Stearnsmic clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit

FloraC (talk | contribs)

4 user groups
27,021 edits
VisualWikitext
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,658 edits
→Decistearn: add very good 23-limit extension (note: i calculated the 23-limit badness as 19-limit badness * 23-limit dirichlet badness / 19-limit dirichlet badness; afaik that should be correct)
m + link to gamelstearn
 
(28 intermediate revisions by 8 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''' 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.
 
== No-5 stearnsmic ==
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.  


== No-five stearnsmic ==
[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7


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


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


[[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, 1322d }}


{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838 }}
[[Badness]] (Sintel): 0.382


=== Overview to extensions ===
=== 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]].  
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. 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. 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 ==
{{See also| Schismatic family #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.  
 
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
[[Subgroup]]: 2.3.5.7
Line 40: Line 49:
{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}
{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}


{{Multival|legend=1| 6 -48 10 -90 -1 158 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~343/243 = 599.9860{{c}}, ~9/7 = 433.8905{{c}}
[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 433.901
: [[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 }}


{{Optimal ET sequence|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 53: 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 tuning (POTE): ~99/70 = 1\2, ~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}}


{{Optimal ET sequence|legend=1| 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 66: 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 tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Optimal tunings:
* WE: ~99/70 = 599.9670{{c}}, ~9/7 = 433.8867{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9112{{c}}


{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354, 578 }}
{{Optimal ET sequence|legend=0| 36, 94, 130, 224, 354, 578 }}


Badness: 0.017514
Badness (Sintel): 0.724


== Supers ==
== Supers ==
Supers tempers out 5120/5103, the [[aberschisma]], and may be described as the {{nowrap| 58 & 94 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 81: Line 98:
{{Mapping|legend=1| 2 1 -12 2 | 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 }}


[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~9/7 = 434.218
{{Optimal ET sequence|legend=1| 36c, 58, 94, 152 }}


{{Optimal ET sequence|legend=1| 58, 94, 152 }}
[[Badness]] (Sintel): 2.35
 
[[Badness]]: 0.092748


=== 11-limit ===
=== 11-limit ===
Line 94: 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 tuning (POTE): ~99/70 = 1\2, ~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 ET sequence|legend=1| 58, 94, 152 }}
{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152 }}


Badness: 0.028240
Badness (Sintel): 0.934


=== 13-limit ===
=== 13-limit ===
Line 107: 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 tuning (POTE): ~99/70 = 1\2, ~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 ET sequence|legend=1| 58, 94, 152f }}
{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152f }}


Badness: 0.021645
Badness (Sintel): 0.894


=== 17-limit ===
=== 17-limit ===
Line 120: 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 tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.181
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=1| 58, 94, 152f }}
{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152f }}


Badness: 0.021316
Badness (Sintel): 1.09


== Stearnscape ==
== Echidna ==
[[Subgroup]]: 2.3.5.7
{{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.


[[Comma list]]: 118098/117649, 250047/250000
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]].


{{Mapping|legend=1| 6 3 2 6 | 0 6 11 10 }}
[[Subgroup]]: 2.3.5.7


: mapping generators: ~2450/2187, ~567/500
[[Comma list]]: 1728/1715, 2048/2025


{{Multival|legend=1| 36 66 60 21 -6 -46 }}
{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
: mapping generators: ~45/32, ~9/7


[[Optimal tuning]] ([[CTE]]): ~2450/2187 = 1\6, ~567/500 = 216.9394 (~245/243 = 16.9394)
[[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| 72, 210, 282, 354 }}
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}


[[Badness]]: 0.090467
[[Badness]] (Sintel): 1.47


=== 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: 176/175, 540/539, 896/891
 
{{Mapping|legend=0| 2 1 9 2 12 | 0 3 -6 5 -7 }}


Mapping: {{mapping| 6 3 2 6 11 | 0 6 11 10 9 }}
Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}


Optimal tuning (CTE): ~55/49 = 1\6, ~567/500 = 216.9242 (~100/99 = 16.9242)
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=1| 72, 210e, 282, 354 }}
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}


Badness: 0.032096
Badness (Sintel): 0.859


=== 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, 34398/34375
Comma list: 176/175, 351/350, 364/363, 540/539


