665edo: Difference between revisions

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~~The ''665 equal temperament'' divides the octave into 665 equal parts of 1.80451 cents each. It~~ is best known for its ~~extremely~~ accurate fifth, only 0.00011 cents ~~flat~~. 665edo is the denominator of a convergent to ~~log2(~~3), after [[41edo]], [[53edo]] and [[306edo]], and before [[15601edo]]. However, it also provides the [[~~Optimal_patent_val|~~optimal patent val]] for the rank ~~four~~ temperament tempering out 4000/3993. ~~In the 3-limit it~~ tempers out the 'satanic' comma, |-1054 665~~>;~~ in the 5-limit the enneadeca, |-14 -19 19~~>~~ and the [[~~Monzismic_temperament|~~monzisma]], |54 -37 2~~>;~~ in the 7-limit, 4375/4374 ~~and~~ the meter, | -~~11 2 7~~ -~~3 >;~~ in the 11-limit, 4000/3993 and ~~| 17~~ -~~5 0 -2 -1 > and | -1 2 -4 5 -2 >~~, providing the optimal patent val for 11-limit [[~~Ragismic_microtemperaments#Brahmagupta|~~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 ~~supports~~ nicolic tempering and hence the [[~~nicolic_tetrad|~~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.

[[Category:~~satanic~~]]

[[Category:~~wizardharry~~]]

[[Category:~~monzismic~~]]

== 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 ===

=== 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: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. It is best known for its extremely accurate fifth, only 0.00011 cents flat. 665edo is the denominator of a convergent to log2(3), after [[41edo]], [[53edo]] and [[306edo]], and before [[15601edo]]. However, it also provides the [[Optimal_patent_val|optimal patent val]] for the rank four temperament tempering out 4000/3993. In the 3-limit it tempers out the 'satanic' comma, |-1054 665&gt;; in the 5-limit the enneadeca, |-14 -19 19&gt; and the [[Monzismic_temperament|monzisma]], |54 -37 2&gt;; in the 7-limit, 4375/4374 and the meter, | -11 2 7 -3 &gt;; in the 11-limit, 4000/3993 and | 17 -5 0 -2 -1 &gt; and | -1 2 -4 5 -2 &gt;, providing the optimal patent val for 11-limit [[Ragismic_microtemperaments#Brahmagupta|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 supports nicolic tempering and hence the [[nicolic_tetrad|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.
+{{Infobox ET}}
-[[Category:satanic]]
+{{ED intro}}
-[[Category:wizardharry]]
-[[Category:monzismic]]
+== 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}}
+=== 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:3-limit record edos|###]] <!-- 3-digit number -->
+[[Category:Satanic]]
+[[Category:Wizardharry]]
+[[Category:Monzismic]]