Rastmic rank-3 clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Move mandos here as a strong extension
→Rabic: Move technical temperament data
 
(28 intermediate revisions by 6 users not shown)
Line 1: Line 1:
{{Technical data page}}
The '''rastmic rank-3 clan''' of temperaments tempers out the rastma, [[243/242]]. Both no-5 rastmic and no-7 rastmic can be the head of this clan. These temperaments divide the fifth in half and use it as an [[11/9]] neutral third.  
The '''rastmic rank-3 clan''' of temperaments tempers out the rastma, [[243/242]]. Both no-5 rastmic and no-7 rastmic can be the head of this clan. These temperaments divide the fifth in half and use it as an [[11/9]] neutral third.  


Temperaments not listed include:  
Temperaments discussed elsewhere include:  
* ''[[Urania]]'' (+81/80 or 121/120) → [[Didymus rank three family #Urania|Didymus rank-3 family]]
* [[Jove]] (+441/440 or 540/539) → [[Breed family #Jove|Breed family]]
* [[Jove]] (+441/440 or 540/539) → [[Breed family #Jove|Breed family]]
* ''[[Hagrid]]'' (+9801/9800) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Hagrid]]'' (+9801/9800) → [[Cataharry family #Hagrid|Cataharry family]]
* ''[[Rabic]]'' (+131769/131072) → [[Alphaxenic rank three clan #Betarabian|Alphaxenic rank-3 clan]]


Considered below are spectacle, cuckoo, mirwomo, mirage, mandos, and parahemif.  
Considered below are spectacle, mirwomo, mandos, cuckoo, parahemif, urania, rabic, and mirage.  


== Spectacle ==
== Spectacle ==
Spectacle, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17700.html Yahoo! Tuning Group | ''Spectacle temperament?'']</ref>, can be described as the 31 & 34 & 41 temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds.  
Spectacle, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17700.html Yahoo! Tuning Group | ''Spectacle temperament?'']</ref>, can be described as the {{nowrap| 31 & 34d & 41 }} temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds. It is therefore generated by octaves, major thirds, and neutral thirds. 3/2 is equated with a stack of two 11/9's as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.
 
It is [[associated temperament|associated]] with the [[marvo]] temperamment.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 17: Line 18:


{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 4 5 | 0 0 1 2 0 }}
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 4 5 | 0 0 1 2 0 }}
: mapping generators: ~2, ~11/9, ~5
: mapping generators: ~2, ~11/9, ~5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 350.0570, ~5/4 = 383.9323
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5486{{c}}, ~11/9 = 350.2171{{c}}, ~5/4 = 384.1078{{c}}
: [[error map]]: {{val| +0.549 -0.972 -1.109 +0.806 +0.864 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.1758{{c}}, ~5/4 = 384.0951{{c}}
: error map: {{val| 0.000 -1.603 -2.219 +0.068 -0.439 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[11-odd-limit]]
* [[11-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 }}, ~11/9 = {{monzo| -2/5 0 0 0 1/5 }}, ~5 = {{monzo| 2/5 -2 1 0 4/5 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1/5 0 0 0 2/5 }}, {{monzo| 2/5 -2 1 0 4/5 }}, {{monzo| -19/5 -4 2 0 12/5 }}, {{monzo| 0 0 0 0 1 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.11


{{Optimal ET sequence|legend=1| 31, 41, 72, 247c, 281, 353c, 425bc, 497bc }}
{{Optimal ET sequence|legend=1| 24d, 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc }}


[[Badness]]: 0.499 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.599


[[Projection pair]]s: 3 242/81 7 366025/52488 11 644204/59049 to 2.5.11/9
[[Projection pair]]s: 3 242/81, 7 366025/52488, 11 644204/59049 to 2.5.11/9


Scales: [[spectacle31]]
Scales: [[spectacle31]]
[[Associated temperament]]: [[marvo]]


