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= Title2 =
= Title2 =
== Octave stretch or compression ==
== Octave compression ==
What follows is a comparison of stretched- and compressed-octave 12edo tunings.
What follows is a comparison of compressed-octave 17edo tunings.


; [[WE|12et, 7-limit WE tuning]]
; 17edo
* Step size: 99.664{{c}}, octave size: 1196.0{{c}}
* Step size: 70.588{{c}}, octave size: 1200.0{{c}}  
Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but a much worse prime 3. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. [[40ed10]] does this as well. An argument could be made that such tunings [[7-limit|harmonies involving the 7th harmonic]] to regular old 12edo without even needing to add any new notes to the octave.
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.
{{Harmonics in cet|99.664|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning}}
{{Harmonics in equal|17|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo}}
{{Harmonics in cet|99.664|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}}
{{Harmonics in equal|17|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 17edo (continued)}}


; [[zpi|34zpi]]  
; [[44ed6]]  
* Step size: 99.807{{c}}, octave size: 1197.7{{c}}
* Step size: NNN{{c}}, octave size: 1198.5{{c}}
Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but a worse prime 3. The tuning 34zpi does this. It might be a good tuning for 5-limit [[meantone]], for composers seeking more pure thirds and sixths than regular 12edo.
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.
{{Harmonics in cet|99.807|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi}}
{{Harmonics in equal|44|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6}}
{{Harmonics in cet|99.807|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi (continued)}}
{{Harmonics in equal|44|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 44ed6 (continued)}}


; [[WE|12et, 5-limit WE tuning]]  
; [[27edt]]  
* Step size: 99.868{{c}}, octave size: 1198.4{{c}}
* Step size: NNN{{c}}, octave size: 1197.5{{c}}
Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but a slightly prime 3. Its 5-limit WE tuning and 5-limit [[TE]] tuning both do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.
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.
{{Harmonics in cet|99.868|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning}}
{{Harmonics in equal|27|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt}}
{{Harmonics in cet|99.868|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}}
{{Harmonics in equal|27|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edt (continued)}}


; 12edo
; [[zpi|56zpi]] / [[WE|17et, 2.3.7.11.13 WE tuning]]
* Step size: 100.000{{c}}, octave size: 1200.0{{c}}  
* Step size: 70.403{{c}}, octave size: 1296.9{{c}}
Pure-octaves 12edo performs well on harmonics 2, 3 and 5 but poorly on harmonics 7, 11 and 13 compared to other edos with a similar number of notes per octave.
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.
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12edo}}
{{Harmonics in cet|70.403|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi}}
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12edo (continued)}}
{{Harmonics in cet|70.403|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 56zpi (continued)}}


; [[31ed6]]
; [[WE|17et, 2.3.7.11 WE tuning]]  
* Step size: 100.063{{c}}, octave size: 1200.8{{c}}
* Step size: 70.392{{c}}, octave size: 1296.7{{c}}
Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse prime 5. This loosely resembles the stretched-octave tunings commonly used on pianos. The tuning 31ed6 does this.
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.
{{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6}}
{{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|31|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6 (continued)}}
{{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)}}
 
; [[19edt]]  
* Step size: 101.103{{c}}, octave size: 1201.2{{c}}
Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse prime 5. The tuning 19edt does this.
{{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt}}
{{Harmonics in equal|19|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt (continued)}}
 
; [[7edf]]
* Step size: 100.3{{c}}, octave size: 1203.35{{c}}
Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 5 and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably won't sound very good here because of the off 5th harmonic. The tuning 7edf does this.
{{Harmonics in equal|7|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf}}
{{Harmonics in equal|7|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf (continued)}}

Latest revision as of 04:16, 22 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 compression

What follows is a comparison of compressed-octave 17edo tunings.

17edo
  • Step size: 70.588 ¢, octave size: 1200.0 ¢

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.

Approximation of harmonics in 17edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +3.9 +0.0 -33.4 +3.9 +19.4 +0.0 +7.9 -33.4 +13.4 +3.9
Relative (%) +0.0 +5.6 +0.0 -47.3 +5.6 +27.5 +0.0 +11.1 -47.3 +19.0 +5.6
Steps
(reduced) 		17
(0) 		27
(10) 		34
(0) 		39
(5) 		44
(10) 		48
(14) 		51
(0) 		54
(3) 		56
(5) 		59
(8) 		61
(10)
Approximation of harmonics in 17edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +6.5 +19.4 -29.4 +0.0 -34.4 +7.9 -15.2 -33.4 +23.3 +13.4 +7.0 +3.9
Relative (%) +9.3 +27.5 -41.7 +0.0 -48.7 +11.1 -21.5 -47.3 +33.1 +19.0 +9.9 +5.6
Steps
(reduced) 		63
(12) 		65
(14) 		66
(15) 		68
(0) 		69
(1) 		71
(3) 		72
(4) 		73
(5) 		75
(7) 		76
(8) 		77
(9) 		78
(10)
44ed6
  • Step size: NNN ¢, octave size: 1198.5 ¢

Compressing the octave of 17edo by around 1.5 ¢ 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.

