540edo - Revision history

ArrowHead294 at 22:46, 20 February 2025

2025-02-20T22:46:23Z

← Older revision		Revision as of 22:46, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|540~~}}		{{ED intro}}

	== Theory ==		== Theory ==
	Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.		Since {{nowrap\|540 {{=}} 2 × 270}} and {{nowrap\|540 {{=}} 45 × 12}}, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.

	The equal temperament [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]] and [[1729/1728]] in the 19-limit; [[1105/1104]] and [[1496/1495]] in the 23-limit. Although it does quite well in these limits, it is not as ''efficient'' as 270edo's original mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied [[essentially tempered chord]]s are worth the load of all the extra notes.		The equal temperament [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]] and [[1729/1728]] in the 19-limit; [[1105/1104]] and [[1496/1495]] in the 23-limit. Although it does quite well in these limits, it is not as ''efficient'' as 270edo's original mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied [[essentially tempered chord]]s are worth the load of all the extra notes.

FloraC: /* Theory */

2024-12-08T10:14:12Z

Theory

← Older revision		Revision as of 10:14, 8 December 2024
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	The equal temperament [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]] and [[1729/1728]] in the 19-limit; [[1105/1104]] and [[1496/1495]] in the 23-limit. Although it does quite well in these limits, it is not as ''efficient'' as 270edo's original mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied [[essentially tempered chord]]s are worth the load of all the extra notes.		The equal temperament [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]] and [[1729/1728]] in the 19-limit; [[1105/1104]] and [[1496/1495]] in the 23-limit. Although it does quite well in these limits, it is not as ''efficient'' as 270edo's original mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied [[essentially tempered chord]]s are worth the load of all the extra notes.

	The ~~approximation to~~ [[29/1\|29]] and [[31/1\|31]] are relatively weak, but [[37/1\|37]], [[41/1\|41]] and [[43/1\|43]] are quite spot on, with the 43 coming from 270edo. For this reason, we may consider it as a full [[43-limit]] system. For all the primes starting with 29, it removes the distinction of otonal and utonal [[superparticular ratio\|superparticular]] pairs (e.g. 29/28 vs 30/29 for prime 29) by tempering out the corresponding [[square superparticular]]s, which is responsible for its slightly flat-tending tuning profile. Prime [[47/1\|47]] does not have that privilege and falls practically halfway between, though the sharp mapping might be preferred to keep [[47/46]] wider than [[48/47]]. As a compensation, you do get a spot-on prime [[53/1\|53]] for free.		The approximated [[29/1\|29]] and [[31/1\|31]] are relatively weak, but [[37/1\|37]], [[41/1\|41]] and [[43/1\|43]] are quite spot on, with the 43 coming from 270edo. For this reason, we may consider it as a full [[43-limit]] system. For all the primes starting with 29, it removes the distinction of otonal and utonal [[superparticular ratio\|superparticular]] pairs (e.g. 29/28 vs 30/29 for prime 29) by tempering out the corresponding [[square superparticular]]s, which is responsible for its slightly flat-tending tuning profile. Prime [[47/1\|47]] does not have that privilege and falls practically halfway between, though the sharp mapping might be preferred to keep [[47/46]] wider than [[48/47]]. As a compensation, you do get a spot-on prime [[53/1\|53]] for free.

	=== Prime harmonics ===		=== Prime harmonics ===
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	540 is a very composite number. The [[prime factorization]] of 540 is {{factorization\|540}}. Its nontrivial divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.		540 is a very composite number. The [[prime factorization]] of 540 is {{factorization\|540}}. Its nontrivial divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.

	A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.		A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.

	== Approximation to JI ==		== Approximation to JI ==

FloraC: +a discussion about the 43- to 53-limit, with the prime table extended to facilitate comparing with 270edo

2024-12-07T15:25:56Z

+a discussion about the 43- to 53-limit, with the prime table extended to facilitate comparing with 270edo

← Older revision		Revision as of 15:25, 7 December 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and ~~perhaps~~ beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.		Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.

	The equal temperament [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]] and [[1729/1728]] in the 19-limit; [[1105/1104]] and [[1496/1495]] in the 23-limit. Although it does quite well in these limits, it is not as ''efficient'' as 270edo's original mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied [[essentially tempered chord]]s are worth the load of all the extra notes.		The equal temperament [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]] and [[1729/1728]] in the 19-limit; [[1105/1104]] and [[1496/1495]] in the 23-limit. Although it does quite well in these limits, it is not as ''efficient'' as 270edo's original mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied [[essentially tempered chord]]s are worth the load of all the extra notes.

