Tetracot - Revision history

Overthink: 34 is more of a central tuning

2026-06-19T20:38:04Z

34 is more of a central tuning

← Older revision		Revision as of 20:38, 19 June 2026
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	\| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13		\| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13
	\| Comma basis = [[20000/19683]] (2.3.5);<br>[[100/99]], [[243/242]] (2.3.5.11)<br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)		\| Comma basis = [[20000/19683]] (2.3.5);<br>[[100/99]], [[243/242]] (2.3.5.11)<br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)
	\| Edo join 1 = 7 \| Edo join 2 = ~~27e~~		\| Edo join 1 = 7 \| Edo join 2 = 34
	\| Mapping = 1; 4 9 10 -2		\| Mapping = 1; 4 9 10 -2
	\| Generators = 10/9		\| Generators = 10/9

Lucius Chiaraviglio: Fix punctuation

2026-05-18T11:25:14Z

Fix punctuation

← Older revision		Revision as of 11:25, 18 May 2026
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	[[Equal temperament]]s that [[support]] tetracot include {{EDOs\| 27, 34, and 41 }}.		[[Equal temperament]]s that [[support]] tetracot include {{EDOs\| 27, 34, and 41 }}.

	Tetracot has four strong [[extension]]s for the 7-, 11-, and 13-limit, which use the same methods of obtaining the [[11/1\|11th]] and [[13/1\|13th]] harmonics (10 generators up and 2 generators down, respectively) but differ in their methods of obtaining the [[7/1\|7th harmonic]].		Tetracot has four strong [[extension]]s for the 7-, 11-, and 13-limit, which use the same methods of obtaining the [[11/1\|11th]] and [[13/1\|13th]] harmonics (10 generators up and 2 generators down, respectively) but differ in their methods of obtaining the [[7/1\|7th harmonic]]:
	* [[Monkey]] (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out [[875/864]] and thereby equating [[7/4]] with ([[6/5]])<sup>3</sup>;		* [[Monkey]] (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out [[875/864]] and thereby equating [[7/4]] with ([[6/5]])<sup>3</sup>;
	* [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])<sup>2</sup>;		* [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])<sup>2</sup>;

Lucius Chiaraviglio: Add summary of how to get the 11th and 13th harmonics

2026-05-18T11:24:10Z

Add summary of how to get the 11th and 13th harmonics

← Older revision		Revision as of 11:24, 18 May 2026
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	[[Equal temperament]]s that [[support]] tetracot include {{EDOs\| 27, 34, and 41 }}.		[[Equal temperament]]s that [[support]] tetracot include {{EDOs\| 27, 34, and 41 }}.

	Tetracot has four strong [[extension]]s for the 7-, 11-, and 13-limit, which ~~differ in their methods of obtaining the [[7/1\|7th harmonic]], but~~ use the same methods of obtaining the [[11/1\|11th]] and [[13/1\|13th]] harmonics.		Tetracot has four strong [[extension]]s for the 7-, 11-, and 13-limit, which use the same methods of obtaining the [[11/1\|11th]] and [[13/1\|13th]] harmonics (10 generators up and 2 generators down, respectively) but differ in their methods of obtaining the [[7/1\|7th harmonic]].
	* [[Monkey]] (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out [[875/864]] and thereby equating [[7/4]] with ([[6/5]])<sup>3</sup>;		* [[Monkey]] (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out [[875/864]] and thereby equating [[7/4]] with ([[6/5]])<sup>3</sup>;
	* [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])<sup>2</sup>;		* [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])<sup>2</sup>;

Lucius Chiaraviglio: Expand the descriptions of the extensions

2026-05-18T11:17:30Z

Expand the descriptions of the extensions

← Older revision		Revision as of 11:17, 18 May 2026
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	[[Equal temperament]]s that [[support]] tetracot include {{EDOs\| 27, 34, and 41 }}.		[[Equal temperament]]s that [[support]] tetracot include {{EDOs\| 27, 34, and 41 }}.

