Xenharmonic Wiki

Stearnsmic clan: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
4,516 edits
mNo edit summary
 
(13 intermediate revisions by 7 users not shown)
Line 1: Line 1:
The '''stearnsmic clan''' tempers out the stearnsma, the no-fives comma {{monzo| 1 10 0 -6 }} = 118098/117649.  
{{Technical data page}}
The '''stearnsmic clan''' tempers out the [[stearnsma]], the no-fives comma {{monzo| 1 10 0 -6 }} = 118098/117649.
 
== No-five stearnsmic ==
[[Subgroup]]: 2.3.7
 
[[Comma list]]: 118098/117649
 
{{Mapping|legend=2| 2 1 2 | 0 3 5 }}
 
: mapping generators: ~343/243, ~9/7
 
{{Mapping|legend=3| 2 1 0 2 | 0 3 0 5 }}
 
[[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:  
Stearnsmic temperaments not listed include:  
* ''[[Hedgehog]]'', {50/49, 245/243} → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Hedgehog]]'' (+50/49 or 245/243) → [[Porcupine family #Hedgehog|Porcupine family]]
* [[Wizard]], {225/224, 118098/117649} → [[Marvel temperaments #Wizard|Marvel temperaments]]
* [[Wizard]] (+225/224) → [[Marvel temperaments #Wizard|Marvel temperaments]]
* ''[[Echidna]]'', {1728/1715, 2048/2025} → [[Diaschismic family #Echidna|Diaschismic family]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Harry]], {2401/2400, 19683/19600} → [[Gravity family #Harry|Gravity family]] and [[Breedsmic temperaments #Harry|Breedsmic temperaments]]
* [[Harry]] (+2401/2400 or 19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Octoid]]'', {4375/4374, 16875/16807} → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]
* ''[[Octoid]]'' (+4375/4374 or 16875/16807) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]
* ''[[Septisuperfourth]]'', {6144/6125, 118098/117649} → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Septisuperfourth]]'' (+6144/6125) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Pogo]]'', {32805/32768, 118098/117649} → [[Schismatic family #Pogo|Schismatic 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 }}


Considered below are supers, stearnscape, garistearn and decistearn.  
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911


== No-five stearnsmic ==
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
Subgroup: 2.3.7
 
Badness: 0.031857
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 118098/117649
Comma list: 540/539, 729/728, 1575/1573, 4096/4095


[[Sval]] [[mapping]]: [{{val| 2 1 2 }}, {{val| 0 3 5 }}]
Mapping: {{mapping| 2 1 22 2 25 -2 | 0 3 -24 5 -25 13 }}


[[Gencom]] [[mapping]]: [{{val| 2 1 0 2 }}, {{val| 0 3 0 5 }}]
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911


[[POTE generator]]: ~9/7 = 433.888
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}


{{Val list|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838 }}
Badness: 0.017514


== Supers ==
== Supers ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 118098/117649
[[Comma list]]: 5120/5103, 118098/117649


[[Mapping]]: [{{val| 2 1 -12 2 }}, {{val| 0 3 23 5 }}]
{{Mapping|legend=1| 2 1 -12 2 | 0 3 23 5 }}
 
{{Multival|legend=1| 6 46 10 59 -1 -106 }}


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


{{Val list|legend=1| 58, 94, 152 }}
{{Optimal ET sequence|legend=1| 58, 94, 152 }}


[[Badness]]: 0.092748
[[Badness]]: 0.092748
Line 45: Line 91:
Comma list: 540/539, 4000/3993, 5120/5103
Comma list: 540/539, 4000/3993, 5120/5103


Mapping: [{{val| 2 1 -12 2 -9 }}, {{val| 0 3 23 5 22 }}]
Mapping: {{mapping| 2 1 -12 2 -9 | 0 3 23 5 22 }}


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


Optimal GPV sequence: {{Val list| 58, 94, 152 }}
{{Optimal ET sequence|legend=1| 58, 94, 152 }}


Badness: 0.028240
Badness: 0.028240
Line 58: Line 104:
Comma list: 352/351, 540/539, 729/728, 1575/1573
Comma list: 352/351, 540/539, 729/728, 1575/1573


Mapping: [{{val| 2 1 -12 2 -9 -2 }}, {{val| 0 3 23 5 22 13 }}]
Mapping: {{mapping| 2 1 -12 2 -9 -2 | 0 3 23 5 22 13 }}


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


Optimal GPV sequence: {{Val list| 58, 94, 152f }}
{{Optimal ET sequence|legend=1| 58, 94, 152f }}


Badness: 0.021645
Badness: 0.021645
Line 71: Line 117:
Comma list: 170/169, 289/288, 352/351, 442/441, 561/560
Comma list: 170/169, 289/288, 352/351, 442/441, 561/560


Mapping: [{{val| 2 1 -12 2 -9 -2 6 }}, {{val| 0 3 23 5 22 13 3 }}]
Mapping: {{mapping| 2 1 -12 2 -9 -2 6 | 0 3 23 5 22 13 3 }}


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


Optimal GPV sequence: {{Val list| 58, 94, 152f }}
{{Optimal ET sequence|legend=1| 58, 94, 152f }}


Badness: 0.021316
Badness: 0.021316


== Stearnscape ==
== Stearnscape ==
Subgroup: 2.3.5.7
[[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 }}


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


{{Multival|legend=1| 36 66 60 21 -6 -46 }}
[[Optimal tuning]] ([[CTE]]): ~2450/2187 = 1\6, ~567/500 = 216.9394 (~245/243 = 16.9394)


[[POTE generator]]: ~567/500 = 216.942
{{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}
 
{{Val list|legend=1| 72, 210, 282, 354 }}


[[Badness]]: 0.090467
[[Badness]]: 0.090467
Line 101: 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: ~567/500 = 216.950
Optimal tuning (CTE): ~55/49 = 1\6, ~567/500 = 216.9242 (~100/99 = 16.9242)


Optimal GPV sequence: {{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) 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 }}


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


{{Val list|legend=1| 94, 282, 658d, 940dd }}
{{Optimal ET sequence|legend=1| 94, 282, 658d, 940dd }}


[[Badness]]: 0.307043
[[Badness]]: 0.307043
Line 131: 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


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


Badness: 0.082255
Badness: 0.082255
Line 144: 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


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


Badness: 0.045941
Badness: 0.045941
Line 157: 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


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


Badness: 0.027608
Badness: 0.027608
Line 168: Line 236:
{{See also| Trisedodge family }}
{{See also| Trisedodge family }}


The ''decistearn'' temperament (80&130) 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]].
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
[[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 189: 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)


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


Badness: 0.038556
Badness: 0.038556
Line 202: 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)


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


Badness: 0.026895
Badness: 0.026895
Line 215: 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)


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


Badness: 0.024234
Badness: 0.024234
Line 228: 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)


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


Badness: 0.018232
Badness: 0.018232


[[Category:Regular temeprament 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:Stearnsmic clan| ]] <!-- main article -->
[[Category:Hades]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Stearnsmic_clan"