624edo: Difference between revisions

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== Theory ==

624edo is [[consistent]] to the [[27-odd-limit]]. ~~The~~ equal temperament [[tempering out|tempers out]] {{monzo| 23 6 -14 }} ([[vishnuzma]]) and {{monzo| -69 45 -1 }} ([[counterschisma]]) in the 5-limit; [[250047/250000]], 2460375/2458624, and 134217728/133984375 in the 7-limit; [[9801/9800]], 46656/46585, [[131072/130977]], and 151263/151250 in the 11-limit; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the 13-limit; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the 17-limit; [[1521/1520]], 2376/2375, [[2432/2431]], and 3328/3325 in the 19-limit; 2024/2023, 2025/2024, and 3888/3887 in the 23-limit.

624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] {{monzo| 23 6 -14 }} ([[vishnuzma]]) and {{monzo| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], [[3520/3519]], and [[3888/3887]] in the [[23-limit]].

It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 2.3.5.11.13.17-[[subgroup]] [[restriction]] thereof.

=== Prime harmonics ===

{{Harmonics in equal|624|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 624edo (continued)}}

=== Subsets and supersets ===

Since 624 factors into {{~~factorization~~|~~624~~}}, 624edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312 }}.

Since 624 factors into primes as {{nowrap| 24 × 3 × 13 }}, 624edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 32:

Line 36:

| {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }}

| {{mapping| 624 989 1449 }}

| -0.0256

| −0.0256

| 0.0510

| 2.65

Line 39:

Line 43:

| 250047/250000, 2460375/2458624, {{monzo| 27 0 -8 -3 }}

| {{mapping| 624 989 1449 1752 }}

| -0.0552

| −0.0552

| 0.0678

| 3.52

Line 46:

Line 50:

| 9801/9800, 46656/46585, 131072/130977, 151263/151250

| {{mapping| 624 989 1449 1752 2159 }}

| -0.0792

| −0.0792

| 0.0772

| 4.02

Line 53:

Line 57:

| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325

| {{mapping| 624 989 1449 1752 2159 2309 }}

| -0.0595

| −0.0595

| 0.0831

| 4.32

Line 60:

Line 64:

| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011

| {{mapping| 624 989 1449 1752 2159 2309 2551 }}

| -0.0795

| −0.0795

| 0.0911

| 4.74

Line 67:

Line 71:

| 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011

| {{mapping| 624 989 1449 1752 2159 2309 2551 2651 }}

| -0.0861

| −0.0861

| 0.0870

| 4.53

|-

| 2.3.5.7.11.13.17.19.23

| 936/935, 1521/1520, 1701/1700, 1716/1715, 2024/2023, 2025/2023, 2376/2375, 2646/2645

| {{mapping| 624 989 1449 1752 2159 2309 2551 2651 2823 }}

| −0.0906

| 0.0830

| 4.32

|}

=== Rank-2 temperaments ===

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

|+Table of rank-2 temperaments by generator

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

! Periods per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

Line 105:

Line 117:

| [[Septichrome]]

|-

|6

| 6

|73\624

| 177\624 (31\624)

|~~140~~.38

| 340.38 (59.62)

|~~425~~/~~392~~

| 162/133 (88/85)

|[[Semiseptichrome]]

| [[Semiseptichrome]]

|-

| 12

| 259\624 (1\624)

| 259\624 (1\624)

| 498.08 (1.92)

| 498.08 (1.92)

| 4/3 (32805/32768)

| 4/3 (32805/32768)

| [[Atomic]]

|-

| 13

| 259\624 (19\624)

| 259\624 (19\624)

| 498.08 (36.54)

| 498.08 (36.54)

| 4/3 (?)

| 4/3 (?)

| [[Aluminium]]

| [[Aluminium]] (5-limit)

|-

| 16

| 259\624 (14\624)

| 259\624 (14\624)

| 498.08 (48.077)

| 498.08 (48.077)

| 4/3 (?)

| 4/3 (?)

| [[Sulfur]]

|-

| 24

| 303\624 (17\624)

| 303\624 (17\624)

| 582.692 (32.692)

| 582.692 (32.692)

| 7/5 (?)

| 7/5 (?)

| [[Chromium]]

|-

| 26

| 259\624 (19\624)

| 259\624 (19\624)

| 498.08 (36.54)

| 498.08 (36.54)

| 4/3 (?)

| 4/3 (?)

| [[Iron]]

|}

<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 forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Music ==

; [[Eliora]]

* [https://www.youtube.com/watch?v=vEDajIHqRUw&pp=ygUGNjI0ZWRv ''Etude in Iron''] (2024)

[[Category:Ainismic]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|624}}
+{{ED intro}}
 == Theory ==
-edo is [[consistent]] to the [[27-odd-limit]]. The equal temperament [[tempering out|tempers out]] {{monzo| 23 6 -14 }} ([[vishnuzma]]) and {{monzo| -69 45 -1 }} ([[counterschisma]]) in the 5-limit; [[250047/250000]], 2460375/2458624, and 134217728/133984375 in the 7-limit; [[9801/9800]], 46656/46585, [[131072/130977]], and 151263/151250 in the 11-limit; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the 13-limit; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the 17-limit; [[1521/1520]], 2376/2375, [[2432/2431]], and 3328/3325 in the 19-limit; 2024/2023, 2025/2024, and 3888/3887 in the 23-limit.
+edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] {{monzo| 23 6 -14 }} ([[vishnuzma]]) and {{monzo| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], [[3520/3519]], and [[3888/3887]] in the [[23-limit]].
+It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 2.3.5.11.13.17-[[subgroup]] [[restriction]] thereof.
 === Prime harmonics ===
 {{Harmonics in equal|624|columns=11}}
+{{Harmonics in equal|624|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 624edo (continued)}}
 === Subsets and supersets ===
-Since 624 factors into {{factorization|624}}, 624edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312 }}.
+Since 624 factors into primes as {{nowrap| 2<sup>4</sup> × 3 × 13 }}, 624edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 32: / Line 36: @@
 | {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }}
 | {{mapping| 624 989 1449 }}
-| -0.0256
+| −0.0256
 | 0.0510
 | 2.65
@@ Line 39: / Line 43: @@
 | 250047/250000, 2460375/2458624, {{monzo| 27 0 -8 -3 }}
 | {{mapping| 624 989 1449 1752 }}
-| -0.0552
+| −0.0552
 | 0.0678
 | 3.52
@@ Line 46: / Line 50: @@
 | 9801/9800, 46656/46585, 131072/130977, 151263/151250
 | {{mapping| 624 989 1449 1752 2159 }}
-| -0.0792
+| −0.0792
 | 0.0772
 | 4.02
@@ Line 53: / Line 57: @@
 | 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325
 | {{mapping| 624 989 1449 1752 2159 2309 }}
-| -0.0595
+| −0.0595
 | 0.0831
 | 4.32
@@ Line 60: / Line 64: @@
 | 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011
 | {{mapping| 624 989 1449 1752 2159 2309 2551 }}
-| -0.0795
+| −0.0795
 | 0.0911
 | 4.74
@@ Line 67: / Line 71: @@
 | 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011
 | {{mapping| 624 989 1449 1752 2159 2309 2551 2651 }}
-| -0.0861
+| −0.0861
 | 0.0870
 | 4.53
+|-
+| 2.3.5.7.11.13.17.19.23
+| 936/935, 1521/1520, 1701/1700, 1716/1715, 2024/2023, 2025/2023, 2376/2375, 2646/2645
+| {{mapping| 624 989 1449 1752 2159 2309 2551 2651 2823 }}
+| −0.0906
+| 0.0830
+| 4.32
 |}
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+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*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 105: / Line 117: @@
 | [[Septichrome]]
 |-
-|6
+| 6
-|73\624
+| 177\624<br />(31\624)
-|140.38
+| 340.38<br />(59.62)
-|425/392
+| 162/133<br />(88/85)
-|[[Semiseptichrome]]
+| [[Semiseptichrome]]
 |-
 | 12
-| 259\624<br>(1\624)
+| 259\624<br />(1\624)
-| 498.08<br>(1.92)
+| 498.08<br />(1.92)
-| 4/3<br>(32805/32768)
+| 4/3<br />(32805/32768)
 | [[Atomic]]
 |-
 | 13
-| 259\624<br>(19\624)
+| 259\624<br />(19\624)
-| 498.08<br>(36.54)
+| 498.08<br />(36.54)
-| 4/3<br>(?)
+| 4/3<br />(?)
-| [[Aluminium]]
+| [[Aluminium]] (5-limit)
 |-
 | 16
-| 259\624<br>(14\624)
+| 259\624<br />(14\624)
-| 498.08<br>(48.077)
+| 498.08<br />(48.077)
-| 4/3<br>(?)
+| 4/3<br />(?)
 | [[Sulfur]]
 |-
 | 24
-| 303\624<br>(17\624)
+| 303\624<br />(17\624)
-| 582.692<br>(32.692)
+| 582.692<br />(32.692)
-| 7/5<br>(?)
+| 7/5<br />(?)
 | [[Chromium]]
 |-
 | 26
-| 259\624<br>(19\624)
+| 259\624<br />(19\624)
-| 498.08<br>(36.54)
+| 498.08<br />(36.54)
-| 4/3<br>(?)
+| 4/3<br />(?)
 | [[Iron]]
 |}
-<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 forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
+== Music ==
+; [[Eliora]]
+* [https://www.youtube.com/watch?v=vEDajIHqRUw&pp=ygUGNjI0ZWRv ''Etude in Iron''] (2024)
+[[Category:Ainismic]]
+[[Category:Listen]]