257edo - Revision history

ArrowHead294 at 16:38, 20 February 2025

2025-02-20T16:38:09Z

← Older revision		Revision as of 16:38, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|257~~}}		{{ED intro}}

	257edo is in[[consistent]] to the [[5-odd-limit]], with significant errors on [[harmonic]]s [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], and [[9/1\|9]].		257edo is in[[consistent]] to the [[5-odd-limit]], with significant errors on [[harmonic]]s [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], and [[9/1\|9]].

FloraC: Ampersand -> ampersand comma to distinguish it from the temp

2025-01-18T09:14:18Z

Ampersand -> ampersand comma to distinguish it from the temp

← Older revision		Revision as of 09:14, 18 January 2025
Line 4:		Line 4:
	257edo is in[[consistent]] to the [[5-odd-limit]], with significant errors on [[harmonic]]s [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], and [[9/1\|9]].		257edo is in[[consistent]] to the [[5-odd-limit]], with significant errors on [[harmonic]]s [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], and [[9/1\|9]].

	In the 7-limit, there is a number of mappings to be considered. First is the {{val\| 257 407 597 721 }} (patent val), where the equal temperament [[tempering out\|tempers out]] 393216/390625 ([[würschmidt comma]]) in the 5-limit and [[1029/1024]] and 177147/175000 in the 7-limit. Using the 257d val, {{val\| 257 407 597 '''722''' }} it tempers out [[1728/1715]], 413343/409600, and [[703125/702464]] in the 7-limit. Using the 257bd val, {{val\| 257 '''408''' 597 '''722''' }}, it tempers out 15625/15552 ([[15625/15552\|kleisma]]) in the 5-limit; [[4000/3969]], [[6144/6125]], and 40353607/39858075 in the 7-limit.		In the 7-limit, there are a number of mappings to be considered. First is the {{val\| 257 407 597 721 }} ([[patent val]]), where the equal temperament [[tempering out\|tempers out]] 393216/390625 ([[würschmidt comma]]) in the 5-limit and [[1029/1024]] and 177147/175000 in the 7-limit. Using the 257d val, {{val\| 257 407 597 '''722''' }} it tempers out [[1728/1715]], 413343/409600, and [[703125/702464]] in the 7-limit. Using the 257bd val, {{val\| 257 '''408''' 597 '''722''' }}, it tempers out 15625/15552 ([[15625/15552\|kleisma]]) in the 5-limit; [[4000/3969]], [[6144/6125]], and 40353607/39858075 in the 7-limit.

	Using the 257c val, {{val\| 257 407 '''596''' 721 }} it tempers out 34171875/33554432 ([[ampersand]]) and {{monzo\| -3 -23 17}} ([[~~Maja~~ family\|maja comma]]) in the 5-limit; [[225/224]], 1029/1024, and {{monzo\| 2 -25 15 1 }} in the 7-limit; [[243/242]], [[385/384]], [[441/440]], and {{monzo\| 0 -2 16 -1 -9 }} in the 11-limit, ~~providing for~~ the 11-limit [[~~Gamelismic clan\|~~miracle ~~temperament~~]].		Using the 257c val, {{val\| 257 407 '''596''' 721 }} it tempers out 34171875/33554432 ([[ampersand comma]]) and {{monzo\| -3 -23 17 }} ([[maja family\|maja comma]]) in the 5-limit; [[225/224]], 1029/1024, and {{monzo\| 2 -25 15 1 }} in the 7-limit; [[243/242]], [[385/384]], [[441/440]], and {{monzo\| 0 -2 16 -1 -9 }} in the 11-limit, supporting the 11-limit [[miracle]] temperament.

	In higher limits, 257edo is a strong 2.11.13.15.27 [[subgroup]] tuning, and it is overall good at the 2.27.15.11.13.37.41.49.53.59.67 subgroup. A [[comma basis]] for the 2.27.15.11.13 subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.		In higher limits, 257edo is a strong 2.11.13.15.27 [[subgroup]] tuning, and it is overall good at the 2.27.15.11.13.37.41.49.53.59.67 subgroup. A [[comma basis]] for the 2.27.15.11.13 subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.

