101ed7: Difference between revisions

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'''101 divisions of the seventh harmonic''' ('''101ed7''') is related to [[36edo]] (sixth-tone tuning), but with the 7/1 rather than the 2/1 being just. The octave is about 1.2347 [[cent]]s stretched and the step size is about 33.3547 cents. It is consistent to the 8-[[integer-limit]]. This tuning tempers out the [[syntonic comma]].

~~Lookalikes:~~ [[~~21edf~~]], [[~~36edo~~]], [[~~57edt~~]], [[~~93ed6~~]]

== 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 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]].

~~== Intervals ==~~

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.

~~{{Interval table}}~~

== Harmonics ==

=== Harmonics ===

~~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.~~

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

~~Overall this means 36edo~~ is ~~still better in~~ the [[~~5-limit~~]], ~~but 101ed7 better in the [[13-limit]]. And the [[7-limit]] and [[11-limit]] could go either way~~.

=== Subsets and supersets ===

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

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 5/1s is less.

== Intervals ==

~~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 in equal|101|7|1|intervals~~=~~prime|columns~~=~~12}}~~

~~36edo for comparsion:~~

== See also ==

* [[21edf]] – relative edf

* [[36edo]] – relative edo

* [[57edt]] – relative edt

* [[93ed6]] – relative ed6

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

~~{{stub}}~~

[[Category:36edo]]

[[Category:~~Edonoi~~]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''101 divisions of the seventh harmonic''' ('''101ed7''') is related to [[36edo]] (sixth-tone tuning), but with the 7/1 rather than the 2/1 being just. The octave is about 1.2347 [[cent]]s stretched and the step size is about 33.3547 cents. It is consistent to the 8-[[integer-limit]]. This tuning tempers out the [[syntonic comma]].
+{{ED intro}}
-Lookalikes: [[21edf]], [[36edo]], [[57edt]], [[93ed6]]
+== 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 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]].
-== Intervals ==
+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.
-{{Interval table}}
-== Harmonics ==
+=== Harmonics ===
-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.
+{{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)}}
-Overall this means 36edo is still better in the [[5-limit]], but 101ed7 better in the [[13-limit]]. And the [[7-limit]] and [[11-limit]] could go either way.
+=== Subsets and supersets ===
+ed7 is the 26th [[prime equal division|prime ed7]], so it does not contain any nontrivial subset ed7's.
-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 5/1s is less.
+== Intervals ==
+{{Interval table}}
-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 in equal|101|7|1|intervals=prime|columns=12}}
-edo for comparsion:
-{{Harmonics in equal|36|2|1|intervals=prime|columns=12}}
+== See also ==
+* [[21edf]] – relative edf
+* [[36edo]] – relative edo
+* [[57edt]] – relative edt
+* [[93ed6]] – relative ed6
+* [[129ed12]] – relative ed12, close to the zeta-optimized tuning for 36edo
-{{stub}}
+[[Category:36edo]]
-[[Category:Edonoi]]