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The '''stearnsmic clan''' tempers out the stearnsma, the no-fives comma {{monzo| 1 10 0 -6 }} = 118098/117649.  
{{Technical data page}}
The '''stearnsmic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[stearnsma]] ({{monzo|legend=1| 1 10 0 -6 }}, [[ratio]]: 118098/117649), a no-5's comma of about 6.59 [[cent]]s.  


Stearnsmic temperaments not listed include:
== No-5 stearnsmic ==
* ''[[Hedgehog]]'', {50/49, 245/243} → [[Porcupine family #Hedgehog|Porcupine family]]
This temperament is generated by a very slightly flat [[~]][[9/7]] major third, three of which minus a [[semi-octave]] period give the [[3/2|perfect fifth]]. Its [[ploidacot]] is diploid alpha-tricot.
* [[Wizard]], {225/224, 118098/117649} → [[Marvel temperaments #Wizard|Marvel temperaments]]
* ''[[Echidna]]'', {1728/1715, 2048/2025} → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Harry]], {2401/2400, 19683/19600} → [[Gravity family #Harry|Gravity family]] and [[Breedsmic temperaments #Harry|Breedsmic temperaments]]
* ''[[Octoid]]'', {4375/4374, 16875/16807} → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]
* ''[[Septisuperfourth]]'', {6144/6125, 118098/117649} → [[Escapade family #Septisuperfourth|Escapade family]]


Considered below are pogo, supers, stearnscape, garistearn and decistearn.  
[[Subgroup]]: 2.3.7


== No-five stearnsmic ==
[[Comma list]]: 118098/117649
Subgroup: 2.3.7
 
{{Mapping|legend=3| 2 1 2 | 0 3 5 }}
 
{{Mapping|legend=5| 2 1 0 2 | 0 3 0 5 }}
: mapping generators: ~343/243, ~9/7


[[Comma list]]: 118098/117649
[[Optimal tuning]]s:  
* [[WE]]: ~343/243 = 599.994{{c}}, ~9/7 = 433.884{{c}}
: [[error map]]: {{val| -0.012 -0.309 +0.582 }}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.885{{c}}
: error map: {{val| 0.000 -0.300 +0.600 }}


[[Sval]] [[mapping]]: [{{val| 2 1 2 }}, {{val| 0 3 5 }}]
{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838, 1322d }}


[[Gencom]] [[mapping]]: [{{val| 2 1 0 2 }}, {{val| 0 3 0 5 }}]
[[Badness]] (Sintel): 0.382


[[POTE generator]]: ~9/7 = 433.888
=== Overview to extensions ===
The second comma in the comma list determines how we extend it to include the [[harmonic]] [[5/1|5]]. Pogo (94 & 130) adds [[32805/32768]], supers (58 & 94) adds [[5120/5103]], echidna (22 & 58) adds [[1728/1715]], and hedgehog (14c & 22) adds [[50/49]]. Those are the [[strong extension]]s.  


{{Val list|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838 }}
Wizard adds [[225/224]]. Harry adds [[2401/2400]]. Those split the generator in two. Bisemidim adds [[126/125]]. Septisuperfourth adds [[6144/6125]]. Those split the generator in three. Stearnscape adds [[250047/250000]] and splits the period in three as well as the generator in two. Octoid adds [[4375/4374]] and splits the period in four. Decistearn adds [[3136/3125]] splits the period in five.


== Pogo ==
All of them extend naturally 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 semi-octave period. For strong extensions, the generator can be taken to be ~11/10, and three ~11/10's then make a ~4/3 as a result of tempering out 4000/3993, making [[10/9]] and [[12/11]] equidistant from 11/10.
{{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.
Temperaments discussed elsewhere are:
* [[Harry]] (+2401/2400 or 19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Septisuperfourth]]'' (+6144/6125) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Decistearn]]'' (+3136/3125) → [[Trisedodge family #Decistearn|Trisedodge family]]
* ''[[Gamelstearn]]'' (+1029/1024) → [[Compton family #Gamelstearn|Compton family]]
* ''[[Garistearn]]'' (+33554432/33480783) → [[94th-octave temperaments #Garistearn|94th-octave temperaments]]


Subgroup: 2.3.5.7
The rest are considered below.  


