Miscellaneous 7-limit temperaments: Difference between revisions

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Move mirkwai here as we're dissolving the mirkwai family
 
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{{Mapping|legend=1| 1 0 -2 -4 | 0 1 1 3 | 0 0 4 3 }}
{{Mapping|legend=1| 1 0 -2 -4 | 0 1 1 3 | 0 0 4 3 }}
: mapping generators: ~2, ~3, ~45/28
: Mapping generators: ~2, ~3, ~45/28


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0258{{c}}, ~3/2 = 701.8709{{c}}, ~45/28 = 821.1067{{c}}
* [[WE]]: ~2 = 1200.0258{{c}}, ~3/2 = 701.8709{{c}}, ~45/28 = 821.1067{{c}}
: [[error map]]: {{val| +0.026 -0.058 -0.042 +0.081 }}
: [[Error map]]: {{val| +0.026 -0.058 -0.042 +0.081 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8671{{c}}, ~45/28 = 821.0917{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8671{{c}}, ~45/28 = 821.0917{{c}}
: error map: {{val| 0.000 -0.088 -0.080 +0.050 }}
: Error map: {{val| 0.000 -0.088 -0.080 +0.050 }}


{{Optimal ET sequence|legend=1| 19, 41, 89, 108, 111, 130, 152, 171, 665, 795, 836, 966, 1137, 1308, 1973, 2144, 3281, 3452 }}
{{Optimal ET sequence|legend=1| 19, 41, 89, 108, 111, 130, 152, 171, 665, 795, 836, 966, 1137, 1308, 1973, 2144, 3281, 3452 }}
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{{Mapping|legend=1| 1 0 -1 -6 | 0 1 1 3 | 0 0 3 7 }}
{{Mapping|legend=1| 1 0 -1 -6 | 0 1 1 3 | 0 0 3 7 }}
: mapping generators: ~2, ~3, ~112/75
: Mapping generators: ~2, ~3, ~112/75


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0384{{c}}, ~3/2 = 701.8990{{c}}, ~112/75 = 694.7610{{c}}
* [[WE]]: ~2 = 1200.0384{{c}}, ~3/2 = 701.8990{{c}}, ~112/75 = 694.7610{{c}}
: [[error map]]: {{val| +0.038 -0.018 -0.132 +0.083 }}
: [[Error map]]: {{val| +0.038 -0.018 -0.132 +0.083 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8998{{c}}, ~112/75 = 694.7370{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8998{{c}}, ~112/75 = 694.7370{{c}}
: error map: {{val| 0.000 -0.055 -0.203 +0.033 }}
: Error map: {{val| 0.000 -0.055 -0.203 +0.033 }}


{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 90, 102d, 109, 121, 140, 152, 171, 665, 836, 1007, 2185, 3192c }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 90, 102d, 109, 121, 140, 152, 171, 665, 836, 1007, 2185, 3192c }}


[[Badness]] (Sintel): 0.661
[[Badness]] (Sintel): 0.661
== Canopic a.k.a. mirkwai ==
: ''For extensions, see [[Swetismic temperaments #Indra]].''
Canopic, a.k.a. mirkwai, tempers out the [[canopic comma]] in the 7-limit.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 16875/16807
{{Mapping|legend=1| 1 0 -5 -4 | 0 1 3 3 | 0 0 5 4 }}
: Mapping generators: ~2, ~3, ~10/7
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9999{{c}}, ~3/2 = 701.7827{{c}}, ~10/7 = 616.0944{{c}}
: [[error map]]: {{val| -0.000 -0.172 -0.493 +0.900 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7827{{c}}, ~10/7 = 616.0945{{c}}
: error map: {{val| 0.000 -0.172 -0.493 +0.900 }}
[[Minimax tuning]]:
* [[7-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 4/7 -4/7 5/7 | 0 -3/7 3/7 5/7 | 0 0 0 1 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3.7
* [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 8/11 -4/11 5/11 | 0 -6/11 3/11 10/11 | 0 0 0 1 }}
: eigenmonzo (unchanged-interval) basis: 2.9/5.7
{{Optimal ET sequence|legend=1| 31, 41, 72, 152, 224 }}
[[Badness]] (Sintel): 1.51
[[Projection pair]]s: <code>5 84375/16807 7 16875/2401</code> to 2.3.7/5


== Greenwoodmic ==
== Greenwoodmic ==
Greenwoodmic tempers out the [[greenwoodma]] in the 7-limit. It equates [[5/2]] with a stack of two [[14/9]]'s. This implies [[prime interval|primes]] [[3/1|3]] and [[5/1|5]] should be tuned flat, and [[7/1|7]] should be tuned sharp. A rank-2 temperament that does that is [[injera]], which introduces little extra [[damage]] over greenwoodmic.  
Greenwoodmic tempers out the [[greenwoodma]] in the 7-limit. It equates [[5/2]] with a stack of two [[14/9]]'s. This implies [[prime interval|primes]] [[3/1|3]] and [[5/1|5]] should be tuned flat, and [[7/1|7]] should be tuned sharp. A rank-2 temperament that does that is [[injera]], which introduces little extra [[damage]] over greenwoodmic.  
In contrast to [[sensamagic]], where two [[9/7]]'s stack to [[5/3]], here two 9/7's stack to [[8/5]]. As such, greenwoodmic induces [[essentially tempered chord]]s in the [[9-odd-limit]]. An obvious 11-limit extension then equates [[5/4]] with [[11/9]] and equates 9/7 with [[14/11]], tempering out [[45/44]] as well as [[99/98]] using the identity 405/392 = (45/44)⋅(99/98).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 53: Line 87:


{{Mapping|legend=1| 1 0 1 -1 | 0 1 0 2 | 0 0 2 1 }}
{{Mapping|legend=1| 1 0 1 -1 | 0 1 0 2 | 0 0 2 1 }}
: mapping generators: ~2, ~3, ~14/9
: Mapping generators: ~2, ~3, ~14/9


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.9369{{c}}, ~3/2 = 693.3783{{c}}, ~14/9 = 790.3845{{c}}
* [[WE]]: ~2 = 1201.9369{{c}}, ~3/2 = 693.3783{{c}}, ~14/9 = 790.3845{{c}}
: [[error map]]: {{val| +1.937 -6.640 -3.608 +10.252 }}
: [[Error map]]: {{val| +1.937 -6.640 -3.608 +10.252 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.3443{{c}}, ~14/9 = 790.0724{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.3443{{c}}, ~14/9 = 790.0724{{c}}
: error map: {{val| 0.000 -8.611 -6.169 +7.935 }}
: Error map: {{val| 0.000 -8.611 -6.169 +7.935 }}


{{Optimal ET sequence|legend=1| 9, 12, 26, 38, 73bc }}
{{Optimal ET sequence|legend=1| 9, 12, 26, 38, 73bc }}
Line 67: Line 101:
== Avicennmic ==
== Avicennmic ==
Avicennmic tempers out the [[avicennma]] in the 7-limit. It equates [[32/21]] with a stack of two [[5/4]]'s, and [[12/7]] with a stack of two [[15/8]]'s octave reduced. This implies [[prime interval|primes]] [[3/1|3]], [[5/1|5]] and [[7/1|7]] should all be tuned flat. A rank-2 temperament that does that is [[flattone]], which introduces little extra [[damage]] over avicennmic.  
Avicennmic tempers out the [[avicennma]] in the 7-limit. It equates [[32/21]] with a stack of two [[5/4]]'s, and [[12/7]] with a stack of two [[15/8]]'s octave reduced. This implies [[prime interval|primes]] [[3/1|3]], [[5/1|5]] and [[7/1|7]] should all be tuned flat. A rank-2 temperament that does that is [[flattone]], which introduces little extra [[damage]] over avicennmic.  
One possible extension of avicennmic to the 11-limit is via [[45/44]] and [[385/384]], using the identity 525/512 = (45/44)⋅(385/384), but the result is somewhat less accurate. Instead, it is more natural to extend it to the [[2.3.5.7.13 subgroup]] by tempering out [[65/64]] and [[105/104]], using the identity 525/512 = (65/64)⋅(105/104).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 73: Line 109:


{{Mapping|legend=1| 1 0 0 9 | 0 1 0 -1 | 0 0 1 -2 }}
{{Mapping|legend=1| 1 0 0 9 | 0 1 0 -1 | 0 0 1 -2 }}
: mapping generators: ~2, ~3, ~5
: Mapping generators: ~2, ~3, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.4446{{c}}, ~3/2 = 697.5230{{c}}, ~5/4 = 375.2486{{c}}
* [[WE]]: ~2 = 1203.4446{{c}}, ~3/2 = 697.5230{{c}}, ~5/4 = 375.2486{{c}}
: [[error map]]: {{val| +3.445 -0.987 -4.176 -3.068 }}
: [[Error map]]: {{val| +3.445 -0.987 -4.176 -3.068 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.1860{{c}}, ~5/4 = 373.6255{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.1860{{c}}, ~5/4 = 373.6255{{c}}
: error map: {{val| 0.000 -5.579 -12.687 -12.265 }}
: Error map: {{val| 0.000 -5.579 -12.687 -12.265 }}


{{Optimal ET sequence|legend=1| 7, 9, 10, 16, 19, 45, 64cd, 93cdd, 119bccdd, 138bccdd }}
{{Optimal ET sequence|legend=1| 7, 9, 10, 16, 19, 45, 64cd, 93cdd, 119bccdd, 138bccdd }}
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{{Mapping|legend=1| 2 0 0 9 | 0 1 0 -4 | 0 0 1 2 }}
{{Mapping|legend=1| 2 0 0 9 | 0 1 0 -4 | 0 0 1 2 }}
: mapping generators: ~567/400, ~3, ~5
: Mapping generators: ~567/400, ~3, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~567/400 = 600.1005{{c}}, ~3/2 = 701.3045{{c}}, ~5/4 = 386.3934{{c}}
* [[WE]]: ~567/400 = 600.1005{{c}}, ~3/2 = 701.3045{{c}}, ~5/4 = 386.3934{{c}}
: [[error map]]: {{val| +0.201 -0.449 +0.482 -0.353 }}
: [[Error map]]: {{val| +0.201 -0.449 +0.482 -0.353 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.2691{{c}}, ~5/4 = 386.5935{{c}}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.2691{{c}}, ~5/4 = 386.5935{{c}}
: error map: {{val| 0.000 -0.686 +0.280 -0.715 }}
: Error map: {{val| 0.000 -0.686 +0.280 -0.715 }}


{{Optimal ET sequence|legend=1| 12, 26, 34, 46, 58, 72, 118, 130, 202, 320, 450, 522, 972bd, 1174bd }}
{{Optimal ET sequence|legend=1| 12, 26, 34, 46, 58, 72, 118, 130, 202, 320, 450, 522, 972bd, 1174bd }}
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{{Mapping|legend=1| 1 0 -5 -1 | 0 1 3 2 | 0 0 4 1 }}
{{Mapping|legend=1| 1 0 -5 -1 | 0 1 3 2 | 0 0 4 1 }}
: mapping generators: ~2, ~3, ~14/9
: Mapping generators: ~2, ~3, ~14/9


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.8917{{c}}, ~3/2 = 700.7235{{c}}, ~14/9 = 770.8371{{c}}
* [[WE]]: ~2 = 1199.8917{{c}}, ~3/2 = 700.7235{{c}}, ~14/9 = 770.8371{{c}}
: [[error map]]: {{val| -0.108 -1.340 -0.578 +3.350 }}
: [[Error map]]: {{val| -0.108 -1.340 -0.578 +3.350 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6977{{c}}, ~14/9 = 770.9417{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.6977{{c}}, ~14/9 = 770.9417{{c}}
: error map: {{val| 0.000 -1.257 -0.454 +3.511 }}
: Error map: {{val| 0.000 -1.257 -0.454 +3.511 }}


{{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84, 137, 221d }}
{{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84, 137, 221d }}
Line 130: Line 166:


[[Projection pair]]: <code>5 2401/486</code> to 2.3.7
[[Projection pair]]: <code>5 2401/486</code> to 2.3.7
== Schismean ==
Schismean tempers out the [[3645/3584|schismean comma]] in the 7-limit. It equates [[7/5]] with a stack of three [[9/8]]'s.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 3645/3584
{{Mapping|legend=1| 1 0 0 -9 | 0 1 0 6 | 0 0 1 1 }}
: Mapping generators: ~2, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.420{{c}}, ~3/2 = 698.145{{c}}, ~5/4 = 382.612{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 697.459{{c}}, ~5/4 = 383.104{{c}}
{{Optimal ET sequence|legend=1| 5c, 7d, 12, 19, 31, 81 }}
[[Badness]] (Sintel): 2.96
== Keegic ==
Keegic tempers out the [[keega]] in the 7-limit, and finds the [[3/1|3rd]] [[harmonic]] by a stack of three [[10/7]]'s.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 1029/1000
{{Mapping|legend=1| 1 0 0 1 | 0 3 0 -1 | 0 0 1 1 }}
: Mapping generators: ~2, ~10/7, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.1181{{c}}, ~10/7 = 633.6603{{c}}, ~5/4 = 390.1534{{c}}
: [[Error map]]: {{val| +1.118 -0.974 +6.076 -8.979 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 633.3435{{c}}, ~5/4 = 391.2567{{c}}
: Error map: {{val| 0.000 -1.924 +4.943 -10.913 }}
{{Optimal ET sequence|legend=1| 15, 19, 53d, 55, 74d }}
[[Badness]] (Sintel): 2.99


