1578edo - Revision history

Lériendil at 02:30, 2 April 2025

2025-04-02T02:30:42Z

← Older revision		Revision as of 02:30, 2 April 2025
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	{{ED intro}}		{{ED intro}}

	1578edo is a very strong higher limit system, and is a [[zeta edo\|zeta peak, peak integer, integral and gap edo]]. It is [[consistency\|distinctly consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]].		1578edo is a very strong higher limit system, and is a [[zeta edo\|zeta peak, peak integer, integral and gap edo]]. It is [[consistency\|distinctly consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]]. It additionally contains an extremely accurate approximation of [[quarter-comma meantone]] inherited from 789edo.

	It ~~is also the most~~ accurate ~~edo below 10000 to approximate~~ [[quarter-comma meantone]]~~, with its approximation of {{monzo\| 0 0 1/4 }} with a relative error of only 0.06%~~.

	Some 31-limit or lower superparticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/26999… .		Some 31-limit or lower superparticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/26999… .

ArrowHead294 at 13:52, 20 February 2025

2025-02-20T13:52:52Z

← Older revision		Revision as of 13:52, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|1578~~}}		{{ED intro}}

	1578edo is a very strong higher limit system, and is a [[zeta edo\|zeta peak, peak integer, integral and gap edo]]. It is [[consistency\|distinctly consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]].		1578edo is a very strong higher limit system, and is a [[zeta edo\|zeta peak, peak integer, integral and gap edo]]. It is [[consistency\|distinctly consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]].

Lériendil: /* Subsets and supersets */

2024-09-04T11:14:26Z

Subsets and supersets

← Older revision		Revision as of 11:14, 4 September 2024
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 1578 factors into {{factorization\|1578}}, 1578edo has subset edos 2, 3, 6, 263, 526, and 789.		Since 1578 factors into {{factorization\|1578}}, 1578edo has subset edos [[2edo\|2]], [[3edo\|3]], [[6edo\|6]], [[263edo\|263]], [[526edo\|526]], and [[789edo\|789]].

FloraC: Adopt template: Factorization; misc. cleanup

2023-10-30T14:01:01Z

Adopt template: Factorization; misc. cleanup

← Older revision		Revision as of 14:01, 30 October 2023
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	{{EDO intro\|1578}}		{{EDO intro\|1578}}

	1578edo is a very strong higher limit system, and is a [[~~The Riemann~~ zeta ~~function and tuning #Zeta EDO lists~~\|zeta peak, peak integer, integral and gap edo]]. It is ~~distinctly~~ [[consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]].		1578edo is a very strong higher limit system, and is a [[zeta edo\|zeta peak, peak integer, integral and gap edo]]. It is [[consistency\|distinctly consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]].

	It is also the most accurate edo below 10000 to approximate [[quarter-comma meantone]], with its approximation of {{monzo\| 0 0 1/4 }} with a relative error of only 0.06%.		It is also the most accurate edo below 10000 to approximate [[quarter-comma meantone]], with its approximation of {{monzo\| 0 0 1/4 }} with a relative error of only 0.06%.
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	{{Harmonics in equal\|1578\|columns=11}}		{{Harmonics in equal\|1578\|columns=11}}

	=== ~~Divisors~~ ===		=== Subsets and supersets ===
	Since 1578 factors into ~~2 × 3 × 263~~, 1578edo has subset edos 2, 3, 6, 263, 526, and 789.		Since 1578 factors into {{factorization\|1578}}, 1578edo has subset edos 2, 3, 6, 263, 526, and 789.

