Ploidacot/Alpha-tricot - Revision history

Xenllium at 12:03, 26 January 2026

2026-01-26T12:03:29Z

← Older revision		Revision as of 12:03, 26 January 2026
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	{{Breadcrumb}}{{Infobox ploidacot\|Ploids=1\|Shears=1\|Cots=3\|Pergen=[P8, P11/3]\|Forms=15, 17, 19, 36\|Title=Alpha-tricot\|Wedgie=3}}		{{Breadcrumb}}{{Infobox ploidacot\|Ploids=1\|Shears=1\|Cots=3\|Pergen=[P8, P11/3]\|Forms=15, 17, 19, 36\|Title=Alpha-tricot\|Wedgie=3}}
	'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about ~~625–640~~{{cent}}, three of which stack to form a perfect twelfth of [[3/1]], and the period is a [[2/1]] octave. Alpha-tricot temperaments generate the [[2L 5s]], [[2L 7s]], and [[2L 9s]] MOS structures. Alpha-tricot temperaments split the diatonic whole tone into three equal parts, producing both supermajor/subminor and supraminor/submajor intervals.		'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about 630–635{{cent}}, three of which stack to form a perfect twelfth of [[3/1]], and the period is a [[2/1]] octave. Alpha-tricot temperaments generate the [[2L 5s]], [[2L 7s]], and [[2L 9s]] MOS structures. Alpha-tricot temperaments split the diatonic whole tone into three equal parts, producing both supermajor/subminor and supraminor/submajor intervals.

	== Intervals and notation ==		== Intervals and notation ==
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	\|+ style="font-size: 105%;" \| Alpha-tricot intervals (assuming pure fifth and octave)		\|+ style="font-size: 105%;" \| Alpha-tricot intervals (assuming pure fifth and octave)
	\|-		\|-
	! #		! rowspan="2" \| #
	! Cents		! rowspan="2" \| Cents
			! colspan="2" \| Wide generator = fifth
			! colspan="2" \| Wide generator = fourth
			\|-
	! Notation		! Notation
	! Name ~~(generator = fifth)~~		! Name
	! Notation		! Notation
	! Name ~~(generator = fourth)~~		! Name
	\|-		\|-
	\| −9		\| −9
	\| 294.14		\| 294.13
	\| Eb		\| Eb
	\| minor third		\| minor third
Line 30:		Line 33:
	\|-		\|-
	\| −7		\| −7
	\| 362.11		\| 362.10
	\| vE		\| vE
	\| submajor third		\| submajor third
Line 44:		Line 47:
	\|-		\|-
	\| −5		\| −5
	\| 430.08		\| 430.07
	\| ^Fb		\| ^Fb
	\| supradiminished fourth		\| supradiminished fourth
Line 58:		Line 61:
	\|-		\|-
	\| −3		\| −3
	\| 498.05		\| 498.04
	\| F		\| F
	\| perfect fourth		\| perfect fourth
Line 72:		Line 75:
	\|-		\|-
	\| −1		\| −1
	\| 566.02		\| 566.01
	\| vF#		\| vF#
	\| subaugmented fourth		\| subaugmented fourth
Line 128:		Line 131:
	\|-		\|-
	\| 7		\| 7
	\| 837.9		\| 837.90
	\| ^Ab		\| ^Ab
	\| supraminor sixth		\| supraminor sixth

Xenllium at 11:00, 7 January 2026

2026-01-07T11:00:08Z

← Older revision		Revision as of 11:00, 7 January 2026
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	[[Liese]] identifies the generator with [[10/7]]. To extend to the full 7-limit, the [[meantone]] mapping of 5 is used (12 generators or 4 fifths up), which has the benefit of flattening the fifth to make the 10/7 more accurate.		[[Liese]] identifies the generator with [[10/7]]. To extend to the full 7-limit, the [[meantone]] mapping of 5 is used (12 generators or 4 fifths up), which has the benefit of flattening the fifth to make the 10/7 more accurate.

	=== ~~12288/12167~~ ===		=== 2.3.23 subgroup temperament ===
	An obvious mapping for the generator is [[23/16]]. This temperament is best tuned by flattening the generator slightly to compromise between the tunings of 23/16 and 3/2. At this point, meantone is a reasonable choice for 5, as it is also tuned with a flattened fifth.		An obvious mapping for the generator is [[23/16]], tempering out the comma [[12288/12167]]. This temperament is best tuned by flattening the generator slightly to compromise between the tunings of 23/16 and 3/2. At this point, meantone is a reasonable choice for 5, as it is also tuned with a flattened fifth.

