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Miscellaneous 7-limit temperaments: Difference between revisions

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See also [[Miscellaneous 5-limit temperaments]].  
See also [[Miscellaneous 5-limit temperaments]].  
== Breeze ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 2460375/2458624
{{Mapping|legend=1| 1 0 -2 -4 | 0 1 1 3 | 0 0 4 3 }}
: Mapping generators: ~2, ~3, ~45/28
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{Optimal ET sequence|legend=1| 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 [[support]]ed by third-comma equal temperaments, and [[171edo]] shows an excellent example of this. 11-limit extensions of this temperament include [[Moctdelismic clan #Mendel|mendel]] and [[Lehmerismic temperaments #Skadi|skadi]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 703125/702464
{{Mapping|legend=1| 1 0 -1 -6 | 0 1 1 3 | 0 0 3 7 }}
: Mapping generators: ~2, ~3, ~112/75
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{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
== 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 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
[[Comma list]]: 405/392
{{Mapping|legend=1| 1 0 1 -1 | 0 1 0 2 | 0 0 2 1 }}
: Mapping generators: ~2, ~3, ~14/9
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{Optimal ET sequence|legend=1| 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 [[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
[[Comma list]]: 525/512
{{Mapping|legend=1| 1 0 0 9 | 0 1 0 -1 | 0 0 1 -2 }}
: Mapping generators: ~2, ~3, ~5
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{Optimal ET sequence|legend=1| 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|legend=1| 2 0 0 9 | 0 1 0 -4 | 0 0 1 2 }}
: Mapping generators: ~567/400, ~3, ~5
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{Optimal ET sequence|legend=1| 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|legend=1| 1 0 -5 -1 | 0 1 3 2 | 0 0 4 1 }}
: Mapping generators: ~2, ~3, ~14/9
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{Optimal ET sequence|legend=1| 8d, 9, 14c, 17c, 22, 31, 53, 84, 137, 221d }}
[[Badness]] (Sintel): 2.29
[[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 ==
: ''For extensions, see [[Keenanismic temperaments #Uniwiz]].''
: ''For extensions, see [[Keenanismic temperaments #Uniwiz]].''
Uniwiz tempers out the [[uniwiz comma]] in the 7-limit, equating the [[9/8|whole tone]] with a stack of four septimal quartertones of [[36/35]], and splits the [[2/1|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
[[Subgroup]]: 2.3.5.7
Line 15: Line 215:


{{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]] ([[CTE]]): ~1225/864 = 600.0000{{c}}, ~35/24 = 651.1974{{c}}, ~5/4 = 385.6847{{c}}
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}


{{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 }}


[[Badness]] (Smith): 0.7042 × 10<sup>-3</sup>
[[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 [[3/1|3rd]] [[harmonic]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 118098/117649
 
{{Mapping|legend=1| 2 1 0 2 | 0 3 0 5 | 0 0 1 0 }}
: Mapping generators: ~343/243, ~9/7, ~5
 
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
 
{{Optimal ET sequence|legend=1| 22, 50, 58, 72, 130, 152, 202, 224, 354 }}
 
[[Badness]] (Sintel): 3.30


== Mirwomo ==
== Mirwomo ==
: ''For extensions, see [[Rastmic rank-3 clan #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
[[Subgroup]]: 2.3.5.7
Line 31: 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}}, ~11/9 = 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}}, ~11/9 = 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 ==
: ''For extensions, see [[Werckismic temperaments #Trimyna]].''
Trimyna tempers out the [[trimyna comma]] in the 7-limit, and finds the [[6/1|6th]] [[harmonic]] by a stack of five [[10/7]]'s.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 50421/50000
{{Mapping|legend=1| 1 -1 0 1 | 0 5 0 -1 | 0 0 1 1 }}
: Mapping generators: ~2, ~10/7, ~5
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{Optimal ET sequence|legend=1| 27, 31, 58, 87, 118, 267d, 385d, 412d }} *
<nowiki>*</nowiki> [[optimal patent val]]: [[294edo|294]]
[[Badness]] (Sintel): 3.84
[[Projection pair]]: <code>3 50000/16807</code> 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)<sup>3</sup> 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)<sup>2</sup>.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 64827/64000
{{Mapping|legend=1| 1 0 1 3 | 0 1 1 0 | 0 0 -4 -3 }}
: Mapping generators: ~2, ~3, ~21/20
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
[[Badness]] (Sintel): 4.16
[[Projection pair]]s: <code>5 320000/64827 7 64000/9261</code> 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|legend=1| 1 0 0 -22 | 0 1 0 1 | 0 0 1 10 }}
: Mapping generators: ~2, ~3, ~5
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{Optimal ET sequence|legend=1| 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 [[5/1|5th]] [[harmonic]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 65536/64827
{{Mapping|legend=1| 1 0 0 4 | 0 4 0 -3 | 0 0 1 0 }}
: Mapping generators: ~2, ~21/16, ~5
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{Optimal ET sequence|legend=1| 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|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 ==
: ''For extensions, see [[Rastmic rank-3 clan #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
[[Subgroup]]: 2.3.5.7
Line 51: 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 ==
: ''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|legend=1| 10 0 0 -11 | 0 1 0 1 | 0 0 1 1 }}
: Mapping generators: ~15/14, ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 119.9964{{c}}, ~3/2 = 702.0734{{c}}, ~5/4 = 386.5626{{c}}
: [[error map]]: {{val| -0.036 +0.082 +0.177 -0.258 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 702.0700{{c}}, ~5/4 = 386.5404{{c}}
: error map: {{val| 0.000 +0.115 +0.227 -0.215 }}
{{Optimal ET sequence|legend=1| 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|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]]
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