[[Comma list]]: 32805/32768, 118098/117649
== Pogo ==
The pogo temperament tempers out the [[schisma]], whose amount of tempering of the fifth is just about right for the stearnsma and vice versa. It may be described as 94 & 130, and is the temperament that reconciles the difference between [[tertiaschis]] and [[countertertiaschis]], using its semi-octave period.


[[Mapping]]: [{{val| 2 1 22 2 }}, {{val| 0 3 -24 5 }}]
[[Subgroup]]: 2.3.5.7


Mapping generators: ~343/243, ~9/7
[[Comma list]]: 32805/32768, 118098/117649


{{Multival|legend=1| 6 -48 10 -90 -1 158 }}
{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}


[[POTE generator]]: ~9/7 = 433.901
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.9860{{c}}, ~9/7 = 433.8905{{c}}
: [[error map]]: {{val| -0.028 -0.298 +0.005 +0.598 }}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.9010{{c}}
: error map: {{val| 0.000 -0.252 +0.062 +0.679 }}


{{Val list|legend=1| 36, 94, 130, 224, 354 }}
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}


[[Badness]]: 0.079635
[[Badness]] (Sintel): 2.02


=== 11-limit ===
=== 11-limit ===
Line 50: Line 64:
Comma list: 540/539, 4000/3993, 32805/32768
Comma list: 540/539, 4000/3993, 32805/32768


Mapping: [{{val| 2 1 22 2 25 }}, {{val| 0 3 -24 5 -25 }}]
{{Mapping|legend=0| 2 1 22 2 25 | 0 3 -24 5 -25 }}
 
Mapping generators: ~99/70, ~9/7


POTE generator: ~9/7 = 433.911
Optimal tunings:  
* WE: ~99/70 = 599.9700{{c}}, ~9/7 = 433.8898{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9125{{c}}


Vals: {{Val list| 36, 94, 130, 224, 354, 578 }}
{{Optimal ET sequence|legend=0| 36, 58ce, 94, 130, 224, 354, 578 }}


Badness: 0.031857
Badness (Sintel): 1.05


=== 13-limit ===
=== 13-limit ===
Line 65: Line 79:
Comma list: 540/539, 729/728, 1575/1573, 4096/4095
Comma list: 540/539, 729/728, 1575/1573, 4096/4095


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


Mapping generators: ~99/70, ~9/7
Optimal tunings:
* WE: ~99/70 = 599.9670{{c}}, ~9/7 = 433.8867{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9112{{c}}


POTE generator: ~9/7 = 433.911
{{Optimal ET sequence|legend=0| 36, 94, 130, 224, 354, 578 }}


Vals: {{Val list| 36, 94, 130, 224, 354, 578 }}
Badness (Sintel): 0.724


Badness: 0.017514
== Supers ==
Supers tempers out 5120/5103, the [[aberschisma]], and may be described as the {{nowrap| 58 & 94 }} temperament.  


== 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 }}
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8062{{c}}, ~9/7 = 434.0778{{c}}
: [[error map]]: {{val| -0.388 +0.085 -0.199 +1.175 }}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 434.2025{{c}}
: error map: {{val| 0.000 +0.653 +0.335 +2.187 }}


[[POTE generator]]: ~9/7 = 434.218
{{Optimal ET sequence|legend=1| 36c, 58, 94, 152 }}


{{Val list|legend=1| 58, 94, 152 }}
[[Badness]] (Sintel): 2.35
 
[[Badness]]: 0.092748


=== 11-limit ===
=== 11-limit ===
Line 95: Line 113:
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|legend=0| 2 1 -12 2 -9 | 0 3 23 5 22 }}


POTE generator: ~9/7 = 434.217
Optimal tunings:  
* WE: ~99/70 = 599.8051{{c}}, ~9/7 = 434.0755{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 434.2002{{c}}


Optimal GPV sequence: {{Val list| 58, 94, 152 }}
{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152 }}


