2520edo: Difference between revisions

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== 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.

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 {[[~~vidar|~~4225/4224, 4375/4374, 6656/6655]], ~~1116491110875/1115001192448~~} and reaches 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, ~~8862938119652501095929/8859136000000000000000~~}.

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 ===

=== 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]]).~~

== Regular temperament properties ==

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

! Periods per 8ve

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Generator

|-

! Cents

! Periods per 8ve

! Associated ~~Ratio~~

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 9

| 663\2520 (103\2520)

| 663\2520 (103\2520)

| 315.714 (49.048)

| 315.714 (49.048)

| 6/5 (36/35)

| 6/5 (36/35)

| [[Ennealimmal]] (2520d)

|-

| 18

| 523\2520 (103\2520)

| 523\2520 (103\2520)

| 249.047 (49.048)

| 249.047 (49.048)

| 231/200 (99/98)

| 231/200 (99/98)

| [[Hemiennealimmal]] (2520de)

|-

| 56

| 1046\2520 (11\2520)

| 1046\2520 (11\2520)

| 498.095 (5.238)

| 498.095 (5.238)

| 4/3 (126/125)

| 4/3 (126/125)

| [[Barium]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Jacobin]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|2520}}
+{{ED intro}}
 == Theory ==
-edo is the 18th [[highly composite edo]]. See Subsets and supersets section for the divisors.
+edo 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.
-edo 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 {[[vidar|4225/4224, 4375/4374, 6656/6655]], 1116491110875/1115001192448} and reaches 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, 8862938119652501095929/8859136000000000000000}.
+edo 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 ===
 {{Harmonics in equal|2520}}
 === 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}}.
-edo 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]]).
 == Regular temperament properties ==
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-! Periods<br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Generator
+|-
-! Cents
+! Periods<br />per 8ve
-! Associated<br>Ratio
+! Generator*
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
 | 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)
 |-
 | 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)
 |-
 | 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]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Jacobin]]