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= Title1 =
= Title1 =
== Octave stretch or compression ==
== Octave stretch or compression ==
64edo's approximations of 3/1, 5/1, 7/1, 11/1 and 17/1 are improved by [[180ed7]], a [[Octave shrinking|compressed-octave]] version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]].


[[149ed5]] can also be used: it is similar to 180ed7 but both the improvements and shortcomings are amplified. Most notably its 2/1 isn’t as accurate as 180ed7's.
What follows is a comparison of stretched-octave 38edo tunings.


If one prefers a ''[[Octave stretch|stretched-octave]]'', 64edo's approximations of 3/1, 5/1, 11/1 and 17/1 are improved by [[221ed11]], a stretched version of 64edo. The trade-off is a slightly worse 2/1 and 13/1.
; 38edo
* Step size: 31.579{{c}}, octave size: 1200.00{{c}}
Pure-octaves 38edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|38|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38edo}}
{{Harmonics in equal|38|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38edo (continued)}}


[[47ed5/3]] can also be used: it is similar to 221ed11 but both the improvements and shortcomings are amplified. Most notably its 2/1 is not as accurate as 221ed11's.
; [[WE|38et, 13-limit WE tuning]]  
* Step size: 31.599{{c}}, octave size: 1200.77{{c}}
Stretching the octave of 38edo by around 1{{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 cet|31.599|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 38et, 13-limit WE tuning}}
{{Harmonics in cet|31.599|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 38et, 13-limit WE tuning (continued)}}


What follows is a comparison of stretched- and compressed-octave 64edo tunings.
; [[ed5|88ed5]]
* Step size: 31.663{{c}}, octave size: 1203.18{{c}}
Stretching the octave of 38edo by around 3{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 88ed5 does this.
{{Harmonics in equal|88|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 88ed5}}
{{Harmonics in equal|88|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 88ed5 (continued)}}


; [[ed7|179ed7]]
; [[zpi|166zpi]]  
* Octave size: 1204.50{{c}}
* Step size: 31.671{{c}}, octave size: 1203.48{{c}}
Stretching the octave of 64edo by around 4.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 8.99{{c}}. The tuning 179ed7 does this. So does the tuning 326zpi whose octave is identical within 0.3{{c}}.
Stretching the octave of 38edo by around 3.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 166zpi does this.
{{Harmonics in equal|179|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 179ed7}}
{{Harmonics in cet|31.671|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 166zpi}}
{{Harmonics in equal|179|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 179ed7 (continued)}}
{{Harmonics in cet|31.671|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 166zpi (continued)}}


; [[ed6|165ed6]]
; [[60edt]]  
* Octave size: 1203.18{{c}}
* Step size: 31.699{{c}}, octave size: 1204.57{{c}}
Stretching the octave of 64edo by around 3{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 9.25{{c}}. The tuning 165ed6 does this.
Stretching the octave of 38edo by around 4.5{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 60edt does this.
{{Harmonics in equal|165|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 165ed6}}
{{Harmonics in equal|60|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edt}}
{{Harmonics in equal|165|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 165ed6 (continued)}}
{{Harmonics in equal|60|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edt (continued)}}
 
; [[ed12|229ed12]]
* Octave size: 1202.29{{c}}
Stretching the octave of 64edo by around 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.17{{c}}. The tuning 229ed12 does this. So does the tuning [[equal tuning|221ed11]] whose octave is identical within 0.1{{c}}.
{{Harmonics in equal|229|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 229ed12}}
{{Harmonics in equal|229|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 229ed12 (continued)}}
 
; [[zpi|327zpi]]
* Step size: 18.767{{c}}, octave size: 1201.09{{c}}
Stretching the octave of 64edo by around 1{{c}} results in improved primes 3 and 11, but worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 9.23{{c}}. The tuning 327zpi does this.
{{Harmonics in cet|18.767|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 327zpi}}
{{Harmonics in cet|18.767|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 327zpi (continued)}}
 
; [[WE|64et, 11-limit WE tuning]]
* Step size: 18.755{{c}}, octave size: 1200.32{{c}}
Stretching the octave of 64edo by around a third of a cent results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 8.50{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
{{Harmonics in cet|18.755|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 64et, 11-limit WE tuning}}
{{Harmonics in cet|18.755|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 64et, 11-limit WE tuning (continued)}}
 
