494edo - Revision history

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:38:11Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:38, 13 March 2026
Line 139:		Line 139:
	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|}		\|}
	<nowiki/>* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[normal ~~lists~~\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

FloraC: Update

2025-03-16T13:40:24Z

Update

← Older revision		Revision as of 13:40, 16 March 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	494 is a very strong [[13-limit\|13]]- and [[17-limit]] equal temperament. 494edo is a [[zeta edo\|zeta peak and zeta peak integer edo]] and [[consistency\|distinctly consistent]] through the [[17-odd-limit]]. It [[tempering out\|tempers out]] the [[enneadeca]], {{monzo\| -14 -19 19 }}, the [[~~tricot~~ comma]], {{monzo\| 39 -29 3 }}, and the [[kwazy comma]], {{monzo\| -53 10 16 }} in the [[5-limit]]. In the [[7-limit]], it tempers out [[4375/4374]] and [[703125/702464]]; in the [[11-limit]] [[3025/3024]] and [[9801/9800]]; in the [[13-limit]] [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and [[6656/6655]]; and in the 17-limit, [[1156/1155]], 1275/1274, 2431/2430, and 2500/2499.		494 is a very strong [[13-limit\|13]]- and [[17-limit]] equal temperament. 494edo is a [[zeta edo\|zeta peak and zeta peak integer edo]] and [[consistency\|distinctly consistent]] through the [[17-odd-limit]]. It [[tempering out\|tempers out]] the [[enneadeca]], {{monzo\| -14 -19 19 }}, the [[alphatricot comma]], {{monzo\| 39 -29 3 }}, and the [[kwazy comma]], {{monzo\| -53 10 16 }} in the [[5-limit]]. In the [[7-limit]], it tempers out [[4375/4374]] and [[703125/702464]]; in the [[11-limit]] [[3025/3024]] and [[9801/9800]]; in the [[13-limit]] [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and [[6656/6655]]; and in the 17-limit, [[1156/1155]], 1275/1274, 2431/2430, and 2500/2499.

	Since the step size is close to [[729/728]], the squbema, the accepted name for 494edo's step is ''squb''.		Since the step size is close to [[729/728]], the squbema, the accepted name for 494edo's step is ''squb''.
Line 101:		Line 101:
	\| 565.99		\| 565.99
	\| 104/75		\| 104/75
	\| [[~~Trillium~~]]		\| [[Alphatrillium]]
	\|-		\|-
	\| 2		\| 2
Line 148:		Line 148:
	[[Category:Kwazy]]		[[Category:Kwazy]]
	[[Category:Listen]]		[[Category:Listen]]
	[[Category:~~Tricot~~]]		[[Category:Alphatricot]]

