239edo - Revision history

Overthink: /* Intervals */ move to separate page to reduce lag

2026-06-14T00:06:25Z

Intervals: move to separate page to reduce lag

← Older revision		Revision as of 00:06, 14 June 2026
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	== Intervals ==		== Intervals ==
	{{~~Interval table~~}}		{{Main\|Table of 239edo intervals}}

	== Regular temperament properties ==		== Regular temperament properties ==

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:30:07Z

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

← Older revision		Revision as of 13:30, 13 March 2026
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	\| [[Neptune]] (7-limit)		\| [[Neptune]] (7-limit)
	\|}		\|}
	<nowiki/>* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms\|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 ==		== Music ==

FloraC: /* Regular temperament properties */ + cutefourths

2026-03-04T10:54:39Z

Regular temperament properties: + cutefourths

← Older revision		Revision as of 10:54, 4 March 2026
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	\| 55.23		\| 55.23
	\| 33/32		\| 33/32
	\| [[Escapade]] / [[alphaquarter]]		\| [[Escapade]] / [[alphaquarter]] (239f)
	\|-		\|-
	\| 1		\| 1
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	\| 14/11		\| 14/11
	\| [[Unthirds]] (239f)		\| [[Unthirds]] (239f)
			\|-
			\| 1
			\| 103\239
			\| 517.15
			\| 66/49
			\| [[Cutefourths]] (239f)
	\|-		\|-
	\| 1		\| 1

FloraC: /* Regular temperament properties */ comma basis

2026-01-17T12:21:13Z

Regular temperament properties: comma basis

← Older revision		Revision as of 12:21, 17 January 2026
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	\|-		\|-
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 352/351, 595/594, 625/624, ~~847~~/~~845~~, 1156/1155, 1575/1573		\| 352/351, 595/594, 625/624, 833/832, 1156/1155, 1575/1573
	\| {{Mapping\| 239 379 555 671 827 885 977 }} (239f)		\| {{Mapping\| 239 379 555 671 827 885 977 }} (239f)
	\| −0.290		\| −0.290

FloraC: /* Regular temperament properties */ ++

2026-01-17T12:05:58Z

Regular temperament properties: ++

← Older revision		Revision as of 12:05, 17 January 2026
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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~~\| 379 -239 }}		\| {{Monzo\| 379 -239 }}
	\| {{~~mapping~~\| 239 379 }}		\| {{Mapping\| 239 379 }}
	\| −0.307		\| −0.307
	\| 0.307		\| 0.307
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	\|-		\|-
	\| 2.3.5		\| 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		\| −0.247
	\| 0.265		\| 0.265
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	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 5120/5103, 29360128/29296875		\| 2401/2400, 5120/5103, 29360128/29296875
	\| {{~~mapping~~\| 239 379 555 671 }}		\| {{Mapping\| 239 379 555 671 }}
	\| −0.204		\| −0.204
	\| 0.241		\| 0.241
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	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 2401/2400, 3025/3024, 4000/3993, 5120/5103		\| 2401/2400, 3025/3024, 4000/3993, 5120/5103
	\| {{~~mapping~~\| 239 379 555 671 827 }}		\| {{Mapping\| 239 379 555 671 827 }}
	\| −0.220		\| −0.220
	\| 0.218		\| 0.218
	\| 4.34		\| 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, 847/845, 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		\|+ 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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	\| [[Neptune]] (7-limit)		\| [[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\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

Lériendil: Text replacement - "rgent temperament" to "rgent tuning"

2025-10-09T14:10:42Z

Text replacement - "rgent temperament" to "rgent tuning"

← Older revision		Revision as of 14:10, 9 October 2025
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	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.		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.

	239 is a convergent to the [[argent ~~temperament~~]], 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.		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.		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.

Lériendil: /* Theory */

2025-09-18T14:34:31Z

Theory

← Older revision		Revision as of 14:34, 18 September 2025
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	== Theory ==		== Theory ==
	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, ~~although the circle it~~ forms ~~is only~~ [[consistent circle~~\|weakly consistent~~]].		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.

	239 is a convergent to the [[argent temperament]], 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.		239 is a convergent to the [[argent temperament]], 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.

Lériendil at 07:22, 18 September 2025

2025-09-18T07:22:52Z

← Older revision		Revision as of 07:22, 18 September 2025
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	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.		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, 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.		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).		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).

Lériendil: updated theory section

2025-09-18T06:58:23Z

updated theory section

← Older revision		Revision as of 06:58, 18 September 2025
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	== Theory ==		== 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, although the circle it forms is only [[consistent circle\|weakly consistent]].

	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 temperament]], 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, 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 ===		=== Prime harmonics ===

Francium: /* Music */ +music

2025-04-16T12:57:53Z

Music: +music

← Older revision		Revision as of 12:57, 16 April 2025
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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]		* "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]		* "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]]		[[Category:Alphaquarter]]