== Uniwiz ==
== Uniwiz ==
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{{Mapping|legend=1| 2 1 0 7 | 0 2 0 3 | 0 0 1 -1 }}
{{Mapping|legend=1| 2 1 0 7 | 0 2 0 3 | 0 0 1 -1 }}
: mapping generators: ~1225/864, ~35/24, ~5
: Mapping generators: ~1225/864, ~35/24, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~1225/864 = 600.1145{{c}}, ~35/24 = 651.0771{{c}}, ~5/4 = 385.4061{{c}}
* [[WE]]: ~1225/864 = 600.1145{{c}}, ~35/24 = 651.0771{{c}}, ~5/4 = 385.4061{{c}}
: [[error map]]: {{val| +0.229 +0.314 -0.450 -0.657 }}
: [[Error map]]: {{val| +0.229 +0.314 -0.450 -0.657 }}
* [[CWE]]: ~1225/864 = 600.1145{{c}}, ~35/24 = 651.0546{{c}}, ~5/4 = 385.4793{{c}}
* [[CWE]]: ~1225/864 = 600.1145{{c}}, ~35/24 = 651.0546{{c}}, ~5/4 = 385.4793{{c}}
: error map: {{val| 0.000 +0.154 -0.834 -1.141 }}
: Error map: {{val| 0.000 +0.154 -0.834 -1.141 }}


{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd }}
{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd }}
Line 163: Line 237:


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~343/243 = 599.9938{{c}}, ~9/7 = 433.8840{{c}}, ~5/4 = 386.3383{{c}}
* [[WE]]: ~343/243 = 599.9938{{c}}, ~9/7 = 433.8840{{c}}, ~5/4 = 386.3383{{c}}
: [[error map]]: {{val| -0.012 -0.309 -0.000 +0.582 }}
: [[Error map]]: {{val| -0.012 -0.309 -0.000 +0.582 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~9/7 = 433.8851{{c}}, ~5/4 = 386.3279{{c}}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~9/7 = 433.8851{{c}}, ~5/4 = 386.3279{{c}}
: error map: {{val| 0.000 -0.300 +0.014 +0.600 }}
: Error map: {{val| 0.000 -0.300 +0.014 +0.600 }}


{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 202, 224, 354 }}
{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 202, 224, 354 }}
Line 185: Line 259:


{{Mapping|legend=1| 1 1 0 6 | 0 2 0 -3 | 0 0 1 -1 }}
{{Mapping|legend=1| 1 1 0 6 | 0 2 0 -3 | 0 0 1 -1 }}
: mapping generators: ~2, ~128/105, ~5
: Mapping generators: ~2, ~128/105, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.8046{{c}}, ~128/105 = 350.3723{{c}}, ~5/4 = 384.1239{{c}}
* [[WE]]: ~2 = 1200.8046{{c}}, ~128/105 = 350.3723{{c}}, ~5/4 = 384.1239{{c}}
: [[error map]]: {{val| +0.805 -0.406 -0.581 -0.848 }}
: [[Error map]]: {{val| +0.805 -0.406 -0.581 -0.848 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 350.1448{{c}}, ~5/4 = 383.8961{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 350.1448{{c}}, ~5/4 = 383.8961{{c}}
: error map: {{val| 0.000 -1.665 -2.418 -3.157 }}
: Error map: {{val| 0.000 -1.665 -2.418 -3.157 }}


{{Optimal ET sequence|legend=1| 17, 21, 24, 31, 41, 72, 281d, 322cd, 353cd, 425bcdd, 497bcdd }}
{{Optimal ET sequence|legend=1| 17, 21, 24, 31, 41, 72, 281d, 322cd, 353cd, 425bcdd, 497bcdd }}


[[Badness]] (Sintel): 3.40
[[Badness]] (Sintel): 3.40
== Decovulture ==
: ''For extensions, see [[Olympic clan #Baffin]].''
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 67108864/66976875
{{Mapping|legend=1| 1 0 0 13 | 0 2 0 -7 | 0 0 1 -2 }}
: mapping generators: ~2, ~8192/4725, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9033{{c}}, ~8192/4725 = 951.0102{{c}}, ~5/4 = 386.5872{{c}}
: [[error map]]: {{val| -0.097 +0.065 +0.080 +0.059 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8192/4725 = 951.0899{{c}}, ~5/4 = 386.6184{{c}}
: error map: {{val| 0.000 +0.225 +0.305 +0.308 }}
{{Optimal ET sequence|legend=1| 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd }}
[[Badness]] (Sintel): 3.82


== Trimyna ==
== Trimyna ==
Line 207: Line 301:


{{Mapping|legend=1| 1 -1 0 1 | 0 5 0 -1 | 0 0 1 1 }}
{{Mapping|legend=1| 1 -1 0 1 | 0 5 0 -1 | 0 0 1 1 }}
: mapping generators: ~2, ~10/7, ~5
: Mapping generators: ~2, ~10/7, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1652{{c}}, ~10/7 = 620.4031{{c}}, ~5/4 = 387.0990{{c}}
* [[WE]]: ~2 = 1200.1652{{c}}, ~10/7 = 620.4031{{c}}, ~5/4 = 387.0990{{c}}
: [[error map]]: {{val| +0.165 -0.105 +1.116 -1.634 }}
: [[Error map]]: {{val| +0.165 -0.105 +1.116 -1.634 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 620.3427{{c}}, ~5/4 = 387.2591{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 620.3427{{c}}, ~5/4 = 387.2591{{c}}
: error map: {{val| 0.000 -0.241 +0.945 -1.910 }}
: Error map: {{val| 0.000 -0.241 +0.945 -1.910 }}


{{Optimal ET sequence|legend=1| 27, 31, 58, 87, 118, 267d, 385d, 412d }} *
{{Optimal ET sequence|legend=1| 27, 31, 58, 87, 118, 267d, 385d, 412d }} *
Line 233: Line 327:


{{Mapping|legend=1| 1 0 1 3 | 0 1 1 0 | 0 0 -4 -3 }}
{{Mapping|legend=1| 1 0 1 3 | 0 1 1 0 | 0 0 -4 -3 }}
: mapping generators: ~2, ~3, ~21/20
: Mapping generators: ~2, ~3, ~21/20


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7175{{c}}, ~3/2 = 700.6331{{c}}, ~21/20 = 78.6164{{c}}
* [[WE]]: ~2 = 1200.7175{{c}}, ~3/2 = 700.6331{{c}}, ~21/20 = 78.6164{{c}}
: [[error map]]: {{val| +0.718 -0.604 +1.289 -2.523 }}
: [[Error map]]: {{val| +0.718 -0.604 +1.289 -2.523 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.5831{{c}}, ~21/20 = 78.4341{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.5831{{c}}, ~21/20 = 78.4341{{c}}
: error map: {{val| 0.000 -1.372 +0.533 -4.128 }}
: Error map: {{val| 0.000 -1.372 +0.533 -4.128 }}


{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
Line 257: Line 351:


{{Mapping|legend=1| 1 0 0 -22 | 0 1 0 1 | 0 0 1 10 }}
{{Mapping|legend=1| 1 0 0 -22 | 0 1 0 1 | 0 0 1 10 }}
: mapping generators: ~2, ~3, ~5
: Mapping generators: ~2, ~3, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9205{{c}}, ~3/2 = 702.0435{{c}}, ~5/4 = 386.6674{{c}}
* [[WE]]: ~2 = 1199.9205{{c}}, ~3/2 = 702.0435{{c}}, ~5/4 = 386.6674{{c}}
: [[error map]]: {{val| -0.079 +0.009 +0.195 -0.029 }}
: [[Error map]]: {{val| -0.079 +0.009 +0.195 -0.029 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0491{{c}}, ~5/4 = 386.6885{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0491{{c}}, ~5/4 = 386.6885{{c}}
: error map: {{val| 0.000 +0.094 +0.375 +0.108 }}
: Error map: {{val| 0.000 +0.094 +0.375 +0.108 }}