=== 13-limit ===
=== 13-limit ===
Line 44: Line 45:
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 4 5 -2 | 0 0 1 2 0 4 }}
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 4 5 -2 | 0 0 1 2 0 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 349.9247, ~5/4 = 384.3505
Optimal tunings:
* WE: ~2 = 1200.6024{{c}}, ~11/9 = 350.1004{{c}}, ~5/4 = 384.5435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.0393{{c}}, ~5/4 = 384.5866{{c}}


{{Optimal ET sequence|legend=1| 31, 72, 103, 175f }} <nowiki>*</nowiki>
{{Optimal ET sequence|legend=0| 31, 65d, 72, 103, 175f, 312bf, 384bcf, 487bceff }} <nowiki>*</nowiki>


<nowiki>*</nowiki> optimal patent val: [[240edo|240]]
<nowiki>*</nowiki> optimal patent val: [[240edo|240]]


Badness: 1.009 × 10<sup>-3</sup>
Badness (Sintel): 0.944
 
== Mirwomo ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Mirwomo]].''
 
Mirwomo tempers out 385/384 and may be described as the {{nowrap| 24 & 31 & 41 }} temperament, equating the undecimal quartertone ~33/32 with the septimal quartertone ~36/35.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 243/242, 385/384
 
{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 -3 5 | 0 0 1 -1 0 }}
: mapping generators: ~2, ~11/9, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7360{{c}}, ~11/9 = 350.1700{{c}}, ~5/4 = 384.3403{{c}}
: [[error map]]: {{val| +0.736 -0.879 -0.501 -0.733 +1.004 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.0035{{c}}, ~5/4 = 384.0785{{c}}
: error map: {{val| 0.000 -1.948 -2.235 -2.915 -1.301 }}
 
{{Optimal ET sequence|legend=1| 17, 21e, 24, 31, 41, 72, 247c, 312bd, 384bcdd, 456bcdde, 528bcdde, 631bbccdde }}
 
[[Badness]] (Sintel): 0.770
 
== Mandos ==
Mandos tempers out 176/175 and may be described as the {{nowrap| 24 & 27e & 31 }} temperament.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 176/175, 243/242
 
{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 5 5 | 0 0 1 -2 0 }}
: mapping generators: ~2, ~11/9, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1949{{c}}, ~11/9 = 350.6135{{c}}, ~5/4 = 390.4090{{c}}
: [[error map]]: {{val| -0.805 -1.533 +2.485 +1.814 +0.139 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5548{{c}}, ~5/4 = 390.2690{{c}}
: error map: {{val| 0.000 -0.845 +3.955 +3.410 +1.456 }}
 
{{Optimal ET sequence|legend=1| 24, 27e, 31, 58, 89, 154d, 181cde, 212cde, 301ccde }}
 
[[Badness]] (Sintel): 0.902
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 176/175, 243/242
 
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 0 5 5 -1 | 0 0 1 -2 0 0 }}
 
Optimal tunings:
* WE: ~2 = 1198.5555{{c}}, ~11/9 = 351.0300{{c}}, ~5/4 = 391.0458{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.1554{{c}}, ~5/4 = 391.1227{{c}}
 
{{Optimal ET sequence|legend=0| 24, 27e, 31, 58, 123df, 181cdeff, 239ccddeefff }}
 
Badness (Sintel): 0.863


== Cuckoo ==
== Cuckoo ==
Cuckoo, named by [[Johannes Werpup]] in 2014<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_106371.html Yahoo! Tuning Group | ''Temperament ideas: A cuckoo, and two oracles'']</ref>, tempers out 126/125 and may be described as the {{nowrap| 24d & 27e & 31 }} temperament.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 58: Line 120:


{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 -4 5 | 0 0 1 3 0 }}
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 -4 5 | 0 0 1 3 0 }}
: mapping generators: ~2, ~11/9, ~5


: mapping generators: ~2, ~11/9, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.8222{{c}}, ~11/9 = 350.4356{{c}}, ~5/4 = 389.8478{{c}}
: [[error map]]: {{val| -0.178 -1.262 +3.178 -1.558 +0.504 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.4213{{c}}, ~5/4 = 389.7308{{c}}
: error map: {{val| 0.000 -1.112 +3.417 -1.318 +0.788 }}


