Rastmic clan: Difference between revisions

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=== Namo ===

Namo requires the harmonically challenging equivalence [[11/9]] = [[16/13]] (which 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 (3/2)<sup>1/2</sup> the 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's only slightly worse). (For these reasons, the recommendations by the various [[optimal tuning]]s to tune flat of 58edo might be misleading, although it's 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

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 351.488

[[Optimal tuning]]s:

* [[CTE]]: 2 = 1\1, ~11/9 = 350.746

* [[CWE]]: 2 = 1\1, ~11/9 = 351.270

* [[POTE]]: 2 = 1\1, ~11/9 = 351.488

[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents

@@ Line 25: / Line 25: @@
 === Namo ===
+Namo requires the harmonically challenging equivalence [[11/9]] = [[16/13]] (which 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 (3/2)<sup>1/2</sup> the 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's only slightly worse). (For these reasons, the recommendations by the various [[optimal tuning]]s to tune flat of 58edo might be misleading, although it's 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
@@ Line 35: / Line 37: @@
 : gencom: [2 11/9; 144/143 243/242]
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/9 = 351.488
+[[Optimal tuning]]s:
+* [[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 }}
+{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 58, 99ef }}
 [[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents