Canopic clan: Difference between revisions

[[Temperament]]s of the '''mirkwai clan''' [[Tempering out|temper out]] the [[mirkwai comma]], {{monzo| 0 3 4 -5 }} = 16875/16807, a no-twos comma.  

{{Mapping|legend=2| 1 3 3 | 0 -5 -4 }}

: sval mapping generators: ~3, ~7/5

[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~7/5 = 583.9584

[[Optimal ET sequence]]: [[13edt|b13]], [[62edt|b62]], [[75edt|b75]], [[88edt|b88]], [[101edt|b101]], [[114edt|b114]], [[355edt|b355]], [[469edt|b469]], [[583edt|b583]], [[697edt|b697]]

The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid.  

The others are weak extensions. Mirkat tempers out [[19683/19600]], splitting the generator in two with a semitwelfth period. Sqrtphi tempers out [[15625/15552]], splitting the period in six. Miracle tempers out [[225/224]]. Pluto tempers out [[4000/3969]]. These split the generator in five. Quanharuk tempers out [[32805/32768]], splitting the generator in three with a 1/5-twelfth period. Semisept tempers out [[1728/1715]] and [[3136/3125]], splitting the generator in six. Kwai tempers out [[5120/5103]], splitting the generator in ten. Grendel tempers out [[6144/6125]], splitting the generator in eleven. Finally, eris tempers out [[65625/65536]], splitting the generator in sixteen.  

Members of the clan considered below are grendel, kwai, pluto, mirkat, eris, subsemifourth, septendesemi, gaster, subsedia, hemiseptisix, browser, and grazer. Discussed elsewhere are:  

* ''[[Nusecond]]'' (+126/125) → [[Starling temperaments #Nusecond|Starling temperaments]]

* ''[[Octoid]]'' (+4375/4374) → [[Ragismic microtemperaments #Octoid|Ragismic microtemperaments]]

For ''no-twos'' extensions, see [[No-twos subgroup temperaments#Canopus]].

: ''For the 5-limit version of this temperament, see [[Syntonic–31 equivalence continuum #Counterwürschmidt]].''

 

 

{{Mapping|legend=1| 1 9 2 7 | 0 -23 1 -13 }}

 

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 386.863

 

{{Optimal ET sequence|legend=1| 31, 90, 121, 152, 335d }}

 

[[Badness]]: 0.051834

 

 

Comma list: 540/539, 1375/1372, 5632/5625

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714

 

 

 

 

 

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kwai]].''

 

Named by [[Gene Ward Smith]] in 2004 for its "bridgeability"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10766.html Yahoo! Tuning Group | ''Kwai'']</ref>, kwai is generated by a fifth, and can be described as 41 & 70.

 

 

 

{{Mapping|legend=1| 1 0 -50 -40 | 0 1 33 27 }}

 

 

 

{{Optimal ET sequence|legend=1| 41, 111, 152, 345, 497d }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the 41 & 80 temperament. It is generated by a sharpened [[7/5]], and [[121edo|59\121]] is about perfect as a tuning.  

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 183.539

[[Badness]]: 0.059376

Mapping: {{mapping| 3 2 1 2 9 | 0 6 13 14 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 183.528

{{Optimal ET sequence|legend=1| 39d, 72, 111, 183, 255 }}

Badness: 0.022126

Mapping: {{mapping| 3 2 1 2 9 1 | 0 6 13 14 3 22 }}

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 183.577

{{Optimal ET sequence|legend=1| 39df, 72, 111, 183 }}

Badness: 0.018632

Mapping: {{mapping| 3 2 1 2 9 1 4 | 0 6 13 14 3 22 18 }}

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 183.579

{{Optimal ET sequence|legend=1| 39dfg, 72, 111, 183 }}

Badness: 0.011775

The [[2.5.7 subgroup]] restriction of this temperament is [[exodia]].