Badness: 0.028240
Badness (Sintel): 0.934


=== 13-limit ===
=== 13-limit ===
Line 108: Line 128:
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|legend=0| 2 1 -12 2 -9 -2 | 0 3 23 5 22 13 }}


POTE generator: ~9/7 = 434.221
Optimal tunings:  
* WE: ~99/70 = 599.7318{{c}}, ~9/7 = 434.0271{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 434.1985{{c}}


Optimal GPV sequence: {{Val list| 58, 94, 152f }}
{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152f }}


Badness: 0.021645
Badness (Sintel): 0.894


=== 17-limit ===
=== 17-limit ===
Line 121: Line 143:
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|legend=0| 2 1 -12 2 -9 -2 6 | 0 3 23 5 22 13 3 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8786{{c}}, ~9/7 = 434.0934{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 434.1726{{c}}
 
{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152f }}
 
Badness (Sintel): 1.09
 
== Echidna ==
{{See also| Chords of echidna }}
 
Echidna adds 1728/1715, the [[orwellisma]], and 2048/2025, the [[diaschisma]], to the commas. It may be described as the {{nowrap| 22 & 58 }} temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp [[7/4]] which some prefer.
 
Echidna becomes more interesting when extended to be an 11-limit temperament by adding [[176/175]], [[540/539]] or [[896/891]] to the commas, where the same tunings can be used as before. It then is able to represent the entire [[11-odd-limit]] [[tonality diamond|diamond]] to within about 6 cents of error within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note 2mos much more. Better yet, it relates three important 11-limit edos: 22edo is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[#Hedgehog|hedgehog]]; 58edo is the smallest tuning that is distinctly consistent in the 11-odd-limit, and 80edo is the third smallest distinctly consistent in the 11-odd-limit.
 
Like most [[diaschismic family #Diaschismic|diaschismic]] extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[diaschismic family #srutal archagall (2.3.5.17)|srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, the last two of which are consistent in the [[17-odd-limit]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 2048/2025
 
{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
: mapping generators: ~45/32, ~9/7
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}
 
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}
 
[[Badness]] (Sintel): 1.47
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 540/539, 896/891
 
{{Mapping|legend=0| 2 1 9 2 12 | 0 3 -6 5 -7 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}
 
Minimax tuning:
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7
 
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}
 
Badness (Sintel): 0.859
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 364/363, 540/539
 
{{Mapping|legend=0| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}
 
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
 
Badness (Sintel): 0.978
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
 
{{Mapping|legend=0| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}
 
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
 
Badness (Sintel): 1.03
 
== Hedgehog ==
{{See also| Sensamagic clan }}
 
Hedgehog is a relatively low-accuracy temperament which tempers out 50/49 ([[jubilisma]]), 245/243 ([[sensamagic comma]]), 250/243 ([[porcupine comma]]), and 2430/2401 ([[nuwell comma]]). It is also a strong extension of [[BPS]].
 
22edo provides an obvious tuning, which happens to be the only [[patent val|patent-val]] edo tuning, but if you are looking for an alternative you could try the {{val| 146 232 338 411 }} (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note [[mos]] gives scope for harmony while stopping well short of 22. A related temperament is [[#Echidna|echidna]], which offers much more accuracy. They merge on 22edo.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 50/49, 245/243
 
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}
 
: mapping generators: ~7/5, ~9/7
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 599.6061{{c}}, ~9/7 = 435.3620{{c}}
: [[error map]]: {{val| -0.788 +3.737 -9.897 +7.197 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4483{{c}}
: error map: {{val| 0.000 +4.390 -9.072 +8.416 }}
 
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
 
[[Badness]] (Sintel): 1.11
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 55/54, 99/98
 
{{Mapping|legend=0| 2 1 1 2 4 | 0 3 5 5 4 }}
 
Optimal tunings:
* WE: ~7/5 = 600.1133{{c}}, ~9/7 = 435.4680{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4431{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}
 
Badness (Sintel): 0.764
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 55/54, 65/63, 99/98
 
