ZPINAME does this.

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Step 44 46 47 48 49 50 51 52 53 54 54 55
EDONOI
  • 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.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Steps
(reduced) 		12
(0) 		19
(7) 		24
(0) 		28
(4) 		31
(7) 		34
(10) 		36
(0) 		38
(2) 		40
(4) 		42
(6) 		43
(7)

{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}

22edo
  • Step size: 54.545 ¢, octave size: 1200.0 ¢

Pure-octaves 22edo approximates all harmonics up to 16 within NNN ¢. The optimal 13-limit WE tuning has octaves only 0.01 ¢ different from pure-octaves 22edo, and the 13-limit TE tuning has octaves only 0.08 ¢ different, so by those metrics pure-octaves 22edo might be considered already optimal.

Approximation of harmonics in 22edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +7.1 +0.0 -4.5 +7.1 +13.0 +0.0 +14.3 -4.5 -5.9 +7.1
Relative (%) +0.0 +13.1 +0.0 -8.2 +13.1 +23.8 +0.0 +26.2 -8.2 -10.7 +13.1
Steps
(reduced) 		22
(0) 		35
(13) 		44
(0) 		51
(7) 		57
(13) 		62
(18) 		66
(0) 		70
(4) 		73
(7) 		76
(10) 		79
(13)
Approximation of harmonics in 22edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -22.3 +13.0 +2.6 +0.0 +4.1 +14.3 -24.8 -4.5 +20.1 -5.9 +26.3 +7.1
Relative (%) -41.0 +23.8 +4.8 +0.0 +7.6 +26.2 -45.4 -8.2 +36.9 -10.7 +48.2 +13.1
Steps
(reduced) 		81
(15) 		84
(18) 		86
(20) 		88
(0) 		90
(2) 		92
(4) 		93
(5) 		95
(7) 		97
(9) 		98
(10) 		100
(12) 		101
(13)
22et, 11-limit WE tuning
  • Step size: 54.494 ¢, octave size: NNN ¢

Compressing the octave of 22edo by around half a cent results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 11-limit WE tuning and 11-limit TE tuning both do this.

Approximation of harmonics in ETNAME, SUBGROUP WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Step 44 46 47 48 49 50 51 52 53 54 54 55
80zpi
  • Step size: 54.483 ¢, octave size: NNN ¢

Compressing the octave of 22edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 80zpi does this.

Approximation of harmonics in 80zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.4 +4.9 -2.7 -7.7 +3.6 +9.1 -4.1 +9.9 -9.1 -10.6 +2.2
Relative (%) -2.5 +9.1 -5.0 -14.1 +6.6 +16.7 -7.6 +18.2 -16.6 -19.5 +4.0
Step 22 35 44 51 57 62 66 70 73 76 79
Approximation of harmonics in 80zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +27.1 +7.7 -2.7 -5.5 -1.5 +8.5 +23.9 -10.4 +14.1 -12.0 +20.0 +0.8
Relative (%) +49.7 +14.2 -5.0 -10.1 -2.7 +15.6 +43.8 -19.1 +25.8 -22.0 +36.8 +1.5
Step 82 84 86 88 90 92 94 95 97 98 100 101
13edf
  • 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.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -12.1 -12.1 -24.2 +21.5 -24.2 -21.0 +17.8 -24.2 +9.4 +6.4 +17.8
Relative (%) -22.4 -22.4 -44.7 +39.8 -44.7 -39.0 +32.9 -44.7 +17.5 +11.9 +32.9
Steps
(reduced) 		22
(9) 		35
(9) 		44
(5) 		52
(0) 		57
(5) 		62
(10) 		67
(2) 		70
(5) 		74
(9) 		77
(12) 		80
(2)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -12.8 +20.9 +9.4 +5.7 +8.7 +17.8 -21.8 -2.6 +20.9 -5.7 +25.4 +5.7
Relative (%) -23.7 +38.7 +17.5 +10.5 +16.2 +32.9 -40.4 -4.9 +38.7 -10.5 +47.0 +10.5
Steps
(reduced) 		82
(4) 		85
(7) 		87
(9) 		89
(11) 		91
(0) 		93
(2) 		94
(3) 		96
(5) 		98
(7) 		99
(8) 		101
(10) 		102
(11)
35edt
  • 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.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.5 +0.0 -9.0 -14.9 -4.5 +0.4 -13.5 +0.0 -19.4 -21.4 -9.0
Relative (%) -8.3 +0.0 -16.5 -27.4 -8.3 +0.6 -24.8 +0.0 -35.7 -39.3 -16.5
Steps
(reduced) 		22
(22) 		35
(0) 		44
(9) 		51
(16) 		57
(22) 		62
(27) 		66
(31) 		70
(0) 		73
(3) 		76
(6) 		79
(9)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +15.5 -4.1 -14.9 -17.9 -14.2 -4.5 +10.6 -23.9 +0.4 -25.8 +5.9 -13.5
Relative (%) +28.5 -7.6 -27.4 -33.0 -26.2 -8.3 +19.5 -43.9 +0.6 -47.6 +10.8 -24.8
Steps
(reduced) 		82
(12) 		84
(14) 		86
(16) 		88
(18) 		90
(20) 		92
(22) 		94
(24) 		95
(25) 		97
(27) 		98
(28) 		100
(30) 		101
(31)
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