FloraC: /* Regular temperament properties */ crazy update

2025-03-06T11:28:09Z

Regular temperament properties: crazy update

← Older revision		Revision as of 11:28, 6 March 2025
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	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal<br />8ve stretch (¢)		! rowspan="2" \| Optimal<br>8ve stretch (¢)
	! colspan="2" \| Tuning error		! colspan="2" \| Tuning error
	\|-		\|-
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	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| 783 -494 }}		\| {{Monzo\| 783 -494 }}
	\| {{~~mapping~~\| 494 783 }}		\| {{Mapping\| 494 783 }}
	\| −0.0219		\| −0.0219
	\| 0.0219		\| 0.0219
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	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| -14 -19 19 }}, {{monzo\| 39 -23 3 }}		\| {{Monzo\| -14 -19 19 }}, {{monzo\| 39 -23 3 }}
	\| {{~~mapping~~\| 494 783 1147 }}		\| {{Mapping\| 494 783 1147 }}
	\| −0.0032		\| −0.0032
	\| 0.0318		\| 0.0318
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	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 703125/702464, {{monzo\| 21 3 1 -10 }}		\| 4375/4374, 703125/702464, {{monzo\| 21 3 1 -10 }}
	\| {{~~mapping~~\| 494 783 1147 1387 }}		\| {{Mapping\| 494 783 1147 1387 }}
	\| −0.0385		\| −0.0385
	\| 0.0670		\| 0.0670
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 3025/3024, 4375/4374, 131072/130977, 234375/234256		\| 3025/3024, 4375/4374, 131072/130977, 234375/234256
	\| {{~~mapping~~\| 494 783 1147 1387 1709 }}		\| {{Mapping\| 494 783 1147 1387 1709 }}
	\| −0.0365		\| −0.0365
	\| 0.0600		\| 0.0600
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213		\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213
	\| {{~~mapping~~\| 494 783 1147 1387 1709 1828 }}		\| {{Mapping\| 494 783 1147 1387 1709 1828 }}
	\| −0.0286		\| −0.0286
	\| 0.0576		\| 0.0576
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095		\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095
	\| {{~~mapping~~\| 494 783 1147 1387 1709 1828 2019 }}		\| {{Mapping\| 494 783 1147 1387 1709 1828 2019 }}
	\| −0.0069		\| −0.0069
	\| 0.0752		\| 0.0752
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	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperaments		! Temperaments
	\|-		\|-
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	\| 565.99		\| 565.99
	\| 104/75		\| 104/75
	\| [[~~Tricot]] / [[trillium~~]]		\| [[Trillium]]
	\|-		\|-
	\| 2		\| 2
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	\| 162.75		\| 162.75
	\| 1125/1024		\| 1125/1024
	\| [[~~Kwazy~~]]		\| [[Crazy]]
	\|-		\|-
	\| 2		\| 2
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	\|-		\|-
	\| 13		\| 13
	\| 205\494<br />(15\494)		\| 205\494<br>(15\494)
	\| 497.98<br />(36.43)		\| 497.98<br/>(36.43)
	\| 4/3<br />(?)		\| 4/3<br>(?)
	\| [[Aluminium]]		\| [[Aluminium]]
	\|-		\|-
	\| 19		\| 19
	\| 205\494<br />(3\494)		\| 205\494<br>(3\494)
	\| 497.98<br />(7.29)		\| 497.98<br>(7.29)
	\| 4/3<br />(225/224)		\| 4/3<br>(225/224)
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|-		\|-
	\| 38		\| 38
	\| 205\494<br />(3\494)		\| 205\494<br>(3\494)
	\| 497.98<br />(7.29)		\| 497.98<br>(7.29)
	\| 4/3<br />(225/224)		\| 4/3<br>(225/224)
	\| [[Hemienneadecal]]		\| [[Hemienneadecal]]
	\|-		\|-
	\| 38		\| 38
	\| 109\494<br />(5\494)		\| 109\494<br>(5\494)
	\| 264.78<br />(12.15)		\| 264.78<br>(12.15)
	\| 500/429<br />(144/143)		\| 500/429<br>(144/143)
	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal~~ lists\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[normal lists\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

ArrowHead294 at 15:05, 20 February 2025

2025-02-20T15:05:53Z

← Older revision		Revision as of 15:05, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|494~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 14:24, 16 January 2025

2025-01-16T14:24:03Z

← Older revision		Revision as of 14:24, 16 January 2025
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	\| {{monzo\| 783 -494 }}		\| {{monzo\| 783 -494 }}
	\| {{mapping\| 494 783 }}		\| {{mapping\| 494 783 }}
	\| ~~−0~~.0219		\| −0.0219
	\| 0.0219		\| 0.0219
	\| 0.90		\| 0.90
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	\| {{monzo\| -14 -19 19 }}, {{monzo\| 39 -23 3 }}		\| {{monzo\| -14 -19 19 }}, {{monzo\| 39 -23 3 }}
	\| {{mapping\| 494 783 1147 }}		\| {{mapping\| 494 783 1147 }}
	\| ~~−0~~.0032		\| −0.0032
	\| 0.0318		\| 0.0318
	\| 1.31		\| 1.31
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	\| 4375/4374, 703125/702464, {{monzo\| 21 3 1 -10 }}		\| 4375/4374, 703125/702464, {{monzo\| 21 3 1 -10 }}
	\| {{mapping\| 494 783 1147 1387 }}		\| {{mapping\| 494 783 1147 1387 }}
	\| ~~−0~~.0385		\| −0.0385
	\| 0.0670		\| 0.0670
	\| 2.76		\| 2.76
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	\| 3025/3024, 4375/4374, 131072/130977, 234375/234256		\| 3025/3024, 4375/4374, 131072/130977, 234375/234256
	\| {{mapping\| 494 783 1147 1387 1709 }}		\| {{mapping\| 494 783 1147 1387 1709 }}
	\| ~~−0~~.0365		\| −0.0365
	\| 0.0600		\| 0.0600
	\| 2.47		\| 2.47
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	\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213		\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213
	\| {{mapping\| 494 783 1147 1387 1709 1828 }}		\| {{mapping\| 494 783 1147 1387 1709 1828 }}
	\| ~~−0~~.0286		\| −0.0286
	\| 0.0576		\| 0.0576
	\| 2.37		\| 2.37
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	\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095		\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095
	\| {{mapping\| 494 783 1147 1387 1709 1828 2019 }}		\| {{mapping\| 494 783 1147 1387 1709 1828 2019 }}
	\| ~~−0~~.0069		\| −0.0069
	\| 0.0752		\| 0.0752
	\| 3.09		\| 3.09
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	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if ~~it is~~ distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

ArrowHead294: /* Intervals */

2025-01-16T14:11:48Z

Intervals

← Older revision		Revision as of 14:11, 16 January 2025
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	== Intervals ==		== Intervals ==
	{{Main\| Table of 494edo intervals }}		{{Main\| Table of 494edo intervals }}
			{{Q-odd-limit intervals\|494}}

	== Regular temperament properties ==		== Regular temperament properties ==

ArrowHead294 at 12:13, 16 November 2024

2024-11-16T12:13:05Z

← Older revision		Revision as of 12:13, 16 November 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.3		\| 2.3
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	\| 0.0752		\| 0.0752
	\| 3.09		\| 3.09
	~~{{comma basis end}~~}		\|}
	* 494et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 13- and 17-limit. It is the first past [[270edo\|270]] with a lower 13-limit relative error, and the first past [[72edo\|72]] with a lower 17-limit relative error. It is narrowly beaten by [[684edo\|684]] in terms of 13-limit absolute error and by [[581edo\|581]] in terms of 17-limit absolute error. Not until [[1506edo\|1506]] do we reach an equal temperament with a lower relative error in either subgroup.		* 494et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 13- and 17-limit. It is the first past [[270edo\|270]] with a lower 13-limit relative error, and the first past [[72edo\|72]] with a lower 17-limit relative error. It is narrowly beaten by [[684edo\|684]] in terms of 13-limit absolute error and by [[581edo\|581]] in terms of 17-limit absolute error. Not until [[1506edo\|1506]] do we reach an equal temperament with a lower relative error in either subgroup.

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 1		\| 1
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	\| 500/429<br />(144/143)		\| 500/429<br />(144/143)
	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Music ==		== Music ==

ArrowHead294 at 05:21, 16 November 2024

2024-11-16T05:21:01Z

← Older revision		Revision as of 05:21, 16 November 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{monzo\| 783 -494 }}		\| {{monzo\| 783 -494 }}
	\| {{mapping\| 494 783 }}		\| {{mapping\| 494 783 }}
	\| -0.0219		\| −0.0219
	\| 0.0219		\| 0.0219
	\| 0.90		\| 0.90
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	\| {{monzo\| -14 -19 19 }}, {{monzo\| 39 -23 3 }}		\| {{monzo\| -14 -19 19 }}, {{monzo\| 39 -23 3 }}
	\| {{mapping\| 494 783 1147 }}		\| {{mapping\| 494 783 1147 }}
	\| -0.0032		\| −0.0032
	\| 0.0318		\| 0.0318
	\| 1.31		\| 1.31
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	\| 4375/4374, 703125/702464, {{monzo\| 21 3 1 -10 }}		\| 4375/4374, 703125/702464, {{monzo\| 21 3 1 -10 }}
	\| {{mapping\| 494 783 1147 1387 }}		\| {{mapping\| 494 783 1147 1387 }}
	\| -0.0385		\| −0.0385
	\| 0.0670		\| 0.0670
	\| 2.76		\| 2.76
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	\| 3025/3024, 4375/4374, 131072/130977, 234375/234256		\| 3025/3024, 4375/4374, 131072/130977, 234375/234256
	\| {{mapping\| 494 783 1147 1387 1709 }}		\| {{mapping\| 494 783 1147 1387 1709 }}
	\| -0.0365		\| −0.0365
	\| 0.0600		\| 0.0600
	\| 2.47		\| 2.47
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	\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213		\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213
	\| {{mapping\| 494 783 1147 1387 1709 1828 }}		\| {{mapping\| 494 783 1147 1387 1709 1828 }}
	\| -0.0286		\| −0.0286
	\| 0.0576		\| 0.0576
	\| 2.37		\| 2.37
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	\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095		\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095
	\| {{mapping\| 494 783 1147 1387 1709 1828 2019 }}		\| {{mapping\| 494 783 1147 1387 1709 1828 2019 }}
	\| -0.0069		\| −0.0069
	\| 0.0752		\| 0.0752
	\| 3.09		\| 3.09
	\|}		{{comma basis end}}
	* 494et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 13- and 17-limit. It is the first past [[270edo\|270]] with a lower 13-limit relative error, and the first past [[72edo\|72]] with a lower 17-limit relative error. It is narrowly beaten by [[684edo\|684]] in terms of 13-limit absolute error and by [[581edo\|581]] in terms of 17-limit absolute error. Not until [[1506edo\|1506]] do we reach an equal temperament with a lower relative error in either subgroup.		* 494et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 13- and 17-limit. It is the first past [[270edo\|270]] with a lower 13-limit relative error, and the first past [[72edo\|72]] with a lower 17-limit relative error. It is narrowly beaten by [[684edo\|684]] in terms of 13-limit absolute error and by [[581edo\|581]] in terms of 17-limit absolute error. Not until [[1506edo\|1506]] do we reach an equal temperament with a lower relative error in either subgroup.

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
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	\|-		\|-
	\| 13		\| 13
	\| 205\494<br>(15\494)		\| 205\494<br />(15\494)
	\| 497.98<br>(36.43)		\| 497.98<br />(36.43)
	\| 4/3<br>(?)		\| 4/3<br />(?)
	\| [[Aluminium]]		\| [[Aluminium]]
	\|-		\|-
	\| 19		\| 19
	\| 205\494<br>(3\494)		\| 205\494<br />(3\494)
	\| 497.98<br>(7.29)		\| 497.98<br />(7.29)
	\| 4/3<br>(225/224)		\| 4/3<br />(225/224)
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|-		\|-
	\| 38		\| 38
	\| 205\494<br>(3\494)		\| 205\494<br />(3\494)
	\| 497.98<br>(7.29)		\| 497.98<br />(7.29)
	\| 4/3<br>(225/224)		\| 4/3<br />(225/224)
	\| [[Hemienneadecal]]		\| [[Hemienneadecal]]
	\|-		\|-
	\| 38		\| 38
	\| 109\494<br>(5\494)		\| 109\494<br />(5\494)
	\| 264.78<br>(12.15)		\| 264.78<br />(12.15)
	\| 500/429<br>(144/143)		\| 500/429<br />(144/143)
	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|}		{{rank-2 end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	== Music ==		== Music ==
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	[[Category:Enneadecal]]		[[Category:Enneadecal]]
	[[Category:Kwazy]]		[[Category:Kwazy]]
			[[Category:Listen]]
	[[Category:Tricot]]		[[Category:Tricot]]
	~~[[Category:Listen]]~~

FloraC: Cleanup; clarify the title row of the rank-2 temp table; -redundant categories

2023-11-02T14:03:37Z

Cleanup; clarify the title row of the rank-2 temp table; -redundant categories

← Older revision		Revision as of 14:03, 2 November 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	~~The '''~~494 equal divisions of the octave''' ('''494edo'''), or the '''494(-tone) equal temperament''' ('''494tet''', '''494et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 494 [[equal]] parts of about 2.43 [[cent]]s each.		{{EDO intro\|494}}

	== Theory ==		== Theory ==
	494 is a very strong [[13-limit\|13]]- and [[17-limit]] equal temperament. 494edo is a [[~~The Riemann~~ zeta ~~function and tuning #Zeta EDO lists~~\|zeta peak and zeta peak integer edo]] and ~~uniquely~~ [[consistent]] through the [[17-odd-limit]]. It [[tempering out\|tempers out]] the [[enneadeca]], {{monzo\| -14 -19 19 }}, the [[tricot comma]], {{monzo\| 39 -29 3 }}, and the [[kwazy comma]], {{monzo\| -53 10 16 }} in the [[5-limit]]. In the [[7-limit]], it tempers out [[4375/4374]] and [[703125/702464]]; in the [[11-limit]] [[3025/3024]] and [[9801/9800]]; in the [[13-limit]] [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and [[6656/6655]]; and in the 17-limit, [[1156/1155]], 1275/1274, 2431/2430, and 2500/2499.		494 is a very strong [[13-limit\|13]]- and [[17-limit]] equal temperament. 494edo is a [[zeta edo\|zeta peak and zeta peak integer edo]] and [[consistency\|distinctly consistent]] through the [[17-odd-limit]]. It [[tempering out\|tempers out]] the [[enneadeca]], {{monzo\| -14 -19 19 }}, the [[tricot comma]], {{monzo\| 39 -29 3 }}, and the [[kwazy comma]], {{monzo\| -53 10 16 }} in the [[5-limit]]. In the [[7-limit]], it tempers out [[4375/4374]] and [[703125/702464]]; in the [[11-limit]] [[3025/3024]] and [[9801/9800]]; in the [[13-limit]] [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and [[6656/6655]]; and in the 17-limit, [[1156/1155]], 1275/1274, 2431/2430, and 2500/2499.

	Since the step size is close to [[729/728]], the squbema, the accepted name for 494edo's step is ''squb''.		Since the step size is close to [[729/728]], the squbema, the accepted name for 494edo's step is ''squb''.
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	{{Harmonics in equal\|494\|prec=3}}		{{Harmonics in equal\|494\|prec=3}}

	=== ~~Divisors~~ and ~~multipliers~~ ===		=== Subsets and supersets ===
	Since 494 ~~= 2 × 13 × 19~~, 494edo has subset edos {{EDOs\| 2, 13, 19, 26, 38, and 247 }}.		Since 494 factors into {{factorization\|494}}, 494edo has subset edos {{EDOs\| 2, 13, 19, 26, 38, and 247 }}.

	[[988edo]], which slices the edostep in two, provides a good correction of the 19th harmonic. [[2964edo]], which slices the edostep in six, provides an extremely precise correction of the 7th harmonic.		[[988edo]], which slices the edostep in two, provides a good correction of the 19th harmonic. [[2964edo]], which slices the edostep in six, provides an extremely precise correction of the 7th harmonic.
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	\| 2.3		\| 2.3
	\| {{monzo\| 783 -494 }}		\| {{monzo\| 783 -494 }}
	\| [{{~~val~~\| 494 783 }}]		\| {{mapping\| 494 783 }}
	\| -0.0219		\| -0.0219
	\| 0.0219		\| 0.0219
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	\| 2.3.5		\| 2.3.5
	\| {{monzo\| -14 -19 19 }}, {{monzo\| 39 -23 3 }}		\| {{monzo\| -14 -19 19 }}, {{monzo\| 39 -23 3 }}
	\| [{{~~val~~\| 494 783 1147 }}]		\| {{mapping\| 494 783 1147 }}
	\| -0.0032		\| -0.0032
	\| 0.0318		\| 0.0318
Line 45:		Line 45:
	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 703125/702464, {{monzo\| 21 3 1 -10 }}		\| 4375/4374, 703125/702464, {{monzo\| 21 3 1 -10 }}
	\| [{{~~val~~\| 494 783 1147 1387 }}]		\| {{mapping\| 494 783 1147 1387 }}
	\| -0.0385		\| -0.0385
	\| 0.0670		\| 0.0670
Line 52:		Line 52:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 3025/3024, 4375/4374, 131072/130977, 234375/234256		\| 3025/3024, 4375/4374, 131072/130977, 234375/234256
	\| [{{~~val~~\| 494 783 1147 1387 1709 }}]		\| {{mapping\| 494 783 1147 1387 1709 }}
	\| -0.0365		\| -0.0365
	\| 0.0600		\| 0.0600
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	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213		\| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 31250/31213
	\| [{{~~val~~\| 494 783 1147 1387 1709 1828 }}]		\| {{mapping\| 494 783 1147 1387 1709 1828 }}
	\| -0.0286		\| -0.0286
	\| 0.0576		\| 0.0576
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	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095		\| 1156/1155, 1275/1274, 1716/1715, 2080/2079, 2431/2430, 4096/4095
	\| [{{~~val~~\| 494 783 1147 1387 1709 1828 2019 }}]		\| {{mapping\| 494 783 1147 1387 1709 1828 2019 }}
	\| -0.0069		\| -0.0069
	\| 0.0752		\| 0.0752
	\| 3.09		\| 3.09
	\|}		\|}
	* 494et has a lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative ~~error~~]] than any previous equal temperaments in the 13- and 17-limit. It is the first past [[270edo\|270]] with a lower 13-limit relative error, and the first past [[72edo\|72]] with a lower 17-limit relative error. It is narrowly beaten by [[684edo\|684]] in terms of 13-limit absolute error and by [[581edo\|581]] in terms of 17-limit absolute error. Not until [[1506edo\|1506]] do we reach an equal temperament with a lower relative error in either subgroup.		* 494et has lower [[Tenney-Euclidean temperament measures #TE simple badness\|relative errors]] than any previous equal temperaments in the 13- and 17-limit. It is the first past [[270edo\|270]] with a lower 13-limit relative error, and the first past [[72edo\|72]] with a lower 17-limit relative error. It is narrowly beaten by [[684edo\|684]] in terms of 13-limit absolute error and by [[581edo\|581]] in terms of 17-limit absolute error. Not until [[1506edo\|1506]] do we reach an equal temperament with a lower relative error in either subgroup.

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
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	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per 8ve		! Periods<br>per 8ve
	! Generator~~<br>(Reduced)~~		! Generator*
	! Cents~~<br>(Reduced)~~		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
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	\| [[Semihemienneadecal]]		\| [[Semihemienneadecal]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Music ==		== Music ==
	* [https://www.youtube.com/watch?v=JGdBFEz7Fq8 Unknown piece in Abigail (~~Op.2, No. 4~~)~~] by [[Eliora]]~~		; [[Eliora]]
			* [https://www.youtube.com/watch?v=JGdBFEz7Fq8 ''Unknown piece in Abigail''] (2023)

	~~[[Category:17-limit]]~~
	[[Category:Enneadecal]]		[[Category:Enneadecal]]
	[[Category:Kwazy]]		[[Category:Kwazy]]
	[[Category:Tricot]]		[[Category:Tricot]]
	[[Category:Listen]]		[[Category:Listen]]

FloraC: Undo

2023-10-30T07:27:30Z

Undo

← Older revision		Revision as of 07:27, 30 October 2023
Line 5:		Line 5:
	494 is a very strong [[13-limit\|13]]- and [[17-limit]] equal temperament. 494edo is a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak and zeta peak integer edo]] and uniquely [[consistent]] through the [[17-odd-limit]]. It [[tempering out\|tempers out]] the [[enneadeca]], {{monzo\| -14 -19 19 }}, the [[tricot comma]], {{monzo\| 39 -29 3 }}, and the [[kwazy comma]], {{monzo\| -53 10 16 }} in the [[5-limit]]. In the [[7-limit]], it tempers out [[4375/4374]] and [[703125/702464]]; in the [[11-limit]] [[3025/3024]] and [[9801/9800]]; in the [[13-limit]] [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and [[6656/6655]]; and in the 17-limit, [[1156/1155]], 1275/1274, 2431/2430, and 2500/2499.		494 is a very strong [[13-limit\|13]]- and [[17-limit]] equal temperament. 494edo is a [[The Riemann zeta function and tuning #Zeta EDO lists\|zeta peak and zeta peak integer edo]] and uniquely [[consistent]] through the [[17-odd-limit]]. It [[tempering out\|tempers out]] the [[enneadeca]], {{monzo\| -14 -19 19 }}, the [[tricot comma]], {{monzo\| 39 -29 3 }}, and the [[kwazy comma]], {{monzo\| -53 10 16 }} in the [[5-limit]]. In the [[7-limit]], it tempers out [[4375/4374]] and [[703125/702464]]; in the [[11-limit]] [[3025/3024]] and [[9801/9800]]; in the [[13-limit]] [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and [[6656/6655]]; and in the 17-limit, [[1156/1155]], 1275/1274, 2431/2430, and 2500/2499.

	Since the step size is close to [[729/728]], the squbema, the accepted name for 494edo's step is ''squb''~~<sup>[why? squbema maps to 7 steps of 494edo]</sup>~~.		Since the step size is close to [[729/728]], the squbema, the accepted name for 494edo's step is ''squb''.

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