Mapping: {{mapping| 6 3 2 6 11 -6 | 0 6 11 10 9 26 }}
{{Mapping|legend=0| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}


Optimal tuning (CTE): ~55/49 = 1\6, ~312/275 = 216.9332 (~105/104 = 16.9332)
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=1| 72, 210ef, 282, 354 }}
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}


Badness: 0.0258
Badness (Sintel): 0.978


=== 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, 936/935, 1156/1155, 1575/1573
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}}


Mapping: {{mapping| 6 3 2 6 11 -6 5 | 0 6 11 10 9 26 18 }}
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}


Optimal tuning (CTE): ~55/49 = 1\6, ~17/15 = 216.9345 (~105/104 = 16.9345)
Badness (Sintel): 1.03


{{Optimal ET sequence|legend=1| 72, 210efg, 282, 354 }}
== Hedgehog ==
{{See also| Sensamagic clan }}


Badness: 0.0154
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]].  


== Garistearn ==
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.  
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]]: 50/49, 245/243


{{Mapping|legend=1| 94 149 0 264 | 0 0 1 0 }}
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}


{{Multival|legend=1| 0 94 0 149 0 -264 }}
: mapping generators: ~7/5, ~9/7


[[Optimal tuning]] ([[POTE]]): ~1029/1024 = 1\94, ~5/4 = 386.7805
[[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| 94, 282, 658d, 940dd }}
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}


[[Badness]]: 0.307043
[[Badness]] (Sintel): 1.11


=== 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: 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}}


Mapping: {{mapping| 94 149 0 264 107 | 0 0 1 0 1 }}
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}


Optimal tuning (POTE): ~1029/1024 = 1\94, ~5/4 = 386.5968
Badness (Sintel): 0.764


{{Optimal ET sequence|legend=1| 94, 282, 376, 658de }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.082255
Comma list: 50/49, 55/54, 65/63, 99/98


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


Comma list: 540/539, 729/728, 1575/1573, 28672/28561
Comma list: 225/224, 325/324, 385/384, 1573/1568


Mapping: {{mapping| 94 149 0 264 107 348 | 0 0 1 0 1 0 }}
{{Mapping|legend=0| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}


Optimal tuning (POTE): ~169/168 = 1\94, ~5/4 = 386.8141
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=1| 94, 282 }}
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


Badness: 0.045941
Badness (Sintel): 0.837


=== 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: 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 }}


Mapping: {{mapping| 94 149 0 264 107 348 166 | 0 0 1 0 1 0 1 }}
Badness (Sintel): 0.694


Optimal tuning (POTE): ~169/168 = 1\94, ~5/4 = 386.8420
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


{{Optimal ET sequence|legend=1| 94, 282 }}
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Badness: 0.027608
{{Mapping|legend=0| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


== Decistearn ==
Optimal tunings:
{{See also| Trisedodge family }}
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


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]].
{{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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3136/3125, 118098/117649
[[Comma list]]: 126/125, 118098/117649
 
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45


{{Mapping|legend=1| 10 2 14 5 | 0 3 2 5 }}
[[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 }}


: mapping generators: ~15/14, ~135/98
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}


{{Multival|legend=1| 30 20 50 -38 -5 60 }}
[[Badness]] (Sintel): 2.47


[[Optimal tuning]] ([[CTE]]): ~15/14 = 1\10, ~135/98 = 553.810 (~36/35 = 46.190)
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Optimal ET sequence|legend=1| 50, 80, 130, 470cd, 600cd, 730cd, 860ccd }}
Comma list: 126/125, 540/539, 1344/1331


[[Badness]]: 0.095543
{{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 tempers out 250047/250000, the [[landscape comma]], with a period of 1/6 octave, and may be described as the {{nowrap| 72 & 282 }} temperament.
 
