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Ragismic microtemperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (9/)
Switch to Sintel's badness, WE & CWE tunings (10/)
Line 1,208: Line 1,208:
: ''For the 5-limit version, see [[8th-octave temperaments #Octoid]].''
: ''For the 5-limit version, see [[8th-octave temperaments #Octoid]].''


The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.
The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai comma]]). In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].
 
The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,215: Line 1,217:


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


[[Optimal tuning]] ([[POTE]]): ~49/45 = 150.000{{c}}, ~7/5 = 583.940{{c}}
[[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]]:  
[[Tuning ranges]]:  
Line 1,225: Line 1,231:
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]


{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}


[[Badness]] (Sintel): 1.080
[[Badness]] (Sintel): 1.08


Scales: [[octoid72]], [[octoid80]]
Scales: [[octoid72]], [[octoid80]]


=== 11-limit ===
=== 11-limit ===
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 minimaxing the 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, if one wants to use 80edo as the tuning, one must use octopus – not octoid – as 80edo does not temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 1,240: Line 1,244:
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.962{{c}}
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:  
Tuning ranges:  
Line 1,246: Line 1,252:
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]


{{Optimal ET sequence|legend=0| 72, 152, 224 }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}


Badness (Sintel): 0.466
Badness (Sintel): 0.466
Line 1,259: Line 1,265:
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.905{{c}}
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 }}
{{Optimal ET sequence|legend=0| 72, 152f, 224 }}
Line 1,268: Line 1,276:


; Music
; Music
* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72] in 224edo tuning
* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – Octoid[72] in 224edo tuning


===== 17-limit =====
===== 17-limit =====
Line 1,277: Line 1,285:
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.842{{c}}
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 }}
{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}
Line 1,290: Line 1,300:
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.932{{c}}
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 }}
{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}
Line 1,297: Line 1,309:


==== Octopus ====
==== 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 = 300{{cent}}.
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 = 300{{c}}.


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,305: Line 1,317:
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.892{{c}}
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| 72, 152, 224f }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}


Badness (Sintel): 0.0896
Badness (Sintel): 0.896


===== 17-limit =====
===== 17-limit =====
Line 1,318: Line 1,332:
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 583.811{{c}}
Optimal tunings:
* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}


{{Optimal ET sequence|legend=0| 72, 152, 224fg, 296ffg }}
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}


Badness (Sintel): 0.795
Badness (Sintel): 0.795
Line 1,331: Line 1,347:
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}


Optimal tuning (POTE): ~12/11 = 150.000{{c}}, ~7/5 = 584.064{{c}}
Optimal tunings:
* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}


{{Optimal ET sequence|legend=0| 72, 152, 224fg, 376ffgh }}
{{Optimal ET sequence|legend=0| 8d, 72, 152 }}


Badness (Sintel): 0.993
Badness (Sintel): 0.993
Line 1,351: Line 1,369:
: mapping generators: ~448/429, ~7/5
: mapping generators: ~448/429, ~7/5


Optimal tuning (POTE): ~448/429 = 75.000{{c}}, ~13/8 = 841.015{{c}}
Optimal tunings:
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}


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


Badness (Sintel): 1.273
Badness (Sintel): 1.27


===== 17-limit =====
===== 17-limit =====
Line 1,364: Line 1,384:
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}


Optimal tuning (POTE): ~117/112 = 75.000{{c}}, ~13/8 = 840.932{{c}}
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 }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}


Badness (Sintel): 1.458
Badness (Sintel): 1.46


===== 19-limit =====
===== 19-limit =====
Line 1,375: Line 1,397:
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444


Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}


Optimal tuning (POTE): ~117/112 = 75.000{{c}}, ~13/8 = 840.896{{c}}
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 }}
{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}


Badness (Sintel): 1.443
Badness (Sintel): 1.44


== Parakleismic ==
== Parakleismic ==
Line 1,387: Line 1,411:
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic (5-limit)]].''
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic (5-limit)]].''


In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat [[6/5]], 13 of which give 32/3, and 14 give 64/5. While 118 no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding [[3136/3125]] and 4375/4374, for which [[99edo]], 118edo, and especially [[217edo]] are accurate tunings.
 
Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,393: Line 1,419:
[[Comma list]]: 3136/3125, 4375/4374
[[Comma list]]: 3136/3125, 4375/4374


{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}
{{Mapping|legend=1| 1 -8 -8 -23 | 0 13 14 35 }}
: mapping generators: ~2, ~6/5
: mapping generators: ~2, ~5/3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 315.181{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7820{{c}}, ~5/3 = 884.6581{{c}}
: [[error map]]: {{val| -0.218 +0.344 +0.644 -0.779 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8088{{c}}
: error map: {{val| 0.000 +0.560 +1.010 -0.516 }}


{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }}
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99, 217, 316, 415 }}


[[Badness]] (Sintel): 0.694
[[Badness]] (Sintel): 0.694
Line 1,407: Line 1,437:
Comma list: 385/384, 3136/3125, 4375/4374
Comma list: 385/384, 3136/3125, 4375/4374


Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}
Mapping: {{mapping| 1 -8 -8 -23 30 | 0 13 14 35 -36 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.251{{c}}
Optimal tunings:
* WE: ~2 = 1200.3296{{c}}, ~5/3 = 884.9921{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7519{{c}}


{{Optimal ET sequence|legend=0| 19, 99, 118 }}
{{Optimal ET sequence|legend=0| 19, 99, 118 }}


Badness (Sintel): 1.643
Badness (Sintel): 1.64


=== Paralytic ===
=== Paralytic ===
The paralytic temperament (118 & 217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.
Paralytic (99e & 118) tempers out [[441/440]], [[5632/5625]], and [[19712/19683]]. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,422: Line 1,454:
Comma list: 441/440, 3136/3125, 4375/4374
Comma list: 441/440, 3136/3125, 4375/4374


Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}
Mapping: {{mapping| 1 -8 -8 -23 -57 | 0 13 14 35 82 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.220{{c}}
Optimal tunings:
* WE: ~2 = 1199.9940{{c}}, ~5/3 = 884.7757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7800{{c}}


{{Optimal ET sequence|legend=0| 19e, 99e, 118, 217, 335, 552d, 887dd }}
{{Optimal ET sequence|legend=0| 19e, …, 99e, 118, 217, 335 }}


Badness (Sintel): 1.191
Badness (Sintel): 1.19


==== 13-limit ====
==== 13-limit ====
Line 1,435: Line 1,469:
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374


Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}
Mapping: {{mapping| 1 -8 -8 -23 -57 59 | 0 13 14 35 82 -75 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.214{{c}}
Optimal tunings:
* WE: ~2 = 1199.9218{{c}}, ~5/3 = 884.7285{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7858{{c}}


{{Optimal ET sequence|legend=0| 99e, 118, 217, 552d, 769de }}
{{Optimal ET sequence|legend=0| 99e, 118, 217 }}


Badness (Sintel): 1.847
Badness (Sintel): 1.85


==== Paraklein ====
==== Paraklein ====
The paraklein temperament (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].
Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 1,450: Line 1,486:
Comma list: 196/195, 352/351, 625/624, 729/728
Comma list: 196/195, 352/351, 625/624, 729/728


Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}
Mapping: {{mapping| 1 -8 -8 -23 -57 -28 | 0 13 14 35 82 43 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.225{{c}}
Optimal tunings:
* WE: ~2 = 1199.8239{{c}}, ~5/3 = 884.6449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7709{{c}}


{{Optimal ET sequence|legend=0| 19e, 99ef, 118, 217ff, 335ff }}
{{Optimal ET sequence|legend=0| 19e, …, 99ef, 118 }}


Badness (Sintel): 1.554
Badness (Sintel): 1.55


=== Parkleismic ===
=== Parkleismic ===
Line 1,463: Line 1,501:
Comma list: 176/175, 1375/1372, 2200/2187
Comma list: 176/175, 1375/1372, 2200/2187


Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}
Mapping: {{mapping| 1 -8 -8 -23 -43 | 0 13 14 35 63 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.060{{c}}
Optimal tunings:
* WE: ~2 = 1199.1848{{c}}, ~5/3 = 884.3386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9158{{c}}


{{Optimal ET sequence|legend=0| 19e, 80, 179, 259cd }}
{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179, 259cd }}


Badness (Sintel): 1.848
Badness (Sintel): 1.85


==== 13-limit ====
==== 13-limit ====
Line 1,476: Line 1,516:
Comma list: 169/168, 176/175, 325/324, 1375/1372
Comma list: 169/168, 176/175, 325/324, 1375/1372


Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}
Mapping: {{mapping| 1 -8 -8 -23 -43 -14 | 0 13 14 35 63 24 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.075{{c}}
Optimal tunings:
* WE: ~2 = 1199.5318{{c}}, ~5/3 = 884.5800{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9118{{c}}


{{Optimal ET sequence|legend=0| 19e, 80, 179 }}
{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179 }}


Badness (Sintel): 1.511
Badness (Sintel): 1.51


=== Paradigmic ===
=== Paradigmic ===
Line 1,489: Line 1,531:
Comma list: 540/539, 896/891, 3136/3125
Comma list: 540/539, 896/891, 3136/3125


Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}
Mapping: {{mapping| 1 -8 -8 -23 16 | 0 13 14 35 -17 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.096{{c}}
Optimal tunings:
* WE: ~2 = 1199.0616{{c}}, ~5/3 = 884.2124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.8877{{c}}


{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e }}
{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e, 457bcddeeee }}


Badness (Sintel): 1.379
Badness (Sintel): 1.38


==== 13-limit ====
==== 13-limit ====
Line 1,502: Line 1,546:
Comma list: 169/168, 325/324, 540/539, 832/825
Comma list: 169/168, 325/324, 540/539, 832/825


Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}
Mapping: {{mapping| 1 -8 -8 -23 16 -14 | 0 13 14 35 -17 24 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.080{{c}}
Optimal tunings:
* WE: ~2 = 1199.2683{{c}}, ~5/3 = 884.3805{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9061{{c}}


{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e }}
{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e }}


Badness (Sintel): 1.479
Badness (Sintel): 1.48


=== Semiparakleismic ===
=== Semiparakleismic ===
Line 1,515: Line 1,561:
Comma list: 3025/3024, 3136/3125, 4375/4374
Comma list: 3025/3024, 3136/3125, 4375/4374


Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}
Mapping: {{mapping| 2 -3 -2 -11 -4 | 0 13 14 35 23 }}
: mapping generators: ~99/70, ~33/28


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 315.181{{c}}
Optimal tunings:
* WE: ~99/70 = 599.9270{{c}}, ~33/28 = 284.7841{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8119{{c}}


{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c, 830c }}
{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c }}


Badness (Sintel): 1.131
Badness (Sintel): 1.13


==== Semiparamint ====
==== Semiparamint ====
Line 1,530: Line 1,579:
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374


Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}
Mapping: {{mapping| 2 -3 -2 -11 -4 15 | 0 13 14 35 23 -16 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 315.156{{c}}
Optimal tunings:
* WE: ~99/70 = 599.8253{{c}}, ~33/28 = 284.7608{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8366{{c}}


{{Optimal ET sequence|legend=0| 80, 118, 198 }}
{{Optimal ET sequence|legend=0| 80, 118, 198 }}


Badness (Sintel): 1.396
Badness (Sintel): 1.40


==== Semiparawolf ====
==== Semiparawolf ====
Line 1,545: Line 1,596:
Comma list: 169/168, 325/324, 364/363, 3136/3125
Comma list: 169/168, 325/324, 364/363, 3136/3125


Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}
Mapping: {{mapping| 2 -3 -2 -11 -4 -4 | 0 13 14 35 23 24 }}


Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~6/5 = 315.184{{c}}
Optimal tunings:
* WE: ~99/70 = 600.0569{{c}}, ~13/11 = 284.8431{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~13/11 = 284.8216{{c}}


{{Optimal ET sequence|legend=0| 80, 118f, 198f }}
{{Optimal ET sequence|legend=0| 80, 118f, 198f }}


Badness (Sintel): 1.672
Badness (Sintel): 1.67


== Counterkleismic ==
== Counterkleismic ==
: ''For the 5-limit temperament, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''


In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
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