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= Title2 =
= Title2 =
== Octave stretch or compression ==
== Octave stretch or compression ==
What follows is a comparison of stretched- and compressed-octave 27edo tunings.
What follows is a comparison of compressed-octave 27edo tunings.
 
; [[zpi|105zpi]]
* Step size: 44.674{{c}}, octave size: NNN{{c}}
Stretching the octave of 27edo 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 105zpi does this.
{{Harmonics in cet|44.674|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 105zpi}}
{{Harmonics in cet|44.674|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 105zpi (continued)}}


; 27edo
; 27edo
* Step size: 44.444{{c}}, octave size: 1200.0{{c}}  
* Step size: 44.444{{c}}, octave size: 1200.0{{c}}  
Pure-octaves 27edo approximates all harmonics up to 16 within NNN{{c}}.
Pure-octaves 27edo approximates all harmonics up to 16 within 18.3{{c}}.
{{Harmonics in equal|27|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo}}
{{Harmonics in equal|27|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo}}
{{Harmonics in equal|27|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo (continued)}}
{{Harmonics in equal|27|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo (continued)}}


; [[WE|27et, 13-limit WE tuning]]  
; [[WE|27et, 13-limit WE tuning]]  
* Step size: 44.375{{c}}, octave size: NNN{{c}}
* Step size: 44.375{{c}}, octave size: 1198.9{{c}}
Compressing the octave of 27edo 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 27edo by around 2{{c}} results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 within 19.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|44.375|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning}}
{{Harmonics in cet|44.375|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning}}
{{Harmonics in cet|44.375|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning (continued)}}
{{Harmonics in cet|44.375|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning (continued)}}
; [[WE|27et, 7-limit WE tuning]]
* Step size: 44.306{{c}}, octave size: NNN{{c}}
Compressing the octave of 27edo by around NNN{{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.
{{Harmonics in cet|44.306|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 7-limit WE tuning}}
{{Harmonics in cet|44.306|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 7-limit WE tuning (continued)}}
; [[zpi|106zpi]]
* Step size: 44.302{{c}}, octave size: NNN{{c}}
Compressing the octave of 27edo 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 106zpi does this.
{{Harmonics in cet|44.302|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi}}
{{Harmonics in cet|44.302|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi (continued)}}


; [[97ed12]]  
; [[97ed12]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: NNN{{c}}, octave size: 1197.5{{c}}
_ing the octave of 27edo 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 97ed12 does this.
Compressing the octave of 27edo by around 2.5{{c}} has the same benefits and drawbacks as the 13-limit tuning, but both are slightly amplified. This approximates all harmonics up to 16 within 17.6{{c}}. The tuning 97ed12 does this.
{{Harmonics in equal|97|12|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12}}
{{Harmonics in equal|97|12|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12}}
{{Harmonics in equal|97|12|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12 (continued)}}
{{Harmonics in equal|97|12|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12 (continued)}}


; [[70ed6]]  
; [[zpi|106zpi]] / [[70ed6]] / [[WE|27et, 7-limit WE tuning]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: ~44.306{{c}}, octave size: ~1196.2{{c}}
_ing the octave of 27edo 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 70ed6 does this.
Compressing the octave of 27edo by around 3.5{{c}} results in even more improvement to primes 3, 5 and 7 than the 13-limit tuning, but now at the cost of moderate damage to 2, 11 and 13. This approximates all harmonics up to 16 within 15.4{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So do the tunings 106zpi and 70ed6.
{{Harmonics in equal|70|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 70ed6}}
{{Harmonics in cet|44.306|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi}}
{{Harmonics in equal|70|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 70ed6 (continued)}}
{{Harmonics in cet|44.306|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi (continued)}}
 
; [[90ed10]]  
* Step size: NNN{{c}}, octave size: 1195.9{{c}}
Compressing the octave of 27edo by around 5.5{{c}} results in improved primes 3, 5, 7 and 11, but a worse prime 2 and much worse 13. This approximates all harmonics up to 16 within 16.4{{c}}. The tuning 90ed10 does this.
{{Harmonics in equal|90|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10}}
{{Harmonics in equal|90|10|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10 (continued)}}


