665edo: Difference between revisions

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== Theory ==

665edo 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]], 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 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 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 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.

=== Prime harmonics ===

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| 2.3

| {{monzo| -1054 665 }}

| [{{~~val~~| 665 1054 }}]

| {{mapping| 665 1054 }}

| +0.0000

| 0.0000

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| 2.3.5

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

| [{{~~val~~| 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 }}

| [{{~~val~~| 665 1054 1544 1867 }}]

| {{mapping| 665 1054 1544 1867 }}

| -0.0015

| 0.0474

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| 2.3.5.7.11

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

| [{{~~val~~| 665 1054 1544 1867 2301 }}]

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

| -0.0511

| 0.1078

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| 2.3.5.7.11.13

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

| [{{~~val~~| 665 1054 1544 1867 2301 2461 }}]

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

| -0.0594

| 0.1002

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Temperaments

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| [[Enneadecal]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Satanic]]

[[Category:Wizardharry]]

[[Category:Monzismic]]

@@ Line 3: / Line 3: @@
 == Theory ==
+edo 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]], 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 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]], 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 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 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.
 === Prime harmonics ===
@@ Line 32: / Line 33: @@
 | 2.3
 | {{monzo| -1054 665 }}
-| [{{val| 665 1054 }}]
+| {{mapping| 665 1054 }}
 | +0.0000
 | 0.0000
@@ Line 39: / Line 40: @@
 | 2.3.5
 | {{monzo| -14 -19 19 }}, {{monzo| 54 -37 2 }}
-| [{{val| 665 1054 1544 }}]
+| {{mapping| 665 1054 1544 }}
 | +0.0213
 | 0.0301
@@ Line 46: / Line 47: @@
 | 2.3.5.7
 | 4375/4374, 703125/702464, {{monzo| 36 -5 0 -10 }}
-| [{{val| 665 1054 1544 1867 }}]
+| {{mapping| 665 1054 1544 1867 }}
 | -0.0015
 | 0.0474
@@ Line 53: / Line 54: @@
 | 2.3.5.7.11
 | 4000/3993, 4375/4374, 117649/117612, 131072/130977
-| [{{val| 665 1054 1544 1867 2301 }}]
+| {{mapping| 665 1054 1544 1867 2301 }}
 | -0.0511
 | 0.1078
@@ Line 60: / Line 61: @@
 | 2.3.5.7.11.13
 | 1575/1573, 2080/2079, 4096/4095, 4375/4374, 31250/31213
-| [{{val| 665 1054 1544 1867 2301 2461 }}]
+| {{mapping| 665 1054 1544 1867 2301 2461 }}
 | -0.0594
 | 0.1002
@@ Line 70: / Line 71: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
 ! Associated<br>Ratio
 ! Temperaments
@@ Line 99: / Line 100: @@
 | [[Enneadecal]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:Satanic]]
 [[Category:Wizardharry]]
 [[Category:Monzismic]]