[[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]]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 }}
 
{{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, 3136/3125, 4000/3993
Comma list: 540/539, 4000/3993, 137781/137500


Mapping: {{mapping| 10 2 14 5 30 | 0 3 2 5 1 }}
{{Mapping|legend=0| 6 3 2 6 11 | 0 6 11 10 9 }}


Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.783 (~36/35 = 46.217)
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 ET sequence|legend=1| 50, 80, 130, 210e, 340ce }}
{{Optimal ET sequence|legend=0| 72, 210e, 282, 354 }}


Badness: 0.038556
Badness (Sintel): 1.06


=== 13-limit ===
=== 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, 729/728, 1575/1573, 34398/34375


Mapping: {{mapping| 10 2 14 5 30 37 | 0 3 2 5 1 0 }}
{{Mapping|legend=0| 6 3 2 6 11 -6 | 0 6 11 10 9 26 }}


Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.783 (~36/35 = 46.217)
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 ET sequence|legend=1| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 72, 210ef, 282, 354 }}


Badness: 0.026895
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: 221/220, 289/288, 351/350, 540/539, 1632/1625
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 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 72, 210efg, 282, 354 }}
 
Badness (Sintel): 0.782
 
== Octoid ==
{{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
 
[[Comma list]]: 4375/4374, 16875/16807
 
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
: mapping generators: ~49/45, ~7/5
 
[[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 }}
 
[[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]
 
{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}
 
[[Badness]] (Sintel): 1.08
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
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|legend=0| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
 
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 ET sequence|legend=0| 72, 152f, 224 }}
 
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: {{mapping| 10 2 14 5 30 37 27 | 0 3 2 5 1 0 3 }}
{{Mapping|legend=0| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}


Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.862 (~36/35 = 46.138)
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=1| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}


Badness: 0.024234
Badness (Sintel): 0.795


=== 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: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399


Mapping: {{mapping| 10 2 14 5 30 37 27 24 | 0 3 2 5 1 0 3 4 }}
{{Mapping|legend=0| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}


Optimal tuning (CTE): ~15/14 = 1\10, ~11/8 = 553.913 (~36/35 = 46.087)
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=1| 50, 80, 130 }}
{{Optimal ET sequence|legend=0| 8d, 72, 152 }}


Badness: 0.018232
Badness (Sintel): 0.993


=== 23-limit ===
Scales: [[Octoid72]], [[Octoid80]]
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
==== Hexadecoid ====
Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.


Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475, 897/896
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
 
{{Mapping|legend=0| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
: mapping generators: ~448/429, ~7/5
 
Optimal tunings:
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}


Mapping: {{mapping| 10 2 14 5 30 37 27 24 36 | 0 3 2 5 1 0 3 4 2 }}
{{Optimal ET sequence|legend=0| 80, 144, 224 }}


Optimal tuning (CTE): ~161/150 = 1\10, ~11/8 = 553.918 (~36/35 = 46.082)
Badness (Sintel): 1.27


{{Optimal ET sequence|legend=1| 30dh, 50, 80, 130 }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.016719
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
 
{{Mapping|legend=0| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
 
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 ET sequence|legend=0| 80, 144, 224, 528dg }}
 
Badness (Sintel): 1.46
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
 
{{Mapping|legend=0| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}
 
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 ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}
 
Badness (Sintel): 1.44


[[Category:Temperament clans]]
[[Category:Stearnsmic clan| ]] <!-- main article -->
[[Category:Stearnsmic clan| ]] <!-- main article -->
[[Category:Hades]] <!-- key article -->
[[Category:Hades]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Temperament clans]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 08:28, 3 October 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The stearnsmic clan of temperaments tempers out the stearnsma (monzo: [1 10 0 -6⟩, ratio: 118098/117649), a no-5's comma of about 6.59 cents.

No-5 stearnsmic

This temperament is generated by a very slightly flat ~9/7 major third, three of which minus a semi-octave period give the perfect fifth. Its ploidacot is diploid alpha-tricot.

Subgroup: 2.3.7

Comma list: 118098/117649

Subgroup-val mapping: [⟨2 1 2], ⟨0 3 5]]

Gencom mapping: [⟨2 1 0 2], ⟨0 3 0 5]]

mapping generators: ~343/243, ~9/7

Optimal tunings:

  • WE: ~343/243 = 599.994 ¢, ~9/7 = 433.884 ¢
error map: ⟨-0.012 -0.309 +0.582]
  • CWE: ~343/243 = 600.000 ¢, ~9/7 = 433.885 ¢
error map: ⟨0.000 -0.300 +0.600]

Optimal ET sequence: 14, 22, 36, 94, 130, 224, 354, 484, 838, 1322d

Badness (Sintel): 0.382

Overview to extensions

The second comma in the comma list determines how we extend it to include the harmonic 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 extensions.