{{Optimal ET sequence|legend=1| 31, 87, 118, 152, 239, 270, 571, 723, 841, 993, 1263, 1564c, 1834c, 2104c }}
{{Optimal ET sequence|legend=1| 31, 87, 118, 152, 239, 270, 571, 723, 841, 993, 1263, 1564c, 1834c, 2104c }}
Line 277: Line 371:


{{Mapping|legend=1| 1 0 0 4 | 0 4 0 -3 | 0 0 1 0 }}
{{Mapping|legend=1| 1 0 0 4 | 0 4 0 -3 | 0 0 1 0 }}
: mapping generators: ~2, ~21/16, ~5
: Mapping generators: ~2, ~21/16, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.2548{{c}}, ~21/16 = 475.5761{{c}}, ~5/4 = 387.8025{{c}}
* [[WE]]: ~2 = 1199.2548{{c}}, ~21/16 = 475.5761{{c}}, ~5/4 = 387.8025{{c}}
: [[error map]]: {{val| -0.745 +0.350 -0.002 +1.465 }}
: [[Error map]]: {{val| -0.745 +0.350 -0.002 +1.465 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 475.8328{{c}}, ~5/4 = 387.5778{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 475.8328{{c}}, ~5/4 = 387.5778{{c}}
: error map: {{val| 0.000 +1.376 +1.264 +3.676 }}
: Error map: {{val| 0.000 +1.376 +1.264 +3.676 }}


{{Optimal ET sequence|legend=1| 5, 10, 15, 33, 38, 43, 53, 111, 121, 164d, 174d, 179, 232d }}
{{Optimal ET sequence|legend=1| 5, 10, 15, 33, 38, 43, 53, 111, 121, 164d, 174d, 179, 232d }}


[[Badness]] (Sintel): 6.18
[[Badness]] (Sintel): 6.18
== Tolerant ==
: ''For extensions, see [[Pentacircle clan #Tolerant]].''
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 179200/177147
{{Mapping|legend=1| 1 0 0 -10 | 0 1 0 11 | 0 0 1 -2 }}
: Mapping generators: ~2, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.539{{c}}, ~3 = 1903.226{{c}}, ~5 = 2785.816{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3 = 1903.805{{c}}, ~5 = 2786.356{{c}}
{{Optimal ET sequence|legend=1| 34d, 39d, 41, 80, 87, 121, 167, 208, 329b, 375b, 496bd }}
[[Badness]] (Sintel): 7.26
== History ==
: ''For extensions, see [[Werckismic temperaments #History]].''
History tempers out the [[historisma]] in the 7-limit, and splits the fourth in six.
[[Comma list]]: 257298363/256000000
{{Mapping|legend=1| 1 2 0 0 | 0 -6 0 7 | 0 0 1 1 }}
: Mapping generators: ~2, ~21/20, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.154{{c}}, ~21/20 = 83.091{{c}}, ~5 = 2786.669{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~21/20 = 83.067{{c}}, ~5 = 2786.515{{c}}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 43, 58, 72, 130, 202}}
[[Badness]] (Sintel): 7.95
== Sensibeta ==
{{See also| Sensibeta comma }}
Sensibeta tempers out the [[sensibeta comma]] in the 7-limit.
[[Comma list]]: 1071875/1062882
{{Mapping|legend=1| 1 0 2 -3 | 0 1 0 4 | 0 0 3 -5 }}
: Mapping generators: ~2, ~3, ~175/162
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.025{{c}}, ~3 = 1902.728{{c}}, ~175/162 = 128.524{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3 = 1902.709{{c}}, ~175/162 = 128.530{{c}}
{{Optimal ET sequence|legend=1| 19, 27, 46, 94, 113, 121, 140 }}
[[Badness]] (Sintel): 7.97


== Parahemif ==
== Parahemif ==
Line 299: Line 447:


{{Mapping|legend=1| 1 1 0 -1 | 0 2 0 13 | 0 0 1 0 }}
{{Mapping|legend=1| 1 1 0 -1 | 0 2 0 13 | 0 0 1 0 }}
: mapping generators: ~2, ~896/729, ~5
: Mapping generators: ~2, ~896/729, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7303{{c}}, ~896/729 = 351.4056{{c}}, ~5/4 = 386.8527{{c}}
* [[WE]]: ~2 = 1199.7303{{c}}, ~896/729 = 351.4056{{c}}, ~5/4 = 386.8527{{c}}
: [[error map]]: {{val| -0.270 +0.586 -0.000 -0.284 }}
: [[Error map]]: {{val| -0.270 +0.586 -0.000 -0.284 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~896/729 = 351.4569{{c}}, ~5/4 = 386.6884{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~896/729 = 351.4569{{c}}, ~5/4 = 386.6884{{c}}
: error map: {{val| 0.000 +0.959 +0.375 +0.114 }}
: Error map: {{val| 0.000 +0.959 +0.375 +0.114 }}


{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99, 239, 338 }}
{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99, 239, 338 }}


[[Badness]] (Sintel): 8.77
[[Badness]] (Sintel): 8.77
== Septimagic ==
Septimagic tempers out the [[septimagic comma]] in the 7-limit and gives the [[2.3.7 subgroup|2.3.7-]][[subgroup]] [[magic]] [[restriction]] an independent generator for the [[5/1|5th]] [[harmonic]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 537824/531441
{{Mapping|legend=1| 1 0 0 -1 | 0 5 0 12 | 0 0 1 0 }}
: Mapping generators: ~2, ~243/196, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8224{{c}}, ~243/196 = 380.6043{{c}}, ~5/4 = 386.6676{{c}}
: [[error map]]: {{val| -0.178 +1.066 -0.001 -1.397 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/196 = 380.6378{{c}}, ~5/4 = 386.5230{{c}}
: error map: {{val| 0.000 +1.234 +0.209 -1.173 }}
{{Optimal ET sequence|legend=1| 19, 41, 104c, 123, 126, 145, 167, 186, 394b, 413 }}
[[Badness]] (Sintel): 11.8
== Compass ==
: ''For extensions, see [[Moctdelismic clan #Compass]].''
Compass tempers out the [[compass comma]] in the 7-limit, and splits the fourth in five. The obvious 11-limit extension tempers out the moctdelisma, [[1375/1372]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 9765625/9680832
{{Mapping|legend=1| 1 2 0 -2 | 0 -5 0 2 | 0 0 1 2 }}
: Mapping generators: ~2, ~625/588, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1156{{c}}, ~625/588 = 99.6359{{c}}, ~5/4 = 385.0414{{c}}
: [[error map]]: {{val| +0.116 +0.097 -1.041 +0.760 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/588 = 99.6080{{c}}, ~5/4 = 385.1108{{c}}
: error map: {{val| 0.000 +0.005 -1.203 +0.612 }}
{{Optimal ET sequence|legend=1| 12, …, 37, 48d, 49, 60, 72, 181, 193, 265 }}
[[Badness]] (Sintel): 13.3


