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[[#Parahemif|Parahemif]]
{{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.


Tempering out the rastma, 243/242, leads to temperaments which divide the fifth in half and use it as an [[11/9|11/9]] neutral third. Temperaments not listed include jove, jovis, jofur, jovial, spectacle, mirwomo, mandos, mirage and hagrid.
Temperaments discussed elsewhere include:
* [[Jove]] (+441/440 or 540/539) → [[Breed family #Jove|Breed family]]
* ''[[Hagrid]]'' (+9801/9800) → [[Cataharry family #Hagrid|Cataharry family]]


=Parahemif=
Considered below are spectacle, mirwomo, mandos, cuckoo, parahemif, urania, rabic, and mirage.
Commas: 243/242, 896/891


Map: [<1 1 0 -1 2|, <0 2 0 13 5|, <0 0 1 0 0|]
== 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 {{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.


EDOs: 17, 24, 41, 58, 99f, 123
It is [[associated temperament|associated]] with the [[marvo]] temperamment.


10^5 * Badness: 134.547
[[Subgroup]]: 2.3.5.7.11


==13-limit==
[[Comma list]]: 225/224, 243/242
Commas: 144/143, 243/242, 364/363


Map: [&lt;1 1 0 -1 2 4|, &lt;0 2 0 13 5 -1|, &lt;0 0 1 0 0 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


EDOs: 17, 24, 34d, 41, 58, 99ef, 157ef
[[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 }}


10^5 * Badness: 119.366
[[Minimax tuning]]:
* [[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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11
 
{{Optimal ET sequence|legend=1| 24d, 31, 41, 65d, 72, 247c, 281, 353c, 425bc, 497bc }}
 
[[Badness]] (Sintel): 0.599
 
[[Projection pair]]s: 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: {{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{{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=0| 31, 65d, 72, 103, 175f, 312bf, 384bcf, 487bceff }} <nowiki>*</nowiki>
 
<nowiki>*</nowiki> optimal patent val: [[240edo|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 {{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, 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
 
[[Comma list]]: 126/125, 243/242
 
{{Mapping|legend=1| 1 1 0 -3 2 | 0 2 0 -4 5 | 0 0 1 3 0 }}
: 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 }}
 
[[Badness]] (Sintel): 1.12
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 243/242
 
Mapping: {{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{{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 }}
 
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|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 }}]
 
Lattice basis:
: 11/9 length = 0.2536, 8/7 length = 2.807
: Angle (11/9, 8/7) = 90 degrees
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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|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 }}
 
[[Badness]] (Sintel): 7.30
 
== 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|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 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 }}
 
{{Optimal ET sequence|legend=1| 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: {{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{{c}}, ~15/14 = 116.6995{{c}}, ~13/8 = 837.1672{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6395{{c}}, ~13/8 = 837.1424{{c}}
 
{{Optimal ET sequence|legend=0| 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|inline=1}}
 
=== 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|inline=1}}
 
=== Prism ===
{{Redirect|Prism|the scale|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|inline=1}}
 
=== 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|inline=1}}
 
=== 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|inline=1}}
 
== References ==
 
[[Category:Temperament clans]]
[[Category:Rastmic rank-3 clan| ]] <!-- main article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Rastmic_rank-3_clan"