665edo - Revision history

Eufalesio: Missing inconsistency

2026-03-24T16:24:09Z

Missing inconsistency

← Older revision		Revision as of 16:24, 24 March 2026
Line 7:		Line 7:
	However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.		However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.

	665edo provides relatively great approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23, with minuscule absolute error. It is considered as an excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the [[27-odd-limit]], and in the full 27-odd-limit, the only inconsistencies are [[11/10]], [[15/11]], [[17/11]], [[23/22]], and their [[octave complement]]s. 665edo provides relatively poor approximations for intervals of 11, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.		665edo provides relatively great approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23, with minuscule absolute error. It is considered as an excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the [[27-odd-limit]], and in the full 27-odd-limit, the only inconsistencies are [[11/10]], [[25/22]], [[15/11]], [[17/11]], [[23/22]], and their [[octave complement]]s. 665edo provides relatively poor approximations for intervals of 11, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.

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

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"

2026-03-13T13:30:08Z

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 107:		Line 107:
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|}		\|}
	<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

	[[Category:3-limit record edos\|###]] <!-- 3-digit number -->		[[Category:3-limit record edos\|###]] <!-- 3-digit number -->

Overthink: /* Theory */ use different wording; 15/11 is inconsistent

2026-03-10T21:43:46Z

Theory: use different wording; 15/11 is inconsistent

← Older revision		Revision as of 21:43, 10 March 2026
Line 7:		Line 7:
	However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.		However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.

	665edo provides relatively great approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23, with minuscule absolute error. It is considered as an excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is ~~almost~~ consistent in the [[27-odd-limit]], ~~failing~~ 11/10, 17/11, 23/22 and [[octave complement]]s. 665edo provides relatively poor approximations for ~~the~~ 11~~-limit intervals~~, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.		665edo provides relatively great approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23, with minuscule absolute error. It is considered as an excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the [[27-odd-limit]], and in the full 27-odd-limit, the only inconsistencies are [[11/10]], [[15/11]], [[17/11]], [[23/22]], and their [[octave complement]]s. 665edo provides relatively poor approximations for intervals of 11, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.

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

FloraC: Formatting

2026-03-10T06:04:27Z

Formatting

← Older revision		Revision as of 06:04, 10 March 2026
Line 7:		Line 7:
	However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.		However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.

	665edo provides relatively great approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23, with minuscule absolute error. It is considered as an excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is almost consistent in the [[27-odd-limit]], failing 11/10, 17/11, 23/22 and [[octave complement]]~~<nowiki/>~~s. 665edo provides relatively poor approximations for the 11-limit intervals, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.		665edo provides relatively great approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23, with minuscule absolute error. It is considered as an excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is almost consistent in the [[27-odd-limit]], failing 11/10, 17/11, 23/22 and [[octave complement]]s. 665edo provides relatively poor approximations for the 11-limit intervals, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 31:		Line 31:
	! 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 38:		Line 38:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| -1054 665 }}		\| {{Monzo\| -1054 665 }}
	\| {{~~mapping~~\| 665 1054 }}		\| {{Mapping\| 665 1054 }}
	\| +0.0000		\| +0.0000
	\| 0.0000		\| 0.0000
Line 45:		Line 45:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| -14 -19 19 }}, {{monzo\| 54 -37 2 }}		\| {{Monzo\| -14 -19 19 }}, {{monzo\| 54 -37 2 }}
	\| {{~~mapping~~\| 665 1054 1544 }}		\| {{Mapping\| 665 1054 1544 }}
	\| +0.0213		\| +0.0213
	\| 0.0301		\| 0.0301
Line 53:		Line 53:
	\| 2.3.5.7		\| 2.3.5.7
	\| 4375/4374, 703125/702464, {{monzo\| 36 -5 0 -10 }}		\| 4375/4374, 703125/702464, {{monzo\| 36 -5 0 -10 }}
	\| {{~~mapping~~\| 665 1054 1544 1867 }}		\| {{Mapping\| 665 1054 1544 1867 }}
	\| −0.0015		\| −0.0015
	\| 0.0474		\| 0.0474
Line 60:		Line 60:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 4000/3993, 4375/4374, 117649/117612, 131072/130977		\| 4000/3993, 4375/4374, 117649/117612, 131072/130977
	\| {{~~mapping~~\| 665 1054 1544 1867 2301 }}		\| {{Mapping\| 665 1054 1544 1867 2301 }}
	\| −0.0511		\| −0.0511
	\| 0.1078		\| 0.1078
Line 67:		Line 67:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 1575/1573, 2080/2079, 4096/4095, 4375/4374, 31250/31213		\| 1575/1573, 2080/2079, 4096/4095, 4375/4374, 31250/31213
	\| {{~~mapping~~\| 665 1054 1544 1867 2301 2461 }}		\| {{Mapping\| 665 1054 1544 1867 2301 2461 }}
	\| −0.0594		\| −0.0594
	\| 0.1002		\| 0.1002
Line 77:		Line 77:
	\|+ 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 92:		Line 92:
	\| 138\665		\| 138\665
	\| 249.02		\| 249.02
	\| {{~~monzo~~\| -26 18 -1 }}		\| {{Monzo\| -26 18 -1 }}
	\| [[Monzismic]]		\| [[Monzismic]]
	\|-		\|-
	\| 7		\| 7
	\| 288\665<br />(3\665)		\| 288\665<br>(3\665)
	\| 519.70<br />(5.41)		\| 519.70<br>(5.41)
	\| 27/20<br />(325/324)		\| 27/20<br>(325/324)
	\| [[Brahmagupta]]		\| [[Brahmagupta]]
	\|-		\|-
	\| 19		\| 19
	\| 276\665<br />(4\665)		\| 276\665<br>(4\665)
	\| 498.05<br />(7.21)		\| 498.05<br>(7.21)
	\| 4/3<br />(225//224)		\| 4/3<br>(225//224)
	\| [[Enneadecal]]		\| [[Enneadecal]]
	\|}		\|}
	<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

