Mint temperaments: Difference between revisions

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Line 32: Line 32:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 231.845{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 231.845{{c}}
: error map: {{val| 0.000 +6.291 +85.850 -24.498 }}
: error map: {{val| 0.000 +6.291 +85.850 -24.498 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~7/6 = 236.0819{{c}}
: [[error map]]: {{val| 0.000 -6.421 +77.376 -41.447 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~7/6 = 239.977{{c}}
: error map: {{val| 0.000 +17.976 +93.640 -8.918 }} -->


{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}
Line 59: Line 55:
* [[CWE]]: ~2 = 1200.000{{c}}, ~15/14 = 139.8991{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~15/14 = 139.8991{{c}}
: error map: {{val| 0.000 -2.460 +33.384 +10.468 }}
: error map: {{val| 0.000 -2.460 +33.384 +10.468 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~15/14 = 138.611{{c}}
: [[error map]]: {{val| 0.000 -8.900 +29.519 +1.451 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~15/14 = 140.927{{c}}
: error map: {{val| 0.000 +2.680 +36.467 +17.663 }} -->


{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
Line 78: Line 70:
* WE: ~2 = 1194.7089{{c}}, ~12/11 = 140.1262{{c}}
* WE: ~2 = 1194.7089{{c}}, ~12/11 = 140.1262{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.8776{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.8776{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~12/11 = 138.556{{c}}
* POTE: ~2 = 1200.000{{c}}, ~12/11 = 140.747{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 9, 17c }}
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}
Line 95: Line 85:
* WE: ~2 = 1195.3694{{c}}, ~13/12 = 140.2080{{c}}
* WE: ~2 = 1195.3694{{c}}, ~13/12 = 140.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.9749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.9749{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~13/12 = 138.741{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/12 = 140.751{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 9, 17c }}
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}
Line 112: Line 100:
* WE: ~2 = 1193.8350{{c}}, ~13/12 = 140.6779{{c}}
* WE: ~2 = 1193.8350{{c}}, ~13/12 = 140.6779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.5885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.5885{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~13/12 = 138.649{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/12 = 141.404{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}
Line 129: Line 115:
* WE: ~2 = 1196.1446{{c}}, ~13/12 = 140.0276{{c}}
* WE: ~2 = 1196.1446{{c}}, ~13/12 = 140.0276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.0301{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.0301{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~13/12 = 138.750{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/12 = 140.479{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}
Line 151: Line 135:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 915.4228{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 915.4228{{c}}
: error map: {{val| 0.000 -9.418 +21.646 +8.288 }}
: error map: {{val| 0.000 -9.418 +21.646 +8.288 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~5/3 = 915.014{{c}}
: [[error map]]: {{val| 0.000 -11.870 +18.785 +6.245 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~5/3 = 915.781{{c}}
: error map: {{val| 0.000 -7.268 +24.154 +10.080 }} -->


{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}
{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}
Line 170: Line 150:
* WE: ~2 = 1196.3345{{c}}, ~5/3 = 913.9469{{c}}
* WE: ~2 = 1196.3345{{c}}, ~5/3 = 913.9469{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.2970{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.2970{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/3 = 915.735{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/3 = 916.747{{c}} -->


{{Optimal ET sequence|legend=0| 4e, …, 13cdee, 17c }}
{{Optimal ET sequence|legend=0| 4e, …, 13cdee, 17c }}
Line 187: Line 165:
* WE: ~2 = 1196.0784{{c}}, ~5/3 = 914.1466{{c}}
* WE: ~2 = 1196.0784{{c}}, ~5/3 = 914.1466{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.6712{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.6712{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/3 = 916.082{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/3 = 917.144{{c}} -->


{{Optimal ET sequence|legend=0| 4ef, …, 13cdeef, 17c }}
{{Optimal ET sequence|legend=0| 4ef, …, 13cdeef, 17c }}
Line 204: Line 180:
* WE: ~2 = 1196.3809{{c}}, ~5/3 = 913.4153{{c}}
* WE: ~2 = 1196.3809{{c}}, ~5/3 = 913.4153{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.8804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.8804{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~5/3 = 915.130{{c}}
* POTE: ~2 = 1200.000{{c}}, ~5/3 = 916.178{{c}} -->


{{Optimal ET sequence|legend=0| 4, 13cd, 17c, 38ce }}
{{Optimal ET sequence|legend=0| 4, 13cd, 17c, 38ce }}
Line 221: Line 195:
* WE: ~2 = 1196.5477{{c}}, ~5/3 = 913.4822{{c}}
* WE: ~2 = 1196.5477{{c}}, ~5/3 = 913.4822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.9416{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.9416{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~7/6 = 284.758{{c}}
* POTE: ~2 = 1200.000{{c}}, ~7/6 = 283.882{{c}} -->


