Flattone - Revision history

FloraC: Cleanup on infobox

2026-02-09T09:11:40Z

Cleanup on infobox

← Older revision		Revision as of 09:11, 9 February 2026
Line 5:		Line 5:
	\| Edo join 1 = 19 \| Edo join 2 = 26		\| Edo join 1 = 19 \| Edo join 2 = 26
	\| Mapping = 1; 1 4 -9 6 -4		\| Mapping = 1; 1 4 -9 6 -4
	\| ~~Generator~~ = 3/2		\| Generators = 3/2
	\| ~~Generator~~ tuning = 693.1		\| Generators tuning = 693.1
	\| Optimization method = CWE		\| Optimization method = CWE
	\| MOS scales = [[5L 2s]], [[7L 5s]], [[7L 12s]]		\| MOS scales = [[5L 2s]], [[7L 5s]], [[7L 12s]]
	\| Pergen = (P8, P5)		\| Pergen = (P8, P5)
	\| Odd limit 1 = 9 ~~\| Odd limit 2 = 13~~		\| Odd limit 1 = 9 \| Mistuning 1 = 15.7 \| Complexity 1 = 19
	\| Mistuning 1 = 15.7 \| Mistuning 2 = 19.3		\| Odd limit 2 = 13 \| Mistuning 2 = 19.3 \| Complexity 2 = 19
	~~\| Complexity 1 = 19~~ \| Complexity 2 = 19
	}}		}}
	'''Flattone''' is an alternative [[extension]] to [[5-limit]] [[meantone]], the [[temperament]] that [[tempering out\|tempers out]] the [[81/80\|syntonic comma (81/80)]]. It is generated by a fifth that is typically flatter than that of [[septimal meantone]], and nine of those reach the [[pitch class]] of [[8/7]], so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full [[9-odd-limit]] [[tonality diamond]] is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.		'''Flattone''' is an alternative [[extension]] to [[5-limit]] [[meantone]], the [[temperament]] that [[tempering out\|tempers out]] the [[81/80\|syntonic comma (81/80)]]. It is generated by a fifth that is typically flatter than that of [[septimal meantone]], and nine of those reach the [[pitch class]] of [[8/7]], so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full [[9-odd-limit]] [[tonality diamond]] is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.

FloraC: tridecimal meantone -> fokkertone

2025-12-17T11:22:33Z

tridecimal meantone -> fokkertone

← Older revision		Revision as of 11:22, 17 December 2025
Line 16:		Line 16:
	'''Flattone''' is an alternative [[extension]] to [[5-limit]] [[meantone]], the [[temperament]] that [[tempering out\|tempers out]] the [[81/80\|syntonic comma (81/80)]]. It is generated by a fifth that is typically flatter than that of [[septimal meantone]], and nine of those reach the [[pitch class]] of [[8/7]], so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full [[9-odd-limit]] [[tonality diamond]] is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.		'''Flattone''' is an alternative [[extension]] to [[5-limit]] [[meantone]], the [[temperament]] that [[tempering out\|tempers out]] the [[81/80\|syntonic comma (81/80)]]. It is generated by a fifth that is typically flatter than that of [[septimal meantone]], and nine of those reach the [[pitch class]] of [[8/7]], so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full [[9-odd-limit]] [[tonality diamond]] is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.

	However, it makes up for that by having simpler 11- and 13-limit interpretations – the whole tone is now flat enough that it can function as [[9/8]], [[10/9]], and [[11/10]], tempering out [[100/99]] and making [[11/8]] an augmented fourth (C–F#). This means the major third functions as both 5/4 and 11/9. Tempering out [[65/64]] means it also represents their [[mediant]] [[16/13]], making [[13/8]] a minor sixth (C–A♭) and a full otonal chord of 8:9:10:11:12:13:14:15:16 accessible with a gamut of 16 notes, compared to 19 for ~~tridecimal meantone~~ or the 29 required by [[meanpop]].		However, it makes up for that by having simpler 11- and 13-limit interpretations – the whole tone is now flat enough that it can function as [[9/8]], [[10/9]], and [[11/10]], tempering out [[100/99]] and making [[11/8]] an augmented fourth (C–F#). This means the major third functions as both 5/4 and 11/9. Tempering out [[65/64]] means it also represents their [[mediant]] [[16/13]], making [[13/8]] a minor sixth (C–A♭) and a full otonal chord of 8:9:10:11:12:13:14:15:16 accessible with a gamut of 16 notes, compared to 19 for [[fokkertone]] or the 29 required by [[meanpop]].
	[[File:45EDO_Otonal.mp3\|none\|thumb\|Harmonic scale 8–16 in 45edo, using the flattone mappings for 13 and 15 rather than the best direct approximations.]]		[[File:45EDO_Otonal.mp3\|none\|thumb\|Harmonic scale 8–16 in 45edo, using the flattone mappings for 13 and 15 rather than the best direct approximations.]]