{{Mapping|legend=0| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}
 
Optimal tunings:
* WE: ~7/5 = 600.3651{{c}}, ~9/7 = 436.1258{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.0483{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}
 
Badness (Sintel): 0.889
 
==== Urchin ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 50/49, 55/54, 66/65
 
{{Mapping|legend=0| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}
 
Optimal tunings:
* WE: ~7/5 = 598.3303{{c}}, ~9/7 = 435.8617{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.3485{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 22f }}
 
Badness (Sintel): 1.04
 
=== Hedgepig ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 245/243, 385/384
 
{{Mapping|legend=0| 2 1 1 2 12 | 0 3 5 5 -7 }}
 
Optimal tunings:
* WE: ~7/5 = 599.7917{{c}}, ~9/7 = 435.2737{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4047{{c}}
 
{{Optimal ET sequence|legend=0| 22 }}
 
Badness (Sintel): 2.26
 
; Music
* [https://web.archive.org/web/20240624173512/http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[hedgehog14|Hedgehog[14]]], 22edo tuning.
 
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


POTE generator: ~9/7 = 434.181
Wizard tempers out the [[marvel comma]] and may be described as {{nowrap| 22 & 72 }}. It splits the ~9/7 in two parts, which can be treated as ~[[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. The [[ploidacot]] signature of wizard is diploid alpha-hexacot. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.  


Optimal GPV sequence: {{Val list| 58, 94, 152f }}
[[Subgroup]]: 2.3.5.7


Badness: 0.021316
[[Comma list]]: 225/224, 118098/117649


== Stearnscape ==
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
Subgroup: 2.3.5.7
: mapping generators: ~1225/864, ~245/216
 
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}
 
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}
 
[[Badness]] (Sintel): 1.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 4000/3993
 
{{Mapping|legend=0| 2 1 5 2 8 | 0 6 -1 10 -3 }}
 
Optimal tunings:
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
 
Badness (Sintel): 0.613
 
==== Lizard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 351/350, 364/363, 385/384
 
{{Mapping|legend=0| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
 
Optimal tunings:
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
 
Badness (Sintel): 0.900
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
 
{{Mapping|legend=0| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
 
Badness (Sintel): 0.741
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
 
{{Mapping|legend=0| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}
 
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
 
Badness (Sintel): 0.955
 
==== Gizzard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 1573/1568
 
{{Mapping|legend=0| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
 
Optimal tunings:
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
 
Badness (Sintel): 0.837
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
 
{{Mapping|legend=0| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
 
Badness (Sintel): 0.694
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
{{Mapping|legend=0| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}
 
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
 
Badness (Sintel): 0.901
 
=== Mage ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 1331/1296
 
{{Mapping|legend=0| 2 1 5 2 4 | 0 6 -1 10 8 }}
 
Optimal tunings:
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
 
Badness (Sintel): 1.91


[[Comma list]]: 118098/117649, 250047/250000
== Bisemidim ==
Bisemidim tempers out [[126/125]] and may be described as the {{nowrap| 50 & 58 }} temperament. It splits the ~9/7 generator into three ~[[49/45]]'s, so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.


[[Mapping]]: [{{val| 6 3 2 6 }}, {{val| 0 6 11 10 }}]
[[Subgroup]]: 2.3.5.7


Mapping generators: ~2450/2187, ~567/500
[[Comma list]]: 126/125, 118098/117649


{{Multival|legend=1| 36 66 60 21 -6 -46 }}
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45


[[POTE generator]]: ~567/500 = 216.942
[[Optimal tuning]]s:  
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}


{{Val list|legend=1| 72, 210, 282, 354 }}
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}


[[Badness]]: 0.090467
[[Badness]] (Sintel): 2.47


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 4000/3993, 137781/137500
Comma list: 126/125, 540/539, 1344/1331
 
{{Mapping|legend=0| 2 1 2 2 5 | 0 9 11 15 8 }}
 
Optimal tunings:
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}
 
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}
 
Badness (Sintel): 1.36


Mapping: [{{val| 6 3 2 6 11 }}, {{val| 0 6 11 10 9 }}]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


POTE generator: ~567/500 = 216.950
Comma list: 126/125, 144/143, 196/195, 364/363


Optimal GPV sequence: {{Val list| 72, 282, 354 }}
{{Mapping|legend=0| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}


Badness: 0.032096
Optimal tunings:  
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}


== Garistearn ==
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}
The ''garistearn'' temperament (94&282) has a period of 1/94 octave and tempers out 118098/117649 and [[garischisma]], 33554432/33480783.


Subgroup: 2.3.5.7
Badness (Sintel): 0.987


[[Comma list]]: 118098/117649, 33554432/33480783
== Stearnscape ==
Stearnscape tempers out 250047/250000, the [[landscape comma]], with a period of 1/6 octave, and may be described as the {{nowrap| 72 & 282 }} temperament.


[[Mapping]]: [{{val|94 149 0 264}}, {{val|0 0 1 0}}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 0 94 0 149 0 -264 }}
[[Comma list]]: 118098/117649, 250047/250000


[[POTE generator]]: ~5/4 = 386.7805
{{Mapping|legend=1| 6 3 2 6 | 0 6 11 10 }}
: mapping generators: ~2450/2187, ~567/500


{{Val list|legend=1| 94, 282, 658d, 940dd }}
[[Optimal tuning]]s:
* [[WE]]: ~2450/2187 = 199.9984{{c}}, ~567/500 = 216.9400{{c}}
: [[error map]]: {{val| -0.010 -0.320 +0.023 +0.564 }}
* [[CWE]]: ~2450/2187 = 200.0000{{c}}, ~567/500 = 216.9406{{c}}
: error map: {{val| 0.000 -0.312 +0.033 +0.580 }}


[[Badness]]: 0.307043
{{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}
 
[[Badness]] (Sintel): 2.29


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 4000/3993, 33554432/33480783
Comma list: 540/539, 4000/3993, 137781/137500


Mapping: [{{val|94 149 0 264 107}}, {{val|0 0 1 0 1}}]
{{Mapping|legend=0| 6 3 2 6 11 | 0 6 11 10 9 }}


POTE generator: ~5/4 = 386.5968
Optimal tunings:  
* WE: ~55/49 = 199.9835{{c}}, ~567/500 = 216.9321{{c}} (~100/99 = 16.9487{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~567/500 = 216.9381{{c}} (~100/99 = 16.9381{{c}})


Optimal GPV sequence: {{Val list| 94, 282, 376, 658de }}
{{Optimal ET sequence|legend=0| 72, 210e, 282, 354 }}


Badness: 0.082255
Badness (Sintel): 1.06


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 540/539, 729/728, 1575/1573, 28672/28561
Comma list: 540/539, 729/728, 1575/1573, 34398/34375


Mapping: [{{val|94 149 0 264 107 348}}, {{val|0 0 1 0 1 0}}]
{{Mapping|legend=0| 6 3 2 6 11 -6 | 0 6 11 10 9 26 }}


POTE generator: ~5/4 = 386.8141
Optimal tunings:  
* WE: ~55/49 = 199.9810{{c}}, ~312/275 = 216.9360{{c}} (~105/104 = 16.9550{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~312/275 = 216.9474{{c}} (~105/104 = 16.9474{{c}})


Optimal GPV sequence: {{Val list| 94, 282 }}
{{Optimal ET sequence|legend=0| 72, 210ef, 282, 354 }}


Badness: 0.045941
Badness (Sintel): 1.06


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 540/539, 729/728, 1156/1155, 1575/1573, 2880/2873
Comma list: 540/539, 729/728, 936/935, 1156/1155, 1575/1573
 
{{Mapping|legend=0| 6 3 2 6 11 -6 5 | 0 6 11 10 9 26 18 }}


Mapping: [{{val|94 149 0 264 107 348 166}}, {{val|0 0 1 0 1 0 1}}]
Optimal tunings:  
* WE: ~55/49 = 199.9814{{c}}, ~17/15 = 216.9376{{c}} (~105/104 = 16.9563{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~17/15 = 216.9487{{c}} (~105/104 = 16.9345{{c}})


POTE generator: ~5/4 = 386.8420
{{Optimal ET sequence|legend=0| 72, 210efg, 282, 354 }}


Optimal GPV sequence: {{Val list| 94, 282 }}
Badness (Sintel): 0.782


Badness: 0.027608
== Octoid ==
{{Main| Octoid }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Octoid]].''


