1920edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1920}}
{{ED intro}}


== Theory ==
== 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-, 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]].
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 ===
=== 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 ===
=== Subsets and supersets ===
Since 1920 factors into {{factorization|1920}}, 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 }}.
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 ==
== Regular temperament properties ==
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| 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*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 30
| 30
| 583\1920<br>(7\1920)
| 583\1920<br />(7\1920)
| 364.375<br>(4.375)
| 364.375<br />(4.375)
| 216/175<br>(385/384)
| 216/175<br />(385/384)
| [[Zinc]]
| [[Zinc]]
|-
|-
| 60
| 60
| 583\1920<br>(7\1920)
| 583\1920<br />(7\1920)
| 364.375<br>(4.375)
| 364.375<br />(4.375)
| 216/175<br>(385/384)
| 216/175<br />(385/384)
| [[Neodymium]] / [[neodymium magnet]]
| [[Neodymium]] / [[neodymium magnet]]
|}
|}
<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


== Music ==
== Music ==

Latest revision as of 07:54, 7 March 2025

← 1919edo 1920edo 1921edo →
Prime factorization 27 × 3 × 5
Step size 0.625 ¢ 
Fifth 1123\1920 (701.875 ¢)
Semitones (A1:m2) 181:145 (113.1 ¢ : 90.63 ¢)
Consistency limit 25
Distinct consistency limit 25

1920 equal divisions of the octave (abbreviated 1920edo or 1920ed2), also called 1920-tone equal temperament (1920tet) or 1920 equal temperament (1920et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1920 equal parts of exactly 0.625 ¢ each. Each step represents a frequency ratio of 21/1920, or the 1920th root of 2.

Theory

1920edo is distinctly consistent through the 25-odd-limit, and in terms of 23-limit relative error, only 1578 and 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.

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

Approximation of prime harmonics in 1920edo
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) +0.000 -0.080 -0.064 -0.076 -0.068 +0.097 +0.045 -0.013 -0.149
Relative (%) +0.0 -12.8 -10.2 -12.1 -10.9 +15.6 +7.1 -2.1 -23.9
Steps
(reduced) 		1920
(0) 		3043
(1123) 		4458
(618) 		5390
(1550) 		6642
(882) 		7105
(1345) 		7848
(168) 		8156
(476) 		8685
(1005)
Approximation of prime harmonics in 1920edo (continued)
Harmonic 29 31 37 41 43 47 53 59 61
Error Absolute (¢) -0.202 -0.036 -0.094 -0.312 -0.268 +0.118 +0.245 +0.203 -0.010
Relative (%) -32.4 -5.7 -15.0 -50.0 -42.8 +18.9 +39.3 +32.5 -1.6
Steps
(reduced) 		9327
(1647) 		9512
(1832) 		10002
(402) 		10286
(686) 		10418
(818) 		10665
(1065) 		10998
(1398) 		11295
(1695) 		11387
(1787)

Subsets and supersets

Since 1920 factors into 27 × 3 × 5, 1920edo has subset 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

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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Eliora
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