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== 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]], 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 ===
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[[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 3-2 telic multiple, and fully consistent to the [[27-odd-limit]] and almost the [[31-odd-limit]].


=== Miscellany ===
=== Miscellany ===
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! 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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| -1054 665 }}
| {{Monzo| -1054 665 }}
| {{mapping| 665 1054 }}
| {{Mapping| 665 1054 }}
| +0.0000
| +0.0000
| 0.0000
| 0.0000
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|-
|-
| 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
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| 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
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| 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
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| 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
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|+ 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
|-
|-
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| 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 #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:Satanic]]
[[Category:Wizardharry]]
[[Category:Wizardharry]]
[[Category:Monzismic]]
[[Category:Monzismic]]
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