== Linus ==
== Linus ==
Line 321: Line 511:


{{Mapping|legend=1| 10 0 0 -11 | 0 1 0 1 | 0 0 1 1 }}
{{Mapping|legend=1| 10 0 0 -11 | 0 1 0 1 | 0 0 1 1 }}
: mapping generators: ~15/14, ~3, ~5
: Mapping generators: ~15/14, ~3, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
Line 332: Line 522:


[[Badness]] (Sintel): 15.7
[[Badness]] (Sintel): 15.7
== Naiad ==
: ''For extensions, see [[Wizardharry clan #Naiad]].''
Naiad tempers out the [[naiadisma]] in the 7-limit. An obvious 13-limit interpretation of one generator (~98/75) is [[13/10]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 161414428/158203125
{{Mapping|legend=1| 1 5 0 2 | 0 -9 0 -4 | 0 0 1 1 }}
: Mapping generators: ~2, ~98/75, ~5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.937{{c}}, ~98/75 = 455.269{{c}}, ~5/4 = 387.946{{c}}
: [[error map]]: {{val| -0.063, +0.311, +1.506, -2.207 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~98/75 = 455.298{{c}}, ~5/4 = 387.896{{c}}
: error map: {{val| 0.000, +0.362, +1.582, -2.123 }}
{{Optimal ET sequence|legend=1| 8d, 21, 29, 37, 50, 58, 87, 145 }}
[[Badness]] (Sintel): 52.5


[[Category:Temperament collections]]
[[Category:Temperament collections]]

Latest revision as of 12:06, 6 June 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.

Below are listed some 7-limit rank-3 temperaments that do not belong to some other temperament collection, the majority of which are restrictions to the 7-limit of temperaments that emerge more fully in higher limits or subgroups; they are sorted by TE logflat badness. Most of these temperaments have low accuracy, high-complexity generators, or large number of generators for simple consonances. This is not an exhaustive list. Only expect to find a temperament here if you have not found it in:

See also Miscellaneous 5-limit temperaments.

Breeze

Subgroup: 2.3.5.7

Comma list: 2460375/2458624

Mapping: [⟨1 0 -2 -4], ⟨0 1 1 3], ⟨0 0 4 3]]

Mapping generators: ~2, ~3, ~45/28

Optimal tunings:

  • WE: ~2 = 1200.0258 ¢, ~3/2 = 701.8709 ¢, ~45/28 = 821.1067 ¢
Error map: ⟨+0.026 -0.058 -0.042 +0.081]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8671 ¢, ~45/28 = 821.0917 ¢
Error map: ⟨0.000 -0.088 -0.080 +0.050]

Optimal ET sequence: 19, 41, 89, 108, 111, 130, 152, 171, 665, 795, 836, 966, 1137, 1308, 1973, 2144, 3281, 3452

Badness (Sintel): 0.520

Metric

Metric tempers out the meter, and splits the syntonic comma into three equal parts, one for the marvel comma, 225/224, and two for the starling comma, 126/125. It is therefore supported by third-comma equal temperaments, and 171edo shows an excellent example of this. 11-limit extensions of this temperament include mendel and skadi.

Subgroup: 2.3.5.7

Comma list: 703125/702464

Mapping: [⟨1 0 -1 -6], ⟨0 1 1 3], ⟨0 0 3 7]]

Mapping generators: ~2, ~3, ~112/75

Optimal tunings:

  • WE: ~2 = 1200.0384 ¢, ~3/2 = 701.8990 ¢, ~112/75 = 694.7610 ¢
Error map: ⟨+0.038 -0.018 -0.132 +0.083]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8998 ¢, ~112/75 = 694.7370 ¢
Error map: ⟨0.000 -0.055 -0.203 +0.033]

Optimal ET sequence: 12, 19, 31, 81, 90, 102d, 109, 121, 140, 152, 171, 665, 836, 1007, 2185, 3192c

Badness (Sintel): 0.661

Canopic a.k.a. mirkwai

For extensions, see Swetismic temperaments #Indra.

Canopic, a.k.a. mirkwai, tempers out the canopic comma in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 16875/16807

Mapping: [⟨1 0 -5 -4], ⟨0 1 3 3], ⟨0 0 5 4]]

Mapping generators: ~2, ~3, ~10/7

Optimal tunings:

  • WE: ~2 = 1199.9999 ¢, ~3/2 = 701.7827 ¢, ~10/7 = 616.0944 ¢
error map: ⟨-0.000 -0.172 -0.493 +0.900]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7827 ¢, ~10/7 = 616.0945 ¢
error map: ⟨0.000 -0.172 -0.493 +0.900]

Minimax tuning:

[[1 0 0 0⟩, [0 4/7 -4/7 5/7⟩, [0 -3/7 3/7 5/7⟩, [0 0 0 1⟩]
eigenmonzo (unchanged-interval) basis: 2.5/3.7
[[1 0 0 0⟩, [0 8/11 -4/11 5/11⟩, [0 -6/11 3/11 10/11⟩, [0 0 0 1⟩]
eigenmonzo (unchanged-interval) basis: 2.9/5.7

Optimal ET sequence: 31, 41, 72, 152, 224

Badness (Sintel): 1.51

Projection pairs: 5 84375/16807 7 16875/2401 to 2.3.7/5

Greenwoodmic

Greenwoodmic tempers out the greenwoodma in the 7-limit. It equates 5/2 with a stack of two 14/9's. This implies primes 3 and 5 should be tuned flat, and 7 should be tuned sharp. A rank-2 temperament that does that is injera, which introduces little extra damage over greenwoodmic.

In contrast to sensamagic, where two 9/7's stack to 5/3, here two 9/7's stack to 8/5. As such, greenwoodmic induces essentially tempered chords in the 9-odd-limit. An obvious 11-limit extension then equates 5/4 with 11/9 and equates 9/7 with 14/11, tempering out 45/44 as well as 99/98 using the identity 405/392 = (45/44)⋅(99/98).

Subgroup: 2.3.5.7

Comma list: 405/392

Mapping: [⟨1 0 1 -1], ⟨0 1 0 2], ⟨0 0 2 1]]

Mapping generators: ~2, ~3, ~14/9

Optimal tunings:

  • WE: ~2 = 1201.9369 ¢, ~3/2 = 693.3783 ¢, ~14/9 = 790.3845 ¢
Error map: ⟨+1.937 -6.640 -3.608 +10.252]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.3443 ¢, ~14/9 = 790.0724 ¢
Error map: ⟨0.000 -8.611 -6.169 +7.935]

Optimal ET sequence: 9, 12, 26, 38, 73bc

Badness (Sintel): 1.82

Avicennmic

Avicennmic tempers out the avicennma in the 7-limit. It equates 32/21 with a stack of two 5/4's, and 12/7 with a stack of two 15/8's octave reduced. This implies primes 3, 5 and 7 should all be tuned flat. A rank-2 temperament that does that is flattone, which introduces little extra damage over avicennmic.

