80ed6: Difference between revisions

(3 intermediate revisions by 2 users not shown)

Line 3:

== Theory ==

80ed6 is related to [[31edo]], but with the 6/1 rather than the [[2/1]] being just. This stretches the octave by about 2 cents.

80ed6 is related to [[31edo]], but with the 6/1 rather than the [[2/1]] being just. This stretches the octave by about 2 cents. Like 31edo, 80ed6 is [[consistent]] to the [[integer limit|12-integer-limit]]. It is pretty well optimized for the [[11-limit]], trading the accuracy of the [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s for an improved [[3/1|3rd harmonic]] and a massively improved [[11/1|11th harmonic]], which is only 2.5 cents flat of just (in comparison, 31edo's 11th harmonic is 9.4 cents flat). Also improved is the [[23/1|23rd harmonic]], which is now only 0.1 cents sharp of just.

The [[13/1|13th]], [[17/1|17th]], and [[19/1|19th harmonics]] are now about halfway between the steps, suggesting the use of [[160ed6]].

=== Harmonics ===

{{Harmonics in equal|80|6|1|intervals=~~prime~~|columns=12|start=12|collapsed=true|Approximation of harmonics in 80ed6 (continued)}}

{{Harmonics in equal|80|6|1|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 80ed6 (continued)}}

=== Subsets and supersets ===

Since 80 factors into primes as {{nowrap| 2<sup>4</sup> × 5 }}, 80ed6 has subset ed6's {{EDs|equave=6| 2, 4, 5, 8, 10, 16, 20, and 40 }}.

Since 80 factors into primes as {{nowrap| 2<sup>4</sup> × 5 }}, 80ed6 has subset ed6's {{EDs|equave=6| 2, 4, 5, 8, 10, 16, 20, and 40 }}. 160ed6, which doubles it, much corrects its 13th, 17th, and 19th harmonics.

== See also ==

Line 18:

Line 20:

* [[72ed5]] – relative ed5

* [[87ed7]] – relative ed7

* [[107ed11]] – relative ed11

* [[111ed12]] – relative ed12

* [[138ed22]] – relative ed22

* [[204ed96]] – close to the zeta-optimized tuning for 31edo

* [[39cET]]

[[Category:31edo]]

@@ Line 3: / Line 3: @@
 == Theory ==
-ed6 is related to [[31edo]], but with the 6/1 rather than the [[2/1]] being just. This stretches the octave by about 2 cents.
+ed6 is related to [[31edo]], but with the 6/1 rather than the [[2/1]] being just. This stretches the octave by about 2 cents. Like 31edo, 80ed6 is [[consistent]] to the [[integer limit|12-integer-limit]]. It is pretty well optimized for the [[11-limit]], trading the accuracy of the [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s for an improved [[3/1|3rd harmonic]] and a massively improved [[11/1|11th harmonic]], which is only 2.5 cents flat of just (in comparison, 31edo's 11th harmonic is 9.4 cents flat). Also improved is the [[23/1|23rd harmonic]], which is now only 0.1 cents sharp of just.
+The [[13/1|13th]], [[17/1|17th]], and [[19/1|19th harmonics]] are now about halfway between the steps, suggesting the use of [[160ed6]].
 === Harmonics ===
-{{Harmonics in equal|80|6|1|intervals=prime|columns=11}}
+{{Harmonics in equal|80|6|1|intervals=integer|columns=11}}
-{{Harmonics in equal|80|6|1|intervals=prime|columns=12|start=12|collapsed=true|Approximation of harmonics in 80ed6 (continued)}}
+{{Harmonics in equal|80|6|1|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 80ed6 (continued)}}
 === Subsets and supersets ===
-Since 80 factors into primes as {{nowrap| 2<sup>4</sup> × 5 }}, 80ed6 has subset ed6's {{EDs|equave=6| 2, 4, 5, 8, 10, 16, 20, and 40 }}.
+Since 80 factors into primes as {{nowrap| 2<sup>4</sup> × 5 }}, 80ed6 has subset ed6's {{EDs|equave=6| 2, 4, 5, 8, 10, 16, 20, and 40 }}. 160ed6, which doubles it, much corrects its 13th, 17th, and 19th harmonics.
 == See also ==
@@ Line 18: / Line 20: @@
 * [[72ed5]] – relative ed5
 * [[87ed7]] – relative ed7
+* [[107ed11]] – relative ed11
 * [[111ed12]] – relative ed12
+* [[138ed22]] – relative ed22
+* [[204ed96]] – close to the zeta-optimized tuning for 31edo
 * [[39cET]]
+[[Category:31edo]]