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Another pass on neutrality, explaining what actually happens with CTE/CWE and giving an alternative that bypasses their mechanics
 
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{{Technical data page}}
{{Technical data page}}
The '''rastmic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[rastma]], {{nowrap|243/242 {{=}} {{monzo| -1 5 0 0 -2 }}}}.
The '''rastmic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[rastma]], 243/242 ({{monzo| -1 5 0 0 -2 }}).


== Neutral ==
== Neutral ==
Neutral is the 2.3.11-[[subgroup]] temperament with a [[generator]] of a neutral third which can be taken to represent {{nowrap|[[11/9]] [[~]] [[27/22]]}}, two of which make up a perfect fifth of [[3/2]]. It can be thought of as the 2.3.11 version of either [[mohajira]] or [[neutrominant]], as well as [[suhajira]] and [[archytas clan #Ringo|ringo]]. Among other things, it is the temperament optimizing the [[neutral tetrad]].
Neutral is the 2.3.11-[[subgroup]] temperament with a [[generator]] of a neutral third which can be taken to represent [[11/9]][[~]][[27/22]], two of which make up a perfect fifth of [[3/2]]. It can be thought of as the 2.3.11 version of either [[mohajira]] or [[neutrominant]], as well as [[suhajira]] and [[ringo]]. Among other things, it is the temperament optimizing the [[neutral tetrad]].


[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11
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{{Mapping|legend=2| 1 1 2 | 0 2 5 }}
{{Mapping|legend=2| 1 1 2 | 0 2 5 }}
: mapping generators: ~2, ~11/9


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


: [[gencom]]: [2 11/9; 243/242]
: mapping generators: ~2, ~11/9


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~11/9 = 350.572
* [[WE]]: ~2 = 1200.0635{{c}}, ~11/9 = 350.5439{{c}}
* [[POTE]]: ~2 = 1\1, ~11/9 = 350.525
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5486{{c}}


{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
{{Optimal ET sequence|legend=1| 7, 17, 24, 65, 89, 202, 291, 380 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
[[Badness]] (Sintel): 0.0700


Scales: [[neutral7]], [[neutral10]], [[neutral17]]
Scales: [[neutral7]], [[neutral10]], [[neutral17]]


=== Namo ===
=== Namo ===
Namo adds [[144/143]] to the comma list and finds ~[[16/13]] at the same neutral third. With {{nowrap|11/9 ~ 16/13}}, it requires a slightly flat ~[[27/22]] as the tuning of the neutral third. [[58edo]] is the largest [[patent val]] tuning for it in the [[optimal ET sequence]], with a tuning between that of [[17edo]] and [[41edo]], so that ~11 and ~13 are practically equally sharp, given that [[29edo]] forms a [[consistent circle]] of [[13/11]]'s with a [[closing error]] of 31.2%. It might be recommended as a tuning for this reason, as having the neutral third much sharper to optimize plausibility of ~16/13 implies that the 11 is extremely sharp because 11/9 must be tuned sharp so that 11 must be sharper than 9, which is thus four times as sharp as however sharp of the (3/2)<sup>1/2</sup> neutral third is, while tuning it much flatter means that you increase the error of 16/13, which in 58edo is already as almost 8{{cent}} off and in [[99edo|99ef]] it is only slightly worse. For these reasons, Godtone is not fond of the recommendations by the various [[optimal tuning]]s to tune flat of 58edo, although it is clear that in an optimal tuning nothing much sharper than 58edo should be used, as making 11 more off than 13 would imply damaging 3 and 11/9 more than necessary. Curiously, [[POTE]] recommends a sharper tuning than both [[CTE]] and [[CWE]] here, but still flat of 58edo.
Namo adds [[144/143]] to the comma list and finds ~[[16/13]] at the same neutral third. With 11/9~16/13, it requires a slightly flat ~[[27/22]] as the tuning of the neutral third. [[58edo]] is the largest [[patent val]] tuning for it in the [[optimal ET sequence]], with a tuning between that of [[17edo]] and [[41edo]], so that ~11 and ~13 are practically equally sharp, given that [[29edo]] forms a [[consistent circle]] of [[13/11]]'s with a [[closing error]] of 31.2%. It might be recommended as a tuning for this reason, as having the neutral third tuned much sharper to optimize the plausibility of prime 13 implies that the 11 is extremely sharp because 11/9 must be tuned sharp so that 11 must be sharper than 9, which is four times as sharp as however sharp of just the (3/2)<sup>1/2</sup> neutral third is, while tuning it much flatter means increasing the error of prime 13, which in 58edo is already almost 8{{cent}} off and in [[99edo|99ef-edo]] it is only slightly worse.  
 
