80ed6: Difference between revisions

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'''[[Ed6|Division of the sixth harmonic]] into 80 equal parts''' (80ED6) is related to [[31edo|31 edo]], but with the 6/1 rather than the 2/1 being just. The octave is about 2 cents stretched and the step size is 38.77 cents.

~~Lookalikes:~~ [[31edo]], [[~~49edt~~]], [[~~72ed5~~]], [[~~18edf~~]]

== 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. 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.

~~==Harmonics==~~

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 in equal~~|~~80|6~~|1|~~prec=2|columns=15}}~~

[[Category:~~Edonoi~~]]

=== Harmonics ===

{{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 }}. 160ed6, which doubles it, much corrects its 13th, 17th, and 19th harmonics.

== See also ==

* [[18edf]] – relative edf

* [[31edo]] – relative edo

* [[49edt]] – relative edt

* [[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 1: / Line 1: @@
 {{Infobox ET}}
-'''[[Ed6|Division of the sixth harmonic]] into 80 equal parts''' (80ED6) is related to [[31edo|31 edo]], but with the 6/1 rather than the 2/1 being just. The octave is about 2 cents stretched and the step size is 38.77 cents.
+{{ED intro}}
-Lookalikes: [[31edo]], [[49edt]], [[72ed5]], [[18edf]]
+== 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. 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.
-==Harmonics==
+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 in equal|80|6|1|prec=2|columns=15}}
-[[Category:Edonoi]]
+=== Harmonics ===
+{{Harmonics in equal|80|6|1|intervals=integer|columns=11}}
+{{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 }}. 160ed6, which doubles it, much corrects its 13th, 17th, and 19th harmonics.
+== See also ==
+* [[18edf]] – relative edf
+* [[31edo]] – relative edo
+* [[49edt]] – relative edt
+* [[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]]