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. 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.

Temperaments discussed elsewhere are:

The rest are considered below.

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.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 118098/117649

Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]

Optimal tunings:

  • WE: ~343/243 = 599.9860 ¢, ~9/7 = 433.8905 ¢
error map: ⟨-0.028 -0.298 +0.005 +0.598]
  • CWE: ~343/243 = 600.000 ¢, ~9/7 = 433.9010 ¢
error map: ⟨0.000 -0.252 +0.062 +0.679]

Optimal ET sequence: 36, 94, 130, 224, 354

Badness (Sintel): 2.02

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 32805/32768

Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]

Optimal tunings:

  • WE: ~99/70 = 599.9700 ¢, ~9/7 = 433.8898 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 433.9125 ¢

Optimal ET sequence: 36, 58ce, 94, 130, 224, 354, 578

Badness (Sintel): 1.05

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573, 4096/4095

Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]

Optimal tunings:

  • WE: ~99/70 = 599.9670 ¢, ~9/7 = 433.8867 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 433.9112 ¢

Optimal ET sequence: 36, 94, 130, 224, 354, 578

Badness (Sintel): 0.724

Supers

Supers tempers out 5120/5103, the aberschisma, and may be described as the 58 & 94 temperament.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 118098/117649

Mapping: [⟨2 1 -12 2], ⟨0 3 23 5]]

Optimal tunings:

  • WE: ~343/243 = 599.8062 ¢, ~9/7 = 434.0778 ¢
error map: ⟨-0.388 +0.085 -0.199 +1.175]
  • CWE: ~343/243 = 600.000 ¢, ~9/7 = 434.2025 ¢
error map: ⟨0.000 +0.653 +0.335 +2.187]

Optimal ET sequence: 36c, 58, 94, 152

Badness (Sintel): 2.35

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 5120/5103

Mapping: [⟨2 1 -12 2 -9], ⟨0 3 23 5 22]]

Optimal tunings:

  • WE: ~99/70 = 599.8051 ¢, ~9/7 = 434.0755 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 434.2002 ¢

Optimal ET sequence: 36ce, 58, 94, 152

Badness (Sintel): 0.934

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1575/1573

Mapping: [⟨2 1 -12 2 -9 -2], ⟨0 3 23 5 22 13]]

Optimal tunings:

  • WE: ~99/70 = 599.7318 ¢, ~9/7 = 434.0271 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 434.1985 ¢

Optimal ET sequence: 36ce, 58, 94, 152f

Badness (Sintel): 0.894

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 289/288, 352/351, 442/441, 561/560

Mapping: [⟨2 1 -12 2 -9 -2 6], ⟨0 3 23 5 22 13 3]]

Optimal tunings:

  • WE: ~99/70 = 599.8786 ¢, ~9/7 = 434.0934 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 434.1726 ¢

Optimal ET sequence: 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 22 & 58 temperament. 58edo or 80edo make for good tunings, or their vals can be added to ⟨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 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; 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 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 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: [⟨2 1 9 2], ⟨0 3 -6 5]]

mapping generators: ~45/32, ~9/7

Optimal tunings:

  • WE: ~45/32 = 599.3056 ¢, ~9/7 = 434.3524 ¢
error map: ⟨-1.389 +0.408 +1.322 +1.547]
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8327 ¢
error map: ⟨0.000 +2.543 +4.690 +5.338]

Optimal ET sequence: 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: [⟨2 1 9 2 12], ⟨0 3 -6 5 -7]]

Optimal tunings:

  • WE: ~45/32 = 599.3085 ¢, ~9/7 = 434.3511 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8647 ¢

Minimax tuning:

  • 11-odd-limit: ~9/7 = [5/12 0 0 1/12 -1/12⟩
[[1 0 0 0 0⟩, [7/4 0 0 1/4 -1/4⟩, [2 0 0 -1/2 1/2⟩, [37/12 0 0 5/12 -5/12⟩, [37/12 0 0 -7/12 7/12⟩]
unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 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: [⟨2 1 9 2 12 19], ⟨0 3 -6 5 -7 -16]]

Optimal tunings:

  • WE: ~45/32 = 599.3397 ¢, ~9/7 = 434.2772 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.7864 ¢

Optimal ET sequence: 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: [⟨2 1 9 2 12 19 6], ⟨0 3 -6 5 -7 -16 3]]

Optimal tunings:

  • WE: ~45/32 = 599.4645 ¢, ~9/7 = 434.4282 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8340 ¢

Optimal ET sequence: 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 edo tuning, but if you are looking for an alternative you could try the ⟨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, which offers much more accuracy. They merge on 22edo.