Approximation of harmonics in 44ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.5 +1.5 -3.0 +33.6 +0.0 +15.1 -4.6 +3.0 +32.1 +8.1 -1.5
Relative (%) -2.2 +2.2 -4.3 +47.7 +0.0 +21.5 -6.5 +4.3 +45.6 +11.5 -2.2
Steps
(reduced) 		17
(17) 		27
(27) 		34
(34) 		40
(40) 		44
(0) 		48
(4) 		51
(7) 		54
(10) 		57
(13) 		59
(15) 		61
(17)
Approximation of harmonics in 44ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +0.9 +13.6 +35.2 -6.1 +30.0 +1.5 -21.6 +30.6 +16.6 +6.6 +0.1 -3.0
Relative (%) +1.3 +19.3 +49.9 -8.6 +42.5 +2.2 -30.6 +43.4 +23.6 +9.4 +0.2 -4.3
Steps
(reduced) 		63
(19) 		65
(21) 		67
(23) 		68
(24) 		70
(26) 		71
(27) 		72
(28) 		74
(30) 		75
(31) 		76
(32) 		77
(33) 		78
(34)
27edt
  • Step size: NNN ¢, octave size: 1197.5 ¢

Compressing the octave of 17edo by around 2.5 ¢ 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.

Approximation of harmonics in 27edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.5 +0.0 -4.9 +31.4 -2.5 +12.4 -7.4 +0.0 +28.9 +4.8 -4.9
Relative (%) -3.5 +0.0 -7.0 +44.6 -3.5 +17.6 -10.5 +0.0 +41.1 +6.8 -7.0
Steps
(reduced) 		17
(17) 		27
(0) 		34
(7) 		40
(13) 		44
(17) 		48
(21) 		51
(24) 		54
(0) 		57
(3) 		59
(5) 		61
(7)
Approximation of harmonics in 27edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.6 +10.0 +31.4 -9.9 +26.0 -2.5 -25.6 +26.5 +12.4 +2.3 -4.2 -7.4
Relative (%) -3.7 +14.1 +44.6 -14.0 +37.0 -3.5 -36.4 +37.6 +17.6 +3.3 -5.9 -10.5
Steps
(reduced) 		63
(9) 		65
(11) 		67
(13) 		68
(14) 		70
(16) 		71
(17) 		72
(18) 		74
(20) 		75
(21) 		76
(22) 		77
(23) 		78
(24)
56zpi / 17et, 2.3.7.11.13 WE tuning
  • Step size: 70.403 ¢, octave size: 1296.9 ¢

Compressing the octave of 17edo by around 3 ¢ results in improved primes NNN, but worse primes NNN. The tunings: 56zpi, 17et, 2.3.7.11.13 TE and 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.

Approximation of harmonics in 56zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.1 -1.1 -6.3 +29.8 -4.2 +10.5 -9.4 -2.1 +26.7 +2.5 -7.4
Relative (%) -4.5 -1.5 -8.9 +42.3 -6.0 +14.9 -13.4 -3.1 +37.9 +3.5 -10.5
Step 17 27 34 40 44 48 51 54 57 59 61
Approximation of harmonics in 56zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.1 +7.4 +28.7 -12.6 +23.3 -5.3 -28.5 +23.5 +9.4 -0.7 -7.2 -10.5
Relative (%) -7.3 +10.5 +40.8 -17.9 +33.0 -7.5 -40.5 +33.4 +13.4 -1.0 -10.3 -14.9
Step 63 65 67 68 70 71 72 74 75 76 77 78
17et, 2.3.7.11 WE tuning
  • Step size: 70.392 ¢, octave size: 1296.7 ¢

Compressing the octave of 17edo by just over 3 ¢ 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.

Approximation of harmonics in 17et, 2.3.7.11 WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.3 -1.4 -6.7 +29.4 -4.7 +10.0 -10.0 -2.7 +26.0 +1.8 -8.0
Relative (%) -4.7 -1.9 -9.5 +41.7 -6.7 +14.2 -14.2 -3.9 +37.0 +2.6 -11.4
Step 17 27 34 40 44 48 51 54 57 59 61
Approximation of harmonics in 17et, 2.3.7.11 WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.8 +6.7 +28.0 -13.3 +22.5 -6.1 -29.3 +22.7 +8.6 -1.5 -8.1 -11.4
Relative (%) -8.3 +9.5 +39.8 -19.0 +31.9 -8.6 -41.6 +32.2 +12.2 -2.2 -11.5 -16.2
Step 63 65 67 68 70 71 72 74 75 76 77 78
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