; [[43edt]]  
; [[43edt]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: NNN{{c}}, octave size: 1204.3{{c}}
_ing the octave of 27edo 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 43edt does this.
Compressing the octave of 27edo by around 5.5{{c}} results in the same benefits and drawbacks as 90ed10, but amplified. This approximates all harmonics up to 16 within 21.2{{c}}. The tuning 43edt does this.
{{Harmonics in equal|43|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt}}
{{Harmonics in equal|43|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt}}
{{Harmonics in equal|43|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt (continued)}}
{{Harmonics in equal|43|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt (continued)}}
; [[90ed10]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of 27edo 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 90ed10 does this.
{{Harmonics in equal|90|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10}}
{{Harmonics in equal|90|10|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10 (continued)}}

Revision as of 00:13, 25 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 compressed-octave 27edo tunings.

27edo
  • Step size: 44.444 ¢, octave size: 1200.0 ¢

Pure-octaves 27edo approximates all harmonics up to 16 within 18.3 ¢.

Approximation of harmonics in 27edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +9.2 +0.0 +13.7 +9.2 +9.0 +0.0 +18.3 +13.7 -18.0 +9.2
Relative (%) +0.0 +20.6 +0.0 +30.8 +20.6 +20.1 +0.0 +41.2 +30.8 -40.5 +20.6
Steps
(reduced) 		27
(0) 		43
(16) 		54
(0) 		63
(9) 		70
(16) 		76
(22) 		81
(0) 		86
(5) 		90
(9) 		93
(12) 		97
(16)
Approximation of harmonics in 27edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +3.9 +9.0 -21.6 +0.0 -16.1 +18.3 +13.6 +13.7 +18.1 -18.0 -6.1 +9.2
Relative (%) +8.8 +20.1 -48.6 +0.0 -36.1 +41.2 +30.6 +30.8 +40.7 -40.5 -13.6 +20.6
Steps
(reduced) 		100
(19) 		103
(22) 		105
(24) 		108
(0) 		110
(2) 		113
(5) 		115
(7) 		117
(9) 		119
(11) 		120
(12) 		122
(14) 		124
(16)
27et, 13-limit WE tuning
  • Step size: 44.375 ¢, octave size: 1198.9 ¢

Compressing the octave of 27edo by around 2 ¢ results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 within 19.9 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 27et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.9 +6.2 -3.7 +9.3 +4.3 +3.7 -5.6 +12.3 +7.4 +19.9 +2.4
Relative (%) -4.2 +13.9 -8.5 +21.0 +9.7 +8.3 -12.7 +27.8 +16.8 +44.9 +5.5
Step 27 43 54 63 70 76 81 86 90 94 97
Approximation of harmonics in 27et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.0 +1.8 +15.5 -7.5 +20.7 +10.5 +5.6 +5.6 +9.8 +18.1 -14.5 +0.5
Relative (%) -6.8 +4.1 +34.9 -16.9 +46.6 +23.6 +12.6 +12.5 +22.2 +40.7 -32.7 +1.2
Step 100 103 106 108 111 113 115 117 119 121 122 124
97ed12
  • Step size: NNN ¢, octave size: 1197.5 ¢

Compressing the octave of 27edo by around 2.5 ¢ has the same benefits and drawbacks as the 13-limit tuning, but both are slightly amplified. This approximates all harmonics up to 16 within 17.6 ¢. The tuning 97ed12 does this.

Approximation of harmonics in 97ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.5 +5.1 -5.1 +7.7 +2.5 +1.8 -7.6 +10.2 +5.2 +17.6 +0.0
Relative (%) -5.7 +11.5 -11.5 +17.5 +5.7 +4.0 -17.2 +23.0 +11.7 +39.7 +0.0
Steps
(reduced) 		27
(27) 		43
(43) 		54
(54) 		63
(63) 		70
(70) 		76
(76) 		81
(81) 		86
(86) 		90
(90) 		94
(94) 		97
(0)
Approximation of harmonics in 97ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.5 -0.8 +12.8 -10.2 +17.9 +7.6 +2.7 +2.6 +6.9 +15.0 -17.6 -2.5
Relative (%) -12.5 -1.7 +28.9 -23.0 +40.4 +17.2 +6.2 +6.0 +15.5 +33.9 -39.6 -5.7
Steps
(reduced) 		100
(3) 		103
(6) 		106
(9) 		108
(11) 		111
(14) 		113
(16) 		115
(18) 		117
(20) 		119
(22) 		121
(24) 		122
(25) 		124
(27)
106zpi / 70ed6 / 27et, 7-limit WE tuning
  • Step size: ~44.306 ¢, octave size: ~1196.2 ¢

Compressing the octave of 27edo by around 3.5 ¢ results in even more improvement to primes 3, 5 and 7 than the 13-limit tuning, but now at the cost of moderate damage to 2, 11 and 13. This approximates all harmonics up to 16 within 15.4 ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this. So do the tunings 106zpi and 70ed6.

Approximation of harmonics in 106zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.7 +3.2 -7.5 +5.0 -0.5 -1.6 -11.2 +6.4 +1.2 +13.4 -4.3
Relative (%) -8.4 +7.2 -16.9 +11.2 -1.2 -3.5 -25.3 +14.5 +2.8 +30.3 -9.6
Step 27 43 54 63 70 76 81 86 90 94 97
Approximation of harmonics in 106zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -9.9 -5.3 +8.2 -15.0 +13.0 +2.7 -2.3 -2.5 +1.6 +9.7 +21.4 -8.0
Relative (%) -22.4 -12.0 +18.4 -33.7 +29.4 +6.0 -5.2 -5.7 +3.7 +21.9 +48.2 -18.1
Step 100 103 106 108 111 113 115 117 119 121 123 124
90ed10
  • Step size: NNN ¢, octave size: 1195.9 ¢

Compressing the octave of 27edo by around 5.5 ¢ results in improved primes 3, 5, 7 and 11, but a worse prime 2 and much worse 13. This approximates all harmonics up to 16 within 16.4 ¢. The tuning 90ed10 does this.

Approximation of harmonics in 90ed10
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.1 +2.6 -8.2 +4.1 -1.5 -2.6 -12.3 +5.2 +0.0 +12.2 -5.6
Relative (%) -9.3 +5.9 -18.5 +9.3 -3.4 -5.9 -27.8 +11.8 +0.0 +27.5 -12.6
Steps
(reduced) 		27
(27) 		43
(43) 		54
(54) 		63
(63) 		70
(70) 		76
(76) 		81
(81) 		86
(86) 		90
(0) 		94
(4) 		97
(7)
Approximation of harmonics in 90ed10 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -11.3 -6.7 +6.7 -16.4 +11.5 +1.1 -3.9 -4.1 +0.0 +8.1 +19.7 -9.7
Relative (%) -25.5 -15.2 +15.2 -37.1 +26.0 +2.5 -8.8 -9.3 +0.0 +18.2 +44.4 -21.9
Steps
(reduced) 		100
(10) 		103
(13) 		106
(16) 		108
(18) 		111
(21) 		113
(23) 		115
(25) 		117
(27) 		119
(29) 		121
(31) 		123
(33) 		124
(34)
43edt
  • Step size: NNN ¢, octave size: 1204.3 ¢

Compressing the octave of 27edo by around 5.5 ¢ results in the same benefits and drawbacks as 90ed10, but amplified. This approximates all harmonics up to 16 within 21.2 ¢. The tuning 43edt does this.

Approximation of harmonics in 43edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.7 +0.0 -11.5 +0.3 -5.7 -7.2 -17.2 +0.0 -5.5 +6.4 -11.5
Relative (%) -13.0 +0.0 -26.0 +0.6 -13.0 -16.3 -39.0 +0.0 -12.4 +14.6 -26.0
Steps
(reduced) 		27
(27) 		43
(0) 		54
(11) 		63
(20) 		70
(27) 		76
(33) 		81
(38) 		86
(0) 		90
(4) 		94
(8) 		97
(11)
Approximation of harmonics in 43edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -17.4 -13.0 +0.3 +21.2 +4.7 -5.7 -10.9 -11.2 -7.2 +0.7 +12.2 -17.2
Relative (%) -39.3 -29.3 +0.6 +48.0 +10.7 -13.0 -24.6 -25.4 -16.3 +1.6 +27.6 -39.0
Steps
(reduced) 		100
(14) 		103
(17) 		106
(20) 		109
(23) 		111
(25) 		113
(27) 		115
(29) 		117
(31) 		119
(33) 		121
(35) 		123
(37) 		124
(38)
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