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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]]s:
* [[WE]]: ~343/243 = 599.994¢, ~9/7 = 433.884¢
* [[CWE]]: ~343/243 = 600.000¢, ~9/7 = 433.885¢
 
{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838 }}
 
[[Badness]] (Sintel): 0.382
 
=== 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]], as well as [[9801/9800]], so that [[11/10]] and [[9/7]] add up to the semioctave.
 
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 }}
 
The pogo temperament (94 & 130) tempers out the [[schisma]], whose amount of tempering of the fifth is just about right for the stearnsma and vice versa. It is also the temperament that reconciles the difference between [[tertiaschis]] and [[countertertiaschis]], using its semi-octave period.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 118098/117649
 
{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}
: mapping generators: ~343/243, ~9/7
 
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.986¢, ~9/7 = 433.890¢
* [[CWE]]: ~343/243 = 600.000¢, ~9/7 = 433.901¢
 
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
 
[[Badness]] (Sintel): 2.015
 
=== 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 = 600.000{{c}}, ~9/7 = 433.911{{c}}
 
{{Optimal ET sequence|legend=0| 36, 94, 130, 224, 354, 578 }}
 
Badness (Smith): 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 = 600.000{{c}}, ~9/7 = 433.911{{c}}
 
{{Optimal ET sequence|legend=0| 36, 94, 130, 224, 354, 578 }}
 
Badness (Smith): 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]]s:
* [[WE]]: ~343/243 = 599.806¢, ~9/7 = 434.078¢
* [[CWE]]: ~343/243 = 600.000¢, ~9/7 = 434.203¢
 
{{Optimal ET sequence|legend=1| 58, 94, 152 }}
 
[[Badness]] (Sintel): 2.347
 
=== 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=0| 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=0| 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=0| 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]]s:  
* [[WE]]: ~2450/2187 = 199.998¢, ~567/500 = 216.940¢
* [[CWE]]: ~2450/2187 = 200.000¢, ~567/500 = 216.941¢


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


[[Badness]]: 0.090467
[[Badness]] (Sintel): 2.289


=== 11-limit ===
=== 11-limit ===
Line 30: Line 156:
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=0| 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=0| 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=0| 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 }}
: mapping generators: ~1029/1024, ~5


[[POTE generator]]: ~5/4 = 386.7805
[[Optimal tuning]]s:  
* [[CTE]]: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.3137¢
* [[CWE]]: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6637¢


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


[[Badness]]: 0.307043
[[Badness]] (Sintel): 7.770


=== 11-limit ===
=== 11-limit ===
Line 60: Line 214:
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 tunings:  
* CTE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.0177¢
* CWE: ~1029/1024 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.4545¢


Vals: {{Val list| 94, 282, 376, 658de }}
{{Optimal ET sequence|legend=0| 94, 282, 376, 658de }}


Badness: 0.082255
Badness (Sintel): 2.719


=== 13-limit ===
=== 13-limit ===
Line 73: Line 229:
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 tunings:  
* CTE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.0177¢
* CWE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6570¢


Vals: {{Val list| 94, 282 }}
{{Optimal ET sequence|legend=0| 94, 282 }}


Badness: 0.045941
Badness (Sintel): 1.898


=== 17-limit ===
=== 17-limit ===
Line 86: Line 244:
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 tunings:  
* CTE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 385.9792¢
* CWE: ~169/168 = 12.7660¢ (1 ⧵ 94), ~5/4 = 386.6751¢


Vals: {{Val list| 94, 282 }}
{{Optimal ET sequence|legend=0| 94, 282 }}


Badness: 0.027608
Badness (Sintel): 1.406


== 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]]s:  
* [[WE]]: ~15/14 = 119.975¢, ~135/98 = 553.845¢
* [[CWE]]: ~15/14 = 120.000¢, ~135/98 = 553.893¢


{{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]] (Sintel): 2.418


=== 11-limit ===
=== 11-limit ===
Line 116: Line 280:
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 tunings:  
* WE: ~15/14 = 119.953¢, ~11/8 = 553.864¢
* CWE: ~15/14 = 120.000¢, ~11/8 = 553.978¢


Vals: {{Val list| 50, 80, 130, 210e, 340ce }}
{{Optimal ET sequence|legend=0| 50, 80, 130, 210e, 340ce }}


Badness: 0.038556
Badness (Sintel): 1.275


=== 13-limit ===
=== 13-limit ===
Line 129: Line 295:
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 tunings:  
* WE: ~15/14 = 119.974¢, ~11/8 = 553.827¢
* CWE: ~15/14 = 120.000¢, ~11/8 = 553.905¢


Vals: {{Val list| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 50, 80, 130 }}


Badness: 0.026895
Badness (Sintel): 1.111


=== 17-limit ===
=== 17-limit ===
Line 142: Line 310:
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 tunings:  
* WE: ~15/14 = 119.990¢, ~11/8 = 553.884¢
* CWE: ~15/14 = 120.000¢, ~11/8 = 553.914¢


Vals: {{Val list| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 50, 80, 130 }}


Badness: 0.024234
Badness (Sintel): 1.235


=== 19-limit ===
=== 19-limit ===
Line 155: Line 325:
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 }}
 
Optimal tunings:
* WE: ~15/14 = 119.995¢, ~11/8 = 553.925¢
* CWE: ~15/14 = 120.000¢, ~11/8 = 553.940¢
 
{{Optimal ET sequence|legend=0| 50, 80, 130 }}
 
Badness (Sintel): 1.109
 
=== 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 }}


POTE generator: ~6/5 = 313.949
Optimal tunings:  
* WE: ~15/14 = 119.997¢, ~11/8 = 553.925¢
* CWE: ~15/14 = 120.000¢, ~11/8 = 553.933¢


Vals: {{Val list| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 30dh, 50, 80, 130 }}


Badness: 0.018232
Badness (Sintel): 1.017


[[Category:Theory]]
[[Category:Temperament clans]]
[[Category:Temperament clan]]
[[Category:Stearnsmic clan| ]] <!-- main article -->
[[Category:Hades]] <!-- key article -->
[[Category:Rank 2]]
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