FloraC: /* Tunings */ + vals

2025-12-17T10:08:40Z

Tunings: + vals

← Older revision		Revision as of 10:08, 17 December 2025
Line 259:		Line 259:
	\|		\|
	\| 690.909		\| 690.909
	\|		\| 33c val
	\|-		\|-
	\|		\|
Line 279:		Line 279:
	\|		\|
	\| 691.304		\| 691.304
	\|		\| 92bccc val
	\|-		\|-
	\|		\|
Line 289:		Line 289:
	\|		\|
	\| 691.525		\| 691.525
	\|		\| 59bc val
	\|-		\|-
	\| [[85edo\|49\85]]		\| [[85edo\|49\85]]
	\|		\|
	\| 691.765		\| 691.765
	\|		\| 85bccf val
	\|-		\|-
	\|		\|
Line 339:		Line 339:
	\|		\|
	\| 692.958		\| 692.958
	\|		\| 71bcf val
	\|-		\|-
	\|		\|
Line 359:		Line 359:
	\|		\|
	\| 693.333		\| 693.333
	\|		\| 45f val
	\|-		\|-
	\|		\|
Line 369:		Line 369:
	\|		\|
	\| 693.750		\| 693.750
	\|		\| 64cdef val
	\|-		\|-
	\|		\|
Line 419:		Line 419:
	\|		\|
	\| 700.000		\| 700.000
	\|		\| 12d val
	\|-		\|-
	\|		\|

FloraC: /* Tunings */ + basic norm-based tunings

2025-12-17T09:58:46Z

Tunings: + basic norm-based tunings

@@ Line 201: / Line 201: @@
 == Tunings ==
 === Tuning spectrum ===
 {| class="wikitable center-all left-4"

FloraC: Make the infobox less wide

2025-12-17T07:40:30Z

Make the infobox less wide

← Older revision		Revision as of 07:40, 17 December 2025
Line 2:		Line 2:
	\| Title = Flattone		\| Title = Flattone
	\| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13		\| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
	\| Comma basis = [[81/80]], [[525/512]] (~~2.3.5.~~7);<br>[[45/44]], [[81/80]], [[385/384]] (~~2.3.5.7.~~11);<br>[[45/44]], [[65/64]], [[78/77]], [[81/80]] (~~2.3.5.7.11.~~13)		\| Comma basis = [[81/80]], [[525/512]] (7-limit);<br>[[45/44]], [[81/80]], [[385/384]] (11-limit);<br>[[45/44]], [[65/64]], [[78/77]], [[81/80]] (13-limit)
	\| Edo join 1 = 19 \| Edo join 2 = 26		\| Edo join 1 = 19 \| Edo join 2 = 26
	\| Mapping = 1; 1 4 -9 6 -4		\| Mapping = 1; 1 4 -9 6 -4

Lucius Chiaraviglio: /* Tuning spectrum */ Make odd-limit diamond monotone bounds specifications grammatically consistent

2025-12-15T08:36:19Z

Tuning spectrum: Make odd-limit diamond monotone bounds specifications grammatically consistent

← Older revision		Revision as of 08:36, 15 December 2025
Line 362:		Line 362:
	\|		\|
	\| 694.737		\| 694.737
	\| Upper bound of 7-, 9-, 11-, 13-odd-limit diamond monotone		\| Upper bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
	\|-		\|-
	\|		\|

