2520edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|2520}} | {{EDO intro|2520}} | ||
== Theory == | == Theory == | ||
2520edo is the 18th [[highly composite edo]]. See Subsets and supersets section for the divisors. | 2520edo is the 18th [[highly composite edo]]. See Subsets and supersets section for the divisors. | ||
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It is a good 2.3.5.11.13 [[subgroup]] tuning where it tempers out [[6656/6655]]. The 2520d val tempers out [[2401/2400]] and [[4375/4374]] and provides a tuning for the [[ennealimmal]] temperament and the rank-3 [[ennealimmic]] temperament. The 2520de val is a tuning for the [[hemiennealimmal]] temperament in the 11-limit. The 2520e val is a member of the [[optimal ET sequence]] for the [[tribilo]] temperament, the 2.3.11 rank-2 temperament tempering out 1771561/1769472. | It is a good 2.3.5.11.13 [[subgroup]] tuning where it tempers out [[6656/6655]]. The 2520d val tempers out [[2401/2400]] and [[4375/4374]] and provides a tuning for the [[ennealimmal]] temperament and the rank-3 [[ennealimmic]] temperament. The 2520de val is a tuning for the [[hemiennealimmal]] temperament in the 11-limit. The 2520e val is a member of the [[optimal ET sequence]] for the [[tribilo]] temperament, the 2.3.11 rank-2 temperament tempering out 1771561/1769472. | ||
2520edo tempers out the [[barium comma]], setting [[81/80]] equal to 1/56th of the octave, and it tunes the [[barium]] temperament on the patent val upwards to the 13-limit. In addition, 2520edo tunes a variation of barium in the 2520d val for which has a comma basis of {[[ | 2520edo tempers out the [[barium comma]], setting [[81/80]] equal to 1/56th of the octave, and it tunes the [[barium]] temperament on the patent val upwards to the 13-limit. In addition, 2520edo tunes a variation of barium in the 2520d val for which has a comma basis of {[[4225/4224]], [[4375/4374]], [[6656/6655]], {{monzo| -22 12 3 5 -2 -3 }}} and reaches the [[7/1|7th harmonic]] in 9 generators instead of 5. [[Eliora]] proposes the name ''baridar'' for this temperament, being a portmanteau of 'barium' and 'vidar'. Overall, barium is best considered in 2520edo as a no-sevens temperament, where it has a comma basis {4225/4224, 6656/6655, {{monzo| -24 46 -15 0 -3 -1 }}}. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|2520}} | {{Harmonics in equal|2520}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
In addition to being a highly composite number, 2520 is the least common multiple of numbers from 1 to 10, meaning 2520edo is the smallest superset of first 10 edos. | In addition to being a highly composite number, 2520 is the least common multiple of numbers from 1 to 10, meaning 2520edo is the smallest superset of first 10 edos. | ||
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Furthermore, one step of 2520edo is 8 pians ([[20160edo|20160/8]]). | Furthermore, one step of 2520edo is 8 pians ([[20160edo|20160/8]]). | ||
== Regular temperament properties == | == Regular temperament properties == | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
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| [[Barium]] | | [[Barium]] | ||
|} | |} | ||
<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:Jacobin]] | [[Category:Jacobin]] | ||
Revision as of 12:07, 15 October 2023
| ← 2519edo | 2520edo | 2521edo → |
Theory
2520edo is the 18th highly composite edo. See Subsets and supersets section for the divisors.
It is a good 2.3.5.11.13 subgroup tuning where it tempers out 6656/6655. The 2520d val tempers out 2401/2400 and 4375/4374 and provides a tuning for the ennealimmal temperament and the rank-3 ennealimmic temperament. The 2520de val is a tuning for the hemiennealimmal temperament in the 11-limit. The 2520e val is a member of the optimal ET sequence for the tribilo temperament, the 2.3.11 rank-2 temperament tempering out 1771561/1769472.
2520edo tempers out the barium comma, setting 81/80 equal to 1/56th of the octave, and it tunes the barium temperament on the patent val upwards to the 13-limit. In addition, 2520edo tunes a variation of barium in the 2520d val for which has a comma basis of {4225/4224, 4375/4374, 6656/6655, [-22 12 3 5 -2 -3⟩} and reaches the 7th harmonic in 9 generators instead of 5. Eliora proposes the name baridar for this temperament, being a portmanteau of 'barium' and 'vidar'. Overall, barium is best considered in 2520edo as a no-sevens temperament, where it has a comma basis {4225/4224, 6656/6655, [-24 46 -15 0 -3 -1⟩}.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.050 | -0.123 | +0.222 | +0.111 | -0.051 | -0.194 | +0.106 | -0.179 | -0.053 | +0.203 |
| Relative (%) | +0.0 | -10.6 | -25.9 | +46.6 | +23.2 | -10.8 | -40.6 | +22.3 | -37.6 | -11.2 | +42.5 | |
| Steps (reduced) |
2520 (0) |
3994 (1474) |
5851 (811) |
7075 (2035) |
8718 (1158) |
9325 (1765) |
10300 (220) |
10705 (625) |
11399 (1319) |
12242 (2162) |
12485 (2405) | |
Subsets and supersets
In addition to being a highly composite number, 2520 is the least common multiple of numbers from 1 to 10, meaning 2520edo is the smallest superset of first 10 edos.
2520edo has subset edos 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260. It is a superabundant edo in addition to being highly composite, with abundancy index of 19/7 = 2.714.
Furthermore, one step of 2520edo is 8 pians (20160/8).
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 9 | 663\2520 (103\2520) |
315.714 (49.048) |
6/5 (36/35) |
Ennealimmal (2520d) |
| 18 | 523\2520 (103\2520) |
249.047 (49.048) |
231/200 (99/98) |
Hemiennealimmal (2520de) |
| 56 | 1046\2520 (11\2520) |
498.095 (5.238) |
4/3 (126/125) |
Barium |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct