Stearnsmic clan: Difference between revisions

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Stearnsmic temperaments temper out the stearnsma, the no-fives comma {{monzo|1 10 0 -6}} = 118098/117649. Sternismic temperaments not listed include [[Gravity family #Harry|harry]], [[Marvel temperaments #Wizard|wizard]], [[Schismatic family #Pogo|pogo]], [[Porcupine family #Hedgehog|hedgehog]], [[Porwell temperaments #Septisuperfourth|septisuperfourth]], [[Hemifamity temperaments #Supers|supers]], and [[Ragismic microtemperaments #Octoid|octoid]].
{{Technical data page}}
The '''stearnsmic clan''' tempers out the [[stearnsma]], the no-fives comma {{monzo| 1 10 0 -6 }} = 118098/117649.


== No-five stearnsmic ==
== No-five stearnsmic ==
Comma: 118098/117649
[[Subgroup]]: 2.3.7


No-five map: [{{val|2 1 0 2}}, {{val|0 3 0 5}}]
[[Comma list]]: 118098/117649


No-five POTE generator: ~9/7 = 433.888
{{Mapping|legend=2| 2 1 2 | 0 3 5 }}


No-five vals: {{Val list| 14, 22, 36, 94, 130, 224, 354, 484, 838 }}
: mapping generators: ~343/243, ~9/7


== Sternscape ==
{{Mapping|legend=3| 2 1 0 2 | 0 3 0 5 }}
Subgroup: 2.3.5.7
 
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.888
 
{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838 }}
 
=== Overview to extensions ===
The second comma in the comma list determines how we extend it to include the [[harmonic]] [[5/1|5]]. Pogo adds [[32805/32768]], supers [[5120/5103]], echidna [[1728/1715]], and hedgehog [[50/49]]. Those are strong extensions. The others are weak. Wizard adds [[225/224]]. Harry adds [[2401/2400]]. Those split the generator in two. Septisuperfourth adds [[6144/6125]] and splits the generator in three. Stearnscape adds [[250047/250000]] and splits the period in three. Octoid adds [[4375/4374]] and splits the period in four. Decistearn adds [[3136/3125]] splits the period in five. They all have neat extensions to the 11-limit via tempering out both [[540/539]] and [[4000/3993]].
 
Stearnsmic temperaments not listed include:
* ''[[Hedgehog]]'' (+50/49 or 245/243) → [[Porcupine family #Hedgehog|Porcupine family]]
* [[Wizard]] (+225/224) → [[Marvel temperaments #Wizard|Marvel temperaments]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Harry]] (+2401/2400 or 19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Octoid]]'' (+4375/4374 or 16875/16807) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]
* ''[[Septisuperfourth]]'' (+6144/6125) → [[Escapade family #Septisuperfourth|Escapade family]]
 
Considered below are pogo, supers, stearnscape, garistearn and decistearn.
 
== Pogo ==
{{See also| Schismatic family #Pogo }}
 
The pogo temperament (94 & 130) tempers out the [[schisma]], whose amount of tempering of the fifth is just about right for the stearnsma.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 118098/117649
 
{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}
 
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.901
 
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
 
[[Badness]]: 0.079635
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4000/3993, 32805/32768
 
Mapping: {{mapping| 2 1 22 2 25 | 0 3 -24 5 -25 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
 
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
 
Badness: 0.031857
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1575/1573, 4096/4095
 
Mapping: {{mapping| 2 1 22 2 25 -2 | 0 3 -24 5 -25 13 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
 
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
 
Badness: 0.017514
 
== Supers ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 118098/117649
 
{{Mapping|legend=1| 2 1 -12 2 | 0 3 23 5 }}
 
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 434.218
 
{{Optimal ET sequence|legend=1| 58, 94, 152 }}
 
[[Badness]]: 0.092748
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4000/3993, 5120/5103
 
Mapping: {{mapping| 2 1 -12 2 -9 | 0 3 23 5 22 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.217
 
{{Optimal ET sequence|legend=1| 58, 94, 152 }}
 
Badness: 0.028240
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 729/728, 1575/1573
 
Mapping: {{mapping| 2 1 -12 2 -9 -2 | 0 3 23 5 22 13 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.221
 
{{Optimal ET sequence|legend=1| 58, 94, 152f }}
 
Badness: 0.021645
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 170/169, 289/288, 352/351, 442/441, 561/560
 
Mapping: {{mapping| 2 1 -12 2 -9 -2 6 | 0 3 23 5 22 13 3 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.181
 