== Decistearn ==
The octoid temperament has a period of 1/8 octave and tempers out 4375/4374, the [[ragisma]], and 16875/16807, the [[canopic comma]]. In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].
{{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 [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 118098/117649
[[Comma list]]: 4375/4374, 16875/16807


[[Mapping]]: [{{val|10 17 24 30}}, {{val|0 -3 -2 -5}}]
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
: mapping generators: ~49/45, ~7/5


{{Multival|legend=1| 30 20 50 -38 -5 60 }}
[[Optimal tuning]]s:
* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}
: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}
* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}
: error map: {{val| 0.000 -0.132 -0.549 +0.880 }}


[[POTE generator]]: ~6/5 = 313.960
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]


{{Val list|legend=1| 50, 80, 130, 470cd, 600cd, 730cd, 860ccd }}
{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}


[[Badness]]: 0.095543
[[Badness]] (Sintel): 1.08


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 540/539, 3136/3125, 4000/3993
Comma list: 540/539, 1375/1372, 4000/3993
 
{{Mapping|legend=0| 8 1 3 3 16 | 0 3 4 5 3 }}
 
Optimal tunings:
* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
 
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}
 
Badness (Sintel): 0.466
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 729/728, 1375/1372


Mapping: [{{val|10 17 24 30 35}}, {{val|0 -3 -2 -5 -1}}]
{{Mapping|legend=0| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}


POTE generator: ~6/5 = 314.079
Optimal tunings:  
* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}


Optimal GPV sequence: {{Val list| 50, 80, 130, 210e, 340ce }}
{{Optimal ET sequence|legend=0| 72, 152f, 224 }}


Badness: 0.038556
Badness (Sintel): 0.631
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
 
{{Mapping|legend=0| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}
 
{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}
 
Badness (Sintel): 0.729
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
 
{{Mapping|legend=0| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
 
Optimal tunings:
* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}
 
{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}
 
Badness (Sintel): 0.975
 
==== Octopus ====
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 364/363, 540/539
 
{{Mapping|legend=0| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}
 
Badness (Sintel): 0.896
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
 
{{Mapping|legend=0| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}
 
Badness (Sintel): 0.795
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
 
{{Mapping|legend=0| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 72, 152 }}
 
Badness (Sintel): 0.993
 
Scales: [[Octoid72]], [[Octoid80]]
 
==== Hexadecoid ====
Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 364/363, 540/539, 3136/3125
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224


Mapping: [{{val|10 17 24 30 35 37}}, {{val|0 -3 -2 -5 -1 0}}]
{{Mapping|legend=0| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
: mapping generators: ~448/429, ~7/5


POTE generator: ~6/5 = 313.947
Optimal tunings:  
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}


Optimal GPV sequence: {{Val list| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 80, 144, 224 }}


Badness: 0.026895
Badness (Sintel): 1.27


=== 17-limit ===
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 289/288, 351/350, 540/539, 1632/1625
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224


Mapping: [{{val|10 17 24 30 35 37 42}}, {{val|0 -3 -2 -5 -1 0 -3}}]
{{Mapping|legend=0| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}


POTE generator: ~6/5 = 313.931
Optimal tunings:  
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}


Optimal GPV sequence: {{Val list| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}


Badness: 0.024234
Badness (Sintel): 1.46


=== 19-limit ===
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444


Mapping: [{{val|10 17 24 30 35 37 42 44}}, {{val|0 -3 -2 -5 -1 0 -3 -4}}]
{{Mapping|legend=0| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}


POTE generator: ~6/5 = 313.949
Optimal tunings:  
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}


Optimal GPV sequence: {{Val list| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}


Badness: 0.018232
Badness (Sintel): 1.44


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