61edo: Difference between revisions

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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.

There are three reasonable vals 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''' '''211''' 226}}). ~~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 & 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]].

There are three reasonable vals 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''' '''211''' 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]].

=== Odd harmonics ===

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 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.
-There are three reasonable vals 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''' '''211''' 226}}). 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 & 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]].
+There are three reasonable vals 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''' '''211''' 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]].
 === Odd harmonics ===