239edo: Difference between revisions

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== Theory ==

239edo ~~has~~ a sharp tendency~~, with~~ [[prime harmonic]]s 3 through 11 all tuned sharp. The ~~equal temperament [[tempering out|tempers out]]~~ [[~~2401~~/~~2400~~]], [[~~5120/5103~~]], ~~and 29360128/29296875 in the 7-limit,~~ [[~~support~~]]~~ing~~ the ~~[[hemififths]] temperament and providing an excellent tuning. It also supports and provides~~ a ~~good tuning for [[quasiorwell]] and [[alphaquarter]]. In the 11-limit, it tempers out [[3025/3024]], [[4000/3993]], [[5632/5625]], and 12005/11979~~.

239edo excels as an [[11-limit]] system with a sharp tendency: [[prime harmonic]]s 3 through 11 are all tuned sharp. The accuracy of [[12/11]] is particularly notable, as 30\239 represents a [[convergent]] to this interval, which forms a barely [[consistent circle]], with the closing error being about 45% of a step.

It also encompasses a large variety of higher primes, specifically commendable in the 2.3.5.7.11.17.29.31.37.43.53.59 subgroup.

239 is a convergent to the [[argent tuning]], where the perfect fourth and fifth are in the logarithmic ratio of 1:√2, and with a fifth in this range, tempers out [[5120/5103]] with great accuracy. This implies that [[81/80]] and [[64/63]] are equated (to 5 steps), that three wholetones ([[9/8]]) stack to [[10/7]], and that the [[2187/2048|apotome]], the [[256/243|limma]], and the [[Pythagorean comma]] are equated with [[15/14]] (24 steps), [[21/20]] (17 steps), and [[50/49]] (7 steps) respectively.

Another notable feature of 239edo is that many of its [[5-limit]] intervals are mapped to composite numbers of steps: [[3/2]] to 140, [[4/3]] to 99, [[5/3]] to 176, [[5/4]] to 77, [[6/5]] to 63, and [[8/5]] to 162. Not only that, but plenty of these form relatively simple logarithmic ratios with each other, as described by [[Don Page comma]]s. In particular: [[gammic]], where 9 units form 6/5, 11 form 5/4, and 20 form 3/2; [[quartonic]], where 7 units form 6/5, 11 form 4/3, and 18 form 8/5; and [[escapade]], where 7 units form 5/4, 9 form 4/3, and 16 form 5/3, are all supported by 239edo with generators 7\239, 9\239, and 11\239 respectively. 239edo is describable as the unique intersection of any two of these.

As a result, 239edo possesses many of the structures associated with temperaments such as [[tetracot]], [[porcupine]], [[bleu]], [[orwell]], and [[sensamagic]], but with interpretations of their generators that are generally much more precise than those temperaments provide (and are thus distinguished from the intervals that those temperaments' simplest generators are mapped to in 239edo). This proves advantageous in terms of adeptness at representing different "flavors" of categories of interval: for example, in the category of submajor or [[equable heptatonic]] seconds, 239edo distinguishes [[11/10]] (the "pine" generator, 1/3 of a perfect fourth) at 33\239, from [[32/29]] (the diminished third on the chain of fifths) at 34\239, from [[31/28]] (the "tetracot" generator, 1/4 of a perfect fifth) at 35\239, from [[10/9]] (the sesquiaugmented unison) at 36\239.

In addition to its 11-limit, 239edo also encompasses a large variety of higher primes, and is specifically commendable in the 2.3.5.7.11.17.29.31.37.43.53.59 subgroup. Primes 23, 67, and 71 are for the most part usable as well, though one should be cautious about pitting 23 against sharp odds (like 9 or 33).

The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[5120/5103]], [[10976/10935]], and 29360128/29296875 in the 7-limit, [[support]]ing the [[hemififths]] temperament and providing an excellent tuning. It also supports and provides a good tuning for [[quasiorwell]], [[neptune]], and [[alphaquarter]]. In the 11-limit, it tempers out [[3025/3024]], [[4000/3993]], [[5632/5625]], [[12005/11979]], and [[41503/41472]], supporting [[quadrafifths]] and [[unthirds]].

=== Prime harmonics ===

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== Intervals ==

== Regular temperament properties ==

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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~~| 379 -239 }}

| {{Monzo| 379 -239 }}

| {{~~mapping~~| 239 379 }}

| {{Mapping| 239 379 }}

| −0.307

| 0.307

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|-

| 2.3.5

| {{~~monzo~~| 3 -18 11}}, {{monzo| 32 -7 -9 }}

| {{Monzo| 3 -18 11 }}, {{monzo| 32 -7 -9 }}

| {{~~mapping~~| 239 379 555 }}

| {{Mapping| 239 379 555 }}

| −0.247

| 0.265

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| 2.3.5.7

| 2401/2400, 5120/5103, 29360128/29296875

| {{~~mapping~~| 239 379 555 671 }}

| {{Mapping| 239 379 555 671 }}

| −0.204

| 0.241

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| 2.3.5.7.11

| 2401/2400, 3025/3024, 4000/3993, 5120/5103

| {{~~mapping~~| 239 379 555 671 827 }}

| {{Mapping| 239 379 555 671 827 }}

| −0.220

| 0.218

| 4.34

|-

| 2.3.5.7.11.17

| 595/594, 1156/1155, 2058/2057, 2401/2400, 5120/5103

| {{Mapping| 239 379 555 671 827 977 }}

| −0.203

| 0.203

| 4.03

|- style="border-top: double;"