; 64edo
* Step size: 18.750{{c}}, octave size: 1200.00{{c}}
Pure-octaves 64edo approximates all harmonics up to 16 within 8.21{{c}}. The octave of 64edo's 13-limit [[WE]] tuning differs by only 0.13{{c}} from pure.
{{Harmonics in equal|64|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 64edo}}
{{Harmonics in equal|64|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 64edo (continued)}}
 
; [[zpi|328zpi]]
* Step size: 18.721{{c}}, octave size: 1198.14{{c}}
Compressing the octave of 64edo by just under 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.02{{c}}. The tuning 328zpi does this.
{{Harmonics in cet|18.721|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 328zpi}}
{{Harmonics in cet|18.721|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 328zpi (continued)}}
 
; [[ed7|180ed7]]
* Octave size: 1197.80{{c}}
Compressing the octave of 64edo by just over 2{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 9.34{{c}}. The tuning 180ed7 does this.
{{Harmonics in equal|180|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 180ed7}}
{{Harmonics in equal|180|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 180ed7 (continued)}}
 
; [[ed12|230ed12]]
* Octave size: 1197.07{{c}}
Compressing the octave of 64edo by around 3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.80{{c}}. The tuning 230ed12 does this.
{{Harmonics in equal|230|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 230ed12}}
{{Harmonics in equal|230|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 230ed12 (continued)}}
 
; [[ed5|149ed5]]
* Step size: Octave size: NNN{{c}}
Compressing the octave of 64edo 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 149ed5 does this.
{{Harmonics in equal|149|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 149ed5}}
{{Harmonics in equal|149|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 149ed5 (continued)}}


= Title2 =
= Title2 =
Line 94: Line 58:


; High-priority
; High-priority
118edo (choose ZPIS)
{{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}}
* 187edt
* 69edf
* 13-limit WE (10.171c)
* Best nearby ZPI(s)
103edo (narrow down edonoi, choose ZPIS)
{{harmonics in equal | 103 | 2 | 1 | intervals=integer | columns=12}}
* 163edt
* 239ed5
* 266ed6
* 289ed7
* 356ed11
* 369ed12
* 381ed13
* 421ed17
* 466ed23
* 13-limit WE (11.658c)
* Best nearby ZPI(s)
111edo (choose ZPIS)
{{harmonics in equal | 111 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
13edo
{{harmonics in equal | 13 | 2 | 1 | intervals=integer | columns=12}}
* Main: "13edo and optimal octave stretching"
* 2.5.11.13 WE (92.483c)
* 2.5.7.13 WE (92.804c)
* 2.3 WE (91.405c) (good for opposite 7 mapping)
* 38zpi (92.531c)
104edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
; Medium-priority


25edo
25edo
Line 118: Line 126:
30edo
30edo
{{harmonics in equal | 30 | 2 | 1 | intervals=integer | columns=12}}
{{harmonics in equal | 30 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
34edo
{{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 151: Line 152:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


38edo
15edo
{{harmonics in equal | 38 | 2 | 1 | intervals=integer | columns=12}}
{{harmonics in equal | 15 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 165: Line 166:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


10edo
18edo
{{harmonics in equal | 10 | 2 | 1 | intervals=integer | columns=12}}
{{harmonics in equal | 18 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 172: Line 173:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


11edo
24edo
{{harmonics in equal | 11 | 2 | 1 | intervals=integer | columns=12}}
{{harmonics in equal | 24 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* 1-2 WE tunings
* Best nearby ZPI(s)
* Best nearby ZPI(
 
15edo
{{harmonics in equal | 15 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
 
18edo
{{harmonics in equal | 18 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)


48edo
48edo
{{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}}
{{harmonics in equal | 48 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
24edo
{{harmonics in equal | 24 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 219: Line 199:
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* 1-2 WE tunings
* Best nearby ZPI(s)
* Best nearby ZPI(s)s)


13edo
10edo
{{harmonics in equal | 13 | 2 | 1 | intervals=integer | columns=12}}
{{harmonics in equal | 10 | 2 | 1 | intervals=integer | columns=12}}
* Main: "13edo and optimal octave stretching"
* Nearby edt, ed6, ed12 and/or edf
* 2.5.11.13 WE (92.483c)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 2.5.7.13 WE (92.804c)
* 1-2 WE tunings
* 2.3 WE (91.405c) (good for opposite 7 mapping)
* 38zpi (92.531c)
 