BudjarnLambeth at 06:55, 9 July 2023

2023-07-09T06:55:40Z

← Older revision		Revision as of 06:55, 9 July 2023
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	~~{{novelty}}{{stub}}~~{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|1578}}		{{EDO intro\|1578}}

BudjarnLambeth at 05:04, 9 July 2023

2023-07-09T05:04:51Z

← Older revision		Revision as of 05:04, 9 July 2023
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	{{Infobox ET}}		{{novelty}}{{stub}}{{Infobox ET}}
	{{EDO intro\|1578}}		{{EDO intro\|1578}}

FloraC: +divisors

2023-01-08T04:27:14Z

+divisors

← Older revision		Revision as of 04:27, 8 January 2023
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	{{EDO intro\|1578}}		{{EDO intro\|1578}}

	~~== Theory ==~~
	1578edo is a very strong higher limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak, peak integer, integral and gap edo]]. It is distinctly [[consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]].		1578edo is a very strong higher limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak, peak integer, integral and gap edo]]. It is distinctly [[consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]].

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	{{Harmonics in equal\|1578\|columns=11}}		{{Harmonics in equal\|1578\|columns=11}}

	~~[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->~~		=== Divisors ===
			Since 1578 factors into 2 × 3 × 263, 1578edo has subset edos 2, 3, 6, 263, 526, and 789.

FloraC: Cleanup

2022-10-29T09:23:46Z

Cleanup

← Older revision		Revision as of 09:23, 29 October 2022
Line 2:		Line 2:
	{{EDO intro\|1578}}		{{EDO intro\|1578}}

	It is a very strong higher limit system, and is a [[~~The_Riemann_Zeta_Function_and_Tuning~~#Zeta EDO lists\|zeta peak, peak integer, integral and gap edo]]. It is distinctly consistent through the [[29-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-~~Euclidean_temperament_measures~~#TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]].		== Theory ==
			1578edo is a very strong higher limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak, peak integer, integral and gap edo]]. It is distinctly [[consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]].

	It is also the most accurate edo below 10000 to approximate [[quarter-comma meantone]], with its approximation of {{monzo\|0 0 1/4}} with a relative error of only 0.06%.		It is also the most accurate edo below 10000 to approximate [[quarter-comma meantone]], with its approximation of {{monzo\| 0 0 1/4 }} with a relative error of only 0.06%.

	~~== Theory ==~~		Some 31-limit or lower superparticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/26999… .
	Some 31 limit or lower superparticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/~~26999...~~ .

	=== Prime harmonics ===		=== Prime harmonics ===

Plumtree: Infobox ET added

2022-10-04T22:11:06Z

Infobox ET added

← Older revision		Revision as of 22:11, 4 October 2022
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			{{Infobox ET}}
	{{EDO intro\|1578}}		{{EDO intro\|1578}}

Fredg999: Misc. edits

2022-08-21T19:11:32Z

Misc. edits

← Older revision		Revision as of 19:11, 21 August 2022
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	~~The '''1578 equal division''' divides the octave into 1578 equal parts of 0.7605 cents each. It is a very strong higher limit system, and is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta~~ EDO ~~lists~~\|zeta peak, peak integer, integral and gap edo]]. It is distinctly consistent through the 29 limit, and is the first edo past 311 with a lower 29-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]]. It is also the lowest past 311 in the 31 limit, the lowest past 581 in the 23 limit, and the lowest past 1178 in the 19 limit. It is also quite strong taken just as an 11-limit system; the only smaller edo with a lower 11-limit relative error is [[342edo\|342]].		{{EDO intro\|1578}}

	It is also the ~~most accurate EDO below 10000 for approximating quarter comma meantone~~, ~~specifically~~ in ~~its m~~-~~chromatic 7L5s MOS scale~~.		It is a very strong higher limit system, and is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists\|zeta peak, peak integer, integral and gap edo]]. It is distinctly consistent through the [[29-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean_temperament_measures#TE simple badness\|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]].

			It is also the most accurate edo below 10000 to approximate [[quarter-comma meantone]], with its approximation of {{monzo\|0 0 1/4}} with a relative error of only 0.06%.

			== Theory ==
	Some 31 limit or lower superparticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/26999... .		Some 31 limit or lower superparticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/26999... .