| 2.3.5.7.11.13

| 352/351, 625/624, 847/845, 1575/1573, 2401/2400

| {{Mapping| 239 379 555 671 827 885 }} (239f)

| −0.318

| 0.296

| 5.89

|-

| 2.3.5.7.11.13.17

| 352/351, 595/594, 625/624, 833/832, 1156/1155, 1575/1573

| {{Mapping| 239 379 555 671 827 885 977 }} (239f)

| −0.290

| 0.282

| 5.63

|}

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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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| 55.23

| 33/32

| [[Escapade]] / [[alphaquarter]]

| [[Escapade]] / [[alphaquarter]] (239f)

|-

| 1

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| 14/11

| [[Unthirds]] (239f)

|-

| 1

| 103\239

| 517.15

| 66/49

| [[Cutefourths]] (239f)

|-

| 1

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| [[Neptune]] (7-limit)

|}

<nowiki />* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<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

== Music ==

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* "thatyounglighthouseboi" from ''albumwithoutspaces'' (2024) – [https://open.spotify.com/track/2dGpklZpWAYSRDHtjc3nDs Spotify] | [https://francium223.bandcamp.com/track/thatyounglighthouseboi Bandcamp] | [https://www.youtube.com/watch?v=nAdJncpsOlk YouTube]

* "Bath And A Nice Dream" from ''You Are A...'' (2024) – [https://open.spotify.com/track/3RJ77kQEsnkjfgjjUFk9DH Spotify] | [https://francium223.bandcamp.com/track/bath-and-a-nice-dream Bandcamp] | [https://www.youtube.com/watch?v=3eBYJM6cRPA YouTube]

* "You Like Frozen Pizza?" from ''Questions, Vol. 2'' (2025) – [https://open.spotify.com/track/09AhftssrVDgwms287FA53 Spotify] | [https://francium223.bandcamp.com/track/you-like-frozen-pizza Bandcamp] | [https://www.youtube.com/watch?v=AVFLQJBLpms YouTube]

