293edo: Difference between revisions

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== Theory ==

293edo does not approximate [[prime harmonic]]s well all the way into the 41st, with none approximated within 30% error. The first harmonic that it approximates well is the 43rd, which is 10% flat compared to the just intonated interval~~. As such, it is only [[consistent]] to the [[5-odd-limit]]~~.

293edo is only [[consistent]] to the [[5-odd-limit]] and it does not approximate [[prime harmonic]]s well all the way into the 41st, with none approximated within 20% [[relative interval error|relative error]], and all primes besides 13 have over 30% error. The first harmonic that it approximates well is the 43rd, which is 10% flat compared to the just intonated interval.

~~Using the [[patent val]]~~, ~~the equal temperament [[tempering out|tempers out]] the [[parakleisma]] and {{monzo| -40 15 7 }}. It also tempers out the [[marvel comma]] in the 7-limit~~.

Nonetheless, a number of mappings can be considered.

The 293bb val, with ~~170\~~293 ~~fifth~~, is a ~~good~~ tuning for the meantone temperament.

Using the [[patent val]], {{val|293 464 680 823}}, 293edo [[tempering out|tempers out]] the [[parakleisma]] and {{monzo| -40 15 7 }} in the 5-limit and the [[marvel comma]] in the 7-limit. The 293bb val, with {{val|293 '''463''' 680 823}}, is a tuning close to the [[POTE]] tuning for the [[meantone]] temperament. 293bcd, {{val|293 '''465 681 822'''}} is a tuning for [[quartonic]].

~~When it comes to~~ the ~~intervals that are not octave~~-~~reduced prime harmonics~~, ~~some which are well-approximated are [[6/5]], [[11/7]]~~, [[~~17/11~~]], [[~~19/17~~]], ~~[[24/23]], [[25/17]], [[25/19]]~~, ~~and respectively their octave inversions.~~ [[~~21/16~~]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmonics have +36 and +37 error respectively, which together cancels out to 1.

In the 11-limit, 293de val, {{val|293 464 680 '''822 1013'''}} provides a tuning close to the [[POTE]] tuning for [[hemiseven]], and and in the 13-limit, 293ef val {{val|293 464 680 823 '''1013 1085'''}} tunes [[merman]].

~~One step of~~ 293edo ~~is at the edge of human pitch perception of 3.~~5 ~~cents~~. ~~When combined with low harmonicity~~, ~~this opens 293edo to~~ a ~~wide range of interpretations~~. ~~For example~~, 293edo ~~also can be interpreted~~ as a ~~dual~~-~~interval tuning~~, with ~~''two notes'' instead of one assigned~~ to ~~a particular interval~~.

293edo nonetheless has good approximations to [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]], and respectively their octave inversions. [[21/16]], which is a composite octave-reduced harmonic, is also well represented.

=== Symmetry454 temperament ===

In the 17-limit, although inconsistent, 293edo in the patent val is a tuning for the [[Symmetry454]] temperament which is described as the 52 & 293 temperament, and constructed from a 52-tone maximal evenness scale which is exactly the distribution of leap years in the 293-year cycle of {{W|Symmetry454}}, the calendar with the same name. 62\293, mapped to [[52/45]], is the generator. See the dedicated page.

=== Odd harmonics ===

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== Regular temperament properties ==

=== Commas ===

~~293edo tempers out the 2.43 {{monzo| 1590 293 }} comma in the [[patent val]], equating a stack of 293 43rd harmonics with 1590 octaves.~~

Using the patent val, it tempers out 225/224, 2500000/2470629, and 344373768/341796875 in the 7-limit; 6250/6237, 8019/8000, 14700/14641, and 16896/16807 in the 11-limit; 351/350, 625/624, 1625/1617, and 13122/13013 in the 13-limit; 715/714, 850/847, 1089/1088, 1377/1375, 2058/2057, and 2880/2873 in the 17-limit.

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|}

=== Scales ===

== Scales ==

The 33L 19s [[Maximal evenness|maximally even]] scale of 293edo is a leap year pattern of a proposed calendar. It employs 62\293 as a generator, described as "accumulator" by the creator of the calendar himself. Likewise, a 71-note cycle with 260\293 generator can be constructed by analogy.

~~The corresponding rank two temperament is therefore called~~ [[Symmetry454]].

* Symmetry454[5]: 45 79 45 79 45

* Symmetry454[19]: 17 11 17 17 17 11 17 17 17 11 17 17 17 11 17 17 17 11 17

* Symmetry454[52]: 6 6 5 6 5 6 6 5 6 6 5 6 6 5 6 5 6 6 5 6 6 5 6 6 5 6 5 6 6 5 6 6 5 6 6 5 6 5 6 6 5 6 6 5 6 6 5 6 5 6 6 5

== Music ==

; [[~~Cinnamon Mavka~~]]

; [[Eliora]]

* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] – using the Symmetry454[52] scale.

* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] (2022) – using the Symmetry454[52] scale.

== External links ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro}}
+{{ED intro}}
 == Theory ==
-edo does not approximate [[prime harmonic]]s well all the way into the 41st, with none approximated within 30% error. The first harmonic that it approximates well is the 43rd, which is 10% flat compared to the just intonated interval. As such, it is only [[consistent]] to the [[5-odd-limit]].
+edo is only [[consistent]] to the [[5-odd-limit]] and it does not approximate [[prime harmonic]]s well all the way into the 41st, with none approximated within 20% [[relative interval error|relative error]], and all primes besides 13 have over 30% error. The first harmonic that it approximates well is the 43rd, which is 10% flat compared to the just intonated interval.
-Using the [[patent val]], the equal temperament [[tempering out|tempers out]] the [[parakleisma]] and {{monzo| -40 15 7 }}. It also tempers out the [[marvel comma]] in the 7-limit.
+Nonetheless, a number of mappings can be considered.
-The 293bb val, with 170\293 fifth, is a good tuning for the meantone temperament.
+Using the [[patent val]], {{val|293 464 680 823}}, 293edo [[tempering out|tempers out]] the [[parakleisma]] and {{monzo| -40 15 7 }} in the 5-limit and the [[marvel comma]] in the 7-limit. The 293bb val, with {{val|293 '''463''' 680 823}}, is a tuning close to the [[POTE]] tuning for the [[meantone]] temperament. 293bcd, {{val|293 '''465 681 822'''}} is a tuning for [[quartonic]].
-When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]], and respectively their octave inversions. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmonics have +36 and +37 error respectively, which together cancels out to 1.
+In the 11-limit, 293de val, {{val|293 464 680 '''822 1013'''}} provides a tuning close to the [[POTE]] tuning for [[hemiseven]], and and in the 13-limit, 293ef val {{val|293 464 680 823 '''1013 1085'''}} tunes [[merman]].
-One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations. For example, 293edo also can be interpreted as a dual-interval tuning, with ''two notes'' instead of one assigned to a particular interval.
+edo nonetheless has good approximations to [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]], and respectively their octave inversions. [[21/16]], which is a composite octave-reduced harmonic, is also well represented.
+=== Symmetry454 temperament ===
+{{Main|Symmetry454}}
+In the 17-limit, although inconsistent, 293edo in the patent val is a tuning for the [[Symmetry454]] temperament which is described as the 52 & 293 temperament, and constructed from a 52-tone maximal evenness scale which is exactly the distribution of leap years in the 293-year cycle of {{W|Symmetry454}}, the calendar with the same name. 62\293, mapped to [[52/45]], is the generator. See the dedicated page.
 === Odd harmonics ===
@@ Line 21: / Line 26: @@
 == Regular temperament properties ==
 === Commas ===
-edo tempers out the 2.43 {{monzo| 1590 293 }} comma in the [[patent val]], equating a stack of 293 43rd harmonics with 1590 octaves.
 Using the patent val, it tempers out 225/224, 2500000/2470629, and 344373768/341796875 in the 7-limit; 6250/6237, 8019/8000, 14700/14641, and 16896/16807 in the 11-limit; 351/350, 625/624, 1625/1617, and 13122/13013 in the 13-limit; 715/714, 850/847, 1089/1088, 1377/1375, 2058/2057, and 2880/2873 in the 17-limit.
@@ Line 74: / Line 77: @@
 |}
-=== Scales ===
+== Scales ==
-The 33L 19s [[Maximal evenness|maximally even]] scale of 293edo is a leap year pattern of a proposed calendar. It employs 62\293 as a generator, described as "accumulator" by the creator of the calendar himself. Likewise, a 71-note cycle with 260\293 generator can be constructed by analogy.
-The corresponding rank two temperament is therefore called [[Symmetry454]].
+* Symmetry454[5]: 45 79 45 79 45
+* Symmetry454[19]: 17 11 17 17 17 11 17 17 17 11 17 17 17 11 17 17 17 11 17
+* Symmetry454[52]: 6 6 5 6 5 6 6 5 6 6 5 6 6 5 6 5 6 6 5 6 6 5 6 6 5 6 5 6 6 5 6 6 5 6 6 5 6 5 6 6 5 6 6 5 6 6 5 6 5 6 6 5
 == Music ==
-; [[Cinnamon Mavka]]
+; [[Eliora]]
-* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] – using the Symmetry454[52] scale.
+* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] (2022) – using the Symmetry454[52] scale.
 == External links ==