{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}
{{Optimal ET sequence|legend=1| 24d, 27e, 31, 58, 89, 154, 185 }}


[[Badness]]: 0.933 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.12


=== 13-limit ===
=== 13-limit ===
Line 72: Line 139:
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 -4 5 -10 | 0 0 1 3 0 5 }}
Mapping: {{mapping| 1 1 0 -3 2 -5 | 0 2 0 -4 5 -10 | 0 0 1 3 0 5 }}


{{Optimal ET sequence|legend=1| 27e, 31, 58, 96d, 154 }}
Optimal tunings:
* WE: ~2 = 1199.7103{{c}}, ~11/9 = 350.5840{{c}}, ~5/4 = 389.8071{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.5682{{c}}, ~5/4 = 389.6104{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 31, 58, 96d, 154 }}
 
Badness (Sintel): 1.23
 
== Parahemif ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Parahemif]].''
 
Parahemif tempers out 896/891 and may be described as the {{nowrap| 24 & 34d & 41 }} temperament. It is related to [[hemif]], the no-5 rank-2 temperament that tempers out the same list of commas. As such, it finds the interval class of 7 at +13 generator steps, as a semi-augmented sixth (C–At). In the 13-limit, it tempers out 144/143, 352/351, 364/363 among others, and finds ~16/13 at the same neutral third as ~11/9.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 243/242, 896/891
 
{{Mapping|legend=1| 1 1 0 -1 2 | 0 2 0 13 5 | 0 0 1 0 0 }}
: mapping generators: ~2, ~11/9, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2633{{c}}, ~11/9 = 351.3189{{c}}, ~5/4 = 387.7835{{c}}
: [[error map]]: {{val| -0.737 -0.054 -0.004 -0.944 +3.803 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 351.4593{{c}}, ~5/4 = 387.4226{{c}}
: error map: {{val| 0.000 +0.964 +1.109 +0.145 +5.979 }}
 
{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99e }} *
 
<nowiki>*</nowiki> [[optimal patent val]]: [[123edo|123]]
 
[[Badness]] (Sintel): 1.62
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 243/242, 364/363
 
Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 0 13 5 -1 | 0 0 1 0 0 0 }}
 
Optimal tunings:
* WE: ~2 = 1198.7603{{c}}, ~11/9 = 351.3275{{c}}, ~5/4 = 388.7872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.6042{{c}}, ~5/4 = 388.4720{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff }}


== Mirwomo ==
Badness (Sintel): 1.12
The mirwomo temperament tunes ~36/35 to exactly half of an apotome ~2187/2048, and tunes ~128/105 to exactly half of a perfect fifth ~3/2.
 
== Urania ==
Urania tempers out 81/80, the syntonic comma. It is essentially [[mohaha]] with an independent generator for prime 7.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 81/80, 121/120
 
{{Mapping|legend=1| 1 1 0 0 2 | 0 2 8 0 5 | 0 0 0 1 0 }}
: mapping generators: ~2, ~11/9, ~7
 
[[Mapping to lattice]]: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]


=== 7-limit ===
Lattice basis:
[[Subgroup]]: 2.3.5.7
: 11/9 length = 0.2536, 8/7 length = 2.807
: Angle (11/9, 8/7) = 90 degrees


[[Comma list]]: 33075/32768
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.8548{{c}}, ~11/9 = 348.6318{{c}}, ~5/4 = 965.0936{{c}}
: [[error map]]: {{val| +1.855 -2.836 +2.741 -0.023 -4.449 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 348.3793{{c}}, ~7/4 = 965.6304{{c}}
: error map: {{val| 0.000 -5.196 +0.721 -3.196 -9.421 }}


{{Mapping|legend=1| 1 1 0 6 | 0 2 0 -3 | 0 0 1 -1 }}
{{Optimal ET sequence|legend=1| 7d, 14c, 17c, 24, 31, 100de, 131bdee, 162bdee }}