[[Category:Alphaquarter]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo has a sharp tendency, with [[prime harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[5120/5103]], and 29360128/29296875 in the 7-limit, [[support]]ing the [[hemififths]] temperament and providing an excellent tuning. It also supports and provides a good tuning for [[quasiorwell]] and [[alphaquarter]]. In the 11-limit, it tempers out [[3025/3024]], [[4000/3993]], [[5632/5625]], and 12005/11979.
+edo excels as an [[11-limit]] system with a sharp tendency: [[prime harmonic]]s 3 through 11 are all tuned sharp. The accuracy of [[12/11]] is particularly notable, as 30\239 represents a [[convergent]] to this interval, which forms a barely [[consistent circle]], with the closing error being about 45% of a step.
-It also encompasses a large variety of higher primes, specifically commendable in the 2.3.5.7.11.17.29.31.37.43.53.59 subgroup.
+is a convergent to the [[argent tuning]], where the perfect fourth and fifth are in the logarithmic ratio of 1:√2, and with a fifth in this range, tempers out [[5120/5103]] with great accuracy. This implies that [[81/80]] and [[64/63]] are equated (to 5 steps), that three wholetones ([[9/8]]) stack to [[10/7]], and that the [[2187/2048|apotome]], the [[256/243|limma]], and the [[Pythagorean comma]] are equated with [[15/14]] (24 steps), [[21/20]] (17 steps), and [[50/49]] (7 steps) respectively.
+Another notable feature of 239edo is that many of its [[5-limit]] intervals are mapped to composite numbers of steps: [[3/2]] to 140, [[4/3]] to 99, [[5/3]] to 176, [[5/4]] to 77, [[6/5]] to 63, and [[8/5]] to 162. Not only that, but plenty of these form relatively simple logarithmic ratios with each other, as described by [[Don Page comma]]s. In particular: [[gammic]], where 9 units form 6/5, 11 form 5/4, and 20 form 3/2; [[quartonic]], where 7 units form 6/5, 11 form 4/3, and 18 form 8/5; and [[escapade]], where 7 units form 5/4, 9 form 4/3, and 16 form 5/3, are all supported by 239edo with generators 7\239, 9\239, and 11\239 respectively. 239edo is describable as the unique intersection of any two of these.
+As a result, 239edo possesses many of the structures associated with temperaments such as [[tetracot]], [[porcupine]], [[bleu]], [[orwell]], and [[sensamagic]], but with interpretations of their generators that are generally much more precise than those temperaments provide (and are thus distinguished from the intervals that those temperaments' simplest generators are mapped to in 239edo). This proves advantageous in terms of adeptness at representing different "flavors" of categories of interval: for example, in the category of submajor or [[equable heptatonic]] seconds, 239edo distinguishes [[11/10]] (the "pine" generator, 1/3 of a perfect fourth) at 33\239, from [[32/29]] (the diminished third on the chain of fifths) at 34\239, from [[31/28]] (the "tetracot" generator, 1/4 of a perfect fifth) at 35\239, from [[10/9]] (the sesquiaugmented unison) at 36\239.
+In addition to its 11-limit, 239edo also encompasses a large variety of higher primes, and is specifically commendable in the 2.3.5.7.11.17.29.31.37.43.53.59 subgroup. Primes 23, 67, and 71 are for the most part usable as well, though one should be cautious about pitting 23 against sharp odds (like 9 or 33).
+The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[5120/5103]], [[10976/10935]], and 29360128/29296875 in the 7-limit, [[support]]ing the [[hemififths]] temperament and providing an excellent tuning. It also supports and provides a good tuning for [[quasiorwell]], [[neptune]], and [[alphaquarter]]. In the 11-limit, it tempers out [[3025/3024]], [[4000/3993]], [[5632/5625]], [[12005/11979]], and [[41503/41472]], supporting [[quadrafifths]] and [[unthirds]].
 === Prime harmonics ===
@@ Line 15: / Line 23: @@
 == Intervals ==
-{{Interval table}}
+{{Main|Table of 239edo intervals}}
 == Regular temperament properties ==
@@ Line 23: / Line 31: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 30: / Line 38: @@
 |-
 | 2.3
-| {{monzo| 379 -239 }}
+| {{Monzo| 379 -239 }}
-| {{mapping| 239 379 }}
+| {{Mapping| 239 379 }}
 | −0.307
 | 0.307
@@ Line 37: / Line 45: @@
 |-
 | 2.3.5
-| {{monzo| 3 -18 11}}, {{monzo| 32 -7 -9 }}
+| {{Monzo| 3 -18 11 }}, {{monzo| 32 -7 -9 }}
-| {{mapping| 239 379 555 }}
+| {{Mapping| 239 379 555 }}
 | −0.247
 | 0.265
@@ Line 45: / Line 53: @@
 | 2.3.5.7
 | 2401/2400, 5120/5103, 29360128/29296875
-| {{mapping| 239 379 555 671 }}
+| {{Mapping| 239 379 555 671 }}
 | −0.204
 | 0.241
@@ Line 52: / Line 60: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 4000/3993, 5120/5103
-| {{mapping| 239 379 555 671 827 }}
+| {{Mapping| 239 379 555 671 827 }}
 | −0.220
 | 0.218
 | 4.34
+|-
+| 2.3.5.7.11.17
+| 595/594, 1156/1155, 2058/2057, 2401/2400, 5120/5103
+| {{Mapping| 239 379 555 671 827 977 }}
+| −0.203
+| 0.203
+| 4.03
+|- style="border-top: double;"
+| 2.3.5.7.11.13
+| 352/351, 625/624, 847/845, 1575/1573, 2401/2400
+| {{Mapping| 239 379 555 671 827 885 }} (239f)
+| −0.318
+| 0.296
+| 5.89
+|-
+| 2.3.5.7.11.13.17
+| 352/351, 595/594, 625/624, 833/832, 1156/1155, 1575/1573
+| {{Mapping| 239 379 555 671 827 885 977 }} (239f)
+| −0.290
+| 0.282
+| 5.63
 |}
@@ Line 62: / Line 91: @@
 |+ 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 90: / Line 119: @@
 | 55.23
 | 33/32
-| [[Escapade]] / [[alphaquarter]]
+| [[Escapade]] / [[alphaquarter]] (239f)
 |-
 | 1
@@ Line 121: / Line 150: @@
 | 14/11
 | [[Unthirds]] (239f)
+|-
+| 1
+| 103\239
+| 517.15
+| 66/49
+| [[Cutefourths]] (239f)
 |-
 | 1
@@ Line 128: / Line 163: @@
 | [[Neptune]] (7-limit)
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<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
 == Music ==
@@ Line 134: / Line 169: @@
 * "thatyounglighthouseboi" from ''albumwithoutspaces'' (2024) – [https://open.spotify.com/track/2dGpklZpWAYSRDHtjc3nDs Spotify] | [https://francium223.bandcamp.com/track/thatyounglighthouseboi Bandcamp] | [https://www.youtube.com/watch?v=nAdJncpsOlk YouTube]
 * "Bath And A Nice Dream" from ''You Are A...'' (2024) – [https://open.spotify.com/track/3RJ77kQEsnkjfgjjUFk9DH Spotify] | [https://francium223.bandcamp.com/track/bath-and-a-nice-dream Bandcamp] | [https://www.youtube.com/watch?v=3eBYJM6cRPA YouTube]
+* "You Like Frozen Pizza?" from ''Questions, Vol. 2'' (2025) – [https://open.spotify.com/track/09AhftssrVDgwms287FA53 Spotify] | [https://francium223.bandcamp.com/track/you-like-frozen-pizza Bandcamp] | [https://www.youtube.com/watch?v=AVFLQJBLpms YouTube]
 [[Category:Alphaquarter]]