61edo: Difference between revisions

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== Theory ==

As ~~an equal temperament~~, ~~61et~~ is characterized by [[tempering out]] 20000/19683 ([[tetracot comma]]) and 262144/253125 ([[passion comma]]) in the 5-limit. In the 7-limit, the [[patent val]] {{val| 61 97 142 '''171''' }} [[support]]s [[valentine]] ({{nowrap|15 &~~amp;~~ 46}}), and is the [[optimal patent val]] for [[freivald]] ({{nowrap|24 &~~amp;~~ 37}}) in the 7-, 11- and 13-limit. The 61d [[val]] {{val| 61 97 142 '''172''' }} is a great tuning for [[modus]] and [[quasisuper]], and is a simple but out-of-tune edo tuning for [[parakleismic]]. ~~{{Harmonics in equal|61}}~~

61edo is only [[consistent]] to the [[5-odd-limit]]. Its [[3/1|3rd]] and [[5/1|5th]] [[harmonic]]s are sharp of just by more than 6 cents, and the [[7/1|7th]] and [[11/1|11th]], though they err by less, are on the flat side. This limits its harmonic inventory. However, it does possess reasonably good approximations of [[21/16]] and [[23/16]], only a bit more than one cent off in each case.

=== Odd harmonics ===

=== As a tuning of other temperaments ===

There are three reasonable [[val]]s in the [[13-limit]]: the [[patent val]] ({{val| 61 97 142 171 211 226 }}), 61d ({{val| 61 97 142 '''172''' 211 226 }}), and 61de ({{val| 61 97 142 '''172''' '''212''' 226 }}). In any case, it is characterized by [[tempering out]] 20000/19683 ([[tetracot comma]]) and 262144/253125 ([[passion comma]]) in the 5-limit. In the 7-limit, the patent val {{val| 61 97 142 '''171''' }} [[support]]s [[valentine]] ({{nowrap| 15 & 46 }}), and is the [[optimal patent val]] for [[freivald]] ({{nowrap| 24 & 37 }}) in the 7-, 11- and 13-limit. The 61d [[val]] {{val| 61 97 142 '''172''' }} is a great tuning for [[modus]] and [[quasisuper]], and is a simple but out-of-tune edo tuning for [[parakleismic]]. The 61d val supports 11-limit [[quasisupra]], and the 61de val supports 11- and 13-limit [[modus]].

=== Subsets and supersets ===

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|[[Freivald]] (61)

|}

<nowiki/>* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave

== Instruments ==

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; [[Bryan Deister]]

* [https://www.youtube.com/shorts/1Ai__APev5M ''microtonal improvisation in 61edo''] (2025)

* [https://www.youtube.com/shorts/S9bJnllI7CI ''61edo prelude''] (2025)

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 == Theory ==
-As an equal temperament, 61et is characterized by [[tempering out]] 20000/19683 ([[tetracot comma]]) and 262144/253125 ([[passion comma]]) in the 5-limit. In the 7-limit, the [[patent val]] {{val| 61 97 142 '''171''' }} [[support]]s [[valentine]] ({{nowrap|15 &amp; 46}}), and is the [[optimal patent val]] for [[freivald]] ({{nowrap|24 &amp; 37}}) in the 7-, 11- and 13-limit. The 61d [[val]] {{val| 61 97 142 '''172''' }} is a great tuning for [[modus]] and [[quasisuper]], and is a simple but out-of-tune edo tuning for [[parakleismic]]. {{Harmonics in equal|61}}
+edo is only [[consistent]] to the [[5-odd-limit]]. Its [[3/1|3rd]] and [[5/1|5th]] [[harmonic]]s are sharp of just by more than 6 cents, and the [[7/1|7th]] and [[11/1|11th]], though they err by less, are on the flat side. This limits its harmonic inventory. However, it does possess reasonably good approximations of [[21/16]] and [[23/16]], only a bit more than one cent off in each case.
+=== Odd harmonics ===
+{{Harmonics in equal|61}}
+=== As a tuning of other temperaments ===
+There are three reasonable [[val]]s in the [[13-limit]]: the [[patent val]] ({{val| 61 97 142 171 211 226 }}), 61d ({{val| 61 97 142 '''172''' 211 226 }}), and 61de ({{val| 61 97 142 '''172''' '''212''' 226 }}). In any case, it is characterized by [[tempering out]] 20000/19683 ([[tetracot comma]]) and 262144/253125 ([[passion comma]]) in the 5-limit. In the 7-limit, the patent val {{val| 61 97 142 '''171''' }} [[support]]s [[valentine]] ({{nowrap| 15 & 46 }}), and is the [[optimal patent val]] for [[freivald]] ({{nowrap| 24 & 37 }}) in the 7-, 11- and 13-limit. The 61d [[val]] {{val| 61 97 142 '''172''' }} is a great tuning for [[modus]] and [[quasisuper]], and is a simple but out-of-tune edo tuning for [[parakleismic]]. The 61d val supports 11-limit [[quasisupra]], and the 61de val supports 11- and 13-limit [[modus]].
 === Subsets and supersets ===
@@ Line 178: / Line 184: @@
 |[[Freivald]] (61)
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave
 == Instruments ==
@@ Line 191: / Line 197: @@
 ; [[Bryan Deister]]
 * [https://www.youtube.com/shorts/1Ai__APev5M ''microtonal improvisation in 61edo''] (2025)
+* [https://www.youtube.com/shorts/S9bJnllI7CI ''61edo prelude''] (2025)