	=== Paralimmal ===		=== Paralimmal ===
	[[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around 715–720{{c}} or so).		[[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around 715–720{{c}} or so).

	[[Category:Ploidacots]]		[[Category:Ploidacots\|Alpha-tricot]]

FloraC: Text replacement - "[[Category:Ploidacot" to "[[Category:Ploidacots"

2026-01-06T09:44:23Z

Text replacement - "[[Category:Ploidacot" to "[[Category:Ploidacots"

← Older revision		Revision as of 09:44, 6 January 2026
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	[[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around 715–720{{c}} or so).		[[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around 715–720{{c}} or so).

	[[Category:~~Ploidacot~~]]		[[Category:Ploidacots]]

Xenllium at 06:43, 4 January 2026

2026-01-04T06:43:26Z

← Older revision		Revision as of 06:43, 4 January 2026
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	{{Breadcrumb}}{{Infobox ploidacot\|Ploids=1\|Shears=1\|Cots=3\|Pergen=[P8, P11/3]\|Forms=5, 7, 12\|Title=Alpha-tricot\|Wedgie=3}}		{{Breadcrumb}}{{Infobox ploidacot\|Ploids=1\|Shears=1\|Cots=3\|Pergen=[P8, P11/3]\|Forms=15, 17, 19, 36\|Title=Alpha-tricot\|Wedgie=3}}
	'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about 625–640{{cent}}, three of which stack to form a perfect twelfth of [[3/1]], and the period is a [[2/1]] octave. Alpha-tricot temperaments generate the [[2L 5s]], [[2L 7s]], and [[2L 9s]] MOS structures. Alpha-tricot temperaments split the diatonic whole tone into three equal parts, producing both supermajor/subminor and supraminor/submajor intervals.		'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about 625–640{{cent}}, three of which stack to form a perfect twelfth of [[3/1]], and the period is a [[2/1]] octave. Alpha-tricot temperaments generate the [[2L 5s]], [[2L 7s]], and [[2L 9s]] MOS structures. Alpha-tricot temperaments split the diatonic whole tone into three equal parts, producing both supermajor/subminor and supraminor/submajor intervals.

Line 167:		Line 167:
	[[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around 715–720{{c}} or so).		[[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around 715–720{{c}} or so).

	~~{{Todo\| unify precision }}~~		[[Category:Ploidacot]]

VectorGraphics at 23:29, 25 October 2025

2025-10-25T23:29:00Z

← Older revision		Revision as of 23:29, 25 October 2025
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	{{Breadcrumb}}{{Infobox ploidacot\|Ploids=1\|Shears=1\|Cots=3\|Pergen=[P8, P11/3]\|Forms=5, 7, 12\|Title=Alpha-tricot}}		{{Breadcrumb}}{{Infobox ploidacot\|Ploids=1\|Shears=1\|Cots=3\|Pergen=[P8, P11/3]\|Forms=5, 7, 12\|Title=Alpha-tricot\|Wedgie=3}}
	'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about 625–640{{cent}}, three of which stack to form a perfect twelfth of [[3/1]], and the period is a [[2/1]] octave. Alpha-tricot temperaments generate the [[2L 5s]], [[2L 7s]], and [[2L 9s]] MOS structures. Alpha-tricot temperaments split the diatonic whole tone into three equal parts, producing both supermajor/subminor and supraminor/submajor intervals.		'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about 625–640{{cent}}, three of which stack to form a perfect twelfth of [[3/1]], and the period is a [[2/1]] octave. Alpha-tricot temperaments generate the [[2L 5s]], [[2L 7s]], and [[2L 9s]] MOS structures. Alpha-tricot temperaments split the diatonic whole tone into three equal parts, producing both supermajor/subminor and supraminor/submajor intervals.