118edo (choose ZPIS)
{{harmonics in equal | 118 | 2 | 1 | intervals=integer | columns=12}}
* 187edt
* 69edf
* 13-limit WE (10.171c)
* Best nearby ZPI(s)
* Best nearby ZPI(s)


103edo (narrow down edonoi, choose ZPIS)
11edo
{{harmonics in equal | 103 | 2 | 1 | intervals=integer | columns=12}}
{{harmonics in equal | 11 | 2 | 1 | intervals=integer | columns=12}}
* 163edt
* 239ed5
* 266ed6
* 289ed7
* 356ed11
* 369ed12
* 381ed13
* 421ed17
* 466ed23
* 13-limit WE (11.658c)
* Best nearby ZPI(s)
 
111edo (choose ZPIS)
{{harmonics in equal | 111 | 2 | 1 | intervals=integer | columns=12}}
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
Line 257: Line 215:
* Best nearby ZPI(s)
* Best nearby ZPI(s)


; Low priority
34edo
 
{{harmonics in equal | 34 | 2 | 1 | intervals=integer | columns=12}}
104edo
* Nearby edt, ed6, ed12 and/or edf
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* 1-2 WE tunings
* Best nearby ZPI(s)
* Best nearby ZPI(s)
; Low priority


125edo
125edo

Latest revision as of 02:39, 3 September 2025

Quick link

User:BudjarnLambeth/Draft related tunings section

Title1

Octave stretch or compression

38edo's approximation of JI can be improved by slightly stretching the octave.

What follows is a comparison of stretched-octave 38edo tunings.

38edo
  • Step size: 31.579 ¢, octave size: 1200.00 ¢

Pure-octaves 38edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 38edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -7.2 +0.0 -7.4 -7.2 +10.1 +0.0 -14.4 -7.4 -14.5 -7.2
Relative (%) +0.0 -22.9 +0.0 -23.3 -22.9 +32.1 +0.0 -45.7 -23.3 -45.8 -22.9
Steps
(reduced) 		38
(0) 		60
(22) 		76
(0) 		88
(12) 		98
(22) 		107
(31) 		114
(0) 		120
(6) 		126
(12) 		131
(17) 		136
(22)
Approximation of harmonics in 38edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +12.1 +10.1 -14.6 +0.0 -10.2 -14.4 -13.3 -7.4 +2.9 -14.5 +3.3 -7.2
Relative (%) +38.3 +32.1 -46.2 +0.0 -32.4 -45.7 -42.1 -23.3 +9.2 -45.8 +10.5 -22.9
Steps
(reduced) 		141
(27) 		145
(31) 		148
(34) 		152
(0) 		155
(3) 		158
(6) 		161
(9) 		164
(12) 		167
(15) 		169
(17) 		172
(20) 		174
(22)
38et, 13-limit WE tuning
  • Step size: 31.599 ¢, octave size: 1200.77 ¢

Stretching the octave of 38edo by around 1 ¢ 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 38et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.8 -6.0 +1.5 -5.6 -5.3 +12.3 +2.3 -12.0 -4.8 -11.8 -4.5
Relative (%) +2.4 -19.0 +4.8 -17.7 -16.6 +38.8 +7.2 -38.1 -15.3 -37.5 -14.2
Step 38 60 76 88 98 107 114 120 126 131 136
Approximation of harmonics in 38et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +14.9 +13.0 -11.6 +3.0 -7.1 -11.3 -10.1 -4.1 +6.3 -11.1 +6.8 -3.7
Relative (%) +47.3 +41.2 -36.8 +9.6 -22.5 -35.7 -31.9 -12.9 +19.8 -35.1 +21.4 -11.8
Step 141 145 148 152 155 158 161 164 167 169 172 174
88ed5
  • Step size: 31.663 ¢, octave size: 1203.18 ¢

Stretching the octave of 38edo by around 3 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 88ed5 does this.

