665edo: Difference between revisions

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== 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 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]].

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.

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]], 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 ===

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=== Subsets and supersets ===

Since 665 factors into {{~~factorization~~|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]]).

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.

=== ~~Miscellaneous properties~~ ===

[[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]].

A [[maximal evenness]] scale deriving from the 118 & 665 temperament, known as [[vavoom]], can also theoretically serve as a calendar leap week cycle corresponding to a year length of 365d 5h 48m 37+17~~/19s~~, 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.

=== 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 ==

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== 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 8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.3

| {{~~monzo~~| -1054 665 }}

| {{Monzo| -1054 665 }}

| {{~~mapping~~| 665 1054 }}

| {{Mapping| 665 1054 }}

| +0.0000

| 0.0000

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|-

| 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

| 0.0301

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| 2.3.5.7

| 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

| 2.63

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| 2.3.5.7.11

| 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

| 5.97

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| 2.3.5.7.11.13

| 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

| 5.55

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| 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

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| 138\665

| 249.02

| {{~~monzo~~| -26 18 -1 }}

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

| [[Monzismic]]

|-

| 7

| 288\665 (3\665)

| 288\665 (3\665)

| 519.70 (5.41)

| 519.70 (5.41)

| 27/20 (325/324)

| 27/20 (325/324)

| [[Brahmagupta]]

|-

| 19

| 276\665 (4\665)

| 276\665 (4\665)

| 498.05 (7.21)

| 498.05 (7.21)

| 4/3 (225//224)

| 4/3 (225//224)

| [[Enneadecal]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<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: @@
 {{Infobox ET}}
-{{EDO intro|665}}
+{{ED intro}}
 == Theory ==
-edo 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]].
+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.
+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 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.
+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 ===
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-Since 665 factors into {{factorization|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]]).
+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.
-=== Miscellaneous properties ===
+[[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]].
-A [[maximal evenness]] scale deriving from the 118 & 665 temperament, known as [[vavoom]], can also theoretically serve as a calendar leap week cycle corresponding to a year length of 365d 5h 48m 37+17/19s, 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.
+=== 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 ==
@@ Line 24: / Line 26: @@
 == 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
-| {{monzo| -1054 665 }}
+| {{Monzo| -1054 665 }}
-| {{mapping| 665 1054 }}
+| {{Mapping| 665 1054 }}
 | +0.0000
 | 0.0000
@@ Line 34: / Line 45: @@
 |-
 | 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
 | 0.0301
@@ Line 42: / Line 53: @@
 | 2.3.5.7
 | 4375/4374, 703125/702464, {{monzo| 36 -5 0 -10 }}
-| {{mapping| 665 1054 1544 1867 }}
+| {{Mapping| 665 1054 1544 1867 }}
-| &minus;0.0015
+| −0.0015
 | 0.0474
 | 2.63
@@ Line 49: / Line 60: @@
 | 2.3.5.7.11
 | 4000/3993, 4375/4374, 117649/117612, 131072/130977
-| {{mapping| 665 1054 1544 1867 2301 }}
+| {{Mapping| 665 1054 1544 1867 2301 }}
-| &minus;0.0511
+| −0.0511
 | 0.1078
 | 5.97
@@ Line 56: / Line 67: @@
 | 2.3.5.7.11.13
 | 1575/1573, 2080/2079, 4096/4095, 4375/4374, 31250/31213
-| {{mapping| 665 1054 1544 1867 2301 2461 }}
+| {{Mapping| 665 1054 1544 1867 2301 2461 }}
-| &minus;0.0594
+| −0.0594
 | 0.1002
 | 5.55
-{{comma basis end}}
+|}
 === 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
@@ Line 74: / Line 92: @@
 | 138\665
 | 249.02
-| {{monzo| -26 18 -1 }}
+| {{Monzo| -26 18 -1 }}
 | [[Monzismic]]
 |-
 | 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]]
 |-
 | 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]]
-{{rank-2 end}}
+|}
-{{orf}}
+<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]]