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= Title2 =
= Title2 =
== Octave compression ==
== Octave stretch ==
What follows is a comparison of compressed-octave 17edo tunings.
What follows is a comparison of stretched-octave 19edo tunings.


; 17edo
; 19edo
* Step size: 70.588{{c}}, octave size: 1200.0{{c}}  
* Step size: 63.158{{c}}, octave size: 1200.0{{c}}  
Pure-octaves 17edo approximates the 2.3.11 subgroup well, it arguably might approximate 7, but not well, and it doesn't really approximate 5. It might make tuning for exploring new harmonies with the 7th, 11th and 13th harmonics not found in 12edo, but its very sharp 7th harmonic might arguably hamper that use case.
Pure-octaves 19edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|17|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo}}
{{Harmonics in equal|19|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edo}}
{{Harmonics in equal|17|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo (continued)}}
{{Harmonics in equal|190|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONAME (continued)}}


; [[44ed6]]  
; [[WE|19et, 2.3.5.11 WE tuning]]  
* Step size: NNN{{c}}, octave size: 1198.5{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 17edo by around 1.5{{c}} results in greatly improved primes 3, 7, 11 and 13, but a slightly worse prime 2. The tuning 44ed6 does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s [[13-limit]] tuning for its size.
_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.
{{Harmonics in equal|44|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6}}
{{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|44|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6 (continued)}}
{{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)}}


; [[27edt]]  
; [[WE|19et, 13-limit WE tuning]]  
* Step size: NNN{{c}}, octave size: 1197.5{{c}}
* Step size: 63.291{{c}}, octave size: NNN{{c}}
Compressing the octave of 17edo by around 2.5{{c}} results in improved primes NNN, but worse primes NNN. The tuning 27edt does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s [[13-limit]] tuning for its size.
_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.
{{Harmonics in equal|27|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt}}
{{Harmonics in cet|63.291|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning}}
{{Harmonics in equal|27|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt (continued)}}
{{Harmonics in cet|63.291|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning (continued)}}


; [[zpi|56zpi]] / [[WE|17et, 2.3.7.11.13 WE tuning]]
; [[zpi|ZPINAME]]  
* Step size: 70.403{{c}}, octave size: 1296.9{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 17edo by around 3{{c}} results in improved primes NNN, but worse primes NNN. The tunings: 56zpi, [[TE|17et, 2.3.7.11.13 TE]] and [[WE|17et, 2.3.7.11.13 WE]] all do this. Their primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s [[13-limit]] tuning for its size.
_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|70.403|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi}}
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|70.403|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi (continued)}}
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME (continued)}}


; [[WE|17et, 2.3.7.11 WE tuning]]  
; [[49ed6]]
* Step size: 70.392{{c}}, octave size: 1296.7{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 17edo by just over 3{{c}} results in improved primes NNN, but worse primes NNN. Its 2.3.7.11 WE tuning and 2.3.7.11 [[TE]] tuning both do this. Their primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s [[13-limit]] tuning for its size.
_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.
{{Harmonics in cet|70.392|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11 WE tuning}}
{{Harmonics in equal|49|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6}}
{{Harmonics in cet|70.392|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17et, 2.3.7.11 WE tuning (continued)}}
{{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6 (continued)}}
 
; [[30edt]]  
* 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.
{{Harmonics in equal|30|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt}}
{{Harmonics in equal|30|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt (continued)}}
 
; [[11edf]]  
* 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.
{{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)}}

Revision as of 00:29, 23 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

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

19edo
  • Step size: 63.158 ¢, octave size: 1200.0 ¢

Pure-octaves 19edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 19edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -7.2 +0.0 -7.4 -7.2 -21.5 +0.0 -14.4 -7.4 +17.1 -7.2
Relative (%) +0.0 -11.4 +0.0 -11.7 -11.4 -34.0 +0.0 -22.9 -11.7 +27.1 -11.4
Steps
(reduced) 		19
(0) 		30
(11) 		38
(0) 		44
(6) 		49
(11) 		53
(15) 		57
(0) 		60
(3) 		63
(6) 		66
(9) 		68
(11)
Approximation of harmonics in EDONAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.53 -2.51 -1.95 +0.00 +2.41 -1.80 -0.67 -1.05 +2.90 -1.84 -3.01 -0.90
Relative (%) -8.4 -39.7 -30.9 +0.0 +38.2 -28.6 -10.6 -16.6 +46.0 -29.2 -47.7 -14.3
Steps
(reduced) 		703
(133) 		723
(153) 		742
(172) 		760
(0) 		777
(17) 		792
(32) 		807
(47) 		821
(61) 		835
(75) 		847
(87) 		859
(99) 		871
(111)
19et, 2.3.5.11 WE tuning
  • 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 ¢. Its 2.3.5.11 WE tuning and 2.3.5.11 TE tuning both do this.

Approximation of harmonics in 19et, 2.3.5.11 WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.6 -6.2 +1.3 -5.9 -5.5 -19.6 +1.9 -12.4 -5.2 +19.4 -4.9
Relative (%) +1.0 -9.8 +2.1 -9.3 -8.8 -31.1 +3.1 -19.6 -8.3 +30.6 -7.8
Step 19 30 38 44 49 53 57 60 63 66 68
Approximation of harmonics in 19et, 2.3.5.11 WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -17.1 -19.0 -12.1 +2.6 +24.0 -11.7 +21.0 -4.6 -25.8 +20.0 +6.2 -4.3
Relative (%) -27.0 -30.1 -19.1 +4.1 +38.0 -18.6 +33.3 -7.2 -40.9 +31.7 +9.9 -6.7
Step 70 72 74 76 78 79 81 82 83 85 86 87
19et, 13-limit WE tuning
  • Step size: 63.291 ¢, 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 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 19et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.5 -3.2 +5.1 -1.5 -0.7 -14.4 +7.6 -6.5 +1.0 +25.9 +1.8
Relative (%) +4.0 -5.1 +8.0 -2.4 -1.1 -22.8 +12.0 -10.2 +1.6 +40.9 +2.9
Step 19 30 38 44 49 53 57 60 63 66 68
Approximation of harmonics in 19et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -10.2 -11.9 -4.7 +10.1 -31.5 -3.9 +29.1 +3.5 -17.6 +28.4 +14.8 +4.4
Relative (%) -16.0 -18.8 -7.5 +16.0 -49.8 -6.2 +45.9 +5.6 -27.9 +44.9 +23.3 +6.9
Step 70 72 74 76 77 79 81 82 83 85 86 87
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
49ed6
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 19edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 49ed6 does this.

Approximation of harmonics in 49ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.8 -2.8 +5.6 -0.9 +0.0 -13.7 +8.4 -5.6 +1.9 +26.8 +2.8
Relative (%) +4.4 -4.4 +8.8 -1.4 +0.0 -21.6 +13.3 -8.8 +3.0 +42.4 +4.4
Steps
(reduced) 		19
(19) 		30
(30) 		38
(38) 		44
(44) 		49
(0) 		53
(4) 		57
(8) 		60
(11) 		63
(14) 		66
(17) 		68
(19)
Approximation of harmonics in 49ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -9.2 -10.9 -3.7 +11.2 -30.5 -2.8 +30.2 +4.7 -16.4 +29.6 +16.0 +5.6
Relative (%) -14.5 -17.1 -5.8 +17.7 -48.1 -4.4 +47.7 +7.4 -26.0 +46.8 +25.2 +8.8
Steps
(reduced) 		70
(21) 		72
(23) 		74
(25) 		76
(27) 		77
(28) 		79
(30) 		81
(32) 		82
(33) 		83
(34) 		85
(36) 		86
(37) 		87
(38)
30edt
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 19edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 30edt does this.

Approximation of harmonics in 30edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.6 +0.0 +9.1 +3.2 +4.6 -8.7 +13.7 +0.0 +7.8 -30.4 +9.1
Relative (%) +7.2 +0.0 +14.4 +5.1 +7.2 -13.7 +21.6 +0.0 +12.3 -48.0 +14.4
Steps
(reduced) 		19
(19) 		30
(0) 		38
(8) 		44
(14) 		49
(19) 		53
(23) 		57
(27) 		60
(0) 		63
(3) 		65
(5) 		68
(8)
Approximation of harmonics in 30edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.6 -4.1 +3.2 +18.3 -23.3 +4.6 -25.6 +12.4 -8.7 -25.8 +24.0 +13.7
Relative (%) -4.2 -6.5 +5.1 +28.8 -36.7 +7.2 -40.4 +19.5 -13.7 -40.8 +37.9 +21.6
Steps
(reduced) 		70
(10) 		72
(12) 		74
(14) 		76
(16) 		77
(17) 		79
(19) 		80
(20) 		82
(22) 		83
(23) 		84
(24) 		86
(26) 		87
(27)
11edf
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 19edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 11edf does this.

Approximation of harmonics in 11edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +12.5 +12.5 +24.9 +21.5 +24.9 +13.3 -26.4 +24.9 -29.8 -3.4 -26.4
Relative (%) +19.5 +19.5 +39.1 +33.7 +39.1 +20.9 -41.4 +39.1 -46.8 -5.3 -41.4
Steps
(reduced) 		19
(8) 		30
(8) 		38
(5) 		44
(0) 		49
(5) 		53
(9) 		56
(1) 		60
(5) 		62
(7) 		65
(10) 		67
(1)
Approximation of harmonics in 11edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +26.5 +25.8 -29.8 -13.9 +8.7 -26.4 +7.6 -17.4 +25.8 +9.1 -4.1 -13.9
Relative (%) +41.5 +40.4 -46.8 -21.8 +13.7 -41.4 +11.9 -27.2 +40.4 +14.2 -6.4 -21.8
Steps
(reduced) 		70
(4) 		72
(6) 		73
(7) 		75
(9) 		77
(0) 		78
(1) 		80
(3) 		81
(4) 		83
(6) 		84
(7) 		85
(8) 		86
(9)
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