Approximation of harmonics in 88ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.2 -2.2 +6.4 +0.0 +1.0 -12.6 +9.5 -4.4 +3.2 -3.5 +4.2
Relative (%) +10.0 -6.9 +20.1 +0.0 +3.1 -39.7 +30.1 -13.9 +10.0 -11.1 +13.2
Steps
(reduced) 		38
(38) 		60
(60) 		76
(76) 		88
(0) 		98
(10) 		106
(18) 		114
(26) 		120
(32) 		126
(38) 		131
(43) 		136
(48)
Approximation of harmonics in 88ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -7.8 -9.4 -2.2 +12.7 +2.8 -1.2 +0.2 +6.4 -14.8 -0.3 -14.0 +7.3
Relative (%) -24.5 -29.7 -6.9 +40.2 +8.7 -3.8 +0.6 +20.1 -46.7 -1.0 -44.1 +23.2
Steps
(reduced) 		140
(52) 		144
(56) 		148
(60) 		152
(64) 		155
(67) 		158
(70) 		161
(73) 		164
(76) 		166
(78) 		169
(81) 		171
(83) 		174
(86)
166zpi
  • Step size: 31.671 ¢, octave size: 1203.48 ¢

Stretching the octave of 38edo by around 3.5 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 166zpi does this.

Approximation of harmonics in 166zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.5 -1.7 +7.0 +0.7 +1.8 -11.7 +10.5 -3.4 +4.2 -2.4 +5.3
Relative (%) +11.0 -5.4 +22.1 +2.3 +5.7 -36.9 +33.1 -10.7 +13.4 -7.6 +16.7
Step 38 60 76 88 98 106 114 120 126 131 136
Approximation of harmonics in 166zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -6.6 -8.2 -1.0 +14.0 +4.0 +0.1 +1.5 +7.7 -13.4 +1.1 -12.5 +8.8
Relative (%) -20.8 -25.9 -3.0 +44.2 +12.8 +0.3 +4.8 +24.4 -42.3 +3.4 -39.6 +27.8
Step 140 144 148 152 155 158 161 164 166 169 171 174
60edt
  • Step size: 31.699 ¢, octave size: 1204.57 ¢

Stretching the octave of 38edo by around 4.5 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 60edt does this.

Approximation of harmonics in 60edt
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 +1.3 +9.1
Relative (%) +14.4 +0.0 +28.8 +10.2 +14.4 -27.5 +43.3 +0.0 +24.6 +4.0 +28.8
Steps
(reduced) 		38
(38) 		60
(0) 		76
(16) 		88
(28) 		98
(38) 		106
(46) 		114
(54) 		120
(0) 		126
(6) 		131
(11) 		136
(16)
Approximation of harmonics in 60edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.6 -4.1 +3.2 -13.4 +8.4 +4.6 +6.1 +12.4 -8.7 +5.9 -7.7 +13.7
Relative (%) -8.3 -13.0 +10.2 -42.3 +26.6 +14.4 +19.1 +39.0 -27.5 +18.5 -24.3 +43.3
Steps
(reduced) 		140
(20) 		144
(24) 		148
(28) 		151
(31) 		155
(35) 		158
(38) 		161
(41) 		164
(44) 		166
(46) 		169
(49) 		171
(51) 		174
(54)

Title2

Lab

Place holder








Approximation of prime harmonics in 1ed300c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0 -102 -86 -69 +49 +59 -105 +2 -28 -130 +55
Relative (%) +0.0 -34.0 -28.8 -22.9 +16.2 +19.8 -35.0 +0.8 -9.4 -43.2 +18.3
Step 4 6 9 11 14 15 16 17 18 19 20


Approximation of prime harmonics in 140ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.6 +3.2 +10.0 +11.3 -3.0 +15.1 +11.6 +3.4 +10.6 +8.8 -14.5
Relative (%) -5.2 +10.4 +32.4 +36.7 -9.8 +49.0 +37.6 +11.0 +34.6 +28.6 -47.1
Steps
(reduced) 		39
(39) 		62
(62) 		91
(91) 		110
(110) 		135
(135) 		145
(5) 		160
(20) 		166
(26) 		177
(37) 		190
(50) 		193
(53)

Possible tunings to be used on each page

You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.

