1920edo: Difference between revisions
m →Regular temperament properties: the ratio is found |
→Theory: +some commas; misc. cleanup |
||
| (5 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{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= | {{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 {{ | 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 == | ||
| Line 16: | Line 19: | ||
=== 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> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 31: | Line 34: | ||
|- | |- | ||
| 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 distinct | ||
== Music == | == Music == | ||