293edo: Difference between revisions
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{{EDO intro|293}} | {{EDO intro|293}} | ||
== 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. | |||
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. | 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. | ||
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Likewise, 293edo also can be interpreted as a dual-interval tuning, with ''two notes'' instead of one assigned to a particular interval. | Likewise, 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 | {{Harmonics in equal|293|columns=10}} | ||
=== Relation to a calendar reform === | |||
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]]. | The corresponding rank two temperament is therefore called [[Symmetry454]]. | ||
== | === Miscellaneous properties === | ||
293edo | 293edo is the 62nd [[prime edo]]. | ||
293edo tempers out | == 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. | |||
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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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? -->. | ||
== Rank | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
!Periods | ! Periods<br>per Octave | ||
per | ! Generator<br>(Reduced) | ||
!Generator | ! Cents<br>(Reduced) | ||
( | ! Associated<br>Ratio | ||
!Cents | ! Temperaments | ||
( | |||
!Associated | |||
!Temperaments | |||
|- | |- | ||
|1 | | 1 | ||
|62\293 | | 62\293 | ||
|253.92 | | 253.92 | ||
|52/45 | | 52/45 | ||
|Symmetry454 | | [[Symmetry454]] | ||
|} | |} | ||
== Music == | == Music == | ||
* [https://www.youtube.com/watch?v=KYcS2hSd93Y Whiplash] by Cinnamon Mavka | * [https://www.youtube.com/watch?v=KYcS2hSd93Y Whiplash] by [[Cinnamon Mavka]] – using the Symmetry454[52] scale. | ||
== | == External links == | ||
* [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52 | * [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:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||