One possible extension of avicennmic to the 11-limit is via 45/44 and 385/384, using the identity 525/512 = (45/44)⋅(385/384), but the result is somewhat less accurate. Instead, it is more natural to extend it to the 2.3.5.7.13 subgroup by tempering out 65/64 and 105/104, using the identity 525/512 = (65/64)⋅(105/104).

Subgroup: 2.3.5.7

Comma list: 525/512

Mapping: [⟨1 0 0 9], ⟨0 1 0 -1], ⟨0 0 1 -2]]

Mapping generators: ~2, ~3, ~5

Optimal tunings:

  • WE: ~2 = 1203.4446 ¢, ~3/2 = 697.5230 ¢, ~5/4 = 375.2486 ¢
Error map: ⟨+3.445 -0.987 -4.176 -3.068]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.1860 ¢, ~5/4 = 373.6255 ¢
Error map: ⟨0.000 -5.579 -12.687 -12.265]

Optimal ET sequence: 7, 9, 10, 16, 19, 45, 64cd, 93cdd, 119bccdd, 138bccdd

Badness (Sintel): 2.10

Varuna

For extensions, see Werckismic temperaments #Varuna.

Varuna tempers out the varunisma in the 7-limit, and splits the octave in two. It then finds 7/4 by a stack of two 10/9's and a semi-octave period. The obvious 11-limit extension tempers out the kalisma, 9801/9800.

Subgroup: 2.3.5.7

Comma list: 321489/320000

Mapping: [⟨2 0 0 9], ⟨0 1 0 -4], ⟨0 0 1 2]]

Mapping generators: ~567/400, ~3, ~5

Optimal tunings:

  • WE: ~567/400 = 600.1005 ¢, ~3/2 = 701.3045 ¢, ~5/4 = 386.3934 ¢
Error map: ⟨+0.201 -0.449 +0.482 -0.353]
  • CWE: ~567/400 = 600.0000 ¢, ~3/2 = 701.2691 ¢, ~5/4 = 386.5935 ¢
Error map: ⟨0.000 -0.686 +0.280 -0.715]

Optimal ET sequence: 12, 26, 34, 46, 58, 72, 118, 130, 202, 320, 450, 522, 972bd, 1174bd

Badness (Sintel): 2.21

Nuwell

For extensions, see Biyatismic clan #Big brother.

Nuwell tempers out the nuwell comma in the 7-limit, and identifies 15/8 by a stack of four 7/6's. An obvious 11-limit extension then finds 11/8~15/11 as an exact half of it, tempering out 99/98 and 121/120.

Subgroup: 2.3.5.7

Comma list: 2430/2401

Mapping: [⟨1 0 -5 -1], ⟨0 1 3 2], ⟨0 0 4 1]]

Mapping generators: ~2, ~3, ~14/9

Optimal tunings:

  • WE: ~2 = 1199.8917 ¢, ~3/2 = 700.7235 ¢, ~14/9 = 770.8371 ¢
Error map: ⟨-0.108 -1.340 -0.578 +3.350]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.6977 ¢, ~14/9 = 770.9417 ¢
Error map: ⟨0.000 -1.257 -0.454 +3.511]

Optimal ET sequence: 8d, 9, 14c, 17c, 22, 31, 53, 84, 137, 221d

Badness (Sintel): 2.29

Projection pair: 5 2401/486 to 2.3.7

Schismean

Schismean tempers out the schismean comma in the 7-limit. It equates 7/5 with a stack of three 9/8's.

Subgroup: 2.3.5.7

Comma list: 3645/3584

Mapping: [⟨1 0 0 -9], ⟨0 1 0 6], ⟨0 0 1 1]]

Mapping generators: ~2, ~3, ~5

Optimal tunings:

  • WE: ~2 = 1201.420 ¢, ~3/2 = 698.145 ¢, ~5/4 = 382.612 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 697.459 ¢, ~5/4 = 383.104 ¢

Optimal ET sequence: 5c, 7d, 12, 19, 31, 81

Badness (Sintel): 2.96

Keegic

Keegic tempers out the keega in the 7-limit, and finds the 3rd harmonic by a stack of three 10/7's.

Subgroup: 2.3.5.7

Comma list: 1029/1000

Mapping: [⟨1 0 0 1], ⟨0 3 0 -1], ⟨0 0 1 1]]

Mapping generators: ~2, ~10/7, ~5

Optimal tunings:

  • WE: ~2 = 1201.1181 ¢, ~10/7 = 633.6603 ¢, ~5/4 = 390.1534 ¢
Error map: ⟨+1.118 -0.974 +6.076 -8.979]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 633.3435 ¢, ~5/4 = 391.2567 ¢
Error map: ⟨0.000 -1.924 +4.943 -10.913]

Optimal ET sequence: 15, 19, 53d, 55, 74d

Badness (Sintel): 2.99

Uniwiz

For extensions, see Keenanismic temperaments #Uniwiz.

Uniwiz tempers out the uniwiz comma in the 7-limit, equating the whole tone with a stack of four septimal quartertones of 36/35, and splits the octave in two. This means the quartertone should be sharpened a bit, leading to the natural 11-limit extension where 385/384 and 9801/9800 are tempered out.

Subgroup: 2.3.5.7

Comma list: 1500625/1492992

Mapping: [⟨2 1 0 7], ⟨0 2 0 3], ⟨0 0 1 -1]]

Mapping generators: ~1225/864, ~35/24, ~5

Optimal tunings:

  • WE: ~1225/864 = 600.1145 ¢, ~35/24 = 651.0771 ¢, ~5/4 = 385.4061 ¢
Error map: ⟨+0.229 +0.314 -0.450 -0.657]
  • CWE: ~1225/864 = 600.1145 ¢, ~35/24 = 651.0546 ¢, ~5/4 = 385.4793 ¢
Error map: ⟨0.000 +0.154 -0.834 -1.141]

Optimal ET sequence: 22, 46, 68, 72, 118, 140, 212, 330, 470, 542d, 872cdd, 1012cdd, 1414ccddd

Badness (Sintel): 3.11

Stearnsmic

For extensions, see Swetismic temperaments #Hades.

Stearnsmic tempers out the stearnsma, and splits the octave in two. A stack of three ~9/7 generators and a semi-octave period give the 3rd harmonic.

Subgroup: 2.3.5.7

Comma list: 118098/117649

Mapping: [⟨2 1 0 2], ⟨0 3 0 5], ⟨0 0 1 0]]

Mapping generators: ~343/243, ~9/7, ~5

Optimal tunings:

  • WE: ~343/243 = 599.9938 ¢, ~9/7 = 433.8840 ¢, ~5/4 = 386.3383 ¢
Error map: ⟨-0.012 -0.309 -0.000 +0.582]
  • CWE: ~343/243 = 600.0000 ¢, ~9/7 = 433.8851 ¢, ~5/4 = 386.3279 ¢
Error map: ⟨0.000 -0.300 +0.014 +0.600]

Optimal ET sequence: 22, 50, 58, 72, 130, 152, 202, 224, 354

Badness (Sintel): 3.30

Mirwomo

For extensions, see Rastmic rank-3 clan #Mirwomo.