Overthink: /* As a detemperament of 7et */ + some missing ratios

2025-12-15T01:50:27Z

As a detemperament of 7et: + some missing ratios

← Older revision		Revision as of 01:50, 15 December 2025
Line 140:		Line 140:
	\|		\|
	\| 270		\| 270
	\| 7/6		\| 7/6, 13/11
	\| 321		\| 321
	\| 6/5		\| 6/5
	\| 372		\| 372
	\| 5/4, 16/13, 26/21		\| 5/4, 11/9, 16/13, 26/21
	\| 423		\| 423
	\| 9/7		\| 9/7
Line 150:		Line 150:
	\| Fourth		\| Fourth
	\| 405		\| 405
	\|		\| 14/11
	\| 456		\| 456
	\| 13/10, 21/16		\| 13/10, 21/16
Line 170:		Line 170:
	\| 20/13, 32/21		\| 20/13, 32/21
	\| 795		\| 795
	\|		\| 11/7
	\|-		\|-
	\| Sixth		\| Sixth
Line 176:		Line 176:
	\| 14/9		\| 14/9
	\| 828		\| 828
	\| 8/5, 13/8, 21/13		\| 8/5, 18/11, 13/8, 21/13
	\| 879		\| 879
	\| 5/3		\| 5/3
	\| 930		\| 930
	\| 12/7		\| 12/7, 22/13
	\| 981		\| 981
	\|		\|

Overthink: 9-odd-limit is better; + minimax errors

2025-12-15T01:35:47Z

9-odd-limit is better; + minimax errors

← Older revision		Revision as of 01:35, 15 December 2025
Line 10:		Line 10:
	\| MOS scales = [[5L 2s]], [[7L 5s]], [[7L 12s]]		\| MOS scales = [[5L 2s]], [[7L 5s]], [[7L 12s]]
	\| Pergen = (P8, P5)		\| Pergen = (P8, P5)
	\| Odd limit 1 = 7 \| Odd limit 2 = 13		\| Odd limit 1 = 9 \| Odd limit 2 = 13
	\| Mistuning 1 = ? \| Mistuning 2 = ?		\| Mistuning 1 = 15.7 \| Mistuning 2 = 19.3
	\| Complexity 1 = 19 \| Complexity 2 = 19		\| Complexity 1 = 19 \| Complexity 2 = 19
	}}		}}

Overthink: expanded infobox (though missing minimax errors)

2025-12-15T01:32:26Z

expanded infobox (though missing minimax errors)

← Older revision		Revision as of 01:32, 15 December 2025
Line 1:		Line 1:
	{{Infobox regtemp		{{Infobox regtemp
	\| Title = Flattone		\| Title = Flattone
	\| Subgroups = 2.3.5.7		\| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
	\| Comma basis = [[81/80]], [[525/512]]		\| Comma basis = [[81/80]], [[525/512]] (2.3.5.7);<br>[[45/44]], [[81/80]], [[385/384]] (2.3.5.7.11);<br>[[45/44]], [[65/64]], [[78/77]], [[81/80]] (2.3.5.7.11.13)
	\| Edo join 1 = 19 \| Edo join 2 = 26		\| Edo join 1 = 19 \| Edo join 2 = 26
	\| Mapping = 1; 1 4 -9		\| Mapping = 1; 1 4 -9 6 -4
	\| Generator = 3/2		\| Generator = 3/2
	\| Generator tuning = 693.1		\| Generator tuning = 693.1
Line 10:		Line 10:
	\| MOS scales = [[5L 2s]], [[7L 5s]], [[7L 12s]]		\| MOS scales = [[5L 2s]], [[7L 5s]], [[7L 12s]]
	\| Pergen = (P8, P5)		\| Pergen = (P8, P5)
			\| Odd limit 1 = 7 \| Odd limit 2 = 13
			\| Mistuning 1 = ? \| Mistuning 2 = ?
			\| Complexity 1 = 19 \| Complexity 2 = 19
	}}		}}
	'''Flattone''' is an alternative [[extension]] to [[5-limit]] [[meantone]], the [[temperament]] that [[tempering out\|tempers out]] the [[81/80\|syntonic comma (81/80)]]. It is generated by a fifth that is typically flatter than that of [[septimal meantone]], and nine of those reach the [[pitch class]] of [[8/7]], so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full [[9-odd-limit]] [[tonality diamond]] is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.		'''Flattone''' is an alternative [[extension]] to [[5-limit]] [[meantone]], the [[temperament]] that [[tempering out\|tempers out]] the [[81/80\|syntonic comma (81/80)]]. It is generated by a fifth that is typically flatter than that of [[septimal meantone]], and nine of those reach the [[pitch class]] of [[8/7]], so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full [[9-odd-limit]] [[tonality diamond]] is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.

