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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 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 almost consistent in the [[27-odd-limit]], failing 11/10, 17/11, 23/22 and [[octave complement]]<nowiki/>s. 665edo provides relatively 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 almost consistent in the [[27-odd-limit]], failing 11/10, 17/11, 23/22 and [[octave complement]]s. 665edo provides relatively 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 ===
=== Prime harmonics ===
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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|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct


[[Category:3-limit record edos|###]] <!-- 3-digit number -->
[[Category:3-limit record edos|###]] <!-- 3-digit number -->

Revision as of 06:04, 10 March 2026

← 664edo 665edo 666edo →
Prime factorization 5 × 7 × 19
Step size 1.80451 ¢ 
Fifth 389\665 (701.955 ¢)
(convergent)
Semitones (A1:m2) 63:50 (113.7 ¢ : 90.23 ¢)
Consistency limit 9
Distinct consistency limit 9

665 equal divisions of the octave (abbreviated 665edo or 665ed2), also called 665-tone equal temperament (665tet) or 665 equal temperament (665et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 665 equal parts of about 1.8 ¢ each. Each step represents a frequency ratio of 21/665, or the 665th root of 2.

Theory

665edo is best known for its unfathomably 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-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, [-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 [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 supports 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 almost consistent in the 27-odd-limit, failing 11/10, 17/11, 23/22 and octave complements. 665edo provides relatively 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

Approximation of prime harmonics in 665edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.000 -0.148 +0.197 +0.863 +0.375 -0.294 +0.231 -0.304 +0.799 +0.829
Relative (%) +0.0 -0.0 -8.2 +10.9 +47.8 +20.8 -16.3 +12.8 -16.9 +44.3 +45.9
Steps
(reduced) 		665
(0) 		1054
(389) 		1544
(214) 		1867
(537) 		2301
(306) 		2461
(466) 		2718
(58) 		2825
(165) 		3008
(348) 		3231
(571) 		3295
(635)

Subsets and supersets

Since 665 factors into 5 × 7 × 19, 665edo has subset 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 (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 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 3717⁄19s, 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.

Intervals

See Table of 665edo intervals.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1054 665⟩ [⟨665 1054]] +0.0000 0.0000 0.00
2.3.5 [-14 -19 19⟩, [54 -37 2⟩ [⟨665 1054 1544]] +0.0213 0.0301 1.67
2.3.5.7 4375/4374, 703125/702464, [36 -5 0 -10⟩ [⟨665 1054 1544 1867]] −0.0015 0.0474 2.63
2.3.5.7.11 4000/3993, 4375/4374, 117649/117612, 131072/130977 [⟨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 [⟨665 1054 1544 1867 2301 2461]] −0.0594 0.1002 5.55

Rank-2 temperaments

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 [-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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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