ArrowHead294 at 15:58, 9 September 2025

2025-09-09T15:58:30Z

Show changes

Lériendil: /* Alphatricot */

2025-09-09T06:39:05Z

Alphatricot

← Older revision		Revision as of 06:39, 9 September 2025
Line 153:		Line 153:

	=== [[Alphatricot]] ===		=== [[Alphatricot]] ===
	Formerly inaccurately just called "tricot", this is a 5-limit microtemperament with the alpha-tricot structure (hence its name). 5/4 is found at ''29'' generators up, the submajor third in the "superaugmented fourth" notation, and the generator can be held to represent ~~either~~ 13/9 (threedic, above), or [[75/52]] ([[140625/140608\|catasmic]], which is far more accurate) in ~~the 2.3.5.13 subgroup~~. As a microtemperament, it is tuned best when the fifth is around just.		Formerly inaccurately just called "tricot", this is a 5-limit microtemperament with the alpha-tricot structure (hence its name). 5/4 is found at ''29'' generators up, the submajor third in the "superaugmented fourth" notation, and in the 2.3.5.13 subgroup the generator can be held to represent 13/9 (threedic, above) in alphatricot, or [[75/52]] ([[140625/140608\|catasmic]], which is far more accurate) in alphatrillium. As a microtemperament, it is tuned best when the fifth is around just, especially in the case of alphatrillium.

	=== Liese ===		=== Liese ===

Lériendil: /* Temperament interpretations */

2025-09-09T06:36:38Z

Temperament interpretations

← Older revision		Revision as of 06:36, 9 September 2025
Line 148:		Line 148:
	== Temperament interpretations ==		== Temperament interpretations ==
	Any valid alpha-tricot temperament assigns a just interpretation to the individual generator.		Any valid alpha-tricot temperament assigns a just interpretation to the individual generator.

			=== Threedic ===
			Probably the most sensical RTT interpretation of the generator is as [[13/9]], tempering out the comma [[2197/2187]], the threedie, in the 2.3.13 subgroup. This temperament is in fact every other step of [[kleismic]] (which splits 13/9 into two [[6/5]]s), and is best tuned with a fifth slightly (0-4 cents) sharp of just.

	=== [[Alphatricot]] ===		=== [[Alphatricot]] ===
	Formerly inaccurately just called "tricot", this is a 5-limit microtemperament with the alpha-tricot structure (hence its name). ~~In the 2.3.5.13 interpretations, the generator represents either 13/9 or 75/52, and~~ 5/4 is found at ''29'' generators up, the submajor third in the "superaugmented fourth" notation. As a microtemperament, it is tuned best when the fifth is around just.		Formerly inaccurately just called "tricot", this is a 5-limit microtemperament with the alpha-tricot structure (hence its name). 5/4 is found at ''29'' generators up, the submajor third in the "superaugmented fourth" notation, and the generator can be held to represent either 13/9 (threedic, above), or [[75/52]] ([[140625/140608\|catasmic]], which is far more accurate) in the 2.3.5.13 subgroup. As a microtemperament, it is tuned best when the fifth is around just.

	=== Liese ===		=== Liese ===
	[[Liese]] identifies the generator with 10/7. To extend to the full 7-limit, the meantone mapping of 5 is used (12 generators or 4 fifths up), which has the benefit of flattening the fifth to make the 10/7 more accurate.		[[Liese]] identifies the generator with [[10/7]]. To extend to the full 7-limit, the [[meantone]] mapping of 5 is used (12 generators or 4 fifths up), which has the benefit of flattening the fifth to make the 10/7 more accurate.

	=== 12288/12167 ===		=== 12288/12167 ===
	An obvious mapping for the generator is 23/16. This temperament is best tuned by flattening the generator slightly to compromise between the tunings of 23/16 and 3/2. At this point, meantone is a reasonable choice for 5, as it is also tuned with a flattened fifth.		An obvious mapping for the generator is [[23/16]]. This temperament is best tuned by flattening the generator slightly to compromise between the tunings of 23/16 and 3/2. At this point, meantone is a reasonable choice for 5, as it is also tuned with a flattened fifth.

	=== Paralimmal ===		=== Paralimmal ===
	[[Paralimmal]] maps the generator to 16/11, meaning it is best tuned with a sharpened fifth (around 715-720 cents or so).		[[Paralimmal]] maps the generator to [[16/11]], meaning it is best tuned with a sharpened fifth (around 715-720 cents or so).

