293edo - Revision history

Eliora: /* Theory */

2026-10-04T23:45:35Z

Theory

← Older revision		Revision as of 23:45, 4 October 2026
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	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]].		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'''}} is a tuning for [[hemiseven]], and and in the 13-limit, 293ef val {{val\|293 464 680 823 '''1013 1085'''}} tunes [[merman]].		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.		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.

Eliora: misc cleanup + emphasize sym454

2026-10-04T23:29:43Z

misc cleanup + emphasize sym454

← Older revision		Revision as of 23:29, 4 October 2026
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	Nonetheless, a number of mappings can be considered.		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.		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]].

	~~The 293bb~~ val, ~~with~~ {{val\|293 '''~~463~~''' ~~680 823~~}}, is a tuning ~~close to the~~ [[~~POTE~~]] ~~tuning for~~ the [[~~meantone~~]] ~~temperament~~.		In the 11-limit, 293de val, {{val\|293 464 680 '''822 1013'''}} is a 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.		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.

	In the 17-limit, although inconsistent, 293edo in the patent val is a tuning for the [[Symmetry454]] temperament which is constructed from a calendar ~~layout by~~ the same name. See the dedicated page.		=== 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.

	=== Odd harmonics ===		=== Odd harmonics ===
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	== 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.		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.

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	== Scales ==		== 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]].		* 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: /* Theory */

2026-10-04T23:11:04Z

Theory

← Older revision		Revision as of 23:11, 4 October 2026
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	== Theory ==		== Theory ==
	293edo 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~~. As such, it is only [[consistent]] to the [[5-odd-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.

	~~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~~.		Nonetheless, a number of mappings can be considered.

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

	~~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.		The 293bb val, with {{val\|293 '''463''' 680 823}}, is a tuning close to the [[POTE]] 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~~. ~~For example~~, 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.

			In the 17-limit, although inconsistent, 293edo in the patent val is a tuning for the [[Symmetry454]] temperament which is constructed from a calendar layout by the same name. See the dedicated page.

	=== Odd harmonics ===		=== Odd harmonics ===

Eliora: /* Scales */

2026-10-04T23:06:58Z

Scales

← Older revision		Revision as of 23:06, 4 October 2026
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	\|}		\|}

	=== Scales ===		== 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 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.

Overthink: /* Theory */ fix info

2026-02-03T00:48:11Z

Theory: fix info

← Older revision		Revision as of 00:48, 3 February 2026
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	== Theory ==		== 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]].		293edo 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. As such, it is only [[consistent]] to the [[5-odd-limit]].

	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.		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.

ArrowHead294 at 19:29, 20 February 2025

2025-02-20T19:29:53Z

← Older revision		Revision as of 19:29, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

Eliora: /* Music */

2024-03-06T13:18:54Z

Music

← Older revision		Revision as of 13:18, 6 March 2024
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	== Music ==		== Music ==
	; [[~~Cinnamon Mavka~~]]		; [[Eliora]]
	* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] (2022) – using the Symmetry454[52] scale.		* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] (2022) – using the Symmetry454[52] scale.

FloraC: /* Music */

2024-03-04T09:04:45Z

Music

← Older revision		Revision as of 09:04, 4 March 2024
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	== Music ==		== Music ==
	; [[Cinnamon Mavka]]		; [[Cinnamon Mavka]]
	* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] – using the Symmetry454[52] scale.		* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] (2022) – using the Symmetry454[52] scale.

	== External links ==		== External links ==

FloraC: Cleanup; move "calendar reform" to the scales section

2024-03-04T09:04:18Z

Cleanup; move "calendar reform" to the scales section

← Older revision		Revision as of 09:04, 4 March 2024
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	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro~~\|293~~}}		{{EDO 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.		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]].

	~~Nonetheless, it is~~ [[~~consistent~~]] in the ~~5-limit, where it~~ tempers out the [[parakleisma]] and {{monzo\| -40 15 7 }}. ~~293edo~~ also tempers out the [[marvel comma]] ~~on the patent val~~ in the 7-limit.		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.

	293bb val, with 170\293 fifth, is ~~an optimal~~ tuning for the meantone temperament.		The 293bb val, with 170\293 fifth, is a good tuning for the meantone temperament.

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

	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.		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.

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

	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|293~~\|columns=10~~}}		{{Harmonics in equal\|293}}

	~~=== 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]].~~

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

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	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 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.		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 ===
			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 ==
			; [[Cinnamon Mavka]]
	* [https://www.youtube.com/watch?v=KYcS2hSd93Y Whiplash~~] by [[Cinnamon Mavka]~~] – using the Symmetry454[52] scale.		* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] – 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]]~~

Eliora: /* Theory */

2022-12-23T01:20:19Z

Theory

← Older revision		Revision as of 01:20, 23 December 2022
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	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.		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, where it tempers out ~~{{monzo\| 8 14 -13 }} (~~[[parakleisma]]) and {{monzo\| -40 15 7 }}. 293edo also tempers out the [[marvel comma]] on the patent val in the 7-limit.		Nonetheless, it is [[consistent]] in the 5-limit, where it tempers out the [[parakleisma]] and {{monzo\| -40 15 7 }}. 293edo also tempers out the [[marvel comma]] on the patent val in the 7-limit.

	293bb val, with 170\293 fifth, is an optimal tuning for the meantone temperament.		293bb val, with 170\293 fifth, is an optimal tuning for the meantone temperament.