509edo: Difference between revisions

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== Theory ==
== Theory ==
509edo has a sharp tendency in lower [[harmonic]]s. It is only consistent to the [[7-odd-limit]] due to harmonic [[3/1|3]] being too sharp, causing harmonic [[9/1|9]] to become inconsistent, but the [[13-limit]] [[TE tuning]] of this temperament is consistent to the 15-integer-limit, so one might want to keep the [[octave stretch|octave compression]] tight.  
509edo has a sharp tendency in lower [[harmonic]]s. It is only consistent to the [[7-odd-limit]] due to harmonic [[3/1|3]] being too sharp, causing harmonic [[9/1|9]] to become inconsistent. However, the [[13-limit]] [[TE tuning]] of this temperament is consistent to the 15-integer-limit, so one might want to keep the [[octave stretch|octave compression]] tight.  


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]].
As an equal temperament, it [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 2401/2400 ([[breedsma]]) and 29360128/29296875 ([[quasiorwellisma]]) 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 ===
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 807 -509 }}
| {{Monzo| 807 -509 }}
| {{mapping| 509 807 }}
| {{Mapping| 509 807 }}
| −0.1890
| −0.1890
| 0.1889
| 0.1889
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|-
|-
| 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 }}
| {{Mapping| 509 807 1182 }}
| −0.1729
| −0.1729
| 0.1559
| 0.1559
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 1600000/1594323, 29360128/29296875
| 2401/2400, 1600000/1594323, 29360128/29296875
| {{mapping| 509 807 1182 1429 }}
| {{Mapping| 509 807 1182 1429 }}
| −0.1415
| −0.1415
| 0.1456
| 0.1456
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 5632/5625, 1600000/1594323
| 2401/2400, 3025/3024, 5632/5625, 1600000/1594323
| {{mapping| 509 807 1182 1429 1761 }}
| {{Mapping| 509 807 1182 1429 1761 }}
| −0.1335
| −0.1335
| 0.1312
| 0.1312
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625
| 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625
| {{mapping| 509 807 1182 1429 1761 1884 }}
| {{Mapping| 509 807 1182 1429 1761 1884 }}
| −0.1618
| −0.1618
| 0.1354
| 0.1354
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913
| 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913
| {{mapping| 509 807 1182 1429 1761 1884 2081 }}
| {{Mapping| 509 807 1182 1429 1761 1884 2081 }}
| −0.1784
| −0.1784
| 0.1318
| 0.1318
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|+ style="font-size: 105%;" | 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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| 115\509
| 115\509
| 271.12
| 271.12
| 1024/875
| 90/77
| [[Quasiorwell]]
| [[Quasiorwell]]
|-
|-
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| 339.49
| 339.49
| 243/200
| 243/200
| [[Amity]]
| [[Amity]] (5-limit)
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Music ==
== Music ==
; [[Francium]]
; [[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]
* "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]
* from ''with our kindness.'' (2026)
** "me do it again." – [https://francium223.bandcamp.com/track/me-do-it-again Bandcamp] | [https://www.youtube.com/watch?v=c8azNIHro_s YouTube]
** "you will get burnt." – [https://francium223.bandcamp.com/track/you-will-get-burnt Bandcamp] | [https://www.youtube.com/watch?v=FoOtGaDix1U YouTube]


[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 14:13, 18 August 2026

← 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. It is only consistent to the 7-odd-limit due to harmonic 3 being too sharp, causing harmonic 9 to become inconsistent. However, the 13-limit TE 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 (breedsma) and 29360128/29296875 (quasiorwellisma) 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 90/77 Quasiorwell
1 144\509 339.49 243/200 Amity (5-limit)

* In minimal-generator form

Music

Francium
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