509edo: Difference between revisions

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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]] (in the 17-limit), [[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]]

Latest revision as of 06:17, 21 February 2025

← 508edo 509edo 510edo →
Prime factorization 509 (prime)
Step size 2.35756 ¢ 
Fifth 298\509 (702.554 ¢)
Semitones (A1:m2) 50:37 (117.9 ¢ : 87.23 ¢)
Consistency limit 7
Distinct consistency limit 7

509 equal divisions of the octave (abbreviated 509edo or 509ed2), also called 509-tone equal temperament (509tet) or 509 equal temperament (509et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 509 equal parts of about 2.36 ¢ each. Each step represents a frequency ratio of 21/509, or the 509th root of 2.

Theory

509edo has a sharp tendency in lower harmonics. The 13-limit optimal tuning of this temperament is consistent to the 15-integer-limit, so one might want to keep the octave compression tight.

As an equal temperament, it 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

Approximation of odd harmonics in 509edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.60 +0.33 +0.13 -1.16 +0.35 +1.12 +0.93 +1.13 -0.46 +0.73 -1.16
Relative (%) +25.4 +13.9 +5.6 -49.2 +14.9 +47.6 +39.3 +48.1 -19.5 +31.0 -49.3
Steps
(reduced) 		807
(298) 		1182
(164) 		1429
(411) 		1613
(86) 		1761
(234) 		1884
(357) 		1989
(462) 		2081
(45) 		2162
(126) 		2236
(200) 		2302
(266)

Subsets and supersets

509edo is the 97th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [807 -509⟩ [⟨509 807]] −0.1890 0.1889 8.01
2.3.5 [9 -13 5⟩, [93 -3 -38⟩ [⟨509 807 1182]] −0.1729 0.1559 6.61
2.3.5.7 2401/2400, 1600000/1594323, 29360128/29296875 [⟨509 807 1182 1429]] −0.1415 0.1456 6.18
2.3.5.7.11 2401/2400, 3025/3024, 5632/5625, 1600000/1594323 [⟨509 807 1182 1429 1761]] −0.1335 0.1312 5.57
2.3.5.7.11.13 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625 [⟨509 807 1182 1429 1761 1884]] −0.1618 0.1354 5.74
2.3.5.7.11.13.17 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913 [⟨509 807 1182 1429 1761 1884 2081]] −0.1784 0.1318 5.59

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 18\509 42.44 40/39 Humorous
1 36\509 84.87 21/20 Amicable
1 115\509 271.12 1024/875 Quasiorwell
1 144\509 339.49 243/200 Amity

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

Music

Francium
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