The [[CWE]] tuning given below takes account of how optimizational resource is best used. As such, the optimum is not raised much in the introduction of prime 13, as each unit increment in the generator implies four and five times more error in the harmonics 9 and 11, respectively, damaging them more than necessary. A sharper pure-octave tuning would be given by CTOP (~351.7142{{c}}), which is still flat of 58edo.


[[Subgroup]]: 2.3.11.13
[[Subgroup]]: 2.3.11.13
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{{Mapping|legend=3| 1 1 0 0 2 4 | 0 2 0 0 5 -1 }}
{{Mapping|legend=3| 1 1 0 0 2 4 | 0 2 0 0 5 -1 }}


: gencom: [2 11/9; 144/143 243/242]
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1198.8553{{c}}, ~11/9 = 351.1523{{c}}
[[Optimal tuning]]s:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 351.2700{{c}}
* [[CTE]]: 2 = 1\1, ~11/9 = 350.746
* [[CWE]]: 2 = 1\1, ~11/9 = 351.270
* [[POTE]]: 2 = 1\1, ~11/9 = 351.488


{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 58, 99ef, 239eefff, 338eeeffff }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 41, 58, 99ef, 239eefff, 338eeeffff }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
[[Badness]] (Sintel): 0.133


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


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


: [[gencom]]: [2 11/9; 64/63 243/242]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1196.3496{{c}}, ~11/9 = 352.8809{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 353.7832{{c}}


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de, 78ddee, 139bdddeeee }}
* [[CTE]]: ~2 = 1\1, ~11/9 = 353.139
* [[POTE]]: ~2 = 1\1, ~11/9 = 353.958


{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de }}
[[Badness]] (Sintel): 0.602


Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
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Comma list: 64/63, 78/77, 144/143
Comma list: 64/63, 78/77, 144/143


Sval mapping: {{mapping| 1 1 4 2 4 | 0 2 -4 5 -1 }}
Subgroup-val mapping: {{mapping| 1 1 4 2 4 | 0 2 -4 5 -1 }}


Gencom mapping: {{mapping| 1 1 0 4 2 4 | 0 2 0 -4 5 -1 }}
Gencom mapping: {{mapping| 1 1 0 4 2 4 | 0 2 0 -4 5 -1 }}


: gencom: [2 11/9; 64/63 78/77 144/143]
Optimal tunings:  
* WE: ~2 = 1196.8114{{c}}, ~11/9 = 352.8351{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 353.6791{{c}}


Optimal tunings:
{{Optimal ET sequence|legend=0| 7, 10, 17, 44e, 61de, 78ddee }}
* CTE: ~2 = 1\1, ~11/9 = 353.219
* POTE: ~2 = 1\1, ~11/9 = 353.775


Optimal ET sequence: {{Optimal ET sequence| 7, 10, 17, 44e, 61de }}
Badness (Sintel): 0.425


Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]
Scales: [[suhajira7]], [[suhajira10]], [[suhajira17]]


