Rastmic clan: Difference between revisions

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[[Comma list]]: 243/242

: mapping generators: ~2, ~11/9

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=== Namo ===

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 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]]~~ (351.~~488~~{{c}}) ~~recommends a sharper tuning than [[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

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[[Comma list]]: 144/143, 243/242

[[Optimal tuning]]s:

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== Suhajira ==

Suhajira can be thought of as "superpyth with quartertones", adding neutral intervals onto the existing pythagorean and septimal ones. Due to the sharper generator, it is now much more natural to add prime 13, with the neutral intervals generally closer to simple intervals of 13 than simple intervals of 11.

[[Subgroup]]: 2.3.7.11

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

[[Optimal tuning]]s:

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

~~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 }}

Optimal tunings:

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[[Comma list]]: 81/80, 121/120

[[Optimal tuning]]s:

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

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 1 0 2 4| 0 2 8 5 -1 }}

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

Optimal tunings:

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[[Badness]] (Sintel): 0.~~401~~

[[Badness]] (Sintel): 0.545

Scales: [[mohaha7]], [[mohaha10]]

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

~~Mapping:~~ {{~~mapping~~| 1 1 0 -3 2 | 0 2 8 20 5 }}

Optimal tunings:

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

~~Mapping:~~ {{~~mapping~~| 1 1 0 -3 2 4 | 0 2 8 20 5 -1 }}

Optimal tunings:

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

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

Optimal tunings:

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

~~Mapping:~~ {{~~mapping~~| 1 1 0 8 2 6 | 0 2 8 -18 5 -8 }}

Optimal tunings:

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

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

Optimal tunings:

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

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

Optimal tunings:

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

: mapping generators: ~77/54, ~7/4

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[[Badness]] (Sintel): 0.776

[[Category:Rastmic clan| ]]

[[Category:Temperament clans]]

[[Category:~~Pages with mostly numerical content]]~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rastmic clan| ]] <!~~-~~- main article -->~~

~~[[Category:Rastmic| ]] ~~

~~[[Category:Rank~~ 2]]

@@ Line 9: / Line 9: @@
 [[Comma list]]: 243/242
-{{Mapping|legend=2| 1 1 2 | 0 2 5 }}
+{{Mapping|legend=3| 1 1 2 | 0 2 5 }}
-{{Mapping|legend=3| 1 1 0 0 2 | 0 2 0 0 5 }}
+{{Mapping|legend=5| 1 1 0 0 2 | 0 2 0 0 5 }}
 : mapping generators: ~2, ~11/9
@@ Line 26: / Line 26: @@
 === Namo ===
-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 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]] (351.488{{c}}) recommends a sharper tuning than [[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
@@ Line 32: / Line 34: @@
 [[Comma list]]: 144/143, 243/242
-{{Mapping|legend=2| 1 1 2 4 | 0 2 5 -1 }}
+{{Mapping|legend=3| 1 1 2 4 | 0 2 5 -1 }}
-{{Mapping|legend=3| 1 1 0 0 2 4 | 0 2 0 0 5 -1 }}
+{{Mapping|legend=5| 1 1 0 0 2 4 | 0 2 0 0 5 -1 }}
 [[Optimal tuning]]s:
@@ Line 45: / Line 47: @@
 == Suhajira ==
+Suhajira can be thought of as "superpyth with quartertones", adding neutral intervals onto the existing pythagorean and septimal ones. Due to the sharper generator, it is now much more natural to add prime 13, with the neutral intervals generally closer to simple intervals of 13 than simple intervals of 11.
 [[Subgroup]]: 2.3.7.11
 [[Comma list]]: 64/63, 243/242
-{{Mapping|legend=2| 1 1 4 2 | 0 2 -4 5 }}
+{{Mapping|legend=3| 1 1 4 2 | 0 2 -4 5 }}
-{{Mapping|legend=3| 1 1 0 4 2 | 0 2 0 -4 5 }}
+{{Mapping|legend=5| 1 1 0 4 2 | 0 2 0 -4 5 }}
 [[Optimal tuning]]s:
@@ Line 68: / Line 72: @@
 Comma list: 64/63, 78/77, 144/143
-Subgroup-val mapping: {{mapping| 1 1 4 2 4 | 0 2 -4 5 -1 }}
+{{Mapping|legend=2| 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 }}
+{{Mapping|legend=4| 1 1 0 4 2 4 | 0 2 0 -4 5 -1 }}
 Optimal tunings:
@@ Line 92: / Line 96: @@
 [[Comma list]]: 81/80, 121/120
-{{Mapping|legend=2| 1 1 0 2 | 0 2 8 5 }}
+{{Mapping|legend=3| 1 1 0 2 | 0 2 8 5 }}
-{{Mapping|legend=3| 1 1 0 0 2 | 0 2 8 0 5 }}
+{{Mapping|legend=5| 1 1 0 0 2 | 0 2 8 0 5 }}
 [[Optimal tuning]]s:
@@ Line 111: / Line 115: @@
 Comma list: 66/65, 81/80, 121/120
-Subgroup-val mapping: {{mapping| 1 1 0 2 4| 0 2 8 5 -1 }}
+{{Mapping|legend=2| 1 1 0 2 4| 0 2 8 5 -1 }}
-Gencom mapping: {{mapping| 1 1 0 0 2 4 | 0 2 8 0 5 -1 }}
+{{Mapping|legend=4| 1 1 0 0 2 4 | 0 2 8 0 5 -1 }}
 Optimal tunings:
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 {{Optimal ET sequence|legend=0| 7, 17c, 24, 31, 55 }}
-[[Badness]] (Sintel): 0.401
+[[Badness]] (Sintel): 0.545
 Scales: [[mohaha7]], [[mohaha10]]
@@ Line 130: / Line 134: @@
 Comma list: 81/80, 121/120, 126/125
-Mapping: {{mapping| 1 1 0 -3 2 | 0 2 8 20 5 }}
+{{Mapping|legend=0| 1 1 0 -3 2 | 0 2 8 20 5 }}
 Optimal tunings:
@@ Line 145: / Line 149: @@
 Comma list: 66/65, 81/80, 121/120, 126/125
-Mapping: {{mapping| 1 1 0 -3 2 4 | 0 2 8 20 5 -1 }}
+{{Mapping|legend=0| 1 1 0 -3 2 4 | 0 2 8 20 5 -1 }}
 Optimal tunings:
@@ Line 162: / Line 166: @@
 Comma list: 81/80, 121/120, 525/512
-Mapping: {{mapping| 1 1 0 8 2 | 0 2 8 -18 5 }}
+{{Mapping|legend=0| 1 1 0 8 2 | 0 2 8 -18 5 }}
 Optimal tunings:
@@ Line 177: / Line 181: @@
 Comma list: 65/64, 81/80, 105/104, 121/120
-Mapping: {{mapping| 1 1 0 8 2 6 | 0 2 8 -18 5 -8 }}
+{{Mapping|legend=0| 1 1 0 8 2 6 | 0 2 8 -18 5 -8 }}
 Optimal tunings:
@@ Line 196: / Line 200: @@
 Comma list: 36/35, 64/63, 121/120
-Mapping: {{mapping| 1 1 0 4 2 | 0 2 8 -4 5 }}
+{{Mapping|legend=0| 1 1 0 4 2 | 0 2 8 -4 5 }}
 Optimal tunings:
@@ Line 211: / Line 215: @@
 Comma list: 36/35, 64/63, 66/65, 121/120
-Mapping: {{mapping| 1 1 0 4 2 4 | 0 2 8 -4 5 -1 }}
+{{Mapping|legend=0| 1 1 0 4 2 4 | 0 2 8 -4 5 -1 }}
 Optimal tunings:
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 [[Comma list]]: 49/48, 243/242
-{{Mapping|legend=2| 2 0 4 -1 | 0 2 1 5 }}
+{{Mapping|legend=3| 2 0 4 -1 | 0 2 1 5 }}
-{{Mapping|legend=3| 2 0 0 4 -1 | 0 2 0 1 5 }}
+{{Mapping|legend=5| 2 0 0 4 -1 | 0 2 0 1 5 }}
 : mapping generators: ~77/54, ~7/4
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 [[Badness]] (Sintel): 0.776
+[[Category:Rastmic clan| ]] <!-- main article -->
 [[Category:Temperament clans]]
-[[Category:Pages with mostly numerical content]]
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rastmic clan| ]] <!-- main article -->
-[[Category:Rastmic| ]] <!-- key article -->
-[[Category:Rank 2]]

Rastmic clan: Difference between revisions

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