	[[Category:3-limit record edos\|###]] <!-- 3-digit number -->		[[Category:3-limit record edos\|###]] <!-- 3-digit number -->

Eufalesio: Little things

2026-03-09T16:32:36Z

Little things

← Older revision		Revision as of 16:32, 9 March 2026
Line 7:		Line 7:
	However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.		However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.

	665edo provides relatively great approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23, with minuscule absolute error. It is considered as an excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the [[27-odd-limit]] ~~(with no elevens)~~. 665edo provides relatively poor approximations for the 11-limit intervals, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.		665edo provides relatively great approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23, with minuscule absolute error. It is considered as an excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is almost consistent in the [[27-odd-limit]], failing 11/10, 17/11, 23/22 and [[octave complement]]<nowiki/>s. 665edo provides relatively poor approximations for the 11-limit intervals, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 17:		Line 17:
	[[1330edo]], which doubles 665edo, provides a good correction of the harmonic 11.		[[1330edo]], which doubles 665edo, provides a good correction of the harmonic 11.

	[[7315edo]], which undecuples 665edo, is the last ~~multiple to be~~ 3-2 telic, ~~also~~ consistent to the [[27-odd-limit]] and almost the [[31-odd-limit]].		[[7315edo]], which undecuples 665edo, is the last 3-2 telic multiple, and fully consistent to the [[27-odd-limit]] and almost the [[31-odd-limit]].

	=== Miscellany ===		=== Miscellany ===

Eufalesio: added 7315 as superset

2026-01-28T11:18:20Z

added 7315 as superset

← Older revision		Revision as of 11:18, 28 January 2026
Line 16:		Line 16:

	[[1330edo]], which doubles 665edo, provides a good correction of the harmonic 11.		[[1330edo]], which doubles 665edo, provides a good correction of the harmonic 11.

			[[7315edo]], which undecuples 665edo, is the last multiple to be 3-2 telic, also consistent to the [[27-odd-limit]] and almost the [[31-odd-limit]].

	=== Miscellany ===		=== Miscellany ===

Eufalesio: I don't think "extremely" doesn't quite do justice to how insane 0.00011 cents of error is

2025-10-27T12:37:45Z

I don't think "extremely" doesn't quite do justice to how insane 0.00011 cents of error is

← Older revision		Revision as of 12:37, 27 October 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	665edo is best known for its ~~extremely~~ accurate [[3/2\|fifth]], only 0.00011 cents compressed. 665edo is the denominator of a convergent to log<sub>2</sub>3, after [[41edo]], [[53edo]], and [[306edo]], and before [[15601edo]], and is the member of this series with the highest 3-2 [[Telicity #k-Strong Telicity\|telicity ''k''-strength]] before being finally surpassed in this regard by [[190537edo]].		665edo is best known for its unfathomably accurate [[3/2\|fifth]], only 0.00011 cents compressed. 665edo is the denominator of a convergent to log<sub>2</sub>3, after [[41edo]], [[53edo]], and [[306edo]], and before [[15601edo]], and is the member of this series with the highest 3-2 [[Telicity #k-Strong Telicity\|telicity ''k''-strength]] before being finally surpassed in this regard by [[190537edo]].