: mapping generators: ~2, ~128/105, ~5
[[Badness]] (Sintel): 1.01


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 350.1375, ~5/4 = 383.8665
[[Complexity spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5


{{Optimal ET sequence|legend=1| 17, 21, 24, 31, 41, 72, 281d }}
Scales: [[urania24]]


[[Badness]]: 0.770 × 10<sup>-3</sup>
== Rabic ==
If the rastma is added to the list of commas along with the [[Alpharabian comma]], you end up with rabic, which splits the octave into 24 equal parts. This temperament is named as such by [[Aura]] in 2022 because tempering out both the Alpharabian comma and the rastma automatically tempers out the [[Betarabian comma]].


=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 243/242, 385/384
[[Comma list]]: 243/242, 131769/131072
 
{{Mapping|legend=1| 24 38 0 0 83 | 0 0 1 0 0 | 0 0 0 1 0 }}
: mapping generators: ~33/32, ~5, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~33/32 = 50.0220{{c}}, ~5/4 = 385.0647{{c}}, ~7/4 = 967.3158{{c}}
: [[error map]]: {{val| +0.538 -1.104 -0.002 -0.002 +0.541 }}
* [[CWE]]: ~33/32 = 50.0000{{c}}, ~5/4 = 385.0647{{c}}, ~7/4 = 967.3158{{c}}
: error map: {{val| 0.000 -1.955 -0.937 -1.133 -1.318 }}
 
{{Optimal ET sequence|legend=1| 24, 48(d), 72, 264, 336b, 408b, 480bcde }}


{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 -3 5 | 0 0 1 -1 0 }}
[[Badness]] (Sintel): 7.30


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 349.9554, ~5/4 = 384.1048
=== Gunn ===
[[Subgroup]]: 2.3.5.7.11.13.17.19


{{Optimal ET sequence|legend=1| 17, 24, 31, 41, 72, 312bd, 384bcd, 456bcde, 528bcde, 631bcde }}
[[Comma list]]: 729/728, 273/272, 153/152, 364/363, 1729/1728


[[Badness]]: 0.641 × 10<sup>-3</sup>
{{Mapping|24 38 48 48 83 108 98 102|0 0 1 0 0 0 0 0|0 0 0 1 0 -1 0 0|legend=1}}


== Mirage ==
[[Optimal tuning|Optimal tunings]]:
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 243/242, 385/384
* [[WE]]: ~33/32 = 50.030{{c}}, ~5/4 = 384.859{{c}}, ~7/4 = 965.679{{c}}


Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 0 | 0 0 0 0 0 1 }}
: [[error map]]: {{val|+0.724 -0.809 -0.007 -1.700 +1.185 -2.949 -2.000 +5.563}}


: mapping generators: ~2, ~15/14, ~13
* [[CWE]]: ~33/32 = 50.000{{c}}, ~5/4 = 384.837{{c}}, ~7/4 = 965.136{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 116.6327, ~13/8 = 837.7040
: error map: {{val|-0.000 -1.955 -1.476 -3.690 -1.318 -5.663 -4.955 2.487}}


{{Optimal ET sequence|legend=1| 10, 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}
Supporting ETs: {{Edos|24p, 72, 48f, 24c, 24df, 48cf, 48d, 96d, 48cd, 24cdf}}


Badness: 0.738 × 10<sup>-3</sup>
[[Badness]] (Sintel): 3.285


== Mandos ==
== Mirage ==
[[Subgroup]]: 2.3.5.7.11
Mirage is [[miracle]] with an independent generator for prime 13.


[[Comma list]]: 176/175, 243/242
[[Subgroup]]: 2.3.5.7.11.13


{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 5 5 | 0 0 1 -2 0 }}
[[Comma list]]: 225/224, 243/242, 385/384


: mapping generators: ~2, ~11/9, ~5
{{Mapping|legend=1| 1 1 3 3 2 0 | 0 6 -7 -2 15 0 | 0 0 0 0 0 1 }}
: mapping generators: ~2, ~15/14, ~13


{{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}}, ~13/8 = 838.2364{{c}}
: [[error map]]: {{val| +0.763 -0.951 -0.974 +0.048 +0.810 -0.004 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}, ~13/8 = 838.2123{{c}}
: error map: {{val| 0.000 -2.074 -2.842 -2.120 -1.615 -2.315 }}


[[Badness]]: 0.751 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=1| 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef }}


=== 13-limit ===
Badness (Sintel): 0.691
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 176/175, 243/242
=== 17-limit ===
Mirage is very naturally a 17-limit temperament, relating 13 and 17 by tempering out [[273/272]], [[715/714]], [[833/832]], and [[936/935]]. Instead of 13/8, the second generator can also be the small comma tempered out by [[miraculous]].


Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 0 5 5 -1 | 0 0 1 -2 0 0 }}
Subgroup: 2.3.5.7.11.13.17


{{Optimal ET sequence|legend=1| 7, 24, 31, 58 }}
Comma list: 225/224, 243/242, 273/272, 385/384


Badness: 0.923 × 10<sup>-3</sup>
Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 0 4 | 0 0 0 0 0 1 1 }}


== Parahemif ==
Optimal tunings:
=== 7-limit ===
* WE: ~2 = 1200.7628{{c}}, ~15/14 = 116.6995{{c}}, ~13/8 = 837.1672{{c}}
[[Subgroup]]: 2.3.5.7
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6395{{c}}, ~13/8 = 837.1424{{c}}


[[Comma list]]: 1605632/1594323
{{Optimal ET sequence|legend=0| 31, 41, 62, 72, 103, 175f, 360bcdff, 463bccdeff }}


{{Mapping|legend=1| 1 1 0 -1 | 0 2 0 13 | 0 0 1 0 }}
Badness (Sintel): 0.715


: mapping generators: ~2, ~896/729, ~5
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~896/729 = 351.4846, ~5/4 = 386.9396
Comma list: 210/209, 225/224, 243/242, 273/272, 343/342


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


[[Badness]]: 0.607 × 10<sup>-3</sup>
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


=== 11-limit ===
Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 385/384
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 243/242, 896/891
{{Todo|complete temperament data|inline=1}}


{{Mapping|legend=1| 1 1 0 -1 2 | 0 2 0 13 5 | 0 0 1 0 0 }}
=== Prism ===
{{Redirect|Prism|the scale|Prism (scale)}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 351.5347, ~5/4 = 388.0217
Subgroup: 2.3.5.7.11.13.17.19


{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99e }} *
Comma list: 225/224, 243/242, 273/272, 324/323, 385/384


<nowiki>*</nowiki> [[optimal patent val]]: [[123edo|123]]
{{Todo|complete temperament data|inline=1}}


[[Badness]]: 1.34547 × 10<sup>-3</sup>
=== Prism 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


=== 13-limit ===
Comma list: 225/224, 243/242, 273/272, 300/299, 324/323, 385/384
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 243/242, 364/363
{{Todo|complete temperament data|inline=1}}


Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 0 13 5 -1 | 0 0 1 0 0 0 }}
=== No-19 23-limit ===
Subgroup: 2.3.5.7.11.13.17.23


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.6908, ~5/4 = 389.1892
Comma list: 225/224, 243/242, 273/272, 300/299, 385/384


{{Optimal ET sequence|legend=1| 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff }}
{{Todo|complete temperament data|inline=1}}


Badness: 1.19366 × 10<sup>-3</sup>
== References ==


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Rastmic rank-3 clan| ]] <!-- main article -->
[[Category:Rastmic rank-3 clan| ]] <!-- main article -->
[[Category:Rastmic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 16:05, 10 August 2026

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

The rastmic rank-3 clan of temperaments tempers out the rastma, 243/242. Both no-5 rastmic and no-7 rastmic can be the head of this clan. These temperaments divide the fifth in half and use it as an 11/9 neutral third.

Temperaments discussed elsewhere include:

Considered below are spectacle, mirwomo, mandos, cuckoo, parahemif, urania, rabic, and mirage.