FloraC: Cleanup +1

2024-03-14T08:24:00Z

Cleanup +1

← Older revision		Revision as of 08:24, 14 March 2024
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|257}}		{{EDO intro\|257}}
	~~== Theory ==~~
	257edo is in[[consistent]] to the [[5-odd-limit]], with significant errors on [[harmonic]]s [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], and [[9/1\|9]].		257edo is in[[consistent]] to the [[5-odd-limit]], with significant errors on [[harmonic]]s [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], and [[9/1\|9]].

	In the 7-limit, there is a number of mappings to be considered. First is the {{val\| 257 407 597 721 }} (patent val), where the equal temperament [[tempering out\|tempers out]] 393216/390625 ([[würschmidt comma]]) in the 5-limit and [[1029/1024]] and 177147/175000 in the 7-limit. Using the 257bd val, {{val\| 257 '''408''' 597 '''722''' }}, it tempers out 15625/15552 ([[15625/15552\|kleisma]]) in the 5-limit; [[4000/3969]], [[6144/6125]], and 40353607/39858075 in the 7-limit.		In the 7-limit, there is a number of mappings to be considered. First is the {{val\| 257 407 597 721 }} (patent val), where the equal temperament [[tempering out\|tempers out]] 393216/390625 ([[würschmidt comma]]) in the 5-limit and [[1029/1024]] and 177147/175000 in the 7-limit. Using the 257d val, {{val\| 257 407 597 '''722''' }} it tempers out [[1728/1715]], 413343/409600, and [[703125/702464]] in the 7-limit. Using the 257bd val, {{val\| 257 '''408''' 597 '''722''' }}, it tempers out 15625/15552 ([[15625/15552\|kleisma]]) in the 5-limit; [[4000/3969]], [[6144/6125]], and 40353607/39858075 in the 7-limit.

	Using the 257c val, {{val\| 257 407 '''596''' 721 }} it tempers out 34171875/33554432 ([[ampersand]]) and {{monzo\| -3 -23 17}} ([[Maja family\|maja comma]]) in the 5-limit; [[225/224]], 1029/1024, and {{monzo\| 2 -25 15 1 }} in the 7-limit; [[243/242]], [[385/384]], [[441/440]], and {{monzo\| 0 -2 16 -1 -9 }} in the 11-limit, providing for the 11-limit [[Gamelismic clan\|miracle temperament]]~~. Using the 257d val, {{val\| 257 407 597 '''722''' }} it tempers out [[1728/1715]], 413343/409600, and [[703125/702464]] in the 7-limit~~.		Using the 257c val, {{val\| 257 407 '''596''' 721 }} it tempers out 34171875/33554432 ([[ampersand]]) and {{monzo\| -3 -23 17}} ([[Maja family\|maja comma]]) in the 5-limit; [[225/224]], 1029/1024, and {{monzo\| 2 -25 15 1 }} in the 7-limit; [[243/242]], [[385/384]], [[441/440]], and {{monzo\| 0 -2 16 -1 -9 }} in the 11-limit, providing for the 11-limit [[Gamelismic clan\|miracle temperament]].

	In higher limits, 257edo is a strong 2.11.13.15.27 [[subgroup]] tuning, and it is overall good at the 2.27.15.11.13.37.41.49.53.59.67 subgroup. A comma basis for the 2.27.15.11.13 subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.		In higher limits, 257edo is a strong 2.11.13.15.27 [[subgroup]] tuning, and it is overall good at the 2.27.15.11.13.37.41.49.53.59.67 subgroup. A [[comma basis]] for the 2.27.15.11.13 subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.

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

FloraC: Cleanup; -properties not related to the edo

2024-03-14T08:22:20Z

Cleanup; -properties not related to the edo

← Older revision		Revision as of 08:22, 14 March 2024
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	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|257}}		{{EDO intro\|257}}
	==Theory==		== Theory ==
	257edo is in[[consistent]] to the 5-limit ~~and higher limit~~, with significant errors on ~~harmonics~~ 3, 5, 7, and 9.		257edo is in[[consistent]] to the [[5-odd-limit]], with significant errors on [[harmonic]]s [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], and [[9/1\|9]].

	In the 7-limit, there is a number of mappings to be considered. First is the {{val\|257 407 597 721}} (patent val), where ~~257edo~~ tempers out 393216/390625 ([[~~Würschmidt~~ comma]]) in the 5-limit and 1029/1024, 177147/175000~~, and 393216/390625~~ in the 7-limit. Using the 257bd val, {{val\|257 '''408''' 597 '''722'''}}, it tempers out 15625/15552 (kleisma) in the 5-limit; 4000/3969, 6144/6125, and 40353607/39858075 in the 7-limit.		In the 7-limit, there is a number of mappings to be considered. First is the {{val\| 257 407 597 721 }} (patent val), where the equal temperament [[tempering out\|tempers out]] 393216/390625 ([[würschmidt comma]]) in the 5-limit and [[1029/1024]] and 177147/175000 in the 7-limit. Using the 257bd val, {{val\| 257 '''408''' 597 '''722''' }}, it tempers out 15625/15552 ([[15625/15552\|kleisma]]) in the 5-limit; [[4000/3969]], [[6144/6125]], and 40353607/39858075 in the 7-limit.

	Using the 257c val, {{~~monzo~~\|257 407 '''596''' 721}} it tempers out 34171875/33554432 (ampersand ~~comma~~) and ~~762939453125/753145430616~~ ([[Maja family\|maja comma]]) in the 5-limit; 225/224, 1029/1024, and ~~854492187500/847288609443~~ in the 7-limit; 243/242, 385/384, 441/440, and ~~152587890625/148550704533~~ in the 11-limit, providing for the 11-limit [[Gamelismic clan\|miracle temperament]]. Using the 257d val, {{~~monzo~~\|257 407 597 '''722'''}} it tempers out 1728/1715, 413343/409600, and 703125/702464 in the 7-limit.		Using the 257c val, {{val\| 257 407 '''596''' 721 }} it tempers out 34171875/33554432 ([[ampersand]]) and {{monzo\| -3 -23 17}} ([[Maja family\|maja comma]]) in the 5-limit; [[225/224]], 1029/1024, and {{monzo\| 2 -25 15 1 }} in the 7-limit; [[243/242]], [[385/384]], [[441/440]], and {{monzo\| 0 -2 16 -1 -9 }} in the 11-limit, providing for the 11-limit [[Gamelismic clan\|miracle temperament]]. Using the 257d val, {{val\| 257 407 597 '''722''' }} it tempers out [[1728/1715]], 413343/409600, and [[703125/702464]] in the 7-limit.

	In higher limits, 257edo is a strong 2.11.13.15.27 subgroup tuning, and it is overall good at the 2.11.13~~.15.27~~.37.41.49.53.59.67 subgroup. A comma basis for the 2.11.13~~.15.27~~ subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.		In higher limits, 257edo is a strong 2.11.13.15.27 [[subgroup]] tuning, and it is overall good at the 2.27.15.11.13.37.41.49.53.59.67 subgroup. A comma basis for the 2.27.15.11.13 subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.

	=== Odd harmonics ===		=== Odd harmonics ===
	{{~~harmonics~~ in equal\|257}}		{{Harmonics in equal\|257}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	257edo is the 55th [[prime ~~EDO~~]].		257edo is the 55th [[prime edo]].
	~~== Trivia ==~~
	~~257 is also the [[wikipedia:257-gon\|number of sides of a polygon]] that is known for being constructed with compass and straightedge.~~

	~~[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->~~
	~~[[Category:Prime EDO]]~~

Eliora: cleanup

2024-01-03T17:20:51Z

cleanup

← Older revision		Revision as of 17:20, 3 January 2024
Line 2:		Line 2:
	{{EDO intro\|257}}		{{EDO intro\|257}}
	==Theory==		==Theory==
	{{~~harmonics~~ in ~~equal~~\|257}}		257edo is in[[consistent]] to the 5-limit and higher limit, with significant errors on harmonics 3, 5, 7, and 9.

