1920edo: Difference between revisions

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~~The '''1920 division''' divides the octave into 1920 equal parts of exactly 0.625 cents each. It~~ is ~~distinctly~~ [[consistent]] through the 25-odd-limit, and in terms of 23-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], only [[1578edo|1578]] and [[1889edo|1889]] are both smaller and with a lower relative error. In the 29-limit, only 1578 beats it, and in the 31-, 37-, 41-, 43- and 47-limit, nothing beats it. Because of this and because it is a very composite number divisible by 12, it is another candidate for [[interval size measure]].

== Theory ==

1920edo is [[consistency|distinctly consistent]] through the [[25-odd-limit]], and in terms of [[23-limit]] [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], only [[1578edo|1578]] and [[1889edo|1889]] are both smaller and with a lower relative error. In the [[29-limit]], only 1578 beats it, and in the [[31-limit|31-]], [[37-limit|37-]], [[41-limit|41-]], [[43-limit|43-]] and [[47-limit]], nothing beats it. Because of this and because it is a very composite number divisible by 12, it is another candidate for [[interval size measure]].

As a micro- (or nano-) temperament, it is a [[landscape]] system in the [[7-limit]], [[tempering out]] [[250047/250000]], and in the [[11-limit]] it tempers out [[9801/9800]]. Beyond that, it tempers out [[10648/10647]] in the [[13-limit]]; [[5832/5831]] and [[14400/14399]] in the [[17-limit]]; [[4200/4199]], [[5985/5984]], and 6860/6859 in the [[19-limit]]; and [[3381/3380]] in the 23-limit.

=== Prime harmonics ===

{{Harmonics in equal|1920|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 1920edo (continued)}}

=== Subsets and supersets ===

Since 1920 factors into {{nowrap| 27 × 3 × 5 }}, 1920edo has subset edos {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 128, 160, 192, 240, 320, 384, 480, 640, 960 }}.

== Regular temperament properties ==

1920edo has the lowest relative error in the 31-, 37-, 41-, and 47-limit.

=== Rank-2 temperaments ===

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

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 179\1920

| 111.875

| 16/15

| [[Vavoom]]

|-

| 30

| 583\1920 (7\1920)

| 364.375 (4.375)

| 216/175 (385/384)

| [[Zinc]]

|-

| 60

| 583\1920 (7\1920)

| 364.375 (4.375)

| 216/175 (385/384)

| [[Neodymium]] / [[neodymium magnet]]

|}

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

==~~= Miscellany =~~==

== Music ==

~~1920 = 27 × 3 × 5~~; ~~some of its divisors are~~ [[~~10edo|10~~]], [~~[12edo|12]], [[15edo|15]], [[16edo|16]], [[24edo|24]], [[60edo|60]], [[80edo|80]], [[96edo|96]~~]~~, [[128edo|128]], [[240edo|240]], [[320edo|320]] and [[640edo|640]].~~

; [[Eliora]]

* [https://www.youtube.com/watch?v=ShbfCHv8Lj0 ''Jazz Improvisation''] (2023)

[[Category:~~Equal divisions of the octave|####~~]] ~~~~

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''1920 division''' divides the octave into 1920 equal parts of exactly 0.625 cents each. It is distinctly [[consistent]] through the 25-odd-limit, and in terms of 23-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], only [[1578edo|1578]] and [[1889edo|1889]] are both smaller and with a lower relative error. In the 29-limit, only 1578 beats it, and in the 31-, 37-, 41-, 43- and 47-limit, nothing beats it. Because of this and because it is a very composite number divisible by 12, it is another candidate for [[interval size measure]].
+{{ED intro}}
+== Theory ==
+edo is [[consistency|distinctly consistent]] through the [[25-odd-limit]], and in terms of [[23-limit]] [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], only [[1578edo|1578]] and [[1889edo|1889]] are both smaller and with a lower relative error. In the [[29-limit]], only 1578 beats it, and in the [[31-limit|31-]], [[37-limit|37-]], [[41-limit|41-]], [[43-limit|43-]] and [[47-limit]], nothing beats it. Because of this and because it is a very composite number divisible by 12, it is another candidate for [[interval size measure]].
+As a micro- (or nano-) temperament, it is a [[landscape]] system in the [[7-limit]], [[tempering out]] [[250047/250000]], and in the [[11-limit]] it tempers out [[9801/9800]]. Beyond that, it tempers out [[10648/10647]] in the [[13-limit]]; [[5832/5831]] and [[14400/14399]] in the [[17-limit]]; [[4200/4199]], [[5985/5984]], and 6860/6859 in the [[19-limit]]; and [[3381/3380]] in the 23-limit.
 === Prime harmonics ===
-{{Harmonics in equal|1920|columns=15}}
+{{Harmonics in equal|1920|columns=9}}
+{{Harmonics in equal|1920|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 1920edo (continued)}}
+=== Subsets and supersets ===
+Since 1920 factors into {{nowrap| 2<sup>7</sup> × 3 × 5 }}, 1920edo has subset edos {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 128, 160, 192, 240, 320, 384, 480, 640, 960 }}.
+== Regular temperament properties ==
+edo has the lowest relative error in the 31-, 37-, 41-, and 47-limit.
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 179\1920
+| 111.875
+| 16/15
+| [[Vavoom]]
+|-
+| 30
+| 583\1920<br />(7\1920)
+| 364.375<br />(4.375)
+| 216/175<br />(385/384)
+| [[Zinc]]
+|-
+| 60
+| 583\1920<br />(7\1920)
+| 364.375<br />(4.375)
+| 216/175<br />(385/384)
+| [[Neodymium]] / [[neodymium magnet]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-=== Miscellany ===
+== Music ==
-= 2<sup>7</sup> × 3 × 5; some of its divisors are [[10edo|10]], [[12edo|12]], [[15edo|15]], [[16edo|16]], [[24edo|24]], [[60edo|60]], [[80edo|80]], [[96edo|96]], [[128edo|128]], [[240edo|240]], [[320edo|320]] and [[640edo|640]].
+; [[Eliora]]
+* [https://www.youtube.com/watch?v=ShbfCHv8Lj0 ''Jazz Improvisation''] (2023)
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
+[[Category:Listen]]