665edo: Difference between revisions
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== Theory == | == Theory == | ||
665edo 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 [[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 === | === Prime harmonics === | ||
| Line 32: | Line 33: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -1054 665 }} | | {{monzo| -1054 665 }} | ||
| | | {{mapping| 665 1054 }} | ||
| +0.0000 | | +0.0000 | ||
| 0.0000 | | 0.0000 | ||
| Line 39: | Line 40: | ||
| 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 }} | ||
| +0.0213 | | +0.0213 | ||
| 0.0301 | | 0.0301 | ||
| Line 46: | Line 47: | ||
| 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 }} | ||
| -0.0015 | | -0.0015 | ||
| 0.0474 | | 0.0474 | ||
| Line 53: | Line 54: | ||
| 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 }} | ||
| -0.0511 | | -0.0511 | ||
| 0.1078 | | 0.1078 | ||
| Line 60: | Line 61: | ||
| 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 }} | ||
| -0.0594 | | -0.0594 | ||
| 0.1002 | | 0.1002 | ||
| Line 70: | Line 71: | ||
|+Table of rank-2 temperaments by generator | |+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 | ||
| Line 99: | Line 100: | ||
| [[Enneadecal]] | | [[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:Satanic]] | ||
[[Category:Wizardharry]] | [[Category:Wizardharry]] | ||
[[Category:Monzismic]] | [[Category:Monzismic]] | ||
Revision as of 08:46, 22 October 2023
| ← 664edo | 665edo | 666edo → |
(convergent)
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-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 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
| 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 = 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.
Miscellaneous properties
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.
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
| 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 it is distinct