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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 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]].


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


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.
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.


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


== Relation to a calendar reform ==
=== Odd harmonics ===
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.
{{Harmonics in equal|293}}


The corresponding rank two temperament is therefore called '''Symmetry454.'''
=== Subsets and supersets ===
293edo is the 62nd [[prime edo]].


== Tempered commas ==
== Regular temperament properties ==
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.  
=== 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.  


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.  
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.
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? -->.


== Scales ==
=== Rank-2 temperaments ===
* LeapWeek[52]
{| class="wikitable center-all left-5"
* LeapDay[71]
! 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 ===
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]].


== 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 - using the LeapWeek[52] scale.
== 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]]
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