Mirwomo tempers out the mirwomo comma in the 7-limit, equating the Pythagorean apotome with a stack of two septimal quartertones of 36/35, and splits the fifth in two. This means the fifth should be flattened a bit and the quartertone should be sharpened, leading to a natural 11-limit extension where 243/242 and 385/384 are tempered out.

Subgroup: 2.3.5.7

Comma list: 33075/32768

Mapping: [⟨1 1 0 6], ⟨0 2 0 -3], ⟨0 0 1 -1]]

Mapping generators: ~2, ~128/105, ~5

Optimal tunings:

  • WE: ~2 = 1200.8046 ¢, ~128/105 = 350.3723 ¢, ~5/4 = 384.1239 ¢
Error map: ⟨+0.805 -0.406 -0.581 -0.848]
  • CWE: ~2 = 1200.0000 ¢, ~128/105 = 350.1448 ¢, ~5/4 = 383.8961 ¢
Error map: ⟨0.000 -1.665 -2.418 -3.157]

Optimal ET sequence: 17, 21, 24, 31, 41, 72, 281d, 322cd, 353cd, 425bcdd, 497bcdd

Badness (Sintel): 3.40

Decovulture

For extensions, see Olympic clan #Baffin.

Subgroup: 2.3.5.7

Comma list: 67108864/66976875

Mapping: [⟨1 0 0 13], ⟨0 2 0 -7], ⟨0 0 1 -2]]

mapping generators: ~2, ~8192/4725, ~5

Optimal tunings:

  • WE: ~2 = 1199.9033 ¢, ~8192/4725 = 951.0102 ¢, ~5/4 = 386.5872 ¢
error map: ⟨-0.097 +0.065 +0.080 +0.059]
  • CWE: ~2 = 1200.0000 ¢, ~8192/4725 = 951.0899 ¢, ~5/4 = 386.6184 ¢
error map: ⟨0.000 +0.225 +0.305 +0.308]

Optimal ET sequence: 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd

Badness (Sintel): 3.82

Trimyna

For extensions, see Werckismic temperaments #Trimyna.

Trimyna tempers out the trimyna comma in the 7-limit, and finds the 6th harmonic by a stack of five 10/7's.

Subgroup: 2.3.5.7

Comma list: 50421/50000

Mapping: [⟨1 -1 0 1], ⟨0 5 0 -1], ⟨0 0 1 1]]

Mapping generators: ~2, ~10/7, ~5

Optimal tunings:

  • WE: ~2 = 1200.1652 ¢, ~10/7 = 620.4031 ¢, ~5/4 = 387.0990 ¢
Error map: ⟨+0.165 -0.105 +1.116 -1.634]
  • CWE: ~2 = 1200.0000 ¢, ~10/7 = 620.3427 ¢, ~5/4 = 387.2591 ¢
Error map: ⟨0.000 -0.241 +0.945 -1.910]

Optimal ET sequence: 27, 31, 58, 87, 118, 267d, 385d, 412d *

* optimal patent val: 294

Badness (Sintel): 3.84

Projection pair: 3 50000/16807 to 2.5.7

Squalentine

For extensions, see Biyatismic clan #Aphrodite.

Squalentine tempers out the squalentine comma in the 7-limit. Its generators can be taken to be 2, 3, and 21/20, and it equates (21/20)3 with 8/7. An obvious 11-limit extension then equates the last generator with 22/21, tempering out 121/120 and 441/440. Notice also 64827/64000 = (121/120)⋅(441/440)2.

Subgroup: 2.3.5.7

Comma list: 64827/64000

Mapping: [⟨1 0 1 3], ⟨0 1 1 0], ⟨0 0 -4 -3]]

Mapping generators: ~2, ~3, ~21/20

Optimal tunings:

  • WE: ~2 = 1200.7175 ¢, ~3/2 = 700.6331 ¢, ~21/20 = 78.6164 ¢
Error map: ⟨+0.718 -0.604 +1.289 -2.523]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.5831 ¢, ~21/20 = 78.4341 ¢
Error map: ⟨0.000 -1.372 +0.533 -4.128]

Optimal ET sequence: 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d

Badness (Sintel): 4.16

Projection pairs: 5 320000/64827 7 64000/9261 to 2.3.7/5

Quasiorwellismic

For extensions, see Lehmerismic temperaments #Ganesha.

Quasiorwellismic tempers out the quasiorwellisma in the 7-limit, and finds 7/6 by a stack of ten 5/4's octave reduced. A natural 11-limit extension thus arises from mapping 11/9 to a stack of four 5/4's octave reduced, leading to ganesha, which tempers out 3025/3024 and 5632/5625.

Subgroup: 2.3.5.7

Comma list: 29360128/29296875

Mapping: [⟨1 0 0 -22], ⟨0 1 0 1], ⟨0 0 1 10]]

Mapping generators: ~2, ~3, ~5

Optimal tunings:

  • WE: ~2 = 1199.9205 ¢, ~3/2 = 702.0435 ¢, ~5/4 = 386.6674 ¢
Error map: ⟨-0.079 +0.009 +0.195 -0.029]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0491 ¢, ~5/4 = 386.6885 ¢
Error map: ⟨0.000 +0.094 +0.375 +0.108]

Optimal ET sequence: 31, 87, 118, 152, 239, 270, 571, 723, 841, 993, 1263, 1564c, 1834c, 2104c

Badness (Sintel): 5.00

Buzzardsmic

Buzzardsmic tempers out the buzzardsma and gives buzzard an independent generator for the 5th harmonic.

Subgroup: 2.3.5.7

Comma list: 65536/64827

Mapping: [⟨1 0 0 4], ⟨0 4 0 -3], ⟨0 0 1 0]]

Mapping generators: ~2, ~21/16, ~5

Optimal tunings:

  • WE: ~2 = 1199.2548 ¢, ~21/16 = 475.5761 ¢, ~5/4 = 387.8025 ¢
Error map: ⟨-0.745 +0.350 -0.002 +1.465]
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.8328 ¢, ~5/4 = 387.5778 ¢
Error map: ⟨0.000 +1.376 +1.264 +3.676]

Optimal ET sequence: 5, 10, 15, 33, 38, 43, 53, 111, 121, 164d, 174d, 179, 232d

Badness (Sintel): 6.18

Tolerant

For extensions, see Pentacircle clan #Tolerant.