Spectacle

Spectacle, named by Gene Ward Smith in 2010[1], can be described as the 31 & 34d & 41 temperament. It tempers out 225/224, making it a sort of marvel infested with neutral thirds. It is therefore generated by octaves, major thirds, and neutral thirds. 3/2 is equated with a stack of two 11/9's as a corollary of 243/242 being tempered out, 7/4 is equated with a stack of four 11/9's and two 5/4's, 11/8 is equated with a stack of five 11/9's, 13/8 is equated with a stack of two 18/11's and four 5/4's, and 17/16 is equated with three 18/11's and three 5/4's. Every harmonic is reached with help of other intervals at most with three 5/4's.

It is associated with the marvo temperamment.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242

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

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • WE: ~2 = 1200.5486 ¢, ~11/9 = 350.2171 ¢, ~5/4 = 384.1078 ¢
error map: ⟨+0.549 -0.972 -1.109 +0.806 +0.864]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.1758 ¢, ~5/4 = 384.0951 ¢
error map: ⟨0.000 -1.603 -2.219 +0.068 -0.439]

Minimax tuning:

  • 11-odd-limit: ~2 = [1 0 0 0 0⟩, ~11/9 = [-2/5 0 0 0 1/5⟩, ~5 = [2/5 -2 1 0 4/5⟩
unchanged-interval (eigenmonzo) basis: 2.9/5.11

Optimal ET sequence: 24d, 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc

Badness (Sintel): 0.599

Projection pairs: 3 242/81, 7 366025/52488, 11 644204/59049 to 2.5.11/9

Scales: spectacle31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350

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

Optimal tunings:

  • WE: ~2 = 1200.6024 ¢, ~11/9 = 350.1004 ¢, ~5/4 = 384.5435 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.0393 ¢, ~5/4 = 384.5866 ¢

Optimal ET sequence: 31, 65d, 72, 103, 175f, 312bf, 384bcf, 487bceff *

* optimal patent val: 240

Badness (Sintel): 0.944

Mirwomo

For the 7-limit version, see Miscellaneous 7-limit temperaments #Mirwomo.

Mirwomo tempers out 385/384 and may be described as the 24 & 31 & 41 temperament, equating the undecimal quartertone ~33/32 with the septimal quartertone ~36/35.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 -3 5], ⟨0 0 1 -1 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • WE: ~2 = 1200.7360 ¢, ~11/9 = 350.1700 ¢, ~5/4 = 384.3403 ¢
error map: ⟨+0.736 -0.879 -0.501 -0.733 +1.004]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.0035 ¢, ~5/4 = 384.0785 ¢
error map: ⟨0.000 -1.948 -2.235 -2.915 -1.301]

Optimal ET sequence: 17, 21e, 24, 31, 41, 72, 247c, 312bd, 384bcdd, 456bcdde, 528bcdde, 631bbccdde

Badness (Sintel): 0.770

Mandos

Mandos tempers out 176/175 and may be described as the 24 & 27e & 31 temperament.

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 5 5], ⟨0 0 1 -2 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • WE: ~2 = 1199.1949 ¢, ~11/9 = 350.6135 ¢, ~5/4 = 390.4090 ¢
error map: ⟨-0.805 -1.533 +2.485 +1.814 +0.139]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5548 ¢, ~5/4 = 390.2690 ¢
error map: ⟨0.000 -0.845 +3.955 +3.410 +1.456]

Optimal ET sequence: 24, 27e, 31, 58, 89, 154d, 181cde, 212cde, 301ccde

Badness (Sintel): 0.902

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242

Mapping: [⟨1 1 0 6 2 4], ⟨0 2 0 5 5 -1], ⟨0 0 1 -2 0 0]]

Optimal tunings:

  • WE: ~2 = 1198.5555 ¢, ~11/9 = 351.0300 ¢, ~5/4 = 391.0458 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.1554 ¢, ~5/4 = 391.1227 ¢

Optimal ET sequence: 24, 27e, 31, 58, 123df, 181cdeff, 239ccddeefff

Badness (Sintel): 0.863

Cuckoo

Cuckoo, named by Johannes Werpup in 2014[2], tempers out 126/125 and may be described as the 24d & 27e & 31 temperament.