Subgroup: 2.3.5.7

Comma list: 50/49, 245/243

Mapping: [⟨2 1 1 2], ⟨0 3 5 5]]

mapping generators: ~7/5, ~9/7

Optimal tunings:

  • WE: ~7/5 = 599.6061 ¢, ~9/7 = 435.3620 ¢
error map: ⟨-0.788 +3.737 -9.897 +7.197]
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4483 ¢
error map: ⟨0.000 +4.390 -9.072 +8.416]

Optimal ET sequence: 8d, 14c, 22

Badness (Sintel): 1.11

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 99/98

Mapping: [⟨2 1 1 2 4], ⟨0 3 5 5 4]]

Optimal tunings:

  • WE: ~7/5 = 600.1133 ¢, ~9/7 = 435.4680 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4431 ¢

Optimal ET sequence: 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: [⟨2 1 1 2 4 3], ⟨0 3 5 5 4 6]]

Optimal tunings:

  • WE: ~7/5 = 600.3651 ¢, ~9/7 = 436.1258 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 436.0483 ¢

Optimal ET sequence: 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: [⟨2 1 1 2 4 6], ⟨0 3 5 5 4 2]]

Optimal tunings:

  • WE: ~7/5 = 598.3303 ¢, ~9/7 = 435.8617 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 436.3485 ¢

Optimal ET sequence: 14c, 22f

Badness (Sintel): 1.04

Hedgepig

Subgroup: 2.3.5.7.11

Comma list: 50/49, 245/243, 385/384

Mapping: [⟨2 1 1 2 12], ⟨0 3 5 5 -7]]

Optimal tunings:

  • WE: ~7/5 = 599.7917 ¢, ~9/7 = 435.2737 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4047 ¢

Optimal ET sequence: 22

Badness (Sintel): 2.26

Music

Wizard

Main article: Wizard
For the 5-limit version, see Miscellaneous 5-limit temperaments #Wizard.

Wizard tempers out the marvel comma and may be described as 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: [⟨2 1 5 2], ⟨0 6 -1 10]]

mapping generators: ~1225/864, ~245/216

Optimal tunings:

  • WE: ~1225/864 = 600.3438 ¢, ~245/216 = 216.8680 ¢
error map: ⟨+0.688 -0.403 -1.463 +0.541]
  • CWE: ~1225/864 = 600.0000 ¢, ~245/216 = 216.7977 ¢
error map: ⟨0.000 -1.169 -3.111 -0.849]

Optimal ET sequence: 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: [⟨2 1 5 2 8], ⟨0 6 -1 10 -3]]

Optimal tunings:

  • WE: ~99/70 = 600.3051 ¢, ~25/22 = 216.8782 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.7961 ¢

Optimal ET sequence: 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: [⟨2 1 5 2 8 11], ⟨0 6 -1 10 -3 -10]]

Optimal tunings:

  • WE: ~55/39 = 600.4824 ¢, ~25/22 = 216.7852 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~25/22 = 216.6247 ¢

Optimal ET sequence: 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: [⟨2 1 5 2 8 11 6], ⟨0 6 -1 10 -3 -10 6]]

Optimal tunings:

  • WE: ~17/12 = 600.5032 ¢, ~17/15 = 216.8002 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.6361 ¢

Optimal ET sequence: 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: [⟨2 1 5 2 8 11 6 2], ⟨0 6 -1 10 -3 -10 6 18]]

Optimal tunings:

  • WE: ~17/12 = 600.4698 ¢, ~17/15 = 216.6925 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.5434 ¢

Optimal ET sequence: 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: [⟨2 1 5 2 8 -2], ⟨0 6 -1 10 -3 26]]

Optimal tunings:

