Gamelismic clan: Difference between revisions

To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.  

Miracle adds [[33075/32768]] and uses the [[secor]], half an ~8/7, as generator. Lemba adds [[525/512]] to the list, and has a half-octave [[period]]. Valentine adds [[6144/6125]] with a generator of ~21/20 and superkleismic adds [[875/864]] with a generator of ~6/5. Unidec adds [[4375/4374]], and has a generator of ~10/9 with a half-octave period. Hemithirds adds [[65625/65536]] with a generator half of a classical major third. Finally, tritikleismic adds [[15625/15552]] and has a generator of 6/5 with a 1/3-octave period.

* ''[[Echidnic]]'' (+686/675} → [[Diaschismic family #Echidnic|Diaschismic family]]

* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]

No-five subgroup extensions of slendric include radon, a 2.3.7.11-subgroup extension that may be viewed as no-five rodan, considered below, euslendric, a 2.3.7.13-subgroup extension, and baladic, a weak 2.3.7.13.17-subgroup extension, considered in [[#Other subgroup extensions]]. Dicussed elsewhere is [[No-fives subgroup temperaments #Gigapyth|gigapyth]] in the 2.3.7.85 subgroup.  

Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap| 26 & 31 }}. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].  

Rodan tempers out 245/243 and can be described as the {{nowrap| 41 & 46 }} temperament. This temperament is more accurate than mothra and extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric. [[87edo]] is excellent for this, with the 17\87 generator missing the 13-limit CWE tuning by less than a microcent.  

Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the {{nowrap| 31 & 41 }} temperament. Its ploidacot is hexacot. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.  

==== Hemisecordite ====

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 273/272, 441/440, 2200/2197

Valentine tempers out [[126/125]] and [[6144/6125]] as well as 1029/1024. It has a generator of [[~]][[21/20]], three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[7-limit symmetrical lattices|lattice of 7-limit tetrads]]. Valentine can be described as the {{nowrap| 31 & 46 }} temperament; its ploidacot is enneacot. [[77edo]], [[108edo]], or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for [[starling]], the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)^1/9 as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)^1/10.

Superkleismic tempers out the keema, [[875/864]], and can be described as the {{nowrap| 15 & 26 }} temperament. It splits the ~7/4 into three ~6/5 generators of around 322 cents. This is noticeably sharper than the [[kleismic]] generator, hence the name.  

Unidec tempers out the ragisma, [[4375/4374]], and may be described as the {{nowrap| 26 & 46 }} temperament. It has a [[semi-octave]] [[period]] and a generator of ~80/63, two of which minus a period make slendric's generator; its [[ploidacot]] is therefore diploid gamma-hexacot. In the 11-limit, the generator represents [[14/11]]. [[190edo]] makes for an excellent tuning in both the 7-limit and 11-limit.  

== Necromanteion ==

Necromanteion, 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> may be described as the {{nowrap| 31 & 51c }} temperament. The generator is a subfifth representing 35/24, four of which minus two octaves make slendric's generator, so its [[ploidacot]] is beta-dodecacot.  

[[Comma list]]: 1029/1024, 5103/5000

{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }}

: mapping generators: ~2, ~35/24

* [[WE]]: ~2 = 1200.2959{{c}}, ~35/24 = 658.3833{{c}}

: [[error map]]: {{val| +0.296 -2.835 +4.130 -0.879 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 658.2313{{c}}

: error map: {{val| 0.000 -3.179 +3.619 -1.751 }}

{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }}

[[Badness]] (Sintel): 2.98

Comma list: 176/175, 243/242, 1029/1024

Mapping: {{mapping| 1 -5 -7 5 -13 | 0 12 17 -4 30 }}

* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}

{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}

Badness (Sintel): 1.77

Comma list: 144/143, 176/175, 243/242, 343/338

Mapping: {{mapping| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }}

* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}

{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }}

Badness (Sintel): 1.94

== Restles ==

{{See also| Lesser tendoneutralic }}

[[Comma list]]: 1029/1024, 153664/151875

{{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }}

: mapping generators: ~2. ~315/256

* [[WE]]: ~2 = 1200.0322{{c}}, ~315/256 = 358.5581{{c}}

: [[error map]]: {{val| +0.032 +0.678 +1.340 -2.930 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~315/256 = 358.5484{{c}}

: error map: {{val| 0.000 +0.626 +1.267 -3.019 }}

{{Optimal ET sequence|legend=1| 77, 87, 164 }}

[[Badness]] (Sintel): 2.73

Comma list: 385/384, 441/440, 153664/151875

Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}

* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}

{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}

Badness (Sintel): 1.81

Comma list: 196/195, 352/351, 385/384, 676/675

Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}

* WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}}

{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}

Badness (Sintel): 1.16

=== Euslendric ===

=== Baladic ===