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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.


== Theory ==
Nonetheless, a number of mappings can be considered.
{{primes in edo|293|columns=14}}
 
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.  
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]].
 
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]].
 
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 ===
{{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.  


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]], and [[25/19]]. [[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.
=== Odd harmonics ===
{{Harmonics in equal|293}}


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.
=== Subsets and supersets ===
{| class="wikitable"
293edo is the 62nd [[prime edo]].
|+Selected intervals
!Degree
!Name
!Cents
!Approximate ratios
|-
|0
|Unison, prime
|0.0000
|1/1 exact
|-
|5
|Minor leap week interval
|
|85/84
|-
|6
|Major leap week interval
|
|71/70
|-
|11
|Bundle of 2
|
|
|-
|17
|Bundle of 3
|
|
|-
|18
|Vicesimotertial quarter-tone
|
|[[24/23]]
|-
|45
|Minor subcycle
|
|
|-
|47
|Undevicesimal meantone
|
|[[19/17]]
|-
|77
|Minor third
|
|[[6/5]]
|-
|79
|Major subcycle
|
|
|-
|115
|21st harmonic
|
|[[21/16]]
|-
|116
|
|
|[[25/19]]
|-
|163
|
|
|[[25/17]]
|-
|191
|
|
|[[11/7]]
|-
|293
|Perfect octave
|
|2/1 exact
|}


== Tempered commas ==
== Regular temperament properties ==
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.  
=== 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.  
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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 [[MOS scale|MOS]] [[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.


* 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 ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] (2022) – using the Symmetry454[52] scale.
 
== External links ==
* [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52\293 Symmetry454 Leap Years]


== Links ==
[[Category:Listen]]
[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]]
Retrieved from "https://en.xen.wiki/w/293edo"