FloraC: CTE -> CWE

2025-11-28T07:50:57Z

CTE -> CWE

← Older revision		Revision as of 07:50, 28 November 2025
Line 1:		Line 1:
	{{Infobox regtemp		{{Infobox regtemp
	\| Comma basis=[[81/80]], [[~~875~~/~~864~~]]		\| Title = Flattone
	\| Edo join 1=19		\| Subgroups = 2.3.5.7
	\| Edo join 2=26		\| Comma basis = [[81/80]], [[525/512]]
	\| Generator=3/2		\| Edo join 1 = 19 \| Edo join 2 = 26
	\| Generator tuning=693.0		\| Mapping = 1; 1 4 -9
	\| ~~Mapping~~=~~1; 1 4 -9~~		\| Generator = 3/2
	\| MOS scales=[[5L 2s]], [[7L 5s]], [[7L 12s]]		\| Generator tuning = 693.1
	~~\| Optimization method=CTE~~		\| Optimization method = CWE
	\| Pergen=(P8, P5)		\| MOS scales = [[5L 2s]], [[7L 5s]], [[7L 12s]]
	~~\| Title=Flattone~~		\| Pergen = (P8, P5)
	~~\| Subgroups=2.3.5.7~~
	}}		}}

	'''Flattone''' is an alternative [[extension]] to [[5-limit]] [[meantone]], the [[temperament]] that [[tempering out\|tempers out]] the [[81/80\|syntonic comma (81/80)]]. It is generated by a fifth that is typically flatter than that of [[septimal meantone]], and nine of those reach the [[pitch class]] of [[8/7]], so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full [[9-odd-limit]] [[tonality diamond]] is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.		'''Flattone''' is an alternative [[extension]] to [[5-limit]] [[meantone]], the [[temperament]] that [[tempering out\|tempers out]] the [[81/80\|syntonic comma (81/80)]]. It is generated by a fifth that is typically flatter than that of [[septimal meantone]], and nine of those reach the [[pitch class]] of [[8/7]], so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a doubly diminished fifth (C–G𝄫). Although 7/4 is simpler than in septimal meantone, the full [[9-odd-limit]] [[tonality diamond]] is more complex as the 5 and 7 are reached by going in opposite directions, while also being less accurate.

Line 35:		Line 33:
	\|-		\|-
	\| 1		\| 1
	\| 693.0		\| 693.1
	\| '''3/2'''		\| '''3/2'''
	\|-		\|-
Line 43:		Line 41:
	\|-		\|-
	\| 3		\| 3
	\| 879.1		\| 879.2
	\| 5/3		\| 5/3
	\|-		\|-
	\| 4		\| 4
	\| 372.1		\| 372.2
	\| '''5/4''', '''16/13''', 26/21		\| '''5/4''', '''16/13''', 26/21
	\|-		\|-
	\| 5		\| 5
	\| 1065.1		\| 1065.3
	\| 11/6, 13/7, 15/8, 24/13		\| 11/6, 13/7, 15/8, 24/13
	\|-		\|-
	\| 6		\| 6
	\| 558.2		\| 558.3
	\| '''11/8''', 18/13		\| '''11/8''', 18/13
	\|-		\|-
	\| 7		\| 7
	\| 51.2		\| 51.4
	\| 25/24, 27/26, 33/32, 36/35, 55/54, 64/63		\| 25/24, 27/26, 33/32, 36/35, 55/54, 64/63
	\|-		\|-
	\| 8		\| 8
	\| 744.2		\| 744.4
	\| 20/13, 32/21		\| 20/13, 32/21
	\|-		\|-
	\| 9		\| 9
	\| 237.3		\| 237.5
	\| '''8/7''', 15/13		\| '''8/7''', 15/13
	\|-		\|-
	\| 10		\| 10
	\| 930.3		\| 930.5
	\| 12/7, 22/13		\| 12/7, 22/13
	\|-		\|-
	\| 11		\| 11
	\| 423.3		\| 423.6
	\| 9/7		\| 9/7
	\|-		\|-
	\| 12		\| 12
	\| 1116.4		\| 1116.6
	\| 27/14, 40/21		\| 27/14, 40/21
	\|-		\|-
	\| 13		\| 13
	\| 609.4		\| 609.7
	\| 10/7		\| 10/7
	\|}		\|}
	<nowiki/>* In 13-limit ~~CTE~~ tuning, octave reduced		<nowiki/>* In 13-limit CWE tuning, octave reduced

	=== As a detemperament of 7et ===		=== As a detemperament of 7et ===