			The approximation to [[29/1\|29]] and [[31/1\|31]] are relatively weak, but [[37/1\|37]], [[41/1\|41]] and [[43/1\|43]] are quite spot on, with the 43 coming from 270edo. For this reason, we may consider it as a full [[43-limit]] system. For all the primes starting with 29, it removes the distinction of otonal and utonal [[superparticular ratio\|superparticular]] pairs (e.g. 29/28 vs 30/29 for prime 29) by tempering out the corresponding [[square superparticular]]s, which is responsible for its slightly flat-tending tuning profile. Prime [[47/1\|47]] does not have that privilege and falls practically halfway between, though the sharp mapping might be preferred to keep [[47/46]] wider than [[48/47]]. As a compensation, you do get a spot-on prime [[53/1\|53]] for free.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|540\|columns=11}}		{{Harmonics in equal\|540\|columns=12}}
			{{Harmonics in equal\|540\|columns=12\|start=13\|collapsed=true\|title=Approximation of prime harmonics in 540edo (continued)}}

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

FloraC: This makes for an interesting comparison with 270edo

2024-08-18T09:21:13Z

This makes for an interesting comparison with 270edo

← Older revision		Revision as of 09:21, 18 August 2024
Line 14:		Line 14:

	A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.		A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.

			== Approximation to JI ==
			{{Q-odd-limit intervals\|540\|23}}

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

FloraC: Compare with 270edo

2024-07-15T10:58:15Z

Compare with 270edo

← Older revision		Revision as of 10:58, 15 July 2024
Line 5:		Line 5:
	Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.		Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.

	The equal temperament [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], 1331/1330, [[1445/1444]] and [[1729/1728]] in the 19-limit; 1105/1104 and 1496/1495 in the 23-limit.		The equal temperament [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1445/1444]] and [[1729/1728]] in the 19-limit; [[1105/1104]] and [[1496/1495]] in the 23-limit. Although it does quite well in these limits, it is not as ''efficient'' as 270edo's original mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied [[essentially tempered chord]]s are worth the load of all the extra notes.

	~~A step of 540edo~~ is ~~known~~ as a ''~~'dexl~~''', ~~proposed by~~ [[~~Joseph Monzo~~]] ~~in April 2023 as an~~ [[~~interval size measure~~]]~~<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>~~.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 14:		Line 12:
	=== Subsets and supersets ===		=== Subsets and supersets ===
	540 is a very composite number. The [[prime factorization]] of 540 is {{factorization\|540}}. Its nontrivial divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.		540 is a very composite number. The [[prime factorization]] of 540 is {{factorization\|540}}. Its nontrivial divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.

			A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.

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

FloraC: Adopt template: Factorization; misc. cleanup

2023-11-02T13:44:48Z

Adopt template: Factorization; misc. cleanup

← Older revision		Revision as of 13:44, 2 November 2023
Line 5:		Line 5:
	Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.		Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.

	It [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], 1331/1330, [[1445/1444]] and [[1729/1728]] in the 19-limit; 1105/1104 and 1496/1495 in the 23-limit.		The equal temperament [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], 1331/1330, [[1445/1444]] and [[1729/1728]] in the 19-limit; 1105/1104 and 1496/1495 in the 23-limit.

	A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.		A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.
Line 13:		Line 13:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	540 is a very composite number. The prime factorization of 540 is ~~2<sup>2</sup> × 3<sup>3</sup> × 5~~. Its nontrivial divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.		540 is a very composite number. The [[prime factorization]] of 540 is {{factorization\|540}}. Its nontrivial divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.

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

FloraC: Cleanup

2023-10-29T11:05:37Z

Cleanup

← Older revision		Revision as of 11:05, 29 October 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	Since 540 = 2 × 270 and 540 = 45 × 12, it contains [[270edo]] and [[12edo]] as subsets, both being important zeta ~~edos~~. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.		Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.