	{{Todo\| unify precision }}		{{Todo\| unify precision }}

Lériendil: /* Alphatricot */

2025-09-09T06:28:40Z

Alphatricot

← Older revision		Revision as of 06:28, 9 September 2025
Line 150:		Line 150:

	=== [[Alphatricot]] ===		=== [[Alphatricot]] ===
	Formerly inaccurately just called "tricot", this is a 5-limit microtemperament with the alpha-tricot structure (hence its name). In the 2.3.5.13 ~~interpretation~~, the generator is 13/9, and 5/4 is found at ''29'' generators up, the submajor third in the "superaugmented fourth" notation. As a microtemperament, it is tuned best when the fifth is around just.		Formerly inaccurately just called "tricot", this is a 5-limit microtemperament with the alpha-tricot structure (hence its name). In the 2.3.5.13 interpretations, the generator represents either 13/9 or 75/52, and 5/4 is found at ''29'' generators up, the submajor third in the "superaugmented fourth" notation. As a microtemperament, it is tuned best when the fifth is around just.

	=== Liese ===		=== Liese ===

VectorGraphics at 00:28, 2 September 2025

2025-09-02T00:28:26Z

← Older revision		Revision as of 00:28, 2 September 2025
Line 1:		Line 1:
	{{Breadcrumb}}		{{Breadcrumb}}{{Infobox ploidacot\|Ploids=1\|Shears=1\|Cots=3\|Pergen=[P8, P11/3]\|Forms=5, 7, 12\|Title=Alpha-tricot}}'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about 625-640 cents, three of which stack to form a perfect twelfth of [[3/1]], and the period is a [[2/1]] octave. Alpha-tricot temperaments generate the [[2L 5s]], [[2L 7s]], and [[2L 9s]] MOS structures. Alpha-tricot temperaments split the diatonic whole tone into three equal parts, producing both supermajor/subminor and supraminor/submajor intervals.

	'''Alpha-tricot''' is a temperament archetype where the generator is a wide tritone of about 625-640 cents, three of which stack to form a perfect twelfth of [[3/1]], and the period is a [[2/1]] octave. Alpha-tricot temperaments generate the [[2L 5s]], [[2L 7s]], and [[2L 9s]] MOS structures. Alpha-tricot temperaments split the diatonic whole tone into three equal parts, producing both supermajor/subminor and supraminor/submajor intervals.

	== Intervals and notation ==		== Intervals and notation ==
Line 15:		Line 13:
	\|-		\|-
	\| -9		\| -9
	\|294.~~135~~		\|294.14
	\|Eb		\|Eb
	\|minor third		\|minor third
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	\|-		\|-
	\| -7		\| -7
	\|362.~~105~~		\|362.11
	\|vE		\|vE
	\|submajor third		\|submajor third
Line 43:		Line 41:
	\|-		\|-
	\| -5		\| -5
	\|430.~~075~~		\|430.08
	\|^Fb		\|^Fb
	\|supradiminished fourth		\|supradiminished fourth
Line 57:		Line 55:
	\|-		\|-
	\| -3		\| -3
	\|498.~~045~~		\|498.05
	\|F		\|F
	\|perfect fourth		\|perfect fourth
Line 71:		Line 69:
	\|-		\|-
	\| -1		\| -1
	\|566.~~015~~		\|566.02
	\|vF#		\|vF#
	\|subaugmented fourth		\|subaugmented fourth
Line 85:		Line 83:
	\|-		\|-
	\|1		\|1
	\|633.~~985~~		\|633.99
	\|^Gb		\|^Gb
	\|supradiminished fifth		\|supradiminished fifth
Line 99:		Line 97:
	\|-		\|-
	\|3		\|3
	\|701.~~955~~		\|701.96
	\|G		\|G
	\|perfect fifth		\|perfect fifth
Line 113:		Line 111:
	\|-		\|-
	\|5		\|5
	\|769.~~925~~		\|769.93
	\|vG#		\|vG#
	\|subaugmented fifth		\|subaugmented fifth
Line 127:		Line 125:
	\|-		\|-
	\|7		\|7
	\|837.~~895~~		\|837.9
	\|^Ab		\|^Ab
	\|supraminor sixth		\|supraminor sixth
Line 141:		Line 139:
	\|-		\|-
	\|9		\|9
	\|905.~~865~~		\|905.87
	\|A		\|A
	\|major sixth		\|major sixth