== Mohaha ==
== Mohaha ==
Mohaha can be thought of, intuitively, as "meantone with quartertones"; as is the 3/2 generator subdivided in half, so is the [[~]][[25/24]] chromatic semitone divided into two equal ~[[33/32]] quarter tones (in the 2.3.5.11 subgroup). Within this paradigm, mohaha is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10~12/11's, and that maps four 3/2's to 5/1. It has a heptatonic mos with three larger steps and four smaller ones, going [[3L 4s|sLsLsLs]]. Taking septimal meantone mapping of 7 leads to [[#Migration]], flattone mapping of 7 leads to [[#Ptolemy]], and dominant mapping of 7 leads to [[#Neutrominant]], while tempering out [[176/175]] gives [[mohajira]] (shown at [[Meantone family#Mohajira|Meantone family]]).
Mohaha can be thought of, intuitively, as meantone with quartertones; as is the 3/2 generator subdivided in half, so is the [[~]][[25/24]] chromatic semitone divided into two equal ~[[33/32]] quartertones (in the 2.3.5.11 subgroup). Within this paradigm, mohaha is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10~12/11's, and that maps four 3/2's to 5/1. It has a heptatonic mos with three larger steps and four smaller ones, going [[3L 4s|sLsLsLs]]. Taking septimal meantone mapping of 7 leads to [[#Migration]], flattone mapping of 7 leads to [[#Ptolemy]], and dominant mapping of 7 leads to [[#Neutrominant]], while tempering out [[176/175]] gives [[mohajira]] (shown at [[Meantone family #Mohajira|Meantone family]]).


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
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{{Mapping|legend=2| 1 1 0 2 | 0 2 8 5 }}
{{Mapping|legend=2| 1 1 0 2 | 0 2 8 5 }}
: sval mapping generators: ~2, ~11/9


{{Mapping|legend=3| 1 1 0 0 2 | 0 2 8 0 5 }}
{{Mapping|legend=3| 1 1 0 0 2 | 0 2 8 0 5 }}
: [[gencom]]: [2 11/9; 81/80 121/120]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~11/9 = 348.8296
* [[WE]]: ~2 = 1201.8548{{c}}, ~11/9 = 348.6318{{c}}
* [[POTE]]: ~2 = 1\1, ~11/9 = 348.0938
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 348.3793{{c}}


{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 69e, 100e, 131bee }}
{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 69e, 100e, 131bee }}
[[Badness]] (Sintel): 0.401


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
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Comma list: 66/65, 81/80, 121/120
Comma list: 66/65, 81/80, 121/120


Sval mapping: {{mapping| 1 1 0 2 4| 0 2 8 5 -1 }}
Subgroup-val mapping: {{mapping| 1 1 0 2 4| 0 2 8 5 -1 }}
 
: sval mapping generators: ~2, ~11/9


Gencom mapping: {{mapping| 1 1 0 0 2 4 | 0 2 8 0 5 -1 }}
Gencom mapping: {{mapping| 1 1 0 0 2 4 | 0 2 8 0 5 -1 }}


: gencom: [2 11/9; 66/65 81/80 121/120]
Optimal tunings:  
* [[WE]]: ~2 = 1199.8846{{c}}, ~11/9 = 348.8820{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 348.9040{{c}}


Optimal tunings:
{{Optimal ET sequence|legend=0| 7, 17c, 24, 31, 55 }}
* CTE: ~2 = 1\1, ~11/9 = 348.8794
* POTE: ~2 = 1\1, ~11/9 = 348.9155


{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 55 }}
[[Badness]] (Sintel): 0.401


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
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Mapping: {{mapping| 1 1 0 -3 2 | 0 2 8 20 5 }}
Mapping: {{mapping| 1 1 0 -3 2 | 0 2 8 20 5 }}
: mapping generators: ~2, ~11/9


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 348.5324
* WE: ~2 = 1201.7423{{c}}, ~11/9 = 348.6879{{c}}
* POTE: ~2 = 1\1, ~11/9 = 348.182
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.2867{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7d, 24d, 31, 100de, 131bdee, 162bdee }}
{{Optimal ET sequence|legend=0| 7d, 24d, 31, 100de, 131bdee, 162bdee }}


Badness: 0.025516
Badness (Sintel): 0.844


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 348.5444
* WE: ~2 = 1200.2590{{c}}, ~11/9 = 348.5648{{c}}
* POTE: ~2 = 1\1, ~11/9 = 348.490
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5009{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7d, 24d, 31 }}
{{Optimal ET sequence|legend=0| 7d, 24d, 31 }}