(Used https://x31eq.com/temper-pyscript/net.html, used WE instead of TE cause it kept defaulting to WE and I kept not remembering to switch it)

High-priority

118edo (choose ZPIS)

Approximation of harmonics in 118edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 -0.26 +0.00 +0.13 -0.26 -2.72 +0.00 -0.52 +0.13 -2.17 -0.26 +3.54
Relative (%) +0.0 -2.6 +0.0 +1.2 -2.6 -26.8 +0.0 -5.1 +1.2 -21.3 -2.6 +34.8
Steps
(reduced) 		118
(0) 		187
(69) 		236
(0) 		274
(38) 		305
(69) 		331
(95) 		354
(0) 		374
(20) 		392
(38) 		408
(54) 		423
(69) 		437
(83)
  • 187edt
  • 69edf
  • 13-limit WE (10.171c)
  • Best nearby ZPI(s)

103edo (narrow down edonoi, choose ZPIS)

Approximation of harmonics in 103edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 -2.93 +0.00 -1.85 -2.93 -1.84 +0.00 +5.80 -1.85 -3.75 -2.93 -1.69
Relative (%) +0.0 -25.1 +0.0 -15.9 -25.1 -15.8 +0.0 +49.8 -15.9 -32.1 -25.1 -14.5
Steps
(reduced) 		103
(0) 		163
(60) 		206
(0) 		239
(33) 		266
(60) 		289
(83) 		309
(0) 		327
(18) 		342
(33) 		356
(47) 		369
(60) 		381
(72)
  • 163edt
  • 239ed5
  • 266ed6
  • 289ed7
  • 356ed11
  • 369ed12
  • 381ed13
  • 421ed17
  • 466ed23
  • 13-limit WE (11.658c)
  • Best nearby ZPI(s)

111edo (choose ZPIS)

Approximation of harmonics in 111edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 +0.75 +0.00 +2.88 +0.75 +4.15 +0.00 +1.50 +2.88 +0.03 +0.75 +2.72
Relative (%) +0.0 +6.9 +0.0 +26.6 +6.9 +38.4 +0.0 +13.8 +26.6 +0.3 +6.9 +25.1
Steps
(reduced) 		111
(0) 		176
(65) 		222
(0) 		258
(36) 		287
(65) 		312
(90) 		333
(0) 		352
(19) 		369
(36) 		384
(51) 		398
(65) 		411
(78)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

13edo

Approximation of harmonics in 13edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +36.5 +0.0 -17.1 +36.5 -45.7 +0.0 -19.3 -17.1 +2.5 +36.5 -9.8
Relative (%) +0.0 +39.5 +0.0 -18.5 +39.5 -49.6 +0.0 -20.9 -18.5 +2.7 +39.5 -10.6
Steps
(reduced) 		13
(0) 		21
(8) 		26
(0) 		30
(4) 		34
(8) 		36
(10) 		39
(0) 		41
(2) 		43
(4) 		45
(6) 		47
(8) 		48
(9)
  • Main: "13edo and optimal octave stretching"
  • 2.5.11.13 WE (92.483c)
  • 2.5.7.13 WE (92.804c)
  • 2.3 WE (91.405c) (good for opposite 7 mapping)
  • 38zpi (92.531c)

104edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)
Medium-priority

25edo

Approximation of harmonics in 25edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 -2.3 +18.0 -8.8 +0.0 -11.9 -2.3 -23.3 +18.0 +23.5
Relative (%) +0.0 +37.6 +0.0 -4.8 +37.6 -18.4 +0.0 -24.8 -4.8 -48.6 +37.6 +48.9
Steps
(reduced) 		25
(0) 		40
(15) 		50
(0) 		58
(8) 		65
(15) 		70
(20) 		75
(0) 		79
(4) 		83
(8) 		86
(11) 		90
(15) 		93
(18)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

26edo

Approximation of harmonics in 26edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -9.6 +0.0 -17.1 -9.6 +0.4 +0.0 -19.3 -17.1 +2.5 -9.6 -9.8
Relative (%) +0.0 -20.9 +0.0 -37.0 -20.9 +0.9 +0.0 -41.8 -37.0 +5.5 -20.9 -21.1
Steps
(reduced) 		26
(0) 		41
(15) 		52
(0) 		60
(8) 		67
(15) 		73
(21) 		78
(0) 		82
(4) 		86
(8) 		90
(12) 		93
(15) 		96
(18)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

