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'''293EDO''' is the [[EDO|equal division of the octave]] into 293 parts of 4.0956 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


293EDO is the 62nd [[prime EDO]].
== Theory ==
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.
 
Nonetheless, a number of mappings can be considered.
 
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]].  


== Theory ==
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]].
{{Harmonics in equal|293|columns=10}}
293 edo does not approximate prime harmonics well all the way into the 41st, unless 30-relative cent 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 cents flat compared to the just intonated interval.  


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.
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.


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.
=== Symmetry454 temperament ===
{{Main|Symmetry454}}


Likewise, 293edo also can be interpreted as a dual-interval tuning, with ''two notes'' instead of one assigned to a particular interval.
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.  


{| class="wikitable mw-collapsible right-3"
=== Odd harmonics ===
|+Selected intervals
{{Harmonics in equal|293}}
|-
! Degree
! Name
! Cents
! Approximate ratios
|-
| 0
| Unison, prime
| 0.0000
| 1/1 exact
|-
|1
|Limit-tone
|4.0596
|423/422
|-
| 5
| Minor leap week interval
| 20.4778
| 85/84
|-
| 6
| Major leap week interval
| 24.5734
| 71/70
|-
| 11
| Bundle of 2
| 45.0512
|
|-
| 17
| Bundle of 3
| 69.6246
|
|-
| 18
| Vicesimotertial quarter-tone
| 73.7201
|[[24/23]]
|-
| 45
| Minor subcycle
| 184.3003
|
|-
| 47
| Undevicesimal meantone
| 192.4915
|[[19/17]]
|-
|56
|Minor septimal second
|229.3515
|8/7, 214/187
|-
|57
|Major septimal second
|233.4471
|8/7, 266/233
|-
|62
|Leap week accumulator
|253.9249
|755/652
|-
| 77
| Minor third
| 315.3584
|[[6/5]]
|-
| 79
| Major subcycle
| 323.5495
|
|-
| 115
| 21st harmonic
| 470.9898
|[[21/16]]
|-
| 116
|25 over 19
| 475.0853
|[[25/19]]
|-
|125
|43rd harmonic
|511.9454
|[[43/32]]
|-
|130
|Vengeance superfourth
|532.4232
|[[34/25]]
|-
| 163
|Vengeance subfifth
| 667.5768
|[[25/17]]
|-
|168
|43rd subharmonic
|688.0546
|64/43
|-
|171
|Perfect fifth
|700.3413
|[[3/2]]
|-
|172
|"Major" fifth
|704.4369
|347/231
|-
| 191
|Undecimal minor sixth
| 782.2526
|[[11/7]]
|-
|216
|Major sixth
|884.6416
|[[5/3]]
|-
|236
|Minor harmonic seventh
|966.5529
|[[7/4]], 187/107
|-
|237
|Major harmonic seventh
|970.6485
|[[7/4]], 233/133
|-
|260
|Leap day accumulator
|1064.8464
|468/253
|-
| 293
| Perfect octave
| 1200.0000
| 2/1 exact
|}


== Tempered commas ==
=== Subsets and supersets ===
293edo tempers out the [1590 0<sup>12</sup> 293⟩ comma in the patent val, equating a stack of 293 43rd harmonics with 1590 octaves.  
293edo is the 62nd [[prime edo]].


293edo tempers out 1224440064/1220703125 (parakleisma) and 1121008359375/1099511627776 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.  
== Regular temperament properties ==
=== Commas ===
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.  
Line 174: Line 35:


Using the 293deg val, it tempers out 385/384, 441/440, 24057/24010, and 234375/234256 in the 11-limit; 625/624, 847/845, 1001/1000, and 1575/1573 in the 13-limit; 561/560, 1225/1224, 1275/1274, and 2025/2023 in the 17-limit.
Using the 293deg val, it tempers out 385/384, 441/440, 24057/24010, and 234375/234256 in the 11-limit; 625/624, 847/845, 1001/1000, and 1575/1573 in the 13-limit; 561/560, 1225/1224, 1275/1274, and 2025/2023 in the 17-limit.
Using the well-approximated intervals, [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]] and [[21/16]], 293edo tempers out 2376/2375, 304175/304128, 2599200/2598977 <!-- can we find a subgroup for this? -->.
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
| 1
| 11\293
| 45.06
| 36/35
| [[Quartonic]] (293bcd)
|-
| 1
| 62\293
| 253.92
| 52/45
| [[Symmetry454]]
|-
| 1
| 118\293
| 483.28
| 320/243
| [[Hemiseven]] (293de)
|-
| 1
| 143\293
| 585.66
| 7/5
| [[Merman]] (293ef)
|-
| 1
| 170\293
| 696.25
| 3/2
| [[Meantone]] (293bb)
|}


== Scales ==
== Scales ==
33L 19s [[Maximal evenness|maximally even]] scale of 293edo has a real life application - it is a leap year pattern of a proposed calendar. Using MOS, 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.


* LeapWeek[52]
* Symmetry454[5]: 45 79 45 79 45
* LeapDay[71]
* 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''] (2022) – using the Symmetry454[52] scale.


* [https://www.youtube.com/watch?v=KYcS2hSd93Y Whiplash] by Cinnamon Mavka
== External links ==
 
* [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52\293 Symmetry454 Leap Years]
== Links ==
* [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52/293 Symmetry454 Leap Years]


[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/293edo"