{{Mapping|legend=1| 1 10 0 6 | 0 -29 8 -11 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~60/49 = 348.216

{{Optimal ET sequence|legend=1| 31, 131, 162, 193, 224, 1823cd, 2271cd }}

[[Badness]]: 0.074719

Mapping: {{mapping| 1 10 0 6 20 | 0 -29 8 -11 -57 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.219

{{Optimal ET sequence|legend=1| 31, 193, 224, 703, 927d, 1151cd }}

Badness: 0.027621

Mapping: {{mapping| 1 10 0 6 20 -14 | 0 -29 8 -11 -57 61 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.213

{{Optimal ET sequence|legend=1| 31, 193, 224 }}

Badness: 0.025137

{{Mapping|legend=1| 1 39 27 45 | 0 -47 -31 -53 }}

: mapping generators: ~2, ~125/72

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~144/125 = 244.719

[[Badness]]: 0.135173

Mapping: {{mapping| 1 39 27 45 56 | 0 -47 -31 -53 -66 }}

Optimal tuning (POTE): ~2 = 1\1, ~121/105 = 244.719

{{Optimal ET sequence|legend=1| 49, 103, 152, 255, 407, 966d }}

Badness: 0.034276

Mapping: {{mapping| 1 39 27 45 56 65 | 0 -47 -31 -53 -66 -77 }}

Optimal tuning (POTE): ~2 = 1\1, ~15/13 = 244.714

{{Optimal ET sequence|legend=1| 49f, 103, 152f, 255, 407f, 662df }}

Badness: 0.028387

The name ''septendesemi'' means a septendecimal semitone ([[17/16]]). The septendesemi temperament (80 &amp; 103) tempers out the mirkwai comma and 1959552/1953125 (''parkleiness'' comma, zotritrigu) in the 7-limit. [[183edo]] provides an excellent tuning for 7, 11, 13, and 17-limit septendesemi.

{{Mapping|legend=1| 1 39 37 53 | 0 -41 -38 -55 }}

 

: mapping generators: ~2, ~648/343

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/324 = 104.916

{{Optimal ET sequence|legend=1| 80, 103, 183 }}

[[Badness]]: 0.146795

Mapping: {{mapping| 1 39 37 53 50 | 0 -41 -38 -55 -51 }}

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 104.916

{{Optimal ET sequence|legend=1| 80, 103, 183 }}

Badness: 0.041554

Mapping: {{mapping| 1 39 37 53 50 11 | 0 -41 -38 -55 -51 -8 }}

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 104.908

{{Optimal ET sequence|legend=1| 80, 103, 183, 469f, 652def }}

Badness: 0.027908

Mapping: {{mapping| 1 39 37 53 50 11 5 | 0 -41 -38 -55 -51 -8 -1 }}

Optimal tuning (POTE): ~2 = 1\1, ~17/16 = 104.909

{{Optimal ET sequence|legend=1| 80, 103, 183, 469f, 652def }}

Badness: 0.020128

: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Gaster]].''

The ''gaster'' temperament (111 &amp; 113) tempers out {{monzo| -70 72 -19 }} (quadbila-negu) in the 5-limit; mirkwai comma (16875/16807) and [[skeetsma]] (14348907/14336000) in the 7-limit. The word "[[Wiktionary: gaster|gaster]]" means [[Wiktionary: abdomen|abdomen]] or [[Wiktionary: stomach|stomach]], but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals [[111edo|111]], [[113edo|113]] and [[224edo|224]] support it even in the 41-limit.

{{Mapping|legend=1| 1 11 38 37 | 0 -19 -72 -69 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~800/567 = 594.641

[[Badness]]: 0.154521

Mapping: {{mapping| 1 11 38 37 -1 | 0 -19 -72 -69 9 }}

Optimal tuning (POTE): ~2 = 1\1, ~512/363 = 594.639

{{Optimal ET sequence|legend=1| 111, 224, 783d, 1007d, 1231dd }}

Badness: 0.054060

Mapping: {{mapping| 1 11 38 37 -1 26 | 0 -19 -72 -69 9 -45 }}

Optimal tuning (POTE): ~2 = 1\1, ~55/39 = 594.639

{{Optimal ET sequence|legend=1| 111, 224, 783df, 1007df, 1231ddf }}

Badness: 0.024882

Mapping: {{mapping| 1 11 38 37 -1 26 14 | 0 -19 -72 -69 9 -45 -20 }}

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.636

{{Optimal ET sequence|legend=1| 111, 224, 559dgg }}

Badness: 0.021436

Mapping: {{mapping| 1 11 38 37 -1 26 14 32 | 0 -19 -72 -69 9 -45 -20 -56 }}

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.636

{{Optimal ET sequence|legend=1| 111, 224 }}

Badness: 0.018370

Mapping: {{mapping| 1 11 38 37 -1 26 14 32 7 | 0 -19 -72 -69 9 -45 -20 -56 -5 }}

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.641

{{Optimal ET sequence|legend=1| 111, 224 }}

Badness: 0.017619

Mapping: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 | 0 -19 -72 -69 9 -45 -20 -56 -5 32 }}