  • WE: ~99/70 = 600.2896 ¢, ~25/22 = 216.9343 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.8501 ¢

Optimal ET sequence: 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: [⟨2 1 5 2 8 -2 6], ⟨0 6 -1 10 -3 26 6]]

Optimal tunings:

  • WE: ~17/12 = 600.3227 ¢, ~17/15 = 216.9414 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8469 ¢

Optimal ET sequence: 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: [⟨2 1 5 2 8 -2 6 15], ⟨0 6 -1 10 -3 26 6 -18]]

Optimal tunings:

  • WE: ~17/12 = 600.2637 ¢, ~17/15 = 216.9570 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8687 ¢

Optimal ET sequence: 72, 94, 166g

Badness (Sintel): 0.901

Mage

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1331/1296

Mapping: [⟨2 1 5 2 4], ⟨0 6 -1 10 8]]

Optimal tunings:

  • WE: ~77/54 = 600.6486 ¢, ~55/48 = 217.1099 ¢
  • CWE: ~77/54 = 600.0000 ¢, ~55/48 = 216.9841 ¢

Optimal ET sequence: 22, 50e, 72ee

Badness (Sintel): 1.91

Bisemidim

Bisemidim tempers out 126/125 and may be described as the 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: [⟨2 1 2 2], ⟨0 9 11 15]]

mapping generators: ~343/243, ~49/45

Optimal tunings:

  • WE: ~343/243 = 599.8915 ¢, ~49/45 = 144.5293 ¢
error map: ⟨-0.217 -1.299 +3.292 -1.103]
  • CWE: ~343/243 = 600.0000 ¢, ~49/45 = 144.5351 ¢
error map: ⟨0.000 -1.139 +3.572 -0.799]

Optimal ET sequence: 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: [⟨2 1 2 2 5], ⟨0 9 11 15 8]]

Optimal tunings:

  • WE: ~99/70 = 599.6360 ¢, ~12/11 = 144.5388 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~12/11 = 144.5623 ¢

Optimal ET sequence: 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: [⟨2 1 2 2 5 5], ⟨0 9 11 15 8 10]]

Optimal tunings:

  • WE: ~55/39 = 599.5217 ¢, ~12/11 = 144.5375 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~12/11 = 144.5698 ¢

Optimal ET sequence: 50, 58, 166cef, 224ceeff

Badness (Sintel): 0.987

Stearnscape

Stearnscape tempers out 250047/250000, the landscape comma, with a period of 1/6 octave, and may be described as the 72 & 282 temperament.

Subgroup: 2.3.5.7

Comma list: 118098/117649, 250047/250000

Mapping: [⟨6 3 2 6], ⟨0 6 11 10]]

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

Optimal tunings:

  • WE: ~2450/2187 = 199.9984 ¢, ~567/500 = 216.9400 ¢
error map: ⟨-0.010 -0.320 +0.023 +0.564]
  • CWE: ~2450/2187 = 200.0000 ¢, ~567/500 = 216.9406 ¢
error map: ⟨0.000 -0.312 +0.033 +0.580]

Optimal ET sequence: 72, 210, 282, 354

Badness (Sintel): 2.29

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨6 3 2 6 11], ⟨0 6 11 10 9]]

Optimal tunings:

  • WE: ~55/49 = 199.9835 ¢, ~567/500 = 216.9321 ¢ (~100/99 = 16.9487 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~567/500 = 216.9381 ¢ (~100/99 = 16.9381 ¢)

Optimal ET sequence: 72, 210e, 282, 354

Badness (Sintel): 1.06

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨6 3 2 6 11 -6], ⟨0 6 11 10 9 26]]

Optimal tunings:

  • WE: ~55/49 = 199.9810 ¢, ~312/275 = 216.9360 ¢ (~105/104 = 16.9550 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~312/275 = 216.9474 ¢ (~105/104 = 16.9474 ¢)

Optimal ET sequence: 72, 210ef, 282, 354

Badness (Sintel): 1.06

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 729/728, 936/935, 1156/1155, 1575/1573

Mapping: [⟨6 3 2 6 11 -6 5], ⟨0 6 11 10 9 26 18]]

Optimal tunings:

  • WE: ~55/49 = 199.9814 ¢, ~17/15 = 216.9376 ¢ (~105/104 = 16.9563 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~17/15 = 216.9487 ¢ (~105/104 = 16.9345 ¢)

Optimal ET sequence: 72, 210efg, 282, 354

Badness (Sintel): 0.782

Octoid

Main article: 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

Comma list: 4375/4374, 16875/16807

Mapping: [⟨8 1 3 3], ⟨0 3 4 5]]

mapping generators: ~49/45, ~7/5

Optimal tunings:

  • WE: ~49/45 = 150.0003 ¢, ~7/5 = 583.9416 ¢
error map: ⟨+0.002 -0.130 -0.547 +0.883]
  • CWE: ~49/45 = 150.0000 ¢, ~7/5 = 583.9411 ¢
error map: ⟨0.000 -0.132 -0.549 +0.880]

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]

Optimal ET sequence: 8d, …, 72, 152, 224

Badness (Sintel): 1.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4000/3993

Mapping: [⟨8 1 3 3 16], ⟨0 3 4 5 3]]

Optimal tunings:

  • WE: ~12/11 = 149.9932 ¢, ~7/5 = 583.9356 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9477 ¢

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: 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: [⟨8 1 3 3 16 -21], ⟨0 3 4 5 3 13]]

Optimal tunings:

  • WE: ~12/11 = 150.0005 ¢, ~7/5 = 583.9066 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9052 ¢

Optimal ET sequence: 72, 152f, 224

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: [⟨8 1 3 3 16 -21 -14], ⟨0 3 4 5 3 13 12]]

Optimal tunings:

  • WE: ~12/11 = 150.0064 ¢, ~7/5 = 583.8666 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.8489 ¢

Optimal ET sequence: 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: [⟨8 1 3 3 16 -21 -14 34], ⟨0 3 4 5 3 13 12 0]]

Optimal tunings:

  • WE: ~12/11 = 149.9785 ¢, ~7/5 = 583.8482 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9138 ¢

Optimal ET sequence: 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: [⟨8 1 3 3 16 14], ⟨0 3 4 5 3 4]]

Optimal tunings:

  • WE: ~12/11 = 150.0313 ¢, ~7/5 = 584.0134 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9583 ¢

Optimal ET sequence: 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: [⟨8 1 3 3 16 14 21], ⟨0 3 4 5 3 4 3]]

Optimal tunings:

  • WE: ~12/11 = 150.0528 ¢, ~7/5 = 584.0161 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9166 ¢

Optimal ET sequence: 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: [⟨8 1 3 3 16 14 21 34], ⟨0 3 4 5 3 4 3 0]]

Optimal tunings:

  • WE: ~12/11 = 150.0049 ¢, ~7/5 = 584.0833 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 584.0712 ¢

Optimal ET sequence: 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.

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224

Mapping: [⟨16 2 6 6 32 67], ⟨0 3 4 5 3 -1]]

mapping generators: ~448/429, ~7/5

Optimal tunings:

  • WE: ~448/429 = 74.9943 ¢, ~7/5 = 583.9408 ¢
  • CWE: ~448/429 = 75.0000 ¢, ~7/5 = 583.9709 ¢

Optimal ET sequence: 80, 144, 224

Badness (Sintel): 1.27

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

Mapping: [⟨16 2 6 6 32 67 81], ⟨0 3 4 5 3 -1 -2]]

Optimal tunings:

  • WE: ~117/112 = 74.9865 ¢, ~7/5 = 583.9626 ¢
  • CWE: ~117/112 = 75.0000 ¢, ~7/5 = 584.0463 ¢

Optimal ET sequence: 80, 144, 224, 528dg

Badness (Sintel): 1.46

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

Mapping: [⟨16 2 6 6 32 67 81 68], ⟨0 3 4 5 3 -1 -2 0]]

Optimal tunings:

  • WE: ~117/112 = 74.9865 ¢, ~7/5 = 583.9642 ¢
  • CWE: ~117/112 = 75.0000 ¢, ~7/5 = 584.0803 ¢

Optimal ET sequence: 80, 144, 224, 304dh, 528dghh

Badness (Sintel): 1.44

Retrieved from "https://en.xen.wiki/index.php?title=Stearnsmic_clan&oldid=238517"