{{Optimal ET sequence|legend=1| 58, 94, 152f }}
 
Badness: 0.021316
 
== Stearnscape ==
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 118098/117649, 250047/250000
[[Comma list]]: 118098/117649, 250047/250000


[[Mapping]]: [{{val|6 3 2 6}}, {{val|0 6 11 10}}]
{{Mapping|legend=1| 6 3 2 6 | 0 6 11 10 }}


{{Multival|legend=1| 36 66 60 21 -6 -46 }}
: mapping generators: ~2450/2187, ~567/500


[[POTE generator]]: ~245/243 = 16.942
[[Optimal tuning]] ([[CTE]]): ~2450/2187 = 1\6, ~567/500 = 216.9394 (~245/243 = 16.9394)


{{Val list|legend=1| 72, 210, 282, 354 }}
{{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}


[[Badness]]: 0.090467
[[Badness]]: 0.090467
Line 30: Line 145:
Comma list: 540/539, 4000/3993, 137781/137500
Comma list: 540/539, 4000/3993, 137781/137500


Mapping: [{{val|6 3 2 6 11}}, {{val|0 6 11 10 9}}]
Mapping: {{mapping| 6 3 2 6 11 | 0 6 11 10 9 }}


POTE generator: ~245/243 = 16.950
Optimal tuning (CTE): ~55/49 = 1\6, ~567/500 = 216.9242 (~100/99 = 16.9242)


Vals: {{Val list| 72, 282, 354 }}
{{Optimal ET sequence|legend=1| 72, 210e, 282, 354 }}


Badness: 0.032096
Badness: 0.032096
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1575/1573, 34398/34375
Mapping: {{mapping| 6 3 2 6 11 -6 | 0 6 11 10 9 26 }}
Optimal tuning (CTE): ~55/49 = 1\6, ~312/275 = 216.9332 (~105/104 = 16.9332)
{{Optimal ET sequence|legend=1| 72, 210ef, 282, 354 }}
Badness: 0.0258
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 540/539, 729/728, 936/935, 1156/1155, 1575/1573
Mapping: {{mapping| 6 3 2 6 11 -6 5 | 0 6 11 10 9 26 18 }}
Optimal tuning (CTE): ~55/49 = 1\6, ~17/15 = 216.9345 (~105/104 = 16.9345)
{{Optimal ET sequence|legend=1| 72, 210efg, 282, 354 }}
Badness: 0.0154


== Garistearn ==
== Garistearn ==
The ''garistearn'' temperament (94&282, named by [[User:Xenllium|Xenllium]]) has a period of 1/94 octave and tempers out 118098/117649 and [[garischisma]], 33554432/33480783.
The garistearn temperament (94 & 282) has a period of 1/94-octave and tempers out 118098/117649 and the [[garischisma]], 33554432/33480783.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 118098/117649, 33554432/33480783
[[Comma list]]: 118098/117649, 33554432/33480783


[[Mapping]]: [{{val|94 149 0 264}}, {{val|0 0 1 0}}]
{{Mapping|legend=1| 94 149 0 264 | 0 0 1 0 }}


{{Multival|legend=1| 0 94 0 149 0 -264 }}
[[Optimal tuning]] ([[POTE]]): ~1029/1024 = 1\94, ~5/4 = 386.7805


[[POTE generator]]: ~5/4 = 386.7805
{{Optimal ET sequence|legend=1| 94, 282, 658d, 940dd }}
 
{{Val list|legend=1| 94, 282, 658d, 940dd }}


[[Badness]]: 0.307043
[[Badness]]: 0.307043
Line 60: Line 199:
Comma list: 540/539, 4000/3993, 33554432/33480783
Comma list: 540/539, 4000/3993, 33554432/33480783


Mapping: [{{val|94 149 0 264 107}}, {{val|0 0 1 0 1}}]
Mapping: {{mapping| 94 149 0 264 107 | 0 0 1 0 1 }}


POTE generator: ~5/4 = 386.5968
Optimal tuning (POTE): ~1029/1024 = 1\94, ~5/4 = 386.5968


Vals: {{Val list| 94, 282, 376, 658de }}
{{Optimal ET sequence|legend=1| 94, 282, 376, 658de }}


Badness: 0.082255
Badness: 0.082255
Line 73: Line 212:
Comma list: 540/539, 729/728, 1575/1573, 28672/28561
Comma list: 540/539, 729/728, 1575/1573, 28672/28561


Mapping: [{{val|94 149 0 264 107 348}}, {{val|0 0 1 0 1 0}}]
Mapping: {{mapping| 94 149 0 264 107 348 | 0 0 1 0 1 0 }}


POTE generator: ~5/4 = 386.8141
Optimal tuning (POTE): ~169/168 = 1\94, ~5/4 = 386.8141