{{Optimal ET sequence|legend=0| 4, 17c, 38ce }}
{{Optimal ET sequence|legend=0| 4, 17c, 38ce }}
Line 243: Line 215:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/49 = 743.9885{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/49 = 743.9885{{c}}
: error map: {{val| 0.000 -6.059 +21.606 +15.047 }}
: error map: {{val| 0.000 -6.059 +21.606 +15.047 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~75/49 = 743.506{{c}}
: [[error map]]: {{val| 0.000, -10.401, +18.228, +9.740 }} -->


{{Optimal ET sequence|legend=1| 8d, 21, 29cd }}
{{Optimal ET sequence|legend=1| 8d, 21, 29cd }}
Line 260: Line 230:
* WE: ~2 = 1194.3937{{c}}, ~75/49 = 741.2117{{c}}
* WE: ~2 = 1194.3937{{c}}, ~75/49 = 741.2117{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 744.2018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 744.2018{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~75/49 = 743.375{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 21, 29cde }}
{{Optimal ET sequence|legend=0| 8d, 21, 29cde }}
Line 276: Line 245:
* WE: ~2 = 1193.8564{{c}}, ~20/13 = 740.9635{{c}}
* WE: ~2 = 1193.8564{{c}}, ~20/13 = 740.9635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2703{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2703{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~20/13 = 743.233{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}
{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}
Line 292: Line 260:
* WE: ~2 = 1194.2330{{c}}, ~20/13 = 741.1242{{c}}
* WE: ~2 = 1194.2330{{c}}, ~20/13 = 741.1242{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2679{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2679{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~20/13 = 743.179{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}
{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}
Line 313: Line 280:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 72.4941{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 72.4941{{c}}
: error map: {{val| 0.000 -9.414 +16.251 +13.692 }}
: error map: {{val| 0.000 -9.414 +16.251 +13.692 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 72.776{{c}}
: [[error map]]: {{val| 0.000 -11.385 +13.153 +12.847 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 72.303{{c}}
: error map: {{val| 0.000 -8.075 +18.355 +14.265 }} -->


{{Optimal ET sequence|legend=1| 16, 17c, 33 }}
{{Optimal ET sequence|legend=1| 16, 17c, 33 }}
Line 332: Line 295:
* WE: ~2 = 1198.4220{{c}}, ~21/20 = 72.2015{{c}}
* WE: ~2 = 1198.4220{{c}}, ~21/20 = 72.2015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4470{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4470{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 72.681{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 72.297{{c}} -->


{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
Line 349: Line 310:
* WE: ~2 = 1199.0238{{c}}, ~21/20 = 72.3506{{c}}
* WE: ~2 = 1199.0238{{c}}, ~21/20 = 72.3506{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4956{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 72.653{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 72.409{{c}} -->


{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
Line 371: Line 330:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 681.2447{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 681.2447{{c}}
: error map: {{val| 0.000 -20.710 +1.239 +6.110 }}
: error map: {{val| 0.000 -20.710 +1.239 +6.110 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 681.278{{c}}
: [[error map]]: {{val| 0.000 -20.677 +0.906 +6.510 }} -->


{{Optimal ET sequence|legend=1| 7, 30b, 37b }}
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}
Line 388: Line 345:
* WE: ~2 = 1202.0285{{c}}, ~3/2 = 682.4922{{c}}
* WE: ~2 = 1202.0285{{c}}, ~3/2 = 682.4922{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.3267{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.3267{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 681.254{{c}} -->


{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
Line 404: Line 360:
* WE: ~2 = 1201.4928{{c}}, ~3/2 = 682.1188{{c}}
* WE: ~2 = 1201.4928{{c}}, ~3/2 = 682.1188{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.2669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.2669{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 681.237{{c}} -->


{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}

Revision as of 09:25, 10 March 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of low complexity, high error, temperaments which temper out the septimal quarter-tone, 36/35. 36 is both a square and a triangular number, and this helps make 36/35 a septimal interval of considerable significance. These temperaments equate 6/5 with 7/6, 5/4 with 9/7, and 7/4 with 9/5, so minor and major thirds and sixths are intervals of 5 and 7 at the same time.

Temperaments discussed elsewhere include

Penta

For the 5-limit version, see Syntonic–diatonic equivalence continuum #University.