29edo

Approximation of harmonics in 29edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +1.5 +0.0 -13.9 +1.5 -17.1 +0.0 +3.0 -13.9 -13.4 +1.5 -12.9
Relative (%) +0.0 +3.6 +0.0 -33.6 +3.6 -41.3 +0.0 +7.2 -33.6 -32.4 +3.6 -31.3
Steps
(reduced) 		29
(0) 		46
(17) 		58
(0) 		67
(9) 		75
(17) 		81
(23) 		87
(0) 		92
(5) 		96
(9) 		100
(13) 		104
(17) 		107
(20)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

30edo

Approximation of harmonics in 30edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 +13.7 +18.0 -8.8 +0.0 -3.9 +13.7 +8.7 +18.0 -0.5
Relative (%) +0.0 +45.1 +0.0 +34.2 +45.1 -22.1 +0.0 -9.8 +34.2 +21.7 +45.1 -1.3
Steps
(reduced) 		30
(0) 		48
(18) 		60
(0) 		70
(10) 		78
(18) 		84
(24) 		90
(0) 		95
(5) 		100
(10) 		104
(14) 		108
(18) 		111
(21)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

35edo

Approximation of harmonics in 35edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -16.2 +0.0 -9.2 -16.2 -8.8 +0.0 +1.8 -9.2 -2.7 -16.2 +16.6
Relative (%) +0.0 -47.4 +0.0 -26.7 -47.4 -25.7 +0.0 +5.3 -26.7 -8.0 -47.4 +48.5
Steps
(reduced) 		35
(0) 		55
(20) 		70
(0) 		81
(11) 		90
(20) 		98
(28) 		105
(0) 		111
(6) 		116
(11) 		121
(16) 		125
(20) 		130
(25)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

36edo

Approximation of harmonics in 36edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 -2.2 +0.0 -3.9 +13.7 +15.3 -2.0 -7.2
Relative (%) +0.0 -5.9 +0.0 +41.1 -5.9 -6.5 +0.0 -11.7 +41.1 +46.0 -5.9 -21.6
Steps
(reduced) 		36
(0) 		57
(21) 		72
(0) 		84
(12) 		93
(21) 		101
(29) 		108
(0) 		114
(6) 		120
(12) 		125
(17) 		129
(21) 		133
(25)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

37edo

Approximation of harmonics in 37edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +11.6 +0.0 +2.9 +11.6 +4.1 +0.0 -9.3 +2.9 +0.0 +11.6 +2.7
Relative (%) +0.0 +35.6 +0.0 +8.9 +35.6 +12.8 +0.0 -28.7 +8.9 +0.1 +35.6 +8.4
Steps
(reduced) 		37
(0) 		59
(22) 		74
(0) 		86
(12) 		96
(22) 		104
(30) 		111
(0) 		117
(6) 		123
(12) 		128
(17) 		133
(22) 		137
(26)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

15edo

Approximation of harmonics in 15edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 +13.7 +18.0 -8.8 +0.0 +36.1 +13.7 +8.7 +18.0 +39.5
Relative (%) +0.0 +22.6 +0.0 +17.1 +22.6 -11.0 +0.0 +45.1 +17.1 +10.9 +22.6 +49.3
Steps
(reduced) 		15
(0) 		24
(9) 		30
(0) 		35
(5) 		39
(9) 		42
(12) 		45
(0) 		48
(3) 		50
(5) 		52
(7) 		54
(9) 		56
(11)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

9edo

Approximation of harmonics in 9edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -35.3 +0.0 +13.7 -35.3 -35.5 +0.0 +62.8 +13.7 -18.0 -35.3 -40.5
Relative (%) +0.0 -26.5 +0.0 +10.3 -26.5 -26.6 +0.0 +47.1 +10.3 -13.5 -26.5 -30.4
Steps
(reduced) 		9
(0) 		14
(5) 		18
(0) 		21
(3) 		23
(5) 		25
(7) 		27
(0) 		29
(2) 		30
(3) 		31
(4) 		32
(5) 		33
(6)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

18edo

Approximation of harmonics in 18edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +31.4 +0.0 +13.7 +31.4 +31.2 +0.0 -3.9 +13.7 -18.0 +31.4 +26.1
Relative (%) +0.0 +47.1 +0.0 +20.5 +47.1 +46.8 +0.0 -5.9 +20.5 -27.0 +47.1 +39.2
Steps
(reduced) 		18
(0) 		29
(11) 		36
(0) 		42
(6) 		47
(11) 		51
(15) 		54
(0) 		57
(3) 		60
(6) 		62
(8) 		65
(11) 		67
(13)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

