665edo: Difference between revisions

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~~The '''665 equal temperament''' divides the octave into 665 equal parts of 1.80451 cents each.~~

== Theory ==

665edo is best known for its unfathomably accurate [[3/2|fifth]], only 0.00011 cents compressed. 665edo is the denominator of a convergent to log23, 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.

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 ===

It is best known for its extremely accurate fifth, only 0.00011 cents compressed. 665edo is the denominator of a convergent to log23, after [[41edo]], [[53edo]] and [[306edo]], and before [[15601edo]]. However, it also provides the [[optimal patent val]] for the rank four temperament tempering out 4000/3993. It tempers out the 'satanic' comma, |-1054 665> in the 3-limit; the enneadeca, |-14 -19 19> and the [[monzisma]], |54 -37 2> in the 5-limit; the ragisma, 4375/4374, the meter, 703125/702464, and 68719476736/68641485507 in the 7-limit; 4000/3993, 46656/46585, 131072/130977 and 151263/151250 in the 11-limit, providing the optimal patent val for 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 tetrad]], 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). Despite its division number of~~ the ~~octave, 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~~.

=== Subsets and supersets ===

Since 665 factors into {{factorisation|665}}, 665edo has subset edos {{EDOs| 5, 7, 19, 35, 95, and 133 }}. One step of 665edo has been proposed as an [[interval size measure]], called a '''Delfi unit'''. A Delfi unit is exactly 48 imps ([[31920edo|48\31920]]).

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

[[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 ===

A [[maximal evenness]] scale deriving from the {{nowrap|118 & 665}} temperament, known as [[vavoom]], can also theoretically serve as a calendar leap week cycle corresponding to a year length of {{nowrap| 365d 5h 48m 37{{frac|17|19}}s}}, about 7 seconds shorter than the average length of the tropical year today. Given the excellence of both 118 and 665 in 5-limit, this is a great point of intersection of solar calendar leap rules and just intonation-based temperaments.

== Intervals ==

See [[Table of 665edo intervals]].

== Regular temperament properties ==

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

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{Monzo| -1054 665 }}

| {{Mapping| 665 1054 }}

| +0.0000

| 0.0000

| 0.00

|-

| 2.3.5

| {{Monzo| -14 -19 19 }}, {{monzo| 54 -37 2 }}

| {{Mapping| 665 1054 1544 }}

| +0.0213

| 0.0301

| 1.67

|-

| 2.3.5.7

| 4375/4374, 703125/702464, {{monzo| 36 -5 0 -10 }}

| {{Mapping| 665 1054 1544 1867 }}

| −0.0015

| 0.0474

| 2.63

|-

| 2.3.5.7.11

| 4000/3993, 4375/4374, 117649/117612, 131072/130977

| {{Mapping| 665 1054 1544 1867 2301 }}

| −0.0511

| 0.1078

| 5.97

|-

| 2.3.5.7.11.13

| 1575/1573, 2080/2079, 4096/4095, 4375/4374, 31250/31213

| {{Mapping| 665 1054 1544 1867 2301 2461 }}

| −0.0594

| 0.1002

| 5.55

|}

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 62\665

| 111.88

| 16/15

| [[Vavoom]]

|-

| 1

| 138\665

| 249.02

| {{Monzo| -26 18 -1 }}

| [[Monzismic]]

|-

| 7

| 288\665 (3\665)

| 519.70 (5.41)

| 27/20 (325/324)

| [[Brahmagupta]]

|-

| 19

| 276\665 (4\665)

| 498.05 (7.21)

| 4/3 (225//224)

| [[Enneadecal]]

|}

<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:~~Equal divisions of the octave~~]]

[[Category:3-limit record edos|###]]

[[Category:Satanic]]

[[Category:Wizardharry]]

[[Category:Monzismic]]

@@ Line 1: / Line 1: @@
-The '''665 equal temperament''' divides the octave into 665 equal parts of 1.80451 cents each.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
+edo 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.
+edo 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 ===
 {{Harmonics in equal|665}}
-It is best known for its extremely accurate 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]]. However, it also provides the [[optimal patent val]] for the rank four temperament tempering out 4000/3993. It tempers out the 'satanic' comma, |-1054 665&gt; in the 3-limit; the enneadeca, |-14 -19 19&gt; and the [[monzisma]], |54 -37 2&gt; in the 5-limit; the ragisma, 4375/4374, the meter, 703125/702464, and 68719476736/68641485507 in the 7-limit; 4000/3993, 46656/46585, 131072/130977 and 151263/151250 in the 11-limit, providing the optimal patent val for 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 tetrad]], 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.
-edo 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). Despite its division number of the octave, 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.
+=== Subsets and supersets ===
+Since 665 factors into {{factorisation|665}}, 665edo has subset edos {{EDOs| 5, 7, 19, 35, 95, and 133 }}. One step of 665edo has been proposed as an [[interval size measure]], called a '''Delfi unit'''. A Delfi unit is exactly 48 imps ([[31920edo|48\31920]]).
+[[1330edo]], which doubles 665edo, provides a good correction of the harmonic 11.
+[[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 ===
+A [[maximal evenness]] scale deriving from the {{nowrap|118 &amp; 665}} temperament, known as [[vavoom]], can also theoretically serve as a calendar leap week cycle corresponding to a year length of {{nowrap| 365d 5h 48m 37{{frac|17|19}}s}}, about 7 seconds shorter than the average length of the tropical year today. Given the excellence of both 118 and 665 in 5-limit, this is a great point of intersection of solar calendar leap rules and just intonation-based temperaments.
+== Intervals ==
+See [[Table of 665edo intervals]].
+== Regular temperament properties ==
+{| 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
+| {{Monzo| -1054 665 }}
+| {{Mapping| 665 1054 }}
+| +0.0000
+| 0.0000
+| 0.00
+|-
+| 2.3.5
+| {{Monzo| -14 -19 19 }}, {{monzo| 54 -37 2 }}
+| {{Mapping| 665 1054 1544 }}
+| +0.0213
+| 0.0301
+| 1.67
+|-
+| 2.3.5.7
+| 4375/4374, 703125/702464, {{monzo| 36 -5 0 -10 }}
+| {{Mapping| 665 1054 1544 1867 }}
+| −0.0015
+| 0.0474
+| 2.63
+|-
+| 2.3.5.7.11
+| 4000/3993, 4375/4374, 117649/117612, 131072/130977
+| {{Mapping| 665 1054 1544 1867 2301 }}
+| −0.0511
+| 0.1078
+| 5.97
+|-
+| 2.3.5.7.11.13
+| 1575/1573, 2080/2079, 4096/4095, 4375/4374, 31250/31213
+| {{Mapping| 665 1054 1544 1867 2301 2461 }}
+| −0.0594
+| 0.1002
+| 5.55
+|}
+=== Rank-2 temperaments ===
+{| 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
+| 62\665
+| 111.88
+| 16/15
+| [[Vavoom]]
+|-
+| 1
+| 138\665
+| 249.02
+| {{Monzo| -26 18 -1 }}
+| [[Monzismic]]
+|-
+| 7
+| 288\665<br>(3\665)
+| 519.70<br>(5.41)
+| 27/20<br>(325/324)
+| [[Brahmagupta]]
+|-
+| 19
+| 276\665<br>(4\665)
+| 498.05<br>(7.21)
+| 4/3<br>(225//224)
+| [[Enneadecal]]
+|}
+<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:Equal divisions of the octave]]
+[[Category:3-limit record edos|###]] <!-- 3-digit number -->
 [[Category:Satanic]]
 [[Category:Wizardharry]]
 [[Category:Monzismic]]