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509edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|509}} == Theory == 509et tempers out 134217728/133984375, 29360128/29296875 and 2401/2400 in the 7-limit and 1073741824/1071794405, 1019215872/1..."
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|509}}
{{ED intro}}
 
== Theory ==
== Theory ==
509et tempers out 134217728/133984375, 29360128/29296875 and [[2401/2400]] in the 7-limit and 1073741824/1071794405, 1019215872/1019046875, 151263/151250, 226492416/226474325, 2359296/2358125, [[1953125/1948617]], 172032/171875, 4302592/4296875, 5767168/5764801, 180224/180075, [[5632/5625]], 422576/421875, [[3025/3024]], [[41503/41472]], 1362944/1361367, 42592/42525, 456533/455625, 322102/321489 and 781258401/781250000 in the 11-limit.
509edo has a sharp tendency in lower [[harmonic]]s. The [[13-limit]] [[TE tuning|optimal tuning]] of this temperament is consistent to the 15-integer-limit, so one might want to keep the [[octave compression]] tight.
It provides the optimal patent val for the temperaments [[Eir]], [[Petrtri]] and Orthocanousmic.
 
As an equal temperament, it [[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]].
 
=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|509}}
{{Harmonics in equal|509}}
=== Subsets and supersets ===
=== Subsets and supersets ===
509 is the 97th [[prime EDO]].
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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|807 -509}}
! rowspan="2" | [[Comma list]]
|{{val|509 807}}
! rowspan="2" | [[Mapping]]
| -0.1890
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 807 -509 }}
| {{mapping| 509 807 }}
| −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 }}
|{{val|509 807 1182}}
| {{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, 4802000/4782969
| 2401/2400, 1600000/1594323, 29360128/29296875
|{{val|509 807 1182 1429}}
| {{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, 41503/41472
| 2401/2400, 3025/3024, 5632/5625, 1600000/1594323
|{{val|509 807 1182 1429 1761}}
| {{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, 4459/4455, 5632/5625
| 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625
|{{val|509 807 1182 1429 1761 1884}}
| {{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, 2431/2430, 7744/7735, 12376/12375, 14161/14157
| 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913
|{{val|509 807 1182 1429 1761 1884 2081}}
| {{mapping| 509 807 1182 1429 1761 1884 2081 }}
| -0.1784
| −0.1784
| 0.1318
| 0.1318
| 5.59
| 5.59
Line 64: Line 70:
=== 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
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|36\509
| 18\509
|84.87
| 42.44
|21/20
| 40/39
|[[Amicable]]
| [[Humorous]]
|-
|-
|1
| 1
|115\509
| 36\509
|271.12
| 84.87
|1024/875
| 21/20
|[[Quasiorwell]]
| [[Amicable]]
|-
|-
|1
| 1
|144\509
| 115\509
|339.49
| 271.12
|243/200
| 1024/875
|[[Amity]]
| [[Quasiorwell]]
|-
| 1
| 144\509
| 339.49
| 243/200
| [[Amity]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Music ==
; [[Francium]]
* "Modern Legends" from ''Mysteries'' (2023) [https://open.spotify.com/track/6GnnrqfYkIcf4xYTivAgRA Spotify] | [https://francium223.bandcamp.com/track/modern-legends Bandcamp] | [https://www.youtube.com/watch?v=A0-37NwYBFM YouTube]
[[Category:Listen]]
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