509edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|509}} | {{EDO intro|509}} | ||
== Theory == | == Theory == | ||
The equal temperament [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) in the 5-limit; [[2401/2400]] and 29360128/29296875 in the 7-limit; and [[3025/3024]], [[5632/5625]], [[41503/41472]], 42592/42525, 151263/151250, 172032/171875, 180224/180075, 322102/321489, 422576/421875, 456533/455625, and [[1953125/1948617]] in the 11-limit. It provides the [[optimal patent val]] for [[petrtri]], the 2.11/5.13/5 subgroup temperament tempering out [[2200/2197]]. | |||
It provides the optimal patent val for | |||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|509}} | {{Harmonics in equal|509}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
509edo is the 97th [[prime edo]]. | |||
==Regular temperament properties== | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! 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]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|807 -509}} | | {{monzo| 807 -509 }} | ||
|{{ | | {{mapping| 509 807 }} | ||
| -0.1890 | | -0.1890 | ||
| 0.1889 | | 0.1889 | ||
| 8.01 | | 8.01 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|9 -13 5}}, {{monzo|93 -3 -38}} | | {{monzo| 9 -13 5 }}, {{monzo| 93 -3 -38 }} | ||
|{{ | | {{mapping| 509 807 1182 }} | ||
| -0.1729 | | -0.1729 | ||
| 0.1559 | | 0.1559 | ||
| 6.61 | | 6.61 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|2401/2400, 1600000/1594323, | | 2401/2400, 1600000/1594323, 29360128/29296875 | ||
|{{ | | {{mapping| 509 807 1182 1429 }} | ||
| -0.1415 | | -0.1415 | ||
| 0.1456 | | 0.1456 | ||
| 6.18 | | 6.18 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|2401/2400, 3025/3024, 5632/5625, | | 2401/2400, 3025/3024, 5632/5625, 1600000/1594323 | ||
|{{ | | {{mapping| 509 807 1182 1429 1761 }} | ||
| -0.1335 | | -0.1335 | ||
| 0.1312 | | 0.1312 | ||
| 5.57 | | 5.57 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|2080/2079, 2200/2197, 3025/3024 | | 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625 | ||
|{{ | | {{mapping| 509 807 1182 1429 1761 1884 }} | ||
| -0.1618 | | -0.1618 | ||
| 0.1354 | | 0.1354 | ||
| 5.74 | | 5.74 | ||
|- | |- | ||
|2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
|1225/1224, 2080/2079, | | 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913 | ||
|{{ | | {{mapping| 509 807 1182 1429 1761 1884 2081 }} | ||
| -0.1784 | | -0.1784 | ||
| 0.1318 | | 0.1318 | ||
| Line 66: | Line 69: | ||
|+Table of rank-2 temperaments by generator | |+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 | ||
|- | |- | ||
|1 | | 1 | ||
|36\509 | | 36\509 | ||
|84.87 | | 84.87 | ||
|21/20 | | 21/20 | ||
|[[Amicable]] | | [[Amicable]] | ||
|- | |- | ||
|1 | | 1 | ||
|115\509 | | 115\509 | ||
|271.12 | | 271.12 | ||
|1024/875 | | 1024/875 | ||
|[[Quasiorwell]] | | [[Quasiorwell]] | ||
|- | |- | ||
|1 | | 1 | ||
|144\509 | | 144\509 | ||
|339.49 | | 339.49 | ||
|243/200 | | 243/200 | ||
|[[Amity]] | | [[Amity]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||