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.646

{{Optimal ET sequence|legend=1| 111, 113, 224 }}

Badness: 0.016815

Mapping: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 0 | 0 -19 -72 -69 9 -45 -20 -56 -5 32 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.644

{{Optimal ET sequence|legend=1| 111, 113, 224 }}

Badness: 0.014790

Mapping: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 0 -27 | 0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 }}

Optimal tuning (POTE): ~2 = 1\1, ~24/17 = 594.644

{{Optimal ET sequence|legend=1| 111, 113, 224 }}

Badness: 0.014377

Mapping: {{mapping| 1 11 38 37 -1 26 14 32 7 -11 0 -27 45 | 0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 -80 }}

 

 

 

 

 

[[Comma list]]: 16875/16807, 65536/64827

{{Mapping|legend=1| 1 0 5 4 | 0 16 -27 -12 }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Badness: 0.031635

 

 

 

 

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 118.968

 

 

 

 

 

 

 

 

Badness: 0.017935

The name ''hemiseptisix'' means a half of septimal major sixth ([[12/7]]). The hemiseptisix temperament (103 &amp; 121) tempers out the mirkwai comma and 95703125/95551488 (''pontiqak'' comma, lazozotritriyo) in the 7-limit. [[224edo]] provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix.

{{Mapping|legend=1| 1 34 17 34 | 0 -53 -24 -51 }}

 

: mapping generators: ~2, ~75/49

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 466.071

[[Badness]]: 0.162826

Mapping: {{mapping| 1 34 17 34 53 | 0 -53 -24 -51 -81 }}

Optimal tuning (POTE): ~2 = 1\1, ~55/42 = 466.070

{{Optimal ET sequence|legend=1| 103, 121, 224 }}

Badness: 0.043381

Mapping: {{mapping| 1 34 17 34 53 30 | 0 -53 -24 -51 -81 -43 }}

Optimal tuning (POTE): ~2 = 1\1, ~55/42 = 466.071

{{Optimal ET sequence|legend=1| 103, 121, 224 }}

Badness: 0.021127

Mapping: {{mapping| 1 34 17 34 53 30 31 | 0 -53 -24 -51 -81 -43 -44 }}

Optimal tuning (POTE): ~2 = 1\1, ~17/13 = 466.074

{{Optimal ET sequence|legend=1| 103, 121, 224 }}

Badness: 0.018611

== Browser ==

{{See also| Sensipent family }}

{{Mapping|legend=1| 1 34 47 58 | 0 -37 -51 -63 }}

 

: mapping generators: ~2, ~90/49

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 148.719

[[Badness]]: 0.217166

Mapping: {{mapping| 1 34 47 58 35 | 0 -37 -51 -63 -36 }}

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 148.729

{{Optimal ET sequence|legend=1| 113, 121, 234, 355e, 589cee }}

Badness: 0.076062

Mapping: {{mapping| 1 34 47 58 35 44 | 0 -37 -51 -63 -36 -46 }}

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 148.729

{{Optimal ET sequence|legend=1| 113, 121, 234, 355e, 589cee }}

Badness: 0.036248

Mapping: {{mapping| 1 34 47 58 35 44 33 | 0 -37 -51 -63 -36 -46 -33 }}

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 148.735

{{Optimal ET sequence|legend=1| 113, 121, 234g, 355eg }}

Badness: 0.025410

Mapping: {{mapping| 1 34 47 58 35 44 33 6 | 0 -37 -51 -63 -36 -46 -33 -2 }}

 

{{Optimal ET sequence|legend=1| 113, 121, 234g, 355eg, 589ceegg }}

Badness: 0.022574

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