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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|293}}
{{ED intro}}


== Theory ==
== Theory ==
293edo does not approximate prime harmonics well all the way into the 41st, unless 30% errors are considered "well", in which case it equally represents all of them. The first harmonic that it approximates within 1 standard deviation of one step is 43rd, which is 10% flat compared to the just intonated interval. Nonetheless, it is [[consistent]] in the 5-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.


293bb val, with 170\293 fifth, is an optimal tuning for the meantone temperament.  
Nonetheless, a number of mappings can be considered.  


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 harmoincs have +36 and +37 error respectively, which together cancels out to 1.
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]].  


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.
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]].


Likewise, 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.


=== Odd harmonics ===
=== Symmetry454 temperament ===
{{Harmonics in equal|293|columns=10}}
{{Main|Symmetry454}}


=== Relation to a calendar reform ===
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.  
The 33L 19s [[Maximal evenness|maximally even]] scale of 293edo has a real life application – it 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]].
=== Odd harmonics ===
{{Harmonics in equal|293}}


=== Miscellaneous properties ===
=== Subsets and supersets ===
293edo is the 62nd [[prime edo]].
293edo is the 62nd [[prime edo]].


== Regular temperament properties ==
== Regular temperament properties ==
=== Commas ===
=== 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.  
 
293edo tempers out {{monzo| 8 14 -13 }} ([[parakleisma]]) and {{monzo| -40 15 7 }} in the 5-limit. 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.  


Using the 293b val, it tempers out 16875/16807, 20000/19683, and 65625/65536 in the 7-limit; 896/891, 6875/6804, 9375/9317, and 12005/11979 in the 11-limit; 352/351, 364/363, 1716/1715, and 8125/8019 in the 13-limit.  
Using the 293b val, it tempers out 16875/16807, 20000/19683, and 65625/65536 in the 7-limit; 896/891, 6875/6804, 9375/9317, and 12005/11979 in the 11-limit; 352/351, 364/363, 1716/1715, and 8125/8019 in the 13-limit.  
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per Octave
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|11\293
| 11\293
|45.06
| 45.06
|36/35
| 36/35
|[[Quartonic]] (293bcd)
| [[Quartonic]] (293bcd)
|-
|-
| 1
| 1
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| [[Symmetry454]]
| [[Symmetry454]]
|-
|-
|1
| 1
|118\293
| 118\293
|483.28
| 483.28
|320/243
| 320/243
|[[Hemiseven]] (293de)
| [[Hemiseven]] (293de)
|-
|-
|1
| 1
|143\293
| 143\293
|585.66
| 585.66
|7/5
| 7/5
|[[Merman]] (293ef)
| [[Merman]] (293ef)
|-
|-
|1
| 1
|170\293
| 170\293
|696.25
| 696.25
|3/2
| 3/2
|[[Meantone]] (293bb)
| [[Meantone]] (293bb)
|}
|}
== Scales ==
* 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 ==
== Music ==
 
; [[Eliora]]
* [https://www.youtube.com/watch?v=KYcS2hSd93Y Whiplash] by [[Cinnamon Mavka]] – using the Symmetry454[52] scale.
* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] (2022) – using the Symmetry454[52] scale.


== External links ==
== External links ==
* [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52\293 Symmetry454 Leap Years]
* [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52\293 Symmetry454 Leap Years]


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Marvel]]
Retrieved from "https://en.xen.wiki/w/293edo"