24edo

Approximation of harmonics in 24edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 -18.8 +0.0 -3.9 +13.7 -1.3 -2.0 +9.5
Relative (%) +0.0 -3.9 +0.0 +27.4 -3.9 -37.7 +0.0 -7.8 +27.4 -2.6 -3.9 +18.9
Steps
(reduced) 		24
(0) 		38
(14) 		48
(0) 		56
(8) 		62
(14) 		67
(19) 		72
(0) 		76
(4) 		80
(8) 		83
(11) 		86
(14) 		89
(17)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(

48edo

Approximation of harmonics in 48edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -2.0 +0.0 -11.3 -2.0 +6.2 +0.0 -3.9 -11.3 -1.3 -2.0 +9.5
Relative (%) +0.0 -7.8 +0.0 -45.3 -7.8 +24.7 +0.0 -15.6 -45.3 -5.3 -7.8 +37.9
Steps
(reduced) 		48
(0) 		76
(28) 		96
(0) 		111
(15) 		124
(28) 		135
(39) 		144
(0) 		152
(8) 		159
(15) 		166
(22) 		172
(28) 		178
(34)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

5edo

Approximation of harmonics in 5edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0 +18 +0 +94 +18 -9 +0 +36 +94 -71 +18 +119
Relative (%) +0.0 +7.5 +0.0 +39.0 +7.5 -3.7 +0.0 +15.0 +39.0 -29.7 +7.5 +49.8
Steps
(reduced) 		5
(0) 		8
(3) 		10
(0) 		12
(2) 		13
(3) 		14
(4) 		15
(0) 		16
(1) 		17
(2) 		17
(2) 		18
(3) 		19
(4)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

6edo

Approximation of harmonics in 6edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +98.0 +0.0 +13.7 +98.0 +31.2 +0.0 -3.9 +13.7 +48.7 +98.0 -40.5
Relative (%) +0.0 +49.0 +0.0 +6.8 +49.0 +15.6 +0.0 -2.0 +6.8 +24.3 +49.0 -20.3
Steps
(reduced) 		6
(0) 		10
(4) 		12
(0) 		14
(2) 		16
(4) 		17
(5) 		18
(0) 		19
(1) 		20
(2) 		21
(3) 		22
(4) 		22
(4)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)s)

10edo

Approximation of harmonics in 10edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 -26.3 +18.0 -8.8 +0.0 +36.1 -26.3 +48.7 +18.0 -0.5
Relative (%) +0.0 +15.0 +0.0 -21.9 +15.0 -7.4 +0.0 +30.1 -21.9 +40.6 +15.0 -0.4
Steps
(reduced) 		10
(0) 		16
(6) 		20
(0) 		23
(3) 		26
(6) 		28
(8) 		30
(0) 		32
(2) 		33
(3) 		35
(5) 		36
(6) 		37
(7)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

11edo

Approximation of harmonics in 11edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -47.4 +0.0 +50.0 -47.4 +13.0 +0.0 +14.3 +50.0 -5.9 -47.4 +32.2
Relative (%) +0.0 -43.5 +0.0 +45.9 -43.5 +11.9 +0.0 +13.1 +45.9 -5.4 -43.5 +29.5
Steps
(reduced) 		11
(0) 		17
(6) 		22
(0) 		26
(4) 		28
(6) 		31
(9) 		33
(0) 		35
(2) 		37
(4) 		38
(5) 		39
(6) 		41
(8)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

34edo

Approximation of harmonics in 34edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +3.9 +0.0 +1.9 +3.9 -15.9 +0.0 +7.9 +1.9 +13.4 +3.9 +6.5
Relative (%) +0.0 +11.1 +0.0 +5.4 +11.1 -45.0 +0.0 +22.3 +5.4 +37.9 +11.1 +18.5
Steps
(reduced) 		34
(0) 		54
(20) 		68
(0) 		79
(11) 		88
(20) 		95
(27) 		102
(0) 		108
(6) 		113
(11) 		118
(16) 		122
(20) 		126
(24)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)
Low priority

125edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

145edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

152edo

  • 241edt
  • 13-limit WE (7.894c)
  • Best nearby ZPI(s)

159edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

166edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

182edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

198edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

212edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

243edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

247edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)
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