Vals: {{Val list| 94, 282 }}
{{Optimal ET sequence|legend=1| 94, 282 }}


Badness: 0.045941
Badness: 0.045941
Line 86: Line 225:
Comma list: 540/539, 729/728, 1156/1155, 1575/1573, 2880/2873
Comma list: 540/539, 729/728, 1156/1155, 1575/1573, 2880/2873


Mapping: [{{val|94 149 0 264 107 348 166}}, {{val|0 0 1 0 1 0 1}}]
Mapping: {{mapping| 94 149 0 264 107 348 166 | 0 0 1 0 1 0 1 }}


POTE generator: ~5/4 = 386.8420
Optimal tuning (POTE): ~169/168 = 1\94, ~5/4 = 386.8420


Vals: {{Val list| 94, 282 }}
{{Optimal ET sequence|legend=1| 94, 282 }}


Badness: 0.027608
Badness: 0.027608


== Decistearn ==
== Decistearn ==
The ''decistearn'' temperament (80&130, named by [[User:Xenllium|Xenllium]]) has a period of 1/10 octave and tempers out the [[3136/3125|hemimean comma]], 3136/3125 as well as the [[15/14ths equal temperament|linus comma]].
{{See also| Trisedodge family }}


Subgroup: 2.3.5.7
The decistearn temperament (80 & 130) has a period of 1/10-octave and tempers out the [[hemimean comma]], 3136/3125 as well as the [[linus comma]].
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 118098/117649
[[Comma list]]: 3136/3125, 118098/117649


[[Mapping]]: [{{val|10 17 24 30}}, {{val|0 -3 -2 -5}}]
{{Mapping|legend=1| 10 2 14 5 | 0 3 2 5 }}


{{Multival|legend=1| 30 20 50 -38 -5 60 }}
: mapping generators: ~15/14, ~135/98


[[POTE generator]]: ~6/5 = 313.960
[[Optimal tuning]] ([[CTE]]): ~15/14 = 1\10, ~135/98 = 553.810 (~36/35 = 46.190)


{{Val list|legend=1| 50, 80, 130, 470cd, 600cd, 730cd, 860ccd }}
{{Optimal ET sequence|legend=1| 50, 80, 130, 470cd, 600cd, 730cd, 860ccd }}


[[Badness]]: 0.095543
[[Badness]]: 0.095543
Line 116: Line 257:
Comma list: 540/539, 3136/3125, 4000/3993
Comma list: 540/539, 3136/3125, 4000/3993


Mapping: [{{val|10 17 24 30 35}}, {{val|0 -3 -2 -5 -1}}]
Mapping: {{mapping| 10 2 14 5 30 | 0 3 2 5 1 }}


POTE generator: ~6/5 = 314.079
Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.783 (~36/35 = 46.217)


Vals: {{Val list| 50, 80, 130, 210e, 340ce }}
{{Optimal ET sequence|legend=1| 50, 80, 130, 210e, 340ce }}


Badness: 0.038556
Badness: 0.038556
Line 129: Line 270:
Comma list: 351/350, 364/363, 540/539, 3136/3125
Comma list: 351/350, 364/363, 540/539, 3136/3125


Mapping: [{{val|10 17 24 30 35 37}}, {{val|0 -3 -2 -5 -1 0}}]
Mapping: {{mapping| 10 2 14 5 30 37 | 0 3 2 5 1 0 }}


POTE generator: ~6/5 = 313.947
Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.783 (~36/35 = 46.217)


Vals: {{Val list| 50, 80, 130 }}
{{Optimal ET sequence|legend=1| 50, 80, 130 }}


Badness: 0.026895
Badness: 0.026895
Line 142: Line 283:
Comma list: 221/220, 289/288, 351/350, 540/539, 1632/1625
Comma list: 221/220, 289/288, 351/350, 540/539, 1632/1625


Mapping: [{{val|10 17 24 30 35 37 42}}, {{val|0 -3 -2 -5 -1 0 -3}}]
Mapping: {{mapping| 10 2 14 5 30 37 27 | 0 3 2 5 1 0 3 }}


POTE generator: ~6/5 = 313.931
Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.862 (~36/35 = 46.138)


Vals: {{Val list| 50, 80, 130 }}
{{Optimal ET sequence|legend=1| 50, 80, 130 }}


Badness: 0.024234
Badness: 0.024234
Line 155: Line 296:
Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475
Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475


Mapping: [{{val|10 17 24 30 35 37 42 44}}, {{val|0 -3 -2 -5 -1 0 -3 -4}}]
Mapping: {{mapping| 10 2 14 5 30 37 27 24 | 0 3 2 5 1 0 3 4 }}


POTE generator: ~6/5 = 313.949
Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.913 (~36/35 = 46.087)


Vals: {{Val list| 50, 80, 130 }}
{{Optimal ET sequence|legend=1| 50, 80, 130 }}


Badness: 0.018232
Badness: 0.018232


[[Category:Theory]]
=== 23-limit ===
[[Category:Temperament clan]]
By equating 16/13 with 69/56 and 85/69 (enabling 56:69:85 chords), we find [[23/16]] at 2 generators underneath 6 periods (which is the [[5edo]] fifth, [[10edo|6\10]]).
 
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475, 897/896
 
Mapping: {{mapping| 10 2 14 5 30 37 27 24 36 | 0 3 2 5 1 0 3 4 2 }}
 
Optimal tuning (CTE): ~161/150 = 1\10, ~11/8 = 553.918 (~36/35 = 46.082)
 
{{Optimal ET sequence|legend=1| 30dh, 50, 80, 130 }}
 
Badness: 0.016719
 
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Stearnsmic clan| ]] <!-- main article -->
[[Category:Hades]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 00:40, 24 June 2025

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.

The stearnsmic clan tempers out the stearnsma, the no-fives comma [1 10 0 -6⟩ = 118098/117649.

No-five stearnsmic

Subgroup: 2.3.7

Comma list: 118098/117649

Sval mapping: [⟨2 1 2], ⟨0 3 5]]

mapping generators: ~343/243, ~9/7

Gencom mapping: [⟨2 1 0 2], ⟨0 3 0 5]]

Optimal tuning (POTE): ~343/243 = 1\2, ~9/7 = 433.888

Optimal ET sequence: 14, 22, 36, 94, 130, 224, 354, 484, 838

Overview to extensions

The second comma in the comma list determines how we extend it to include the harmonic 5. Pogo adds 32805/32768, supers 5120/5103, echidna 1728/1715, and hedgehog 50/49. Those are strong extensions. The others are weak. Wizard adds 225/224. Harry adds 2401/2400. Those split the generator in two. Septisuperfourth adds 6144/6125 and splits the generator in three. Stearnscape adds 250047/250000 and splits the period in three. Octoid adds 4375/4374 and splits the period in four. Decistearn adds 3136/3125 splits the period in five. They all have neat extensions to the 11-limit via tempering out both 540/539 and 4000/3993.

Stearnsmic temperaments not listed include:

Considered below are pogo, supers, stearnscape, garistearn and decistearn.

Pogo

See also: Schismatic family § Pogo

The pogo temperament (94 & 130) tempers out the schisma, whose amount of tempering of the fifth is just about right for the stearnsma.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 118098/117649

Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]

Optimal tuning (POTE): ~343/243 = 1\2, ~9/7 = 433.901

Optimal ET sequence: 36, 94, 130, 224, 354

Badness: 0.079635

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 32805/32768

Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911

Optimal ET sequence: 36, 94, 130, 224, 354, 578

Badness: 0.031857

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573, 4096/4095

Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911

Optimal ET sequence: 36, 94, 130, 224, 354, 578

Badness: 0.017514

Supers

Subgroup: 2.3.5.7

Comma list: 5120/5103, 118098/117649

Mapping: [⟨2 1 -12 2], ⟨0 3 23 5]]

Optimal tuning (POTE): ~343/243 = 1\2, ~9/7 = 434.218

Optimal ET sequence: 58, 94, 152

Badness: 0.092748

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 5120/5103

Mapping: [⟨2 1 -12 2 -9], ⟨0 3 23 5 22]]

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.217

Optimal ET sequence: 58, 94, 152

Badness: 0.028240

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1575/1573

Mapping: [⟨2 1 -12 2 -9 -2], ⟨0 3 23 5 22 13]]

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 434.221

Optimal ET sequence: 58, 94, 152f

Badness: 0.021645

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 289/288, 352/351, 442/441, 561/560

Mapping: [⟨2 1 -12 2 -9 -2 6], ⟨0 3 23 5 22 13 3]]

Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.181

Optimal ET sequence: 58, 94, 152f

Badness: 0.021316

Stearnscape

Subgroup: 2.3.5.7

Comma list: 118098/117649, 250047/250000

Mapping: [⟨6 3 2 6], ⟨0 6 11 10]]

mapping generators: ~2450/2187, ~567/500

Optimal tuning (CTE): ~2450/2187 = 1\6, ~567/500 = 216.9394 (~245/243 = 16.9394)

