509edo: Difference between revisions

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== 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 ===

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! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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|-

| 2.3

| {{~~monzo~~| 807 -509 }}

| {{Monzo| 807 -509 }}

| {{~~mapping~~| 509 807 }}

| {{Mapping| 509 807 }}

| −0.1890

| 0.1889

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|-

| 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

| 0.1559

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| 2.3.5.7

| 2401/2400, 1600000/1594323, 29360128/29296875

| {{~~mapping~~| 509 807 1182 1429 }}

| {{Mapping| 509 807 1182 1429 }}

| −0.1415

| 0.1456

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| 2.3.5.7.11

| 2401/2400, 3025/3024, 5632/5625, 1600000/1594323

| {{~~mapping~~| 509 807 1182 1429 1761 }}

| {{Mapping| 509 807 1182 1429 1761 }}

| −0.1335

| 0.1312

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| 2.3.5.7.11.13

| 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

| 0.1354

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| 2.3.5.7.11.13.17

| 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

| 0.1318

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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

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| 115\509

| 271.12

| ~~1024~~/~~875~~

| 90/77

| [[Quasiorwell]]

|-

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| 339.49

| 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 ==

; [[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]

* ~~"me do it again."~~ from ''with our kindness.'' (2026) – [https://francium223.bandcamp.com/track/me-do-it-again Bandcamp] | [https://www.youtube.com/watch?v=c8azNIHro_s 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]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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.
+edo 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 ===
@@ Line 19: / Line 19: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 26: / Line 26: @@
 |-
 | 2.3
-| {{monzo| 807 -509 }}
+| {{Monzo| 807 -509 }}
-| {{mapping| 509 807 }}
+| {{Mapping| 509 807 }}
 | −0.1890
 | 0.1889
@@ Line 33: / Line 33: @@
 |-
 | 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
 | 0.1559
@@ Line 41: / Line 41: @@
 | 2.3.5.7
 | 2401/2400, 1600000/1594323, 29360128/29296875
-| {{mapping| 509 807 1182 1429 }}
+| {{Mapping| 509 807 1182 1429 }}
 | −0.1415
 | 0.1456
@@ Line 48: / Line 48: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 5632/5625, 1600000/1594323
-| {{mapping| 509 807 1182 1429 1761 }}
+| {{Mapping| 509 807 1182 1429 1761 }}
 | −0.1335
 | 0.1312
@@ Line 55: / Line 55: @@
 | 2.3.5.7.11.13
 | 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
 | 0.1354
@@ Line 62: / Line 62: @@
 | 2.3.5.7.11.13.17
 | 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
 | 0.1318
@@ Line 72: / Line 72: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 93: / Line 93: @@
 | 115\509
 | 271.12
-| 1024/875
+| 90/77
 | [[Quasiorwell]]
 |-
@@ Line 100: / Line 100: @@
 | 339.49
 | 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 ==
 ; [[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]
-* "me do it again." from ''with our kindness.'' (2026) – [https://francium223.bandcamp.com/track/me-do-it-again Bandcamp] | [https://www.youtube.com/watch?v=c8azNIHro_s 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]]