	Tetracot has ~~many~~ [[extension]]s for the 7-, 11-, and 13-limit. ~~These are~~ [[~~monkey~~]] (34 & 41), [[~~bunya~~]] (34d & 41), [[~~modus~~]] (27e & 34d) and [[~~wollemia~~]] (27e & 34).		Tetracot has four strong [[extension]]s for the 7-, 11-, and 13-limit, which differ in their methods of obtaining the [[7/1\|7th harmonic]], but use the same methods of obtaining the [[11/1\|11th]] and [[13/1\|13th]] harmonics.
			* [[Monkey]] (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out [[875/864]] and thereby equating [[7/4]] with ([[6/5]])<sup>3</sup>;
			* [[Bunya]] (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out [[225/224]] and thereby equating [[7/2]] with ([[15/8]])<sup>2</sup>;
			* [[Modus]] (27e & 34d) obtains the 7th harmonic at 8 generators down, tempering out [[64/63]] and thereby equating 7/4 with [[16/9]];
			* [[Wollemia]] (27e & 34) obtains the 7th harmonic at 19 generators up, tempering out [[126/125]] and thereby equating [[7/1]] with ([[5/3]])<sup>3</sup>([[3/2]]).

	See [[Tetracot family]] for technical data.		See [[Tetracot family]] for technical data.

FloraC: Page split followup

2026-04-16T15:00:29Z

Page split followup

← Older revision		Revision as of 15:00, 16 April 2026
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	'''Tetracot''', in this article, is the [[rank-2 temperament\|rank-2]] [[regular temperament\|temperament]] in the 2.3.5.11.13 [[subgroup]] [[generator\|generated]] by a submajor second of about 174–178{{cent}} which represents both [[10/9]] and [[11/10]]. It is so named because the generator is a quarter of fifth: four such generators make a perfect fifth which approximates [[3/2]], which cannot occur in [[12edo]], resulting in [[100/99]], [[144/143]], and [[243/242]] being [[tempering out\|tempered out]]. This is in contrast to [[meantone]], where 10/9 is tuned sharper than or equal to just in order to be equated with [[9/8]].		'''Tetracot''', in this article, is the [[rank-2 temperament\|rank-2]] [[regular temperament\|temperament]] in the 2.3.5.11.13 [[subgroup]] [[generator\|generated]] by a submajor second of about 174–178{{cent}} which represents both [[10/9]] and [[11/10]]. It is so named because the generator is a quarter of fifth: four such generators make a perfect fifth which approximates [[3/2]], which cannot occur in [[12edo]], resulting in [[100/99]], [[144/143]], and [[243/242]] being [[tempering out\|tempered out]]. This is in contrast to [[meantone]], where 10/9 is tuned sharper than or equal to just in order to be equated with [[9/8]].

	~~Tetracot has many~~ [[~~extension~~]]s ~~for the 7-, 11-, and 13-limit. See~~ [[~~Tetracot extensions~~]]~~. Equal temperaments that support~~ tetracot include {{EDOs\| 27, 34, and 41 }}.		[[Equal temperament]]s that [[support]] tetracot include {{EDOs\| 27, 34, and 41 }}.

	See [[Tetracot family]] for ~~more~~ technical data.		Tetracot has many [[extension]]s for the 7-, 11-, and 13-limit. These are [[monkey]] (34 & 41), [[bunya]] (34d & 41), [[modus]] (27e & 34d) and [[wollemia]] (27e & 34).

			See [[Tetracot family]] for technical data.

	== Intervals ==		== Intervals ==

FloraC: /* Tunings */ note vals

2026-04-15T11:04:45Z

Tunings: note vals

← Older revision		Revision as of 11:04, 15 April 2026
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	\|		\|
	\| 177.778		\| 177.778
	\| ~~Upper~~ bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone		\| 27e val, upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
	\|-		\|-
	\|		\|
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	\|		\|
	\| 180.000		\| 180.000
	\| ~~Upper~~ bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone		\| 20ce val, upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone
	\|-		\|-
	\|		\|

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

2026-04-15T11:03:01Z

Tunings: + norm-based tunings

@@ Line 115: / Line 115: @@
 == Tunings ==
 === Tuning spectrum ===
 {| class="wikitable center-all left-4"

