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Stearnsmic clan: Difference between revisions

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Stearnsmic temperaments temper out the stearnsma, the no-fives comma |1 10 0 -6> = 118098/117649. Sternismic temperaments not listed include [[Gravity_family#Harry|harry]], [[Marvel_temperaments#Wizard|wizard]], [[Schismatic_family#Pogo|pogo]] and [[Porcupine_family#Hedgehog|hedgehog]].
{{Technical data page}}
The '''stearnsmic clan''' tempers out the [[stearnsma]], the no-fives comma {{monzo| 1 10 0 -6 }} = 118098/117649.


=Sternscape=
== No-five stearnsmic ==
Commas: 118098/117649, 250047/250000
[[Subgroup]]: 2.3.7


POTE generator: ~245/243 = 16.942
[[Comma list]]: 118098/117649


Map: [<6 3 2 6|, <0 6 11 10|]
{{Mapping|legend=2| 2 1 2 | 0 3 5 }}


Wedgie: <<36 66 60 21 -6 -46||
: mapping generators: ~343/243, ~9/7


EDOs: 72, 210, 282, 354
{{Mapping|legend=3| 2 1 0 2 | 0 3 0 5 }}


Badness: 0.0905
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.888


==11-limit==
{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838 }}
Commas: 540/539, 4000/3993, 137781/137500


POTE generator: ~245/243 = 16.950
=== 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]].  


Map: [<6 3 2 6 11|, <0 6 11 10 9|]
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]]


EDOs: 72, 282, 354
Considered below are pogo, supers, stearnscape, garistearn and decistearn.


Badness: 0.0321
== 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
 
{{Mapping|legend=1| 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|legend=1| 72, 210, 282, 354 }}
 
[[Badness]]: 0.090467
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4000/3993, 137781/137500
 
Mapping: {{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|legend=1| 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: {{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 ==
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|legend=1| 94 149 0 264 | 0 0 1 0 }}
 
[[Optimal tuning]] ([[POTE]]): ~1029/1024 = 1\94, ~5/4 = 386.7805
 
{{Optimal ET sequence|legend=1| 94, 282, 658d, 940dd }}
 
[[Badness]]: 0.307043
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4000/3993, 33554432/33480783
 
Mapping: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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|legend=1| 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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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: {{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|legend=1| 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, [[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]]
Retrieved from "https://en.xen.wiki/w/Stearnsmic_clan"