Rastmic clan: Difference between revisions

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Rework intro. The first sentence in mohaha can serve as a good general description for neutral. +CTE tunings (so you see namo's fifth is indeed sharper than neutral's)
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The '''rastmic clan''' of temperaments [[tempers out]] [[243/242]].
The '''rastmic clan''' of [[regular temperament|temperaments]] [[tempers out]] the [[rastma]], 243/242 = {{monzo| -1 5 0 0 -2 }}.


== Neutral ==
== Neutral ==
Neutral 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 [[Archytas clan#Ringo|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=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]
: [[gencom]]: [2 11/9; 243/242]


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 350.525
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~11/9 = 350.572
* [[POTE]]: ~2 = 1\1, ~11/9 = 350.525


{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
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=== Namo ===
=== Namo ===
Namo has [[11/9]]~[[16/13]], an equivalence which [[User:Godtone|Godtone]] considers harmonically challenging, and 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, an equivalence which [[User:Godtone|Godtone]] considers harmonically challenging, 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.


[[Subgroup]]: 2.3.11.13
[[Subgroup]]: 2.3.11.13
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[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: 2 = 1\1, ~11/9 = 350.7455
* [[CTE]]: 2 = 1\1, ~11/9 = 350.746
* [[CWE]]: 2 = 1\1, ~11/9 = 351.2700
* [[CWE]]: 2 = 1\1, ~11/9 = 351.270
* [[POTE]]: 2 = 1\1, ~11/9 = 351.488
* [[POTE]]: 2 = 1\1, ~11/9 = 351.488


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== Suhajira ==
== Suhajira ==
Subgroup: 2.3.7.11
[[Subgroup]]: 2.3.7.11


Comma list: 64/63, 243/242
[[Comma list]]: 64/63, 243/242


Sval mapping: {{mapping| 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
: sval mapping generators: ~2, ~11/9


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


: gencom: [2 3/2; 64/63 99/98]
: [[gencom]]: [2 3/2; 64/63 99/98]


Optimal tuning (POTE): ~11/9 = 353.958
[[Optimal tuning]]s:
* [[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 }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de }}
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: gencom: [2 3/2; 64/63 78/77 99/98]
: gencom: [2 3/2; 64/63 78/77 99/98]


Optimal tuning (POTE): ~11/9 = 353.775
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 353.219
* POTE: ~2 = 1\1, ~11/9 = 353.775


{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de }}
Optimal ET sequence: {{Optimal ET sequence| 7, 10, 17, 44e, 61de }}


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


== Mohaha ==
== Mohaha ==
Mohaha is the 2.3.5.11 subgroup temperament with a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 11/9. 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 [[4L 3s|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]] 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 [[4L 3s|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=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]
: [[gencom]]: [2 11/9; 81/80 121/120]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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* POTE: ~2 = 1\1, ~11/9 = 348.182
* POTE: ~2 = 1\1, ~11/9 = 348.182


{{Optimal ET sequence|legend=1| 7d, 24d, 31, 100de, 131bdee, 162bdee }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 24d, 31, 100de, 131bdee, 162bdee }}


Badness: 0.025516
Badness: 0.025516
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* POTE: ~2 = 1\1, ~11/9 = 348.490
* POTE: ~2 = 1\1, ~11/9 = 348.490


{{Optimal ET sequence|legend=1| 7d, 24d, 31 }}
Optimal ET sequence: {{Optimal ET sequence| 7d, 24d, 31 }}


Badness: 0.028071
Badness: 0.028071
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: mapping generators: ~2, ~11/9
: mapping generators: ~2, ~11/9


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 346.922
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 346.905
* POTE: ~2 = 1\1, ~11/9 = 346.922


{{Optimal ET sequence|legend=1| 7, 31dd, 38d, 45e, 83bcddee }}
Optimal ET sequence: {{Optimal ET sequence| 7, 31dd, 38d, 45e, 83bcddee }}


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 1 0 8 2 6 | 0 2 8 -18 5 -8 }}
Mapping: {{mapping| 1 1 0 8 2 6 | 0 2 8 -18 5 -8 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 346.910
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 346.754
* POTE: ~2 = 1\1, ~11/9 = 346.910


{{Optimal ET sequence|legend=1| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
Optimal ET sequence: {{Optimal ET sequence| 7, 31ddf, 38df, 45ef, 83bcddeeff }}


Badness: 0.034316
Badness: 0.034316
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: mapping generators: ~2, ~11/9
: mapping generators: ~2, ~11/9


Wedgie: {{Multival| 2 8 -4 5 8 -12 1 -32 -16 28 }}
Wedgie: {{multival| 2 8 -4 5 8 -12 1 -32 -16 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.934
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 349.865
* POTE: ~2 = 1\1, ~11/9 = 350.934


{{Optimal ET sequence|legend=1| 7, 17c, 24d, 41cd }}
Optimal ET sequence: {{Optimal ET sequence| 7, 17c, 24d, 41cd }}


Badness: 0.040240
Badness: 0.040240
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Mapping: {{mapping| 1 1 0 4 2 4 | 0 2 8 -4 5 -1 }}
Mapping: {{mapping| 1 1 0 4 2 4 | 0 2 8 -4 5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.816
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 349.904
* POTE: ~2 = 1\1, ~11/9 = 350.816


{{Optimal ET sequence|legend=1| 7, 17c, 24d, 41cd }}
Optimal ET sequence: {{Optimal ET sequence| 7, 17c, 24d, 41cd }}


Badness: 0.027214
Badness: 0.027214

Revision as of 12:50, 11 October 2024

The rastmic clan of temperaments tempers out the rastma, 243/242 = [-1 5 0 0 -2⟩.

