Ploidacot/Alpha-tricot - Revision history

FloraC: /* Temperament interpretations */ - "inaccurately" (don't rewrite history); - links from section titles

2026-10-28T08:13:04Z

Temperament interpretations: - "inaccurately" (don't rewrite history); - links from section titles

← Older revision		Revision as of 08:13, 28 October 2026
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	=== Threedic ===		=== 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{{c}}) sharp of just.		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{{c}}) sharp of just.

	=== [[Kangaroo]] ===		=== Kangaroo ===
	Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with paralimmal (see below). This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.		[[Kangaroo]] is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with paralimmal (see below). This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.

	=== [[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 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.		Formerly just called ''tricot'', [[alphatricot]] 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 ===
	[[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.

	=== 2.3.23 subgroup temperament ===		=== 2.3.23-subgroup temperament ===
	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.		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).

CompactStar: /* Kangaroo */

2026-10-27T22:52:06Z

Kangaroo

← Older revision		Revision as of 22:52, 27 October 2026
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	=== [[Kangaroo]] ===		=== [[Kangaroo]] ===
	Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with ~~[[#~~paralimmal]] below. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.		Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with paralimmal (see below). This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.

	=== [[Alphatricot]] ===		=== [[Alphatricot]] ===

CompactStar: /* = Kangaroo */

2026-10-27T22:51:46Z

= Kangaroo

← Older revision		Revision as of 22:51, 27 October 2026
Line 158:		Line 158:
	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{{c}}) sharp of just.		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{{c}}) sharp of just.

	==== [[Kangaroo]] ===		=== [[Kangaroo]] ===
	Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with [[#paralimmal]] below. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.		Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with [[#paralimmal]] below. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.

CompactStar: /* = Kangaroo */

2026-10-27T22:51:41Z

= Kangaroo

← Older revision		Revision as of 22:51, 27 October 2026
Line 158:		Line 158:
	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{{c}}) sharp of just.		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{{c}}) sharp of just.

	=== [[Kangaroo]] ==		==== [[Kangaroo]] ===
	Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with [[#paralimmal]] below. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.		Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with [[#paralimmal]] below. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.

CompactStar: /* = Kangaroo */

2026-10-27T22:51:34Z

= Kangaroo

← Older revision		Revision as of 22:51, 27 October 2026
Line 158:		Line 158:
	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{{c}}) sharp of just.		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{{c}}) sharp of just.

	==== [[Kangaroo]] ===		=== [[Kangaroo]] ==
	Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with [[#paralimmal]] below. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.		Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with [[#paralimmal]] below. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.

CompactStar: /* Temperament interpretations */

2026-10-27T22:51:27Z

Temperament interpretations

← Older revision		Revision as of 22:51, 27 October 2026
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	=== Threedic ===		=== 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{{c}}) sharp of just.		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{{c}}) sharp of just.

			==== [[Kangaroo]] ===
			Kangaroo is a weak extension of [[archy]] that. interprets the alpha-tricot generator, somewhat bizarrely, as [[35/24]][[~]][[40/27]], which maps [[5/4]] to +10 generators and [[7/4]] to -6 generators. The 11-limit extension simply equates the generator to [[16/11]], merging it with [[#paralimmal]] below. This favors a particularly sharp diatonic fifth (around 710–720 cents) because otherwise the ~[[5/4]] will be too flat. This temperament is not very accurate, but it might be interesting to anyone who has used [[15edo]], as the 15-not can be seen as a sort of [[detemperament]] of 15edo. It also provides a significantly less complex avenue to the 5-limit compared to most other interpretations.

	=== [[Alphatricot]] ===		=== [[Alphatricot]] ===

Fredg999 category edits: Removing from Category:Ploidacots using Cat-a-lot

2026-07-16T06:38:45Z

Removing from Category:Ploidacots using Cat-a-lot

← Older revision		Revision as of 06:38, 16 July 2026
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	=== 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\|Alpha-tricot]]~~

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
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	\|-		\|-
	\| −7		\| −7
	\| 362.11		\| 362.10
	\| vE		\| vE
	\| submajor third		\| submajor third
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	\|-		\|-
	\| −5		\| −5
	\| 430.08		\| 430.07
	\| ^Fb		\| ^Fb
	\| supradiminished fourth		\| supradiminished fourth
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	\|-		\|-
	\| −3		\| −3
	\| 498.05		\| 498.04
	\| F		\| F
	\| perfect fourth		\| perfect fourth
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	\|-		\|-
	\| −1		\| −1
	\| 566.02		\| 566.01
	\| vF#		\| vF#
	\| subaugmented fourth		\| subaugmented fourth
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	\|-		\|-
	\| 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]]