User:BudjarnLambeth/Sandbox2: Difference between revisions

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


Line 26: Line 28:
{{Harmonics in cet|63.291|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning}}
{{Harmonics in cet|63.291|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning}}
{{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 cet|63.291|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19et, 13-limit WE tuning (continued)}}
; [[zpi|ZPINAME]]
* 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}}. The tuning ZPINAME does this.
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME (continued)}}


; [[49ed6]]  
; [[49ed6]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: NNN{{c}}, octave size: 1202.8{{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.
_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 equal|49|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6}}
{{Harmonics in equal|49|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6}}
{{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6 (continued)}}
{{Harmonics in equal|49|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 49ed6 (continued)}}
; [[zpi|65zpi]]
* Step size: 63.331{{c}}, octave size: 1203.3{{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.
{{Harmonics in cet|63.331|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi}}
{{Harmonics in cet|63.331|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 65zpi (continued)}}


; [[30edt]]  
; [[30edt]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: NNN{{c}}, octave size: 1204.6{{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 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=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 30edt}}
Line 46: Line 48:


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

Revision as of 00:34, 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

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 ¢, 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-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
49ed6
  • Step size: NNN ¢, octave size: 1202.8 ¢

_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)
65zpi
  • Step size: 63.331 ¢, octave size: 1203.3 ¢

_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 65zpi does this.

Approximation of harmonics in 65zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.3 -2.0 +6.6 +0.3 +1.3 -12.3 +9.9 -4.1 +3.5 +28.5 +4.6
Relative (%) +5.2 -3.2 +10.4 +0.4 +2.0 -19.4 +15.6 -6.4 +5.6 +45.0 +7.2
Step 19 30 38 44 49 53 57 60 63 66 68
Approximation of harmonics in 65zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -7.4 -9.0 -1.8 +13.2 -28.5 -0.8 -31.0 +6.8 -14.3 -31.5 +18.2 +7.8
Relative (%) -11.6 -14.2 -2.8 +20.8 -45.0 -1.2 -49.0 +10.8 -22.6 -49.8 +28.7 +12.4
Step 70 72 74 76 77 79 80 82 83 84 86 87
30edt
  • Step size: NNN ¢, octave size: 1204.6 ¢

_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: 1212.5 ¢

_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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