ArrowHead294 at 02:55, 30 March 2026

2026-03-30T02:55:07Z

← Older revision		Revision as of 02:55, 30 March 2026
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	\| Title = Tetracot		\| Title = Tetracot
	\| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13		\| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13
	\| Comma basis = [[20000/19683]] (2.3.5); <br>[[100/99]], [[243/242]] (2.3.5.11) <br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)		\| Comma basis = [[20000/19683]] (2.3.5);<br>[[100/99]], [[243/242]] (2.3.5.11)<br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)
	\| Edo join 1 = 7 \| Edo join 2 = 27e		\| Edo join 1 = 7 \| Edo join 2 = 27e
	\| Mapping = 1; 4 9 10 -2		\| Mapping = 1; 4 9 10 -2
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	[[File: Tetracot 7et Detempering.png\|thumb\|Tetracot as a 34-tone 7et detempering]]		[[File: Tetracot 7et Detempering.png\|thumb\|Tetracot as a 34-tone 7et detempering]]

	Tetracot is considered as a [[cluster temperament]] with 7 clusters of notes in an octave, so it is naturally a [[detemperament]] of the [[7edo\|7 equal temperament]]. The diagram on the right shows a 34-tone detempered scale, with a generator range of ~~-16~~ to +17, which covers all the intervals in the no-7 13-odd-limit. Each category is divided into four or five qualities separated by 7 generator steps, which represent [[40/39]], [[45/44]], [[55/54]], [[65/64]], [[66/65]], [[81/80]], and [[121/120]] all at once.		Tetracot is considered as a [[cluster temperament]] with 7 clusters of notes in an octave, so it is naturally a [[detemperament]] of the [[7edo\|7 equal temperament]]. The diagram on the right shows a 34-tone detempered scale, with a generator range of −16 to +17, which covers all the intervals in the no-7 13-odd-limit. Each category is divided into four or five qualities separated by 7 generator steps, which represent [[40/39]], [[45/44]], [[55/54]], [[65/64]], [[66/65]], [[81/80]], and [[121/120]] all at once.

	== Scales ==		== Scales ==
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	\|		\|
	\| 171.429		\| 171.429
	\| Lower bound of 2.3.5.11 subgroup 11-odd-limit, <br />2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone		\| Lower bound of 2.3.5.11 subgroup 11-odd-limit,<br />2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
	\|-		\|-
	\|		\|

FloraC: + as a detemp

2026-03-29T12:00:29Z

+ as a detemp

← Older revision		Revision as of 12:00, 29 March 2026
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	See [[Tetracot family]] for more technical data.		See [[Tetracot family]] for more technical data.

	== Interval chain ==		== Intervals ==
	Tetracot is considered as a [[cluster temperament]] with seven clusters of notes in an octave. The chroma interval between adjacent notes in each cluster represents [[40/39]], [[45/44]], [[55/54]], [[65/64]], [[66/65]], [[81/80]], and [[121/120]] all at once. In the following table, odd harmonics and subharmonics 1–15 are in '''bold'''.		=== Interval chain ===
			In the following table, odd harmonics and subharmonics 1–15 are in '''bold'''.

	{\| class="wikitable right-1 right-2"		{\| class="wikitable right-1 right-2"
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	\| 15/13		\| 15/13
	\|}		\|}
	<nowiki />* In 2.3.5.11.13 subgroup CTE tuning		<nowiki/>* In 2.3.5.11.13 subgroup CTE tuning

			=== As a detemperament of 7et ===
			[[File: Tetracot 7et Detempering.png\|thumb\|Tetracot as a 34-tone 7et detempering]]

			Tetracot is considered as a [[cluster temperament]] with 7 clusters of notes in an octave, so it is naturally a [[detemperament]] of the [[7edo\|7 equal temperament]]. The diagram on the right shows a 34-tone detempered scale, with a generator range of -16 to +17, which covers all the intervals in the no-7 13-odd-limit. Each category is divided into four or five qualities separated by 7 generator steps, which represent [[40/39]], [[45/44]], [[55/54]], [[65/64]], [[66/65]], [[81/80]], and [[121/120]] all at once.

