93ed6: Difference between revisions

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== Theory ==

93ed6 is nearly identical to [[36edo]], but with the 6th harmonic rather than the [[~~2/1|~~octave]] being just. The octave is stretched by about 0.757 [[cent]]s.

93ed6 is nearly identical to [[36edo]], but with the 6th harmonic rather than the [[octave]] being just. The octave is stretched by about 0.757 [[cent]]s (almost identical to [[101ed7]], where the octave is stretched by about 0.770 cents). Like 36edo, 93ed6 is [[consistent]] to the [[integer limit|8-integer-limit]].

Compared to 36edo, 93ed6 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 93ed6 is better in the [[13-limit|13-]] and [[17-limit]], especially when treating it as a dual-5 dual-11 tuning.

The local [[Riemann zeta function #Optimal octave stretch|zeta peak]] around 36 is located at 35.982388, which has a step size of 33.3496{{c}} and has octaves stretched by 0.587{{c}}, making 93ed6 very close to optimal for 36edo.

=== Harmonics ===

{{Harmonics in equal|93|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 93ed6 (continued)}}

=== Subsets and supersets ===

Since 93 factors into primes as {{nowrap| 3 × 31 }}, 93ed6 contains subset ed6's [[3ed6]] and [[31ed6]].

== See also ==

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* [[57edt]] – relative edt

* [[101ed7]] – relative ed7

* [[129ed12]] – relative ed12

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

[[Category:36edo]]

@@ Line 3: / Line 3: @@
 == Theory ==
-ed6 is nearly identical to [[36edo]], but with the 6th harmonic rather than the [[2/1|octave]] being just. The octave is stretched by about 0.757 [[cent]]s.
+ed6 is nearly identical to [[36edo]], but with the 6th harmonic rather than the [[octave]] being just. The octave is stretched by about 0.757 [[cent]]s (almost identical to [[101ed7]], where the octave is stretched by about 0.770 cents). Like 36edo, 93ed6 is [[consistent]] to the [[integer limit|8-integer-limit]].
+Compared to 36edo, 93ed6 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 93ed6 is better in the [[13-limit|13-]] and [[17-limit]], especially when treating it as a dual-5 dual-11 tuning.
+The local [[Riemann zeta function #Optimal octave stretch|zeta peak]] around 36 is located at 35.982388, which has a step size of 33.3496{{c}} and has octaves stretched by 0.587{{c}}, making 93ed6 very close to optimal for 36edo.
 === Harmonics ===
 {{Harmonics in equal|93|6|1|intervals=integer|columns=11}}
 {{Harmonics in equal|93|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 93ed6 (continued)}}
+=== Subsets and supersets ===
+Since 93 factors into primes as {{nowrap| 3 × 31 }}, 93ed6 contains subset ed6's [[3ed6]] and [[31ed6]].
 == See also ==
@@ Line 14: / Line 21: @@
 * [[57edt]] – relative edt
 * [[101ed7]] – relative ed7
-* [[129ed12]] – relative ed12
+* [[129ed12]] – relative ed12, close to the zeta-optimized tuning for 36edo
+[[Category:36edo]]