Subgroup: 2.3.5.7

Comma list: 28/25, 36/35

Mapping: [⟨1 1 2 2], ⟨0 3 2 4]]

mapping generators: ~2, ~7/6

Optimal tunings:

  • WE: ~2 = 1186.8093 ¢, ~7/6 = 237.3390 ¢
error map: ⟨-13.191 -3.129 +61.983 -45.851]
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 231.845 ¢
error map: ⟨0.000 +6.291 +85.850 -24.498]

Optimal ET sequence: 1bd, …, 4bcd, 5

Badness (Sintel): 1.19

Progression

Not to be confused with Progress.
For the 5-limit version, see Miscellaneous 5-limit temperaments #Lafayette.

Progression, named by Gene Ward Smith in 2011[1] can be described as the 8d & 9 temperament, and 17edo is an obvious tuning for it.

Subgroup: 2.3.5.7

Comma list: 36/35, 392/375

Mapping: [⟨1 1 2 2], ⟨0 5 3 7]]

mapping generators: ~2, ~15/14

Optimal tunings:

  • WE: ~2 = 1193.9544 ¢, ~15/14 = 140.2169 ¢
error map: ⟨-6.046 -6.916 +22.246 +0.601]
  • CWE: ~2 = 1200.000 ¢, ~15/14 = 139.8991 ¢
error map: ⟨0.000 -2.460 +33.384 +10.468]

Optimal ET sequence: 8d, 9, 17c

Badness (Sintel): 1.22

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 77/75

Mapping: [⟨1 1 2 2 3], ⟨0 5 3 7 4]]

Optimal tunings:

  • WE: ~2 = 1194.7089 ¢, ~12/11 = 140.1262 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 139.8776 ¢

Optimal ET sequence: 8d, 9, 17c

Badness (Sintel): 0.861

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 66/65

Mapping: [⟨1 1 2 2 3 3], ⟨0 5 3 7 4 6]]

Optimal tunings:

  • WE: ~2 = 1195.3694 ¢, ~13/12 = 140.2080 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.9749 ¢

Optimal ET sequence: 8d, 9, 17c

Badness (Sintel): 0.750

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 26/25, 36/35, 51/50, 56/55, 66/65

Mapping: [⟨1 1 2 2 3 3 4], ⟨0 5 3 7 4 6 1]]

Optimal tunings:

  • WE: ~2 = 1193.8350 ¢, ~13/12 = 140.6779 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.5885 ¢

Optimal ET sequence: 8d, 9, 17cg

Badness (Sintel): 0.853

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65

Mapping: [⟨1 1 2 2 3 3 4 4], ⟨0 5 3 7 4 6 1 2]]

Optimal tunings:

  • WE: ~2 = 1196.1446 ¢, ~13/12 = 140.0276 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.0301 ¢

Optimal ET sequence: 8d, 9, 17cg

Badness (Sintel): 1.02

Subklei

For the 5-limit version, see Miscellaneous 5-limit temperaments #Delorean.

Subgroup: 2.3.5.7

Comma list: 36/35, 1029/1000

Mapping: [⟨1 -3 -3 -1], ⟨0 6 7 5]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1197.5285 ¢, ~5/3 = 915.8950 ¢
error map: ⟨-2.471 -11.171 +18.366 +3.121]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.4228 ¢
error map: ⟨0.000 -9.418 +21.646 +8.288]

Optimal ET sequence: 4, 13cd, 17c, 21, 38c

Badness (Sintel): 1.55

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 77/75, 352/343

Mapping: [⟨1 -3 -3 -1 -8], ⟨0 6 7 5 15]]

Optimal tunings:

  • WE: ~2 = 1196.3345 ¢, ~5/3 = 913.9469 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 916.2970 ¢

Optimal ET sequence: 4e, …, 13cdee, 17c

Badness (Sintel): 1.48

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 66/65, 352/343

Mapping: [⟨1 3 4 4 7 7], ⟨0 6 7 5 15 14]]

Optimal tunings:

  • WE: ~2 = 1196.0784 ¢, ~5/3 = 914.1466 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 916.6712 ¢

Optimal ET sequence: 4ef, …, 13cdeef, 17c

Badness (Sintel): 1.34

Subkla

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 1029/1000

Mapping: [⟨1 -3 -3 -1 5], ⟨0 6 7 5 -2]]

Optimal tunings:

  • WE: ~2 = 1196.3809 ¢, ~5/3 = 913.4153 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.8804 ¢

Optimal ET sequence: 4, 13cd, 17c, 38ce

Badness (Sintel): 1.56

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 66/65, 640/637

Mapping: [⟨1 -3 -3 -1 5 6], ⟨0 6 7 5 -2 -3]]

Optimal tunings:

  • WE: ~2 = 1196.5477 ¢, ~5/3 = 913.4822 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.9416 ¢

Optimal ET sequence: 4, 17c, 38ce

Badness (Sintel): 1.52

Naian

For the 5-limit version, see Miscellaneous 5-limit temperaments #Naian.

Subgroup: 2.3.5.7

Comma list: 36/35, 9604/9375

Mapping: [⟨1 -4 -2 -4], ⟨0 9 7 11]]

mapping generators: ~2, ~75/49

Optimal tunings:

  • WE: ~2 = 1195.8807 ¢, ~75/49 = 741.8522 ¢
error map: ⟨-4.119 -8.808 +14.890 +8.026]
  • CWE: ~2 = 1200.0000 ¢, ~75/49 = 743.9885 ¢
error map: ⟨0.000 -6.059 +21.606 +15.047]

Optimal ET sequence: 8d, 21, 29cd

Badness (Sintel): 2.89

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 2541/2500

Mapping: [⟨1 -4 -2 -4 1], ⟨0 9 7 11 4]]

Optimal tunings:

  • WE: ~2 = 1194.3937 ¢, ~75/49 = 741.2117 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/49 = 744.2018 ¢

Optimal ET sequence: 8d, 21, 29cde

Badness (Sintel): 1.95

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 66/65, 507/500

Mapping: [⟨1 -4 -2 -4 1 0], ⟨0 9 7 11 4 6]]

Optimal tunings:

  • WE: ~2 = 1193.8564 ¢, ~20/13 = 740.9635 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.2703 ¢

Optimal ET sequence: 8d, 21, 29cdef

Badness (Sintel): 1.55

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 36/35, 51/50, 56/55, 66/65, 170/169

Mapping: [⟨1 -4 -2 -4 1 0 1], ⟨0 9 7 11 4 6 5]]

Optimal tunings:

  • WE: ~2 = 1194.2330 ¢, ~20/13 = 741.1242 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.2679 ¢

Optimal ET sequence: 8d, 21, 29cdef

Badness (Sintel): 1.43

Slurpee

For the 5-limit version, see Miscellaneous 5-limit temperaments #Slurpee.

Subgroup: 2.3.5.7

Comma list: 36/35, 51200/50421

Mapping: [⟨1 2 3 3], ⟨0 -7 -11 -3]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • WE: ~2 = 1197.9281 ¢, ~21/20 = 72.1780 ¢
error map: ⟨-2.072 -11.345 +13.512 +8.424]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4941 ¢
error map: ⟨0.000 -9.414 +16.251 +13.692]

Optimal ET sequence: 16, 17c, 33

Badness (Sintel): 2.91

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 121/120, 352/343

Mapping: [⟨1 2 3 3 4], ⟨0 -7 -11 -3 -9]]

Optimal tunings:

  • WE: ~2 = 1198.4220 ¢, ~21/20 = 72.2015 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4470 ¢

Optimal ET sequence: 16, 17c, 33

Badness (Sintel): 1.67

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 66/65, 143/140, 352/343

Mapping: [⟨1 2 3 3 4 4], ⟨0 -7 -11 -3 -9 -5]]

Optimal tunings:

  • WE: ~2 = 1199.0238 ¢, ~21/20 = 72.3506 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4956 ¢

Optimal ET sequence: 16, 17c, 33

Badness (Sintel): 1.37

Shallowtone

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Shallowtone (5-limit).

Subgroup: 2.3.5.7

Comma list: 36/35, 295245/262144

Mapping: [⟨1 0 18 -16], ⟨0 1 -10 12]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1202.4397 ¢, ~3/2 = 682.6219 ¢
error map: ⟨+2.440 -16.893 +6.986 +12.877]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.2447 ¢
error map: ⟨0.000 -20.710 +1.239 +6.110]

Optimal ET sequence: 7, 30b, 37b

Badness (Sintel): 7.79

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 72171/65536

Mapping: [⟨1 0 18 -16 16], ⟨0 1 -10 12 -8]]

Optimal tunings:

  • WE: ~2 = 1202.0285 ¢, ~3/2 = 682.4922 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.3267 ¢

Optimal ET sequence: 7, 30b, 37b

Badness (Sintel): 4.29

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 16731/16384

Mapping: [⟨1 0 18 -16 16 -1], ⟨0 1 -10 12 -8 3]]

Optimal tunings:

  • WE: ~2 = 1201.4928 ¢, ~3/2 = 682.1188 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.2669 ¢

Optimal ET sequence: 7, 30b, 37b

Badness (Sintel): 3.19

References

  1. ↑ Yahoo! Tuning Group | Progression temperament
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