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242

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

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • WE: ~2 = 1199.8222 ¢, ~11/9 = 350.4356 ¢, ~5/4 = 389.8478 ¢
error map: ⟨-0.178 -1.262 +3.178 -1.558 +0.504]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.4213 ¢, ~5/4 = 389.7308 ¢
error map: ⟨0.000 -1.112 +3.417 -1.318 +0.788]

Optimal ET sequence: 24d, 27e, 31, 58, 89, 154, 185

Badness (Sintel): 1.12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242

Mapping: [⟨1 1 0 -3 2 -5], ⟨0 2 0 -4 5 -10], ⟨0 0 1 3 0 5]]

Optimal tunings:

  • WE: ~2 = 1199.7103 ¢, ~11/9 = 350.5840 ¢, ~5/4 = 389.8071 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.5682 ¢, ~5/4 = 389.6104 ¢

Optimal ET sequence: 27e, 31, 58, 96d, 154

Badness (Sintel): 1.23

Parahemif

For the 7-limit version, see Miscellaneous 7-limit temperaments #Parahemif.

Parahemif tempers out 896/891 and may be described as the 24 & 34d & 41 temperament. It is related to hemif, the no-5 rank-2 temperament that tempers out the same list of commas. As such, it finds the interval class of 7 at +13 generator steps, as a semi-augmented sixth (C–At). In the 13-limit, it tempers out 144/143, 352/351, 364/363 among others, and finds ~16/13 at the same neutral third as ~11/9.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 896/891

Mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5], ⟨0 0 1 0 0]]

mapping generators: ~2, ~11/9, ~5

Optimal tunings:

  • WE: ~2 = 1199.2633 ¢, ~11/9 = 351.3189 ¢, ~5/4 = 387.7835 ¢
error map: ⟨-0.737 -0.054 -0.004 -0.944 +3.803]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.4593 ¢, ~5/4 = 387.4226 ¢
error map: ⟨0.000 +0.964 +1.109 +0.145 +5.979]

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99e *

* optimal patent val: 123

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 243/242, 364/363

Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1], ⟨0 0 1 0 0 0]]

Optimal tunings:

  • WE: ~2 = 1198.7603 ¢, ~11/9 = 351.3275 ¢, ~5/4 = 388.7872 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.6042 ¢, ~5/4 = 388.4720 ¢

Optimal ET sequence: 17c, 24, 34d, 41, 58, 99ef, 157eff, 290cdeeefff

Badness (Sintel): 1.12

Urania

Urania tempers out 81/80, the syntonic comma. It is essentially mohaha with an independent generator for prime 7.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120

Mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5], ⟨0 0 0 1 0]]

mapping generators: ~2, ~11/9, ~7

Mapping to lattice: [⟨0 2 8 0 5], ⟨0 0 0 -1 0]]

Lattice basis:

11/9 length = 0.2536, 8/7 length = 2.807
Angle (11/9, 8/7) = 90 degrees

Optimal tunings:

  • WE: ~2 = 1201.8548 ¢, ~11/9 = 348.6318 ¢, ~5/4 = 965.0936 ¢
error map: ⟨+1.855 -2.836 +2.741 -0.023 -4.449]
  • CWE: ~2 = 1200.0000 ¢, ~11/9 = 348.3793 ¢, ~7/4 = 965.6304 ¢
error map: ⟨0.000 -5.196 +0.721 -3.196 -9.421]

Optimal ET sequence: 7d, 14c, 17c, 24, 31, 100de, 131bdee, 162bdee

Badness (Sintel): 1.01

Complexity spectrum: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5

Scales: urania24

Rabic

If the rastma is added to the list of commas along with the Alpharabian comma, you end up with rabic, which splits the octave into 24 equal parts. This temperament is named as such by Aura in 2022 because tempering out both the Alpharabian comma and the rastma automatically tempers out the Betarabian comma.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 131769/131072

