742edo - Revision history

Overthink: /* Prime harmonics */ another table

2026-04-25T22:24:36Z

Prime harmonics: another table

← Older revision		Revision as of 22:24, 25 April 2026
Line 6:		Line 6:

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|742}}		{{Harmonics in equal\|742\|columns=11}}
			{{Harmonics in equal\|742\|columns=11\|start=12\|collapsed=1\|title=Approximation of prime harmonics in 742edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

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"

2026-03-13T13:30:06Z

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 13:30, 13 March 2026
Line 127:		Line 127:
	\| [[Iodine]]		\| [[Iodine]]
	\|}		\|}
	<nowiki/>* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms\|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

	== Scales ==		== Scales ==
	* Silicon[28]: 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43		* Silicon[28]: 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43

FloraC: /* Regular temperament properties */ emmthird

2025-09-04T15:17:42Z

Regular temperament properties: emmthird

← Older revision		Revision as of 15:17, 4 September 2025
Line 17:		Line 17:
	! 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
	\|-		\|-
Line 24:		Line 24:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| 23 6 -14 }}, {{monzo\| -84 53 }}		\| {{Monzo\| 23 6 -14 }}, {{monzo\| -84 53 }}
	\| {{~~mapping~~\| 742 1176 1723 }}		\| {{Mapping\| 742 1176 1723 }}
	\| −0.0157		\| −0.0157
	\| 0.0555		\| 0.0555
Line 32:		Line 32:
	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 14348907/14336000, {{monzo\| 23 6 -14 }}		\| 2401/2400, 14348907/14336000, {{monzo\| 23 6 -14 }}
	\| {{~~mapping~~\| 742 1176 1723 2083 }}		\| {{Mapping\| 742 1176 1723 2083 }}
	\| −0.0035		\| −0.0035
	\| 0.0525		\| 0.0525
Line 39:		Line 39:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040		\| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040
	\| {{~~mapping~~\| 742 1176 1723 2083 2567 }}		\| {{Mapping\| 742 1176 1723 2083 2567 }}
	\| −0.0123		\| −0.0123
	\| 0.0501		\| 0.0501
Line 46:		Line 46:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325		\| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325
	\| {{~~mapping~~\| 742 1176 1723 2083 2567 2746 }}		\| {{Mapping\| 742 1176 1723 2083 2567 2746 }}
	\| −0.0302		\| −0.0302
	\| 0.0608		\| 0.0608
Line 53:		Line 53:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655		\| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655
	\| {{~~mapping~~\| 742 1176 1723 2083 2567 2746 3033 }}		\| {{Mapping\| 742 1176 1723 2083 2567 2746 3033 }}
	\| −0.0317		\| −0.0317
	\| 0.0564		\| 0.0564
Line 60:		Line 60:
	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211		\| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211
	\| {{~~mapping~~\| 742 1176 1723 2083 2567 2746 3033 3152 }}		\| {{Mapping\| 742 1176 1723 2083 2567 2746 3033 3152 }}
	\| −0.0295		\| −0.0295
	\| 0.0531		\| 0.0531
Line 67:		Line 67:
	\| 2.3.5.7.11.13.17.19.23		\| 2.3.5.7.11.13.17.19.23
	\| 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211		\| 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211
	\| {{~~mapping~~\| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)		\| {{Mapping\| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)
	\| −0.0468		\| −0.0468
	\| 0.0699		\| 0.0699
Line 79:		Line 79:
	\|+ 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
	\|-		\|-
Line 90:		Line 90:
	\| 8388608/7381125		\| 8388608/7381125
	\| [[Fortune]]		\| [[Fortune]]
			\|-
			\| 1
			\| 243\742
			\| 392.992
			\| 2744/2187
			\| [[Emmthird]] (7-limit)
	\|-		\|-
	\| 1		\| 1
Line 104:		Line 110:
	\|-		\|-
	\| 14		\| 14
	\| 434\742<br />(10\742)		\| 434\742<br>(10\742)
	\| 701.886<br />(16.173)		\| 701.886<br>(16.173)
	\| 3/2<br />(105/104)		\| 3/2<br>(105/104)
	\| [[Silicon]]		\| [[Silicon]]
	\|-		\|-
	\| 53		\| 53
	\| 239\742<br />(1\742)		\| 239\742<br>(1\742)
	\| 386.523<br />(1.617)		\| 386.523<br>(1.617)
	\| 5/4<br />(32805/32768)		\| 5/4<br>(32805/32768)
	\| [[Mercator]]		\| [[Mercator]]
	\|-		\|-
	\| 53		\| 53
	\| 565\742<br />(5\742)		\| 565\742<br>(5\742)
	\| 913.746<br />(8.086)		\| 913.746<br>(8.086)
	\| 441/260<br />(196/195)		\| 441/260<br>(196/195)
	\| [[Iodine]]		\| [[Iodine]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms\|minimal form]] in parentheses if distinct

	== Scales ==		== Scales ==
	* Silicon[28]: 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43		* Silicon[28]: 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43

BudjarnLambeth: Add scales section

2025-05-27T10:07:18Z

Add scales section

← Older revision		Revision as of 10:07, 27 May 2025
Line 122:		Line 122:
	\|}		\|}
	<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

			== Scales ==
			* Silicon[28]: 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43 10 43

ArrowHead294 at 16:11, 20 February 2025

2025-02-20T16:11:46Z

← Older revision		Revision as of 16:11, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T16:35:41Z

Theory: cleanup

← Older revision		Revision as of 16:35, 17 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	742edo is a very strong 19-limit system and a [[zeta peak edo]], and is [[consistency\|distinctly consistent]] in the [[21-odd-limit]]. It [[tempers out]] the [[vishnuzma]] and the fortune comma in the 5-limit, [[support]]ing [[vishnu]] and [[fortune]]; [[2401/2400]] in the 7-limit, [[9801/9800]] in the 11-limit, [[4096/4095]], [[6656/6655]], [[10648/10647]] in the 13-limit, [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]], [[5832/5831]] in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.		742edo is a very strong 19-limit system and a [[zeta peak edo]], and is [[consistency\|distinctly consistent]] in the [[21-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] the [[vishnuzma]] and the fortune comma in the 5-limit, [[support]]ing [[vishnu]] and [[fortune]]; [[2401/2400]] in the 7-limit, [[9801/9800]] in the 11-limit, [[4096/4095]], [[6656/6655]], [[10648/10647]] in the 13-limit, [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]], [[5832/5831]] in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 742 factors into ~~{{factorization\|742}}~~, 742edo has subset edos {{EDOs\| 2, 7, 14, 53, 106, and 371 }}, of which [[7edo]], [[14edo]] and [[53edo]] are very notable. It supports [[silicon]] ({{nowrap\|224 &~~amp;~~ 518}}) with ~~period~~ 14 in the 13-limit, and [[iodine]] ({{nowrap\|159 &~~amp;~~ 583f}}) with ~~period~~ 53 in the 17-limit.		Since 742 factors into 2 × 7 × 53, 742edo has subset edos {{EDOs\| 2, 7, 14, 53, 106, and 371 }}, of which [[7edo]], [[14edo]] and [[53edo]] are very notable. It supports [[silicon]] ({{nowrap\|224 & 518}}) with 14 periods per octave in the 13-limit, and [[iodine]] ({{nowrap\|159& 583f}}) with 53 periods per octave in the 17-limit.

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

ArrowHead294 at 14:04, 16 January 2025

2025-01-16T14:04:47Z

← Older revision		Revision as of 14:04, 16 January 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro~~\|742~~}}		{{EDO intro}}

	== Theory ==		== Theory ==
	742edo is a very strong 19-limit system and a [[zeta peak edo]], and is [[consistency\|distinctly consistent]] in the [[21-odd-limit]]. ~~The equal temperament~~ [[~~tempering out\|~~tempers out]] the [[vishnuzma]] and the fortune comma in the 5-limit, [[support]]ing [[vishnu]] and [[fortune]]; [[2401/2400]] in the 7-limit, [[9801/9800]] in the 11-limit, [[4096/4095]], [[6656/6655]], [[10648/10647]] in the 13-limit, [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]], [[5832/5831]] in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.		742edo is a very strong 19-limit system and a [[zeta peak edo]], and is [[consistency\|distinctly consistent]] in the [[21-odd-limit]]. It [[tempers out]] the [[vishnuzma]] and the fortune comma in the 5-limit, [[support]]ing [[vishnu]] and [[fortune]]; [[2401/2400]] in the 7-limit, [[9801/9800]] in the 11-limit, [[4096/4095]], [[6656/6655]], [[10648/10647]] in the 13-limit, [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]], [[5832/5831]] in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 742 factors into {{factorization\|742}}, 742edo has subset edos {{EDOs\| 2, 7, 14, 53, 106, and 371 }}, of which [[7edo]], [[14edo]] and [[53edo]] are very notable. It supports [[silicon]] (224 & 518) with period 14 in the 13-limit, and [[iodine]] (159 & 583f) with period 53 in the 17-limit.		Since 742 factors into {{factorization\|742}}, 742edo has subset edos {{EDOs\| 2, 7, 14, 53, 106, and 371 }}, of which [[7edo]], [[14edo]] and [[53edo]] are very notable. It supports [[silicon]] ({{nowrap\|224 & 518}}) with period 14 in the 13-limit, and [[iodine]] ({{nowrap\|159 & 583f}}) with period 53 in the 17-limit.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 26:		Line 26:
	\| {{monzo\| 23 6 -14 }}, {{monzo\| -84 53 }}		\| {{monzo\| 23 6 -14 }}, {{monzo\| -84 53 }}
	\| {{mapping\| 742 1176 1723 }}		\| {{mapping\| 742 1176 1723 }}
	\| ~~−0~~.0157		\| −0.0157
	\| 0.0555		\| 0.0555
	\| 3.43		\| 3.43
Line 33:		Line 33:
	\| 2401/2400, 14348907/14336000, {{monzo\| 23 6 -14 }}		\| 2401/2400, 14348907/14336000, {{monzo\| 23 6 -14 }}
	\| {{mapping\| 742 1176 1723 2083 }}		\| {{mapping\| 742 1176 1723 2083 }}
	\| ~~−0~~.0035		\| −0.0035
	\| 0.0525		\| 0.0525
	\| 3.24		\| 3.24
Line 40:		Line 40:
	\| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040		\| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040
	\| {{mapping\| 742 1176 1723 2083 2567 }}		\| {{mapping\| 742 1176 1723 2083 2567 }}
	\| ~~−0~~.0123		\| −0.0123
	\| 0.0501		\| 0.0501
	\| 3.10		\| 3.10
Line 47:		Line 47:
	\| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325		\| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325
	\| {{mapping\| 742 1176 1723 2083 2567 2746 }}		\| {{mapping\| 742 1176 1723 2083 2567 2746 }}
	\| ~~−0~~.0302		\| −0.0302
	\| 0.0608		\| 0.0608
	\| 3.76		\| 3.76
Line 54:		Line 54:
	\| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655		\| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655
	\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 }}		\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 }}
	\| ~~−0~~.0317		\| −0.0317
	\| 0.0564		\| 0.0564
	\| 3.49		\| 3.49
Line 61:		Line 61:
	\| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211		\| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211
	\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 3152 }}		\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 3152 }}
	\| ~~−0~~.0295		\| −0.0295
	\| 0.0531		\| 0.0531
	\| 3.28		\| 3.28
Line 68:		Line 68:
	\| 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211		\| 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211
	\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)		\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)
	\| ~~−0~~.0468		\| −0.0468
	\| 0.0699		\| 0.0699
	\| 4.32		\| 4.32
Line 121:		Line 121:
	\| [[Iodine]]		\| [[Iodine]]
	\|}		\|}
	<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

ArrowHead294: Partial undo

2024-11-16T12:15:38Z

Partial undo

← Older revision		Revision as of 12:15, 16 November 2024
Line 12:		Line 12:

	== 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]] (%)
	\|-		\|-
	\| 2.3.5		\| 2.3.5
Line 62:		Line 71:
	\| 0.0699		\| 0.0699
	\| 4.32		\| 4.32
	~~{{comma basis end}~~}		\|}
	* 742et has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any previous equal temperaments. It is only bettered by [[935edo\|935]] in terms of absolute error, and by [[1178edo\|1178]] in terms of relative error.		* 742et has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any previous equal temperaments. It is only bettered by [[935edo\|935]] in terms of absolute error, and by [[1178edo\|1178]] in terms of relative error.
	* 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats [[581edo\|581]] and is bettered by [[764edo\|764]]; in the 23-limit it beats [[718edo\|718]] and is bettered by [[814edo\|814]].		* 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats [[581edo\|581]] and is bettered by [[764edo\|764]]; in the 23-limit it beats [[718edo\|718]] and is bettered by [[814edo\|814]].

	=== 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
Line 104:		Line 120:
	\| 441/260<br />(196/195)		\| 441/260<br />(196/195)
	\| [[Iodine]]		\| [[Iodine]]
	~~{{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

ArrowHead294 at 05:11, 16 November 2024

2024-11-16T05:11:29Z

← Older revision		Revision as of 05:11, 16 November 2024
Line 12:		Line 12:

	== 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]] (%)~~
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{monzo\| 23 6 -14 }}, {{monzo\| -84 53 }}		\| {{monzo\| 23 6 -14 }}, {{monzo\| -84 53 }}
	\| {{mapping\| 742 1176 1723 }}		\| {{mapping\| 742 1176 1723 }}
	\| -0.0157		\| −0.0157
	\| 0.0555		\| 0.0555
	\| 3.43		\| 3.43
Line 32:		Line 24:
	\| 2401/2400, 14348907/14336000, {{monzo\| 23 6 -14 }}		\| 2401/2400, 14348907/14336000, {{monzo\| 23 6 -14 }}
	\| {{mapping\| 742 1176 1723 2083 }}		\| {{mapping\| 742 1176 1723 2083 }}
	\| -0.0035		\| −0.0035
	\| 0.0525		\| 0.0525
	\| 3.24		\| 3.24
Line 39:		Line 31:
	\| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040		\| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040
	\| {{mapping\| 742 1176 1723 2083 2567 }}		\| {{mapping\| 742 1176 1723 2083 2567 }}
	\| -0.0123		\| −0.0123
	\| 0.0501		\| 0.0501
	\| 3.10		\| 3.10
Line 46:		Line 38:
	\| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325		\| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325
	\| {{mapping\| 742 1176 1723 2083 2567 2746 }}		\| {{mapping\| 742 1176 1723 2083 2567 2746 }}
	\| -0.0302		\| −0.0302
	\| 0.0608		\| 0.0608
	\| 3.76		\| 3.76
Line 53:		Line 45:
	\| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655		\| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655
	\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 }}		\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 }}
	\| -0.0317		\| −0.0317
	\| 0.0564		\| 0.0564
	\| 3.49		\| 3.49
Line 60:		Line 52:
	\| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211		\| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211
	\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 3152 }}		\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 3152 }}
	\| -0.0295		\| −0.0295
	\| 0.0531		\| 0.0531
	\| 3.28		\| 3.28
Line 67:		Line 59:
	\| 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211		\| 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211
	\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)		\| {{mapping\| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i)
	\| -0.0468		\| −0.0468
	\| 0.0699		\| 0.0699
	\| 4.32		\| 4.32
	\|}		{{comma basis end}}
	* 742et has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any previous equal temperaments. It is only bettered by [[935edo\|935]] in terms of absolute error, and by [[1178edo\|1178]] in terms of relative error.		* 742et has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness\|relative error]] than any previous equal temperaments. It is only bettered by [[935edo\|935]] in terms of absolute error, and by [[1178edo\|1178]] in terms of relative error.
	* 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats [[581edo\|581]] and is bettered by [[764edo\|764]]; in the 23-limit it beats [[718edo\|718]] and is bettered by [[814edo\|814]].		* 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats [[581edo\|581]] and is bettered by [[764edo\|764]]; in the 23-limit it beats [[718edo\|718]] and is bettered by [[814edo\|814]].

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center~~-~~all left-5"~~		{{rank-2 begin}}
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
Line 101:		Line 88:
	\|-		\|-
	\| 14		\| 14
	\| 434\742<br>(10\742)		\| 434\742<br />(10\742)
	\| 701.886<br>(16.173)		\| 701.886<br />(16.173)
	\| 3/2<br>(105/104)		\| 3/2<br />(105/104)
	\| [[Silicon]]		\| [[Silicon]]
	\|-		\|-
	\| 53		\| 53
	\| 239\742<br>(1\742)		\| 239\742<br />(1\742)
	\| 386.523<br>(1.617)		\| 386.523<br />(1.617)
	\| 5/4<br>(32805/32768)		\| 5/4<br />(32805/32768)
	\| [[Mercator]]		\| [[Mercator]]
	\|-		\|-
	\| 53		\| 53
	\| 565\742<br>(5\742)		\| 565\742<br />(5\742)
	\| 913.746<br>(8.086)		\| 913.746<br />(8.086)
	\| 441/260<br>(196/195)		\| 441/260<br />(196/195)
	\| [[Iodine]]		\| [[Iodine]]
	\|}		{{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}}

FloraC: Adopt template: Factorization; misc. cleanup

2023-11-02T12:21:15Z

Adopt template: Factorization; misc. cleanup

← Older revision		Revision as of 12:21, 2 November 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	742edo is a very strong 19-limit system and a [[zeta peak edo]], and is ~~uniquely~~ [[consistent]] in the [[21-odd-limit]]. It tempers out the vishnuzma and the fortune comma in the 5-limit, ~~supporting~~ [[vishnu]] and [[fortune]]; [[2401/2400]] in the 7-limit, [[9801/9800]] in the 11-limit, [[4096/4095]], [[6656/6655]], [[10648/10647]] in the 13-limit, [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]], [[5832/5831]] in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.		742edo is a very strong 19-limit system and a [[zeta peak edo]], and is [[consistency\|distinctly consistent]] in the [[21-odd-limit]]. The equal temperament [[tempering out\|tempers out]] the [[vishnuzma]] and the fortune comma in the 5-limit, [[support]]ing [[vishnu]] and [[fortune]]; [[2401/2400]] in the 7-limit, [[9801/9800]] in the 11-limit, [[4096/4095]], [[6656/6655]], [[10648/10647]] in the 13-limit, [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]], [[5832/5831]] in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	742 = 2 × 7 × 53, ~~so it notably contains~~ [[~~53edo~~]] and [[~~14edo~~]]. It supports [[silicon]] ~~temperament~~ (224 & ~~742~~) with period 14 in the 13-limit, and [[iodine]] ~~temperament~~ (159 & ~~742~~) with period 53 in the 17-limit.		Since 742 factors into {{factorization\|742}}, 742edo has subset edos {{EDOs\| 2, 7, 14, 53, 106, and 371 }}, of which [[7edo]], [[14edo]] and [[53edo]] are very notable. It supports [[silicon]] (224 & 518) with period 14 in the 13-limit, and [[iodine]] (159 & 583f) with period 53 in the 17-limit.

	== Regular temperament properties ==		== Regular temperament properties ==
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	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-