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[[Comma list]]: 118098/117649
[[Comma list]]: 118098/117649


{{Mapping|legend=2| 2 1 2 | 0 3 5 }}
{{Mapping|legend=3| 2 1 2 | 0 3 5 }}


{{Mapping|legend=3| 2 1 0 2 | 0 3 0 5 }}
{{Mapping|legend=5| 2 1 0 2 | 0 3 0 5 }}
: mapping generators: ~343/243, ~9/7
: mapping generators: ~343/243, ~9/7


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


{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838, 1322d }}
{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838, 1322d }}
Line 27: Line 27:
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.  
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.  


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.  
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.  


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.  
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.  


Stearnsmic temperaments not listed include:  
Temperaments discussed elsewhere are:  
* ''[[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]]
* [[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]]
* ''[[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]]


Considered below are pogo, supers, stearnscape, garistearn and decistearn.
The rest are considered below.  


== Pogo ==
== Pogo ==
Line 52: Line 51:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~343/243 = 599.9860{{c}}, ~9/7 = 433.8905{{c}}
* [[WE]]: ~343/243 = 599.9860{{c}}, ~9/7 = 433.8905{{c}}
: [[error map]]: {[val| -0.028 -0.298 +0.005 +0.598 }}
: [[error map]]: {{val| -0.028 -0.298 +0.005 +0.598 }}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.9010{{c}}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.9010{{c}}
: error map: {[val| 0.000 -0.252 +0.062 +0.679 }}
: error map: {{val| 0.000 -0.252 +0.062 +0.679 }}


{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
Line 65: Line 64:
Comma list: 540/539, 4000/3993, 32805/32768
Comma list: 540/539, 4000/3993, 32805/32768


Mapping: {{mapping| 2 1 22 2 25 | 0 3 -24 5 -25 }}
{{Mapping|legend=0| 2 1 22 2 25 | 0 3 -24 5 -25 }}


Optimal tunings:  
Optimal tunings:  
Line 80: Line 79:
Comma list: 540/539, 729/728, 1575/1573, 4096/4095
Comma list: 540/539, 729/728, 1575/1573, 4096/4095


Mapping: {{mapping| 2 1 22 2 25 -2 | 0 3 -24 5 -25 13 }}
{{Mapping|legend=0| 2 1 22 2 25 -2 | 0 3 -24 5 -25 13 }}


Optimal tunings:  
Optimal tunings:  
Line 101: Line 100:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~343/243 = 599.8062{{c}}, ~9/7 = 434.0778{{c}}
* [[WE]]: ~343/243 = 599.8062{{c}}, ~9/7 = 434.0778{{c}}
: [[error map]]: {[val| -0.388 +0.085 -0.199 +1.175 }}
: [[error map]]: {{val| -0.388 +0.085 -0.199 +1.175 }}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 434.2025{{c}}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 434.2025{{c}}
: error map: {[val| 0.000 +0.653 +0.335 +2.187 }}
: error map: {{val| 0.000 +0.653 +0.335 +2.187 }}


{{Optimal ET sequence|legend=1| 36c, 58, 94, 152 }}
{{Optimal ET sequence|legend=1| 36c, 58, 94, 152 }}
Line 114: Line 113:
Comma list: 540/539, 4000/3993, 5120/5103
Comma list: 540/539, 4000/3993, 5120/5103


Mapping: {{mapping| 2 1 -12 2 -9 | 0 3 23 5 22 }}
{{Mapping|legend=0| 2 1 -12 2 -9 | 0 3 23 5 22 }}


Optimal tunings:  
Optimal tunings:  
Line 129: Line 128:
Comma list: 352/351, 540/539, 729/728, 1575/1573
Comma list: 352/351, 540/539, 729/728, 1575/1573


Mapping: {{mapping| 2 1 -12 2 -9 -2 | 0 3 23 5 22 13 }}
{{Mapping|legend=0| 2 1 -12 2 -9 -2 | 0 3 23 5 22 13 }}


Optimal tunings:  
Optimal tunings:  
Line 144: 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: {{mapping| 2 1 -12 2 -9 -2 6 | 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:  
Optimal tunings:  
Line 153: Line 152:


Badness (Sintel): 1.09
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]].''
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.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 225/224, 118098/117649
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: 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
== 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.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 126/125, 118098/117649
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45
[[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 }}
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}
[[Badness]] (Sintel): 2.47
=== 11-limit ===
Subgroup: 2.3.5.7.11
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
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 144/143, 196/195, 364/363
{{Mapping|legend=0| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
Optimal tunings:
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}
Badness (Sintel): 0.987


== Stearnscape ==
== Stearnscape ==
Line 166: Line 522:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2450/2187 = 199.9984{{c}}, ~567/500 = 216.9400{{c}}
* [[WE]]: ~2450/2187 = 199.9984{{c}}, ~567/500 = 216.9400{{c}}
: [[error map]]: {[val| -0.010 -0.320 +0.023 +0.564 }}
: [[error map]]: {{val| -0.010 -0.320 +0.023 +0.564 }}
* [[CWE]]: ~2450/2187 = 200.0000{{c}}, ~567/500 = 216.9406{{c}}
* [[CWE]]: ~2450/2187 = 200.0000{{c}}, ~567/500 = 216.9406{{c}}
: error map: {[val| 0.000 -0.312 +0.033 +0.580 }}
: error map: {{val| 0.000 -0.312 +0.033 +0.580 }}


{{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}
{{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}
Line 179: Line 535:
Comma list: 540/539, 4000/3993, 137781/137500
Comma list: 540/539, 4000/3993, 137781/137500


Mapping: {{mapping| 6 3 2 6 11 | 0 6 11 10 9 }}
{{Mapping|legend=0| 6 3 2 6 11 | 0 6 11 10 9 }}


Optimal tunings:  
Optimal tunings:  
Line 194: Line 550:
Comma list: 540/539, 729/728, 1575/1573, 34398/34375
Comma list: 540/539, 729/728, 1575/1573, 34398/34375


Mapping: {{mapping| 6 3 2 6 11 -6 | 0 6 11 10 9 26 }}
{{Mapping|legend=0| 6 3 2 6 11 -6 | 0 6 11 10 9 26 }}


Optimal tunings:  
Optimal tunings:  
Line 209: Line 565:
Comma list: 540/539, 729/728, 936/935, 1156/1155, 1575/1573
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 }}
{{Mapping|legend=0| 6 3 2 6 11 -6 5 | 0 6 11 10 9 26 18 }}


Optimal tunings:  
Optimal tunings:  
Line 219: Line 575:
Badness (Sintel): 0.782
Badness (Sintel): 0.782


== Garistearn ==
== Octoid ==
The garistearn temperament (94 & 282) tempers out the same commas as [[94edo]] in the [[2.3.7 subgroup]] while adding an independent generator for [[prime interval|prime]] [[5/1|5]].  
{{Main| Octoid }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Octoid]].''
 
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]].
 
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]]: 118098/117649, 33554432/33480783
[[Comma list]]: 4375/4374, 16875/16807


{{Mapping|legend=1| 94 149 0 264 | 0 0 1 0 }}
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
: mapping generators: ~1029/1024, ~5
: mapping generators: ~49/45, ~7/5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~1029/1024 = 12.7638{{c}}, ~5/4 = 386.7156{{c}} (~5120/5103 = 3.8011{{c}})
* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}
: [[error map]]: {[val| -0.201 -0.146 -0.000 +0.822 }}
: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}
* [[CWE]]: ~1029/1024 = 12.7660{{c}}, ~5/4 = 386.6637{{c}} (~5120/5103 = 3.6850{{c}})
* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}
: error map: {[val| 0.000 +0.173 +0.350 +1.387 }}
: error map: {{val| 0.000 -0.132 -0.549 +0.880 }}


{{Optimal ET sequence|legend=1| 94, 188, 282, 658d, 940dd }}
[[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]


[[Badness]] (Sintel): 7.77
{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}
 
[[Badness]] (Sintel): 1.08


=== 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, 1375/1372, 4000/3993


Mapping: {{mapping| 94 149 0 264 107 | 0 0 1 0 1 }}
{{Mapping|legend=0| 8 1 3 3 16 | 0 3 4 5 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~1029/1024 = 12.7637{{c}}, ~5/4 = 386.5270{{c}} (~5120/5103 = 3.6174{{c}})
* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
* CWE: ~1029/1024 = 12.7660{{c}}, ~5/4 = 386.4545{{c}} (~5120/5103 = 3.4758{{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| 94, 188e, 282, 376, 658de }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}


Badness (Sintel): 2.72
Badness (Sintel): 0.466


=== 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, 625/624, 729/728, 1375/1372


Mapping: {{mapping| 94 149 0 264 107 348 | 0 0 1 0 1 0 }}
{{Mapping|legend=0| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~169/168 = 12.7628{{c}}, ~5/4 = 386.7181{{c}} (~352/351 = 3.8344{{c}})
* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
* CWE: ~169/168 = 12.7660{{c}}, ~5/4 = 386.6570{{c}} (~352/351 = 3.6783{{c}})
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}


{{Optimal ET sequence|legend=0| 94, 188e, 282, 658deff, 940ddeefff }}
{{Optimal ET sequence|legend=0| 72, 152f, 224 }}


Badness (Sintel): 1.90
Badness (Sintel): 0.631


=== 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: 375/374, 540/539, 625/624, 715/714, 729/728


Mapping: {{mapping| 94 149 0 264 107 348 166 | 0 0 1 0 1 0 1 }}
{{Mapping|legend=0| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~169/168 = 12.7628{{c}}, ~5/4 = 386.7471{{c}} (~352/351 = 3.8623{{c}})
* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
* CWE: ~169/168 = 12.7660{{c}}, ~5/4 = 386.6751{{c}} (~352/351 = 3.6963{{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


{{Optimal ET sequence|legend=0| 94, 188eg, 282, 658deff, 940ddeefffg }}
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714


Badness (Sintel): 1.41
{{Mapping|legend=0| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}


== Decistearn ==
Optimal tunings:
{{See also| Trisedodge family }}
* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}


The decistearn temperament has a period of 1/10-octave and tempers out the [[hemimean comma]], 3136/3125 as well as the [[linus comma]]. It may be described as 80 & 130.
{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}


[[Subgroup]]: 2.3.5.7
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]]: 3136/3125, 118098/117649
Comma list: 169/168, 325/324, 364/363, 540/539


{{Mapping|legend=1| 10 2 14 5 | 0 3 2 5 }}
{{Mapping|legend=0| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
: mapping generators: ~15/14, ~135/98


[[Optimal tuning]]s:  
Optimal tunings:  
* [[WE]]: ~15/14 = 119.9752{{c}}, ~135/98 = 553.8455{{c}}
* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
: [[error map]]: {[val| -0.248 -0.468 +1.030 +0.277 }}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~135/98 = 553.8935{{c}}
: error map: {[val| 0.000 -0.275 +1.473 +0.642 }}


{{Optimal ET sequence|legend=1| 50, 80, 130, 470cd, 600cd, 730cd }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}


[[Badness]] (Sintel): 2.42
Badness (Sintel): 0.896


=== 11-limit ===
===== 17-limit =====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 540/539, 3136/3125, 4000/3993
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539


Mapping: {{mapping| 10 2 14 5 30 | 0 3 2 5 1 }}
{{Mapping|legend=0| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~15/14 = 119.9534{{c}}, ~11/8 = 553.8639{{c}}
* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9779{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}


{{Optimal ET sequence|legend=0| 50, 80, 130, 210e, 340ce, 470cdee, 810ccddeee }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}


Badness (Sintel): 1.27
Badness (Sintel): 0.795


=== 13-limit ===
===== 19-limit =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 351/350, 364/363, 540/539, 3136/3125
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399


Mapping: {{mapping| 10 2 14 5 30 37 | 0 3 2 5 1 0 }}
{{Mapping|legend=0| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~15/14 = 119.9741{{c}}, ~11/8 = 553.8275{{c}}
* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9051{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}


{{Optimal ET sequence|legend=0| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 8d, 72, 152 }}


Badness (Sintel): 1.11
Badness (Sintel): 0.993


=== 17-limit ===
Scales: [[Octoid72]], [[Octoid80]]
Subgroup: 2.3.5.7.11.13.17
 
==== Hexadecoid ====
Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.
 
Subgroup: 2.3.5.7.11.13


Comma list: 221/220, 289/288, 351/350, 540/539, 1632/1625
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224


Mapping: {{mapping| 10 2 14 5 30 37 27 | 0 3 2 5 1 0 3 }}
{{Mapping|legend=0| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
: mapping generators: ~448/429, ~7/5


Optimal tunings:  
Optimal tunings:  
* WE: ~15/14 = 119.9899{{c}}, ~11/8 = 553.8840{{c}}
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9141{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}


{{Optimal ET sequence|legend=0| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 80, 144, 224 }}


Badness (Sintel): 1.23
Badness (Sintel): 1.27


=== 19-limit ===
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17


Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224


Mapping: {{mapping| 10 2 14 5 30 37 27 24 | 0 3 2 5 1 0 3 4 }}
{{Mapping|legend=0| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~15/14 = 119.9949{{c}}, ~11/8 = 553.9253{{c}}
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9402{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}


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


Badness (Sintel): 1.11
Badness (Sintel): 1.46


=== 23-limit ===
===== 19-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
 
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
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444


Mapping: {{mapping| 10 2 14 5 30 37 27 24 36 | 0 3 2 5 1 0 3 4 2 }}
{{Mapping|legend=0| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~15/14 = 119.9973{{c}}, ~11/8 = 553.9247{{c}}
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9331{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}


{{Optimal ET sequence|legend=0| 30dh, 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}


Badness (Sintel): 1.02
Badness (Sintel): 1.44


[[Category:Stearnsmic clan| ]] <!-- main article -->
[[Category:Stearnsmic clan| ]] <!-- main article -->
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