Badness: 0.028071
Badness (Sintel): 1.16


=== Ptolemy ===
=== Ptolemy ===
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Mapping: {{mapping| 1 1 0 8 2 | 0 2 8 -18 5 }}
Mapping: {{mapping| 1 1 0 8 2 | 0 2 8 -18 5 }}


: mapping generators: ~2, ~11/9
Optimal tunings:  
* WE: ~2 = 1203.8812{{c}}, ~11/9 = 348.0444{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 346.9194{{c}}


Optimal tunings:
{{Optimal ET sequence|legend=0| 7, 31dd, 38d, 45e, 83bcddee }}
* CTE: ~2 = 1\1, ~11/9 = 346.905
* POTE: ~2 = 1\1, ~11/9 = 346.922


Optimal ET sequence: {{Optimal ET sequence| 7, 31dd, 38d, 45e, 83bcddee }}
Badness (Sintel): 1.94


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 346.754
* WE: ~2 = 1203.9456{{c}}, ~11/9 = 348.0510{{c}}
* POTE: ~2 = 1\1, ~11/9 = 346.910
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 346.8878{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
{{Optimal ET sequence|legend=0| 7, 31ddf, 38df, 45ef, 83bcddeeff, 128bccddeeefff }}


Badness: 0.034316
Badness (Sintel): 1.42


=== Neutrominant ===
=== Neutrominant ===
<span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span>
{{Main| Neutrominant }}
{{Main| Neutrominant }}


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Mapping: {{mapping| 1 1 0 4 2 | 0 2 8 -4 5 }}
Mapping: {{mapping| 1 1 0 4 2 | 0 2 8 -4 5 }}
: mapping generators: ~2, ~11/9


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 349.865
* WE: ~2 = 1195.9529{{c}}, ~11/9 = 349.7505{{c}}
* POTE: ~2 = 1\1, ~11/9 = 350.934
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.7031{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 17c, 24d, 41cd }}
{{Optimal ET sequence|legend=0| 7, 17c, 24d, 41cd }}


Badness: 0.040240
Badness (Sintel): 1.33


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/9 = 349.904
* WE: ~2 = 1196.3143{{c}}, ~11/9 = 349.7387{{c}}
* POTE: ~2 = 1\1, ~11/9 = 350.816
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6573{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 17c, 24d, 41cd }}
{{Optimal ET sequence|legend=0| 7, 17c, 24d, 41cd }}


Badness: 0.027214
Badness (Sintel): 1.12


== Semisema ==
== Semisema ==
In addition to dividing the perfect fifth into two equal parts of {{nowrap|11/9 ~ 27/22}}, semisema, being an extension of [[semaphore]], also divides the perfect fourth into two equal parts of {{nowrap|7/6 ~ 8/7}}.
In addition to dividing the perfect fifth into two equal parts of 11/9~27/22, semisema, being an extension of [[semaphore]], also divides the perfect fourth into two equal parts of 7/6~8/7.


Subgroup: 2.3.7.11
[[Subgroup]]: 2.3.7.11


Comma list: 49/48, 243/242
[[Comma list]]: 49/48, 243/242


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


: sval mapping generators: ~77/54, ~11/9
{{Mapping|legend=3| 2 0 0 4 -1 | 0 2 0 1 5 }}


{{Mapping|2 2 0 5 4|0 2 0 1 5|legend=3}}
: mapping generators: ~77/54, ~7/4


: [[gencom]]: [77/54 11/9; 49/48 243/242]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 601.5167{{c}}, ~7/4 = 951.1267{{c}}
* [[CWE]]: ~2 = 600.0000{{c}}, ~7/4 = 949.6883{{c}}


Optimal tunings:
{{Optimal ET sequence|legend=1| 10, 14, 24, 62dd }}
* CTE: ~77/54 = 1\2, ~11/9 = 351.181
* POTE: ~77/54 = 1\2, ~11/9 = 348.728


Optimal ET Sequence: {{Optimal ET sequence|10, 14, 24, 62dd}}
[[Badness]] (Sintel): 0.776


[[Category:Temperament clans]]
[[Category:Temperament clans]]
Retrieved from "https://en.xen.wiki/w/Rastmic_clan"