Trisedodge family - Revision history

FloraC: + overview to extensions; review intro to septimal trisedodge extensions

2026-09-05T12:18:49Z

+ overview to extensions; review intro to septimal trisedodge extensions

← Older revision		Revision as of 12:18, 5 September 2026
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	Named by [[Petr Pařízek]] in 2011, ''trisedodge'' (originally spelt ''trisedoge'') means that three semidiminished [[octave]]s add up to [[7/1]], and that an octave is made of 5 [[period]]s<ref name="naming">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group \| ''Suggested names for the unclasified temperaments'']</ref>.		Named by [[Petr Pařízek]] in 2011, ''trisedodge'' (originally spelt ''trisedoge'') means that three semidiminished [[octave]]s add up to [[7/1]], and that an octave is made of 5 [[period]]s<ref name="naming">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group \| ''Suggested names for the unclasified temperaments'']</ref>.

	~~Temperaments discussed elsewhere include [[15th-octave temperaments #Quindecic\|quindecic]] and [[Stearnsmic clan #Decistearn\|decistearn]]. Considered below are trisedodge and coblack.~~

	== Trisedodge ==		== Trisedodge ==
	The generator of trisedodge is ~864/625 at around 554 ~~cents~~, which in all 11-limit ~~extensions~~ is used to represent [[11/8]], and three of them and a period is equal to [[3/1]]. This generator, when reduced to the minimal size, represents [[25/24]]. However, another possible generator is ~6/5, reached by a period plus 25/24, that is, ~~(144/125)(25/24) =~~ 6/5.		The generator of trisedodge is [[~]][[864/625]] at around 554 [[cent]]s, which in all [[11-limit]] [[extension]]s is used to represent [[11/8]], and three of them and a period is equal to [[3/1]]. This generator, when reduced to the minimal size, represents [[25/24]]. However, another possible generator is ~[[6/5]], reached by a period plus 25/24, that is, 6/5 = (144/125)⋅(25/24).

	~~In the 11-limit the generator can be taken to be ~11/10, reached as a period minus 