Neutral

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

Comma list: 243/242

Sval mapping: [⟨1 1 2], ⟨0 2 5]]

mapping generators: ~2, ~11/9

Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5]]

gencom: [2 11/9; 243/242]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 350.572
  • POTE: ~2 = 1\1, ~11/9 = 350.525

Optimal ET sequence: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380

RMS error: 0.3021 cents

Scales: neutral7, neutral10, neutral17

Namo

Namo adds 144/143 to the comma list and finds ~16/13 at the same neutral third. With 11/9~16/13, an equivalence which Godtone considers harmonically challenging, 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)1/2 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 ¢ off and in 99ef it is only slightly worse. For these reasons, Godtone is not fond of the recommendations by the various optimal tunings 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.

Subgroup: 2.3.11.13

Comma list: 144/143, 243/242

Sval mapping: [⟨1 1 2 4], ⟨0 2 5 -1]]

Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 0 0 5 -1]]

gencom: [2 11/9; 144/143 243/242]

Optimal tunings:

  • 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: 7, 10, 17, 24, 41, 58, 99ef, 239eefff, 338eeeffff

RMS error: 0.7038 cents

Suhajira

Subgroup: 2.3.7.11

Comma list: 64/63, 243/242

Sval mapping: [⟨1 1 4 2], ⟨0 2 -4 5]]

sval mapping generators: ~2, ~11/9

Gencom mapping: [⟨1 1 0 4 2], ⟨0 1 0 -4 5]]

gencom: [2 3/2; 64/63 99/98]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 353.139
  • POTE: ~2 = 1\1, ~11/9 = 353.958

Optimal ET sequence: 7, 10, 17, 44e, 61de

Scales: suhajira7, suhajira10, suhajira17

2.3.7.11.13 subgroup

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 144/143

Sval mapping: [⟨1 1 4 2 4], ⟨0 2 -4 5 -1]]

Gencom mapping: [⟨1 1 0 4 2 4], ⟨0 1 0 -4 5 -1]]

gencom: [2 3/2; 64/63 78/77 99/98]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 353.219
  • POTE: ~2 = 1\1, ~11/9 = 353.775

Optimal ET sequence: 7, 10, 17, 44e, 61de

Scales: suhajira7, suhajira10, suhajira17

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 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).

2.3.5.11 subgroup

The S-expression-based comma list of this temperament is {S6/S8 = S9, S11}.

Subgroup: 2.3.5.11

Comma list: 81/80, 121/120

Sval mapping: [⟨1 1 0 2], ⟨0 2 8 5]]

sval mapping generators: ~2, ~11/9

Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5]]

gencom: [2 11/9; 81/80 121/120]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 348.8296
  • POTE: ~2 = 1\1, ~11/9 = 348.0938

Optimal ET sequence: 7, 17c, 24, 31, 69e, 100e, 131bee

Scales: mohaha7, mohaha10

Mohoho

Subgroup: 2.3.5.11.13

Comma list: 66/65, 81/80, 121/120

Sval mapping: [⟨1 1 0 2 4], ⟨0 2 8 5 -1]]

sval mapping generators: ~2, ~11/9

Gencom 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:

  • CTE: ~2 = 1\1, ~11/9 = 348.8794
  • POTE: ~2 = 1\1, ~11/9 = 348.9155

Optimal ET sequence: 7, 17c, 24, 31, 55

Scales: mohaha7, mohaha10

Migration

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120, 126/125

Mapping: [⟨1 1 0 -3 2], ⟨0 2 8 20 5]]

mapping generators: ~2, ~11/9

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 348.5324
  • POTE: ~2 = 1\1, ~11/9 = 348.182

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

Badness: 0.025516

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 121/120, 126/125

Mapping: [⟨1 1 0 -3 2 4], ⟨0 2 8 20 5 -1]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 348.5444
  • POTE: ~2 = 1\1, ~11/9 = 348.490

Optimal ET sequence: 7d, 24d, 31

Badness: 0.028071

Ptolemy

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120, 525/512

Mapping: [⟨1 1 0 8 2], ⟨0 2 8 -18 5]]

mapping generators: ~2, ~11/9

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 346.905
  • POTE: ~2 = 1\1, ~11/9 = 346.922

Optimal ET sequence: 7, 31dd, 38d, 45e, 83bcddee

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 81/80, 105/104, 121/120

Mapping: [⟨1 1 0 8 2 6], ⟨0 2 8 -18 5 -8]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 346.754
  • POTE: ~2 = 1\1, ~11/9 = 346.910

Optimal ET sequence: 7, 31ddf, 38df, 45ef, 83bcddeeff

Badness: 0.034316

Neutrominant

Deutsch

Main article: Neutrominant

The neutrominant temperament (formerly maqamic temperament) has a hemififth generator (~11/9) and tempers out 36/35 and 121/120. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.

Subgroup: 2.3.5.7.11

Comma list: 36/35, 64/63, 121/120

Mapping: [⟨1 1 0 4 2], ⟨0 2 8 -4 5]]

mapping generators: ~2, ~11/9

Wedgie: ⟨⟨ 2 8 -4 5 8 -12 1 -32 -16 28 ]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 349.865
  • POTE: ~2 = 1\1, ~11/9 = 350.934

Optimal ET sequence: 7, 17c, 24d, 41cd

Badness: 0.040240

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 64/63, 66/65, 121/120

Mapping: [⟨1 1 0 4 2 4], ⟨0 2 8 -4 5 -1]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~11/9 = 349.904
  • POTE: ~2 = 1\1, ~11/9 = 350.816

Optimal ET sequence: 7, 17c, 24d, 41cd

Badness: 0.027214

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