Optimal ET sequence: 72, 210, 282, 354

Badness: 0.090467

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 137781/137500

Mapping: [⟨6 3 2 6 11], ⟨0 6 11 10 9]]

Optimal tuning (CTE): ~55/49 = 1\6, ~567/500 = 216.9242 (~100/99 = 16.9242)

Optimal ET sequence: 72, 210e, 282, 354

Badness: 0.032096

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573, 34398/34375

Mapping: [⟨6 3 2 6 11 -6], ⟨0 6 11 10 9 26]]

Optimal tuning (CTE): ~55/49 = 1\6, ~312/275 = 216.9332 (~105/104 = 16.9332)

Optimal ET sequence: 72, 210ef, 282, 354

Badness: 0.0258

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 729/728, 936/935, 1156/1155, 1575/1573

Mapping: [⟨6 3 2 6 11 -6 5], ⟨0 6 11 10 9 26 18]]

Optimal tuning (CTE): ~55/49 = 1\6, ~17/15 = 216.9345 (~105/104 = 16.9345)

Optimal ET sequence: 72, 210efg, 282, 354

Badness: 0.0154

Garistearn

The garistearn temperament (94 & 282) has a period of 1/94-octave and tempers out 118098/117649 and the garischisma, 33554432/33480783.

Subgroup: 2.3.5.7

Comma list: 118098/117649, 33554432/33480783

Mapping: [⟨94 149 0 264], ⟨0 0 1 0]]

Optimal tuning (POTE): ~1029/1024 = 1\94, ~5/4 = 386.7805

Optimal ET sequence: 94, 282, 658d, 940dd

Badness: 0.307043

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 33554432/33480783

Mapping: [⟨94 149 0 264 107], ⟨0 0 1 0 1]]

Optimal tuning (POTE): ~1029/1024 = 1\94, ~5/4 = 386.5968

Optimal ET sequence: 94, 282, 376, 658de

Badness: 0.082255

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573, 28672/28561

Mapping: [⟨94 149 0 264 107 348], ⟨0 0 1 0 1 0]]

Optimal tuning (POTE): ~169/168 = 1\94, ~5/4 = 386.8141

Optimal ET sequence: 94, 282

Badness: 0.045941

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 729/728, 1156/1155, 1575/1573, 2880/2873

Mapping: [⟨94 149 0 264 107 348 166], ⟨0 0 1 0 1 0 1]]

Optimal tuning (POTE): ~169/168 = 1\94, ~5/4 = 386.8420

Optimal ET sequence: 94, 282

Badness: 0.027608

Decistearn

See also: Trisedodge family

The decistearn temperament (80 & 130) has a period of 1/10-octave and tempers out the hemimean comma, 3136/3125 as well as the linus comma.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 118098/117649

Mapping: [⟨10 2 14 5], ⟨0 3 2 5]]

mapping generators: ~15/14, ~135/98

Optimal tuning (CTE): ~15/14 = 1\10, ~135/98 = 553.810 (~36/35 = 46.190)

Optimal ET sequence: 50, 80, 130, 470cd, 600cd, 730cd, 860ccd

Badness: 0.095543

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 4000/3993

Mapping: [⟨10 2 14 5 30], ⟨0 3 2 5 1]]

Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.783 (~36/35 = 46.217)

Optimal ET sequence: 50, 80, 130, 210e, 340ce

Badness: 0.038556

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 540/539, 3136/3125

Mapping: [⟨10 2 14 5 30 37], ⟨0 3 2 5 1 0]]

Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.783 (~36/35 = 46.217)

Optimal ET sequence: 50, 80, 130

Badness: 0.026895

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 289/288, 351/350, 540/539, 1632/1625

Mapping: [⟨10 2 14 5 30 37 27], ⟨0 3 2 5 1 0 3]]

Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.862 (~36/35 = 46.138)

Optimal ET sequence: 50, 80, 130

Badness: 0.024234

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475

Mapping: [⟨10 2 14 5 30 37 27 24], ⟨0 3 2 5 1 0 3 4]]

Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.913 (~36/35 = 46.087)

Optimal ET sequence: 50, 80, 130

Badness: 0.018232

23-limit

By equating 16/13 with 69/56 and 85/69 (enabling 56:69:85 chords), we find 23/16 at 2 generators underneath 6 periods (which is the 5edo fifth, 6\10).

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475, 897/896

Mapping: [⟨10 2 14 5 30 37 27 24 36], ⟨0 3 2 5 1 0 3 4 2]]

Optimal tuning (CTE): ~161/150 = 1\10, ~11/8 = 553.918 (~36/35 = 46.082)

Optimal ET sequence: 30dh, 50, 80, 130

Badness: 0.016719

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