	However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.		However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.

	665edo provides ~~excellent~~ approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23. It is considered as ~~the~~ excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the [[27-odd-limit]] (with no elevens). 665edo provides poor approximations for the 11-limit intervals, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.		665edo provides relatively great approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23, with minuscule absolute error. It is considered as an excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the [[27-odd-limit]] (with no elevens). 665edo provides relatively poor approximations for the 11-limit intervals, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.

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

FloraC: + category

2025-07-12T12:48:22Z

+ category

← Older revision		Revision as of 12:48, 12 July 2025
Line 107:		Line 107:
	<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

			[[Category:3-limit record edos\|###]] <!-- 3-digit number -->
	[[Category:Satanic]]		[[Category:Satanic]]
	[[Category:Wizardharry]]		[[Category:Wizardharry]]
	[[Category:Monzismic]]		[[Category:Monzismic]]

FloraC: No need to remind readers of what a regular temperament is everywhere

2025-04-11T11:19:10Z

No need to remind readers of what a regular temperament is everywhere

← Older revision		Revision as of 11:19, 11 April 2025
Line 5:		Line 5:
	665edo is best known for its extremely accurate [[3/2\|fifth]], only 0.00011 cents compressed. 665edo is the denominator of a convergent to log<sub>2</sub>3, after [[41edo]], [[53edo]], and [[306edo]], and before [[15601edo]], and is the member of this series with the highest 3-2 [[Telicity #k-Strong Telicity\|telicity ''k''-strength]] before being finally surpassed in this regard by [[190537edo]].		665edo is best known for its extremely accurate [[3/2\|fifth]], only 0.00011 cents compressed. 665edo is the denominator of a convergent to log<sub>2</sub>3, after [[41edo]], [[53edo]], and [[306edo]], and before [[15601edo]], and is the member of this series with the highest 3-2 [[Telicity #k-Strong Telicity\|telicity ''k''-strength]] before being finally surpassed in this regard by [[190537edo]].

	However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736. ~~(''See [[regular temperament]] for more about what all this means and how to use it.'')~~		However, it also provides the [[optimal patent val]] for the rank-4 temperament tempering out [[4000/3993]]. It [[tempering out\|tempers out]] the [[satanic comma]], {{monzo\| -1054 665 }} in the 3-limit; the [[enneadeca]], {{monzo\| -14 -19 19 }}, and the [[monzisma]], {{monzo\| 54 -37 2 }} in the 5-limit; the [[ragisma]], 4375/4374, the [[meter]], 703125/702464, and {{monzo\| 36 -5 0 -10 }} in the 7-limit; [[4000/3993]], 46656/46585, [[131072/130977]] and 151263/151250 in the 11-limit, providing the optimal patent val for the 11-limit [[brahmagupta]] temperament. In the 13-limit, it tempers out [[1575/1573]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]; since it tempers out 1575/1573, the nicola, it [[support]]s nicolic tempering and hence the [[nicolic chords]], for which it provides an excellent tuning. In the 17-limit it tempers out [[1156/1155]], [[1275/1274]], [[2058/2057]], [[2500/2499]] and [[5832/5831]]; in the 19-limit it tempers out 969/968, [[1445/1444]], 2432/2431, 3136/3135, 3250/3249, and 4200/4199; in the 23-limit it tempers out 1288/1287, 1863/1862, 2025/2024, 2185/2184, and 2737/2736.

	665edo provides excellent approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23. It is considered as the excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the [[27-odd-limit]] (with no elevens). 665edo provides poor approximations for the 11-limit intervals, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.		665edo provides excellent approximations for the 7-limit intervals and harmonics 13, 17, 19, and 23. It is considered as the excellent 2.3.5.7.13.17.19.23 subgroup temperament, on which it is consistent in the [[27-odd-limit]] (with no elevens). 665edo provides poor approximations for the 11-limit intervals, with two mappings possible for the [[11/8]] fourth: a sharp one from the [[patent val]], and a flat one from the 665e val. Using the 665e val, [[41503/41472]], 42592/42525, 160083/160000, and 539055/537824 are tempered out in the 11-limit.
Line 24:		Line 24:

	== Regular temperament properties ==		== Regular temperament properties ==
	~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	\|-		\|-

BudjarnLambeth: /* Regular temperament properties */

2025-04-11T07:47:45Z

Regular temperament properties

← Older revision		Revision as of 07:47, 11 April 2025
Line 24:		Line 24:

	== Regular temperament properties ==		== Regular temperament properties ==
			''See [[regular temperament]] for more about what all this means and how to use it.''

	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	\|-		\|-