	== Scales ==		== Scales ==
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	\|		\|
	\|-		\|-
	\|		\|
	\|243/200		\| 243/200
	\|168.574		\| 168.574
	\|1/2-comma		\| 1/2-comma
	\|-		\|-
	\| 1\7		\| 1\7
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	\| Lower bound of 2.3.5.11 subgroup 11-odd-limit, <br />2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone		\| Lower bound of 2.3.5.11 subgroup 11-odd-limit, <br />2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
	\|-		\|-
	\|		\|
	\|27/20		\| 27/20
	\|173.184		\| 173.184
	\|1/3-comma		\| 1/3-comma
	\|-		\|-
	\|		\|
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	\|		\|
	\|-		\|-
	\|		\|
	\|81/80		\| 81/80
	\|174.501		\| 174.501
	\|2/7-comma		\| 2/7-comma
	\|-		\|-
	\|		\|
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	\| 11/8		\| 11/8
	\| 175.132		\| 175.132
	\| 2.3.5.11 subgroup 11-odd-limit minimax		\| 2.3.5.11-subgroup 11-odd-limit minimax
	\|-		\|-
	\|		\|
	\| 3/2		\| 3/2
	\| 175.489		\| 175.489
	\|1/4-comma		\| 1/4-comma
	\|-		\|-
	\| 6\41		\| 6\41
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	\| 13/11		\| 13/11
	\| 175.899		\| 175.899
	\| 2.3.5.11.13 subgroup 13- and 15-odd-limit minimax		\| 2.3.5.11.13-subgroup 13- and 15-odd-limit minimax
	\|-		\|-
	\|		\|
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	\| 5/3		\| 5/3
	\| 176.872		\| 176.872
	\|1/5-comma		\| 1/5-comma
	\|-		\|-
	\|		\|
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	\| Upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone		\| Upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
	\|-		\|-
	\|		\|
	\|27/25		\| 27/25
	\|177.794		\| 177.794
	\|1/6-comma		\| 1/6-comma
	\|-		\|-
	\|		\|
	\|243/125		\| 243/125
	\|178.452		\| 178.452
	\|1/7-comma		\| 1/7-comma
	\|-		\|-
	\|		\|
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	\|		\|
	\| 180.000		\| 180.000
	\| Upper bound of 2.3.5.11 subgroup 11-odd-limit diamond monotone		\| Upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone
	\|-		\|-
	\|		\|

FloraC: Cleanup on infobox

2026-02-09T09:48:15Z

Cleanup on infobox

← Older revision		Revision as of 09:48, 9 February 2026
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	{{~~interwiki~~		{{Interwiki
			\| en = Tetracot
	\| de = Tetracot		\| de = Tetracot
	~~\| en = Tetracot~~
	\| es =		\| es =
	\| ja =		\| ja =
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	\| Title = Tetracot		\| Title = Tetracot
	\| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13		\| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13
	\| Comma basis = [[20000/19683]] (2.3.5); <br> [[100/99]], [[243/242]] (2.3.5.11) <br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)		\| Comma basis = [[20000/19683]] (2.3.5); <br>[[100/99]], [[243/242]] (2.3.5.11) <br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)
	\| ~~Generator~~ = ~~10/9~~		\| Edo join 1 = 7 \| Edo join 2 = 27e
	\| Mapping = 1; 4 9 10 -2		\| Mapping = 1; 4 9 10 -2
			\| Generators = 10/9
			\| Generators tuning = 176.1
			\| Optimization method = CWE
			\| MOS scales = [[6L 1s]], [[7L 6s]], [[7L 13s]]
	\| Pergen = (P8, P5/4)		\| Pergen = (P8, P5/4)
	\| Color name = Saquadyo		\| Color name = Saquadyo
	~~\| Edo join 1 = 7 \| Edo join 2 = 27e~~
	~~\| Optimization method = CWE~~
	~~\| Generator tuning = 176.1~~
	~~\| MOS scales = [[6L 1s]], [[7L 6s]], [[7L 13s]]~~
	\| Odd limit 1 = 5 \| Mistuning 1 = 3.07 \| Complexity 1 = 13		\| Odd limit 1 = 5 \| Mistuning 1 = 3.07 \| Complexity 1 = 13
	\| Odd limit 2 = (2.3.5.11.13) 15 \| Mistuning 2 = 10.9 \| Complexity 2 = 20		\| Odd limit 2 = 2.3.5.11.13 15 \| Mistuning 2 = 10.9 \| Complexity 2 = 20
	}}		}}
	{{About\|the regular temperament\|the ploidacot signature\|Ploidacot/Tetracot}}		{{About\|the regular temperament\|the ploidacot signature\|Ploidacot/Tetracot}}