			In the 7-limit, there is a number of mappings to be considered. First is the {{val\|257 407 597 721}} (patent val), where 257edo tempers out 393216/390625 ([[Würschmidt comma]]) in the 5-limit and 1029/1024, 177147/175000, and 393216/390625 in the 7-limit. Using the 257bd val, {{val\|257 '''408''' 597 '''722'''}}, it tempers out 15625/15552 (kleisma) in the 5-limit; 4000/3969, 6144/6125, and 40353607/39858075 in the 7-limit.

	~~257edo is good at 2.~~11.~~13.37.41.53.59.67 subgroup~~.		Using the 257c val, {{monzo\|257 407 '''596''' 721}} it tempers out 34171875/33554432 (ampersand comma) and 762939453125/753145430616 ([[Maja family\|maja comma]]) in the 5-limit; 225/224, 1029/1024, and 854492187500/847288609443 in the 7-limit; 243/242, 385/384, 441/440, and 152587890625/148550704533 in the 11-limit, providing for the 11-limit [[Gamelismic clan\|miracle temperament]]. Using the 257d val, {{monzo\|257 407 597 '''722'''}} it tempers out 1728/1715, 413343/409600, and 703125/702464 in the 7-limit.

	~~It is inconsistent to the 5-limit and~~ higher ~~limit~~, ~~with four mappings possible for the 7-limit: <257 407 597 721\| (patent val), <257 408 597 722\| (257bd), <257 407 596 721\| (257c)~~, and ~~<257 407 597 722\| (257d). Using the patent val,~~ it ~~tempers out 393216/390625 ([[Würschmidt comma]]) and \|-36 33 -7> in~~ the ~~5-limit; 1029/1024, 177147/175000, and 393216/390625 in the 7-limit~~. ~~Using the 257bd val, it tempers out 15625/15552 (kleisma) and \|69 -42 -1> in the 5-limit; 4000/3969, 6144/6125, and 40353607/39858075 in the 7-limit~~. ~~Using the 257c val, it tempers out 34171875/33554432 (ampersand~~ comma) and 762939453125/753145430616 ([[Maja family\|maja comma]]) in the 5-limit; 225/224, 1029/1024, and 854492187500/847288609443 in the 7-limit; 243/242, 385/384, 441/440, and 152587890625/148550704533 in the 11-limit, providing for the 11~~-limit [[Gamelismic clan\|miracle temperament]]~~. ~~Using the 257d val~~, ~~it tempers out 1728~~/~~1715~~, ~~413343~~/~~409600~~, ~~and 703125~~/~~702464 in the 7-limit~~.		In higher limits, 257edo is a strong 2.11.13.15.27 subgroup tuning, and it is overall good at the 2.11.13.15.27.37.41.49.53.59.67 subgroup. A comma basis for the 2.11.13.15.27 subgroup is {4225/4224, 256000/255879, 225000/224939, 4159375/4153344}.

			=== Odd harmonics ===
			{{harmonics in equal\|257}}

			=== Subsets and supersets ===
	257edo is the 55th [[prime EDO]].		257edo is the 55th [[prime EDO]].
			== Trivia ==
	257 is also the [[wikipedia:257-gon\|number of sides of a polygon]] that is known for being constructed with compass and straightedge.		257 is also the [[wikipedia:257-gon\|number of sides of a polygon]] that is known for being constructed with compass and straightedge.

	[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->		[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->
	[[Category:Prime EDO]]		[[Category:Prime EDO]]

Plumtree: Infobox ET added

2022-10-04T21:28:45Z

Infobox ET added

← Older revision		Revision as of 21:28, 4 October 2022
Line 1:		Line 1:
			{{Infobox ET}}
	{{EDO intro\|257}}		{{EDO intro\|257}}
	==Theory==		==Theory==

Eliora at 11:52, 19 July 2022

2022-07-19T11:52:12Z

← Older revision		Revision as of 11:52, 19 July 2022
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	~~'''257edo''' is the [[~~EDO\|equal ~~division of the octave]] into~~ 257 ~~parts of 4~~.~~66926 [[cent]]s each~~. It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: <257 407 597 721\| (patent val), <257 408 597 722\| (257bd), <257 407 596 721\| (257c), and <257 407 597 722\| (257d). Using the patent val, it tempers out 393216/390625 ([[Würschmidt comma]]) and \|-36 33 -7> in the 5-limit; 1029/1024, 177147/175000, and 393216/390625 in the 7-limit. Using the 257bd val, it tempers out 15625/15552 (kleisma) and \|69 -42 -1> in the 5-limit; 4000/3969, 6144/6125, and 40353607/39858075 in the 7-limit. Using the 257c val, it tempers out 34171875/33554432 (ampersand comma) and 762939453125/753145430616 ([[Maja family\|maja comma]]) in the 5-limit; 225/224, 1029/1024, and 854492187500/847288609443 in the 7-limit; 243/242, 385/384, 441/440, and 152587890625/148550704533 in the 11-limit, providing for the 11-limit [[Gamelismic clan\|miracle temperament]]. Using the 257d val, it tempers out 1728/1715, 413343/409600, and 703125/702464 in the 7-limit.		{{EDO intro\|257}}
			==Theory==
			{{harmonics in equal\|257}}

			257edo is good at 2.11.13.37.41.53.59.67 subgroup.

			It is inconsistent to the 5-limit and higher limit, with four mappings possible for the 7-limit: <257 407 597 721\| (patent val), <257 408 597 722\| (257bd), <257 407 596 721\| (257c), and <257 407 597 722\| (257d). Using the patent val, it tempers out 393216/390625 ([[Würschmidt comma]]) and \|-36 33 -7> in the 5-limit; 1029/1024, 177147/175000, and 393216/390625 in the 7-limit. Using the 257bd val, it tempers out 15625/15552 (kleisma) and \|69 -42 -1> in the 5-limit; 4000/3969, 6144/6125, and 40353607/39858075 in the 7-limit. Using the 257c val, it tempers out 34171875/33554432 (ampersand comma) and 762939453125/753145430616 ([[Maja family\|maja comma]]) in the 5-limit; 225/224, 1029/1024, and 854492187500/847288609443 in the 7-limit; 243/242, 385/384, 441/440, and 152587890625/148550704533 in the 11-limit, providing for the 11-limit [[Gamelismic clan\|miracle temperament]]. Using the 257d val, it tempers out 1728/1715, 413343/409600, and 703125/702464 in the 7-limit.

	257edo is the 55th [[prime EDO]].		257edo is the 55th [[prime EDO]].

Fredg999 category edits: Sort key

2022-07-03T19:05:27Z

Sort key

← Older revision		Revision as of 19:05, 3 July 2022
Line 5:		Line 5:
	257 is also the [[wikipedia:257-gon\|number of sides of a polygon]] that is known for being constructed with compass and straightedge.		257 is also the [[wikipedia:257-gon\|number of sides of a polygon]] that is known for being constructed with compass and straightedge.

	[[Category:Equal divisions of the octave]]		[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->
	[[Category:Prime EDO]]		[[Category:Prime EDO]]

Eliora at 17:31, 5 January 2022

2022-01-05T17:31:47Z

← Older revision		Revision as of 17:31, 5 January 2022
Line 2:		Line 2:

	257edo is the 55th [[prime EDO]].		257edo is the 55th [[prime EDO]].

			257 is also the [[wikipedia:257-gon\|number of sides of a polygon]] that is known for being constructed with compass and straightedge.

	[[Category:Equal divisions of the octave]]		[[Category:Equal divisions of the octave]]
	[[Category:Prime EDO]]		[[Category:Prime EDO]]

Keenan Pepper category edits: Moving from Category:Edo to Category:Equal divisions of the octave using Cat-a-lot

2020-12-04T23:13:22Z

Moving from Category:Edo to Category:Equal divisions of the octave using Cat-a-lot

← Older revision		Revision as of 23:13, 4 December 2020
Line 3:		Line 3:
	257edo is the 55th [[prime EDO]].		257edo is the 55th [[prime EDO]].

	[[Category:~~Edo~~]]		[[Category:Equal divisions of the octave]]
	[[Category:Prime EDO]]		[[Category:Prime EDO]]