Subgroup: 2.3.5.7

Comma list: 179200/177147

Mapping: [⟨1 0 0 -10], ⟨0 1 0 11], ⟨0 0 1 -2]]

Mapping generators: ~2, ~3, ~5

Optimal tunings:

  • WE: ~2 = 1199.539 ¢, ~3 = 1903.226 ¢, ~5 = 2785.816 ¢
  • CWE: ~2 = 1200.000 ¢, ~3 = 1903.805 ¢, ~5 = 2786.356 ¢

Optimal ET sequence: 34d, 39d, 41, 80, 87, 121, 167, 208, 329b, 375b, 496bd

Badness (Sintel): 7.26

History

For extensions, see Werckismic temperaments #History.

History tempers out the historisma in the 7-limit, and splits the fourth in six.

Comma list: 257298363/256000000

Mapping: [⟨1 2 0 0], ⟨0 -6 0 7], ⟨0 0 1 1]]

Mapping generators: ~2, ~21/20, ~5

Optimal tunings:

  • WE: ~2 = 1200.154 ¢, ~21/20 = 83.091 ¢, ~5 = 2786.669 ¢
  • CWE: ~2 = 1200.000 ¢, ~21/20 = 83.067 ¢, ~5 = 2786.515 ¢

Optimal ET sequence: 14c, 15, 29, 43, 58, 72, 130, 202

Badness (Sintel): 7.95

Sensibeta

See also: Sensibeta comma

Sensibeta tempers out the sensibeta comma in the 7-limit.

Comma list: 1071875/1062882

Mapping: [⟨1 0 2 -3], ⟨0 1 0 4], ⟨0 0 3 -5]]

Mapping generators: ~2, ~3, ~175/162

Optimal tunings:

  • WE: ~2 = 1200.025 ¢, ~3 = 1902.728 ¢, ~175/162 = 128.524 ¢
  • CWE: ~2 = 1200.000 ¢, ~3 = 1902.709 ¢, ~175/162 = 128.530 ¢

Optimal ET sequence: 19, 27, 46, 94, 113, 121, 140

Badness (Sintel): 7.97

Parahemif

For extensions, see Rastmic rank-3 clan #Parahemif.

Parahemif tempers out the parahemif comma in the 7-limit, equating a Pythagorean apotome with a stack of two septimal third-tones of 28/27, and splits the fifth in two. It also equates the large septimal diesis of 49/48 with the Pythagorean comma. This means the fifth should be tuned sharp and the septimal third-tone should be flattened to a somewhat large quartertone which can be used as the undecimal quartertone of 33/32, leading to a natural 11-limit extension where 243/242 and 896/891 are tempered out.

Subgroup: 2.3.5.7

Comma list: 1605632/1594323

Mapping: [⟨1 1 0 -1], ⟨0 2 0 13], ⟨0 0 1 0]]

Mapping generators: ~2, ~896/729, ~5

Optimal tunings:

  • WE: ~2 = 1199.7303 ¢, ~896/729 = 351.4056 ¢, ~5/4 = 386.8527 ¢
Error map: ⟨-0.270 +0.586 -0.000 -0.284]
  • CWE: ~2 = 1200.0000 ¢, ~896/729 = 351.4569 ¢, ~5/4 = 386.6884 ¢
Error map: ⟨0.000 +0.959 +0.375 +0.114]

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99, 239, 338

Badness (Sintel): 8.77

Septimagic

Septimagic tempers out the septimagic comma in the 7-limit and gives the 2.3.7-subgroup magic restriction an independent generator for the 5th harmonic.

Subgroup: 2.3.5.7

Comma list: 537824/531441

Mapping: [⟨1 0 0 -1], ⟨0 5 0 12], ⟨0 0 1 0]]

Mapping generators: ~2, ~243/196, ~5

Optimal tunings:

  • WE: ~2 = 1199.8224 ¢, ~243/196 = 380.6043 ¢, ~5/4 = 386.6676 ¢
error map: ⟨-0.178 +1.066 -0.001 -1.397]
  • CWE: ~2 = 1200.0000 ¢, ~243/196 = 380.6378 ¢, ~5/4 = 386.5230 ¢
error map: ⟨0.000 +1.234 +0.209 -1.173]

Optimal ET sequence: 19, 41, 104c, 123, 126, 145, 167, 186, 394b, 413

Badness (Sintel): 11.8

Compass

For extensions, see Moctdelismic clan #Compass.

Compass tempers out the compass comma in the 7-limit, and splits the fourth in five. The obvious 11-limit extension tempers out the moctdelisma, 1375/1372.

Subgroup: 2.3.5.7

Comma list: 9765625/9680832

Mapping: [⟨1 2 0 -2], ⟨0 -5 0 2], ⟨0 0 1 2]]

Mapping generators: ~2, ~625/588, ~5

Optimal tunings:

  • WE: ~2 = 1200.1156 ¢, ~625/588 = 99.6359 ¢, ~5/4 = 385.0414 ¢
error map: ⟨+0.116 +0.097 -1.041 +0.760]
  • CWE: ~2 = 1200.0000 ¢, ~625/588 = 99.6080 ¢, ~5/4 = 385.1108 ¢
error map: ⟨0.000 +0.005 -1.203 +0.612]

Optimal ET sequence: 12, …, 37, 48d, 49, 60, 72, 181, 193, 265

Badness (Sintel): 13.3

Linus

For extensions, see Kalismic temperaments #Linus.

Linus tempers out the linus comma in the 7-limit, and splits the octave into twelve equal parts of ~15/14. The obvious 11-limit extension tempers out the kalisma, 9801/9800.

Subgroup: 2.3.5.7

Comma list: 578509309952/576650390625

Mapping: [⟨10 0 0 -11], ⟨0 1 0 1], ⟨0 0 1 1]]

Mapping generators: ~15/14, ~3, ~5

Optimal tunings:

  • WE: ~15/14 = 119.9964 ¢, ~3/2 = 702.0734 ¢, ~5/4 = 386.5626 ¢
error map: ⟨-0.036 +0.082 +0.177 -0.258]
  • CWE: ~15/14 = 120.0000 ¢, ~3/2 = 702.0700 ¢, ~5/4 = 386.5404 ¢
error map: ⟨0.000 +0.115 +0.227 -0.215]

Optimal ET sequence: 50, 60, 80, 130, 270, 1270, 1540, 1810, 1940, 2080, 2210c, 2480c

Badness (Sintel): 15.7

Naiad

For extensions, see Wizardharry clan #Naiad.

Naiad tempers out the naiadisma in the 7-limit. An obvious 13-limit interpretation of one generator (~98/75) is 13/10.

Subgroup: 2.3.5.7

Comma list: 161414428/158203125

Mapping: [⟨1 5 0 2], ⟨0 -9 0 -4], ⟨0 0 1 1]]

Mapping generators: ~2, ~98/75, ~5

Optimal tunings:

  • WE: ~2 = 1199.937 ¢, ~98/75 = 455.269 ¢, ~5/4 = 387.946 ¢
error map: ⟨-0.063, +0.311, +1.506, -2.207]
  • CWE: ~2 = 1200.000 ¢, ~98/75 = 455.298 ¢, ~5/4 = 387.896 ¢
error map: ⟨0.000, +0.362, +1.582, -2.123]

Optimal ET sequence: 8d, 21, 29, 37, 50, 58, 87, 145

Badness (Sintel): 52.5

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