2520edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|2520}}
{{ED intro}}


== 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.  


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.
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=== 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. Its subset edos are {{EDOs| 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 {{nowrap|19/7 {{=}} 2.714}}.
 
2520edo has subset edos {{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 ([[20160edo|20160/8]]).
Furthermore, one step of 2520edo is 8 pians ([[20160edo|20160/8]]).
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
| 9
| 9
| 663\2520<br>(103\2520)
| 663\2520<br />(103\2520)
| 315.714<br>(49.048)
| 315.714<br />(49.048)
| 6/5<br>(36/35)
| 6/5<br />(36/35)
| [[Ennealimmal]] (2520d)
| [[Ennealimmal]] (2520d)
|-
|-
| 18
| 18
| 523\2520<br>(103\2520)
| 523\2520<br />(103\2520)
| 249.047<br>(49.048)
| 249.047<br />(49.048)
| 231/200<br>(99/98)
| 231/200<br />(99/98)
| [[Hemiennealimmal]] (2520de)
| [[Hemiennealimmal]] (2520de)
|-
|-
| 56
| 56
| 1046\2520<br>(11\2520)
| 1046\2520<br />(11\2520)
| 498.095<br>(5.238)
| 498.095<br />(5.238)
| 4/3<br>(126/125)
| 4/3<br />(126/125)
| [[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
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Jacobin]]
[[Category:Jacobin]]

Latest revision as of 06:07, 21 February 2025

← 2519edo 2520edo 2521edo →
Prime factorization 23 × 32 × 5 × 7
Step size 0.47619 ¢ 
Fifth 1474\2520 (701.905 ¢) (→ 737\1260)
Semitones (A1:m2) 238:190 (113.3 ¢ : 90.48 ¢)
Consistency limit 5
Distinct consistency limit 5
Special properties

2520 equal divisions of the octave (abbreviated 2520edo or 2520ed2), also called 2520-tone equal temperament (2520tet) or 2520 equal temperament (2520et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2520 equal parts of about 0.476 ¢ each. Each step represents a frequency ratio of 21/2520, or the 2520th root of 2.

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

Approximation of prime harmonics in 2520edo
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. Its subset edos are 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

Table of rank-2 temperaments by generator
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 distinct

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