Mapping: [⟨24 38 0 0 83], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

mapping generators: ~33/32, ~5, ~7

Optimal tunings:

  • WE: ~33/32 = 50.0220 ¢, ~5/4 = 385.0647 ¢, ~7/4 = 967.3158 ¢
error map: ⟨+0.538 -1.104 -0.002 -0.002 +0.541]
  • CWE: ~33/32 = 50.0000 ¢, ~5/4 = 385.0647 ¢, ~7/4 = 967.3158 ¢
error map: ⟨0.000 -1.955 -0.937 -1.133 -1.318]

Optimal ET sequence: 24, 48(d), 72, 264, 336b, 408b, 480bcde

Badness (Sintel): 7.30

Gunn

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 729/728, 273/272, 153/152, 364/363, 1729/1728

Mapping: [⟨24 38 48 48 83 108 98 102], ⟨0 0 1 0 0 0 0 0], ⟨0 0 0 1 0 -1 0 0]]

Optimal tunings:

  • WE: ~33/32 = 50.030 ¢, ~5/4 = 384.859 ¢, ~7/4 = 965.679 ¢
error map: ⟨+0.724 -0.809 -0.007 -1.700 +1.185 -2.949 -2.000 +5.563]
  • CWE: ~33/32 = 50.000 ¢, ~5/4 = 384.837 ¢, ~7/4 = 965.136 ¢
error map: ⟨-0.000 -1.955 -1.476 -3.690 -1.318 -5.663 -4.955 2.487]

Supporting ETs: 24p, 72, 48f, 24c, 24df, 48cf, 48d, 96d, 48cd, 24cdf

Badness (Sintel): 3.285

Mirage

Mirage is miracle with an independent generator for prime 13.

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384

Mapping: [⟨1 1 3 3 2 0], ⟨0 6 -7 -2 15 0], ⟨0 0 0 0 0 1]]

mapping generators: ~2, ~15/14, ~13

Optimal tunings:

  • WE: ~2 = 1200.7626 ¢, ~15/14 = 116.7069 ¢, ~13/8 = 838.2364 ¢
error map: ⟨+0.763 -0.951 -0.974 +0.048 +0.810 -0.004]
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.6469 ¢, ~13/8 = 838.2123 ¢
error map: ⟨0.000 -2.074 -2.842 -2.120 -1.615 -2.315]

Optimal ET sequence: 31, 41, 62, 72, 103, 175f, 216c, 288cdf, 391bcdef

Badness (Sintel): 0.691

17-limit

Mirage is very naturally a 17-limit temperament, relating 13 and 17 by tempering out 273/272, 715/714, 833/832, and 936/935. Instead of 13/8, the second generator can also be the small comma tempered out by miraculous.

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 385/384

Mapping: [⟨1 1 3 3 2 0 0], ⟨0 6 -7 -2 15 0 4], ⟨0 0 0 0 0 1 1]]

Optimal tunings:

  • WE: ~2 = 1200.7628 ¢, ~15/14 = 116.6995 ¢, ~13/8 = 837.1672 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 116.6395 ¢, ~13/8 = 837.1424 ¢

Optimal ET sequence: 31, 41, 62, 72, 103, 175f, 360bcdff, 463bccdeff

Badness (Sintel): 0.715

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 210/209, 225/224, 243/242, 273/272, 343/342

Todo: complete temperament data

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 385/384

Todo: complete temperament data

Prism

"Prism" redirects here. For the scale, see Prism (scale).

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 243/242, 273/272, 324/323, 385/384

Todo: complete temperament data

Prism 23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 225/224, 243/242, 273/272, 300/299, 324/323, 385/384

Todo: complete temperament data

No-19 23-limit

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 225/224, 243/242, 273/272, 300/299, 385/384

Todo: complete temperament data

References

  1. ↑ Yahoo! Tuning Group | Spectacle temperament?
  2. ↑ Yahoo! Tuning Group | Temperament ideas: A cuckoo, and two oracles
Retrieved from "https://en.xen.wiki/index.php?title=Rastmic_rank-3_clan&oldid=235541"