101ed7: Difference between revisions

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== Theory ==

101ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[~~2/1|~~octave]] being just. The octave is stretched by about 1.23 [[cent]]s. Like 36edo, 101ed7 is [[consistent]] to the [[integer limit|8-integer-limit]].

101ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[octave]] being just. The octave is stretched by about 0.770 [[cent]]s (almost identical to [[93ed6]], where the octave is stretched by about 0.757 cents). Like 36edo, 101ed7 is [[consistent]] to the [[integer limit|8-integer-limit]].

Compared to 36edo, 101ed7~~'s harmonics are almost exactly~~ the ~~same~~, ~~but it has a~~ slightly better [[3/1]], [[7/1]], and [[13/1]], and a slightly worse 2~~/1 and~~ [[5/1]] ~~versus 36edo~~. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit~~]]. (The [[7~~-~~limit~~]] and [[~~11-limit]] could go either way.)~~

Compared to 36edo, 101ed7 is pretty well optimized for the 2.3.7.13.17 [[subgroup]], with slightly better [[3/1|3]], [[7/1|7]], [[13/1|13]] and [[17/1|17]], and a slightly worse 2 versus 36edo. Using the [[patent val]], the [[5/1|5]] is also less accurate. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit|13-]] and [[17-limit]], especially when treating it as a dual-5 dual-11 tuning.

~~36edo's 5~~-limit ~~dominance flips on its head, though, if one approaches it as a [[dual-n|dual-5~~]] ~~tuning. In that case~~, ~~the fact that 101ed7's 5/1 is closer to 50% relative error is actually a ''good'' thing, because~~ it ~~means the error on the worse of the two 5/1's is less. So as a single-5 5-limit tuning, 36edo is better. But as a dual-5 5-limit tuning, 101ed7 is better, and~~ as a dual-5, dual-11 ~~[[31-limit]]~~ tuning~~, 101ed7 is exceptional for its size. It is very accurate~~.

=== Harmonics ===

{{Harmonics in equal|101|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 101ed7 (continued)}}

=== Subsets and supersets ===

101ed7 is the 26th [[prime equal division|prime ed7]], so it does not contain any nontrivial subset ed7's.

== Intervals ==

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* [[57edt]] – relative edt

* [[93ed6]] – relative ed6

* [[129ed12]] – relative ed12

* [[129ed12]] – relative ed12, close to the zeta-optimized tuning for 36edo

[[Category:36edo]]

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 == Theory ==
-ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[2/1|octave]] being just. The octave is stretched by about 1.23 [[cent]]s. Like 36edo, 101ed7 is [[consistent]] to the [[integer limit|8-integer-limit]].
+ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[octave]] being just. The octave is stretched by about 0.770 [[cent]]s (almost identical to [[93ed6]], where the octave is stretched by about 0.757 cents). Like 36edo, 101ed7 is [[consistent]] to the [[integer limit|8-integer-limit]].
-Compared to 36edo, 101ed7's harmonics are almost exactly the same, but it has a slightly better [[3/1]], [[7/1]], and [[13/1]], and a slightly worse 2/1 and [[5/1]] versus 36edo. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit]]. (The [[7-limit]] and [[11-limit]] could go either way.)
+Compared to 36edo, 101ed7 is pretty well optimized for the 2.3.7.13.17 [[subgroup]], with slightly better [[3/1|3]], [[7/1|7]], [[13/1|13]] and [[17/1|17]], and a slightly worse 2 versus 36edo. Using the [[patent val]], the [[5/1|5]] is also less accurate. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit|13-]] and [[17-limit]], especially when treating it as a dual-5 dual-11 tuning.
-edo's 5-limit dominance flips on its head, though, if one approaches it as a [[dual-n|dual-5]] tuning. In that case, the fact that 101ed7's 5/1 is closer to 50% relative error is actually a ''good'' thing, because it means the error on the worse of the two 5/1's is less. So as a single-5 5-limit tuning, 36edo is better. But as a dual-5 5-limit tuning, 101ed7 is better, and as a dual-5, dual-11 [[31-limit]] tuning, 101ed7 is exceptional for its size. It is very accurate.
 === Harmonics ===
 {{Harmonics in equal|101|7|1|intervals=integer|columns=11}}
 {{Harmonics in equal|101|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 101ed7 (continued)}}
+=== Subsets and supersets ===
+ed7 is the 26th [[prime equal division|prime ed7]], so it does not contain any nontrivial subset ed7's.
 == Intervals ==
@@ Line 21: / Line 22: @@
 * [[57edt]] – relative edt
 * [[93ed6]] – relative ed6
-* [[129ed12]] – relative ed12
+* [[129ed12]] – relative ed12, close to the zeta-optimized tuning for 36edo
 [[Category:36edo]]