	It tempers out [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], 1331/1330, [[1445/1444]] and [[1729/1728]] in the 19-limit; 1105/1104 and 1496/1495 in the 23-limit.		It [[tempering out\|tempers out]] [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], 1331/1330, [[1445/1444]] and [[1729/1728]] in the 19-limit; 1105/1104 and 1496/1495 in the 23-limit.

	~~Its~~ step is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.		A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 13:		Line 13:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	540 is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. Its divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.		540 is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. Its nontrivial divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 28:		Line 28:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 4096/4095		\| 676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 4096/4095
	\| [{{~~val~~\| 540 856 1254 1516 1868 1998 2207 }}]		\| {{mapping\| 540 856 1254 1516 1868 1998 2207 }}
	\| -0.0022		\| -0.0022
	\| 0.1144		\| 0.1144
Line 35:		Line 35:
	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 676/675, 1001/1000, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1729/1728		\| 676/675, 1001/1000, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1729/1728
	\| [{{~~val~~\| 540 856 1254 1516 1868 1998 2207 2294 }}]		\| {{mapping\| 540 856 1254 1516 1868 1998 2207 2294 }}
	\| -0.0098		\| -0.0098
	\| 0.1088		\| 0.1088
Line 42:		Line 42:
	\| 2.3.5.7.11.13.17.19.23		\| 2.3.5.7.11.13.17.19.23
	\| 676/675, 1001/1000, 1105/1104, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1496/1495		\| 676/675, 1001/1000, 1105/1104, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1496/1495
	\| [{{~~val~~\| 540 856 1254 1516 1868 1998 2207 2294 2443 }}]		\| {{mapping\| 540 856 1254 1516 1868 1998 2207 2294 2443 }}
	\| -0.024		\| -0.024
	\| 0.1100		\| 0.1100
	\| 4.95		\| 4.95
	\|}		\|}

			== Notes ==

BudjarnLambeth: Corrected Joe Monzo to Joseph Monzo

2023-08-09T08:44:25Z

Corrected Joe Monzo to Joseph Monzo

← Older revision		Revision as of 08:44, 9 August 2023
Line 7:		Line 7:
	It tempers out [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], 1331/1330, [[1445/1444]] and [[1729/1728]] in the 19-limit; 1105/1104 and 1496/1495 in the 23-limit.		It tempers out [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], 1331/1330, [[1445/1444]] and [[1729/1728]] in the 19-limit; 1105/1104 and 1496/1495 in the 23-limit.

	Its step is known as a '''dexl''', proposed by [[~~Joe~~ Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.		Its step is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.

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

FloraC: Expansion

2023-07-09T06:32:05Z

Expansion

← Older revision		Revision as of 06:32, 9 July 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	Since 540 = 2 × 270 and 540 = 45 × 12, it contains [[270edo]] and [[12edo]] as subsets, both ~~belonging to [[The Riemann Zeta Function and Tuning #Zeta EDO lists\|the ''zeta peak edos'', ''zeta integral edos'' and ''~~zeta ~~gap~~ edos~~'' sequences]]~~. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.		Since 540 = 2 × 270 and 540 = 45 × 12, it contains [[270edo]] and [[12edo]] as subsets, both being important zeta edos. It is [[enfactoring\|enfactored]] in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, no longer [[consistent]] in the [[15-odd-limit]], all because of [[15/13]] being 1.14 cents sharp of just.

			It tempers out [[1156/1155]] and [[2601/2600]] in the 17-limit; [[1216/1215]], 1331/1330, [[1445/1444]] and [[1729/1728]] in the 19-limit; 1105/1104 and 1496/1495 in the 23-limit.

			Its step is known as a '''dexl''', proposed by [[Joe Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia \| Dexl / 540edo]</ref>.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|540\|columns=11}}		{{Harmonics in equal\|540\|columns=11}}

	=== ~~Divisors~~ ===		=== Subsets and supersets ===
	540 is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. Its divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.		540 is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. Its divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.

FloraC: Formatting and -redundant categories

2023-07-09T06:23:47Z

Formatting and -redundant categories

← Older revision		Revision as of 06:23, 9 July 2023
Line 9:		Line 9:

	=== Divisors ===		=== Divisors ===
	540 is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. Its divisors are 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270.		540 is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. Its divisors are {{EDOs\| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \| Subgroup		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list\|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
	\|-		\|-
	! [[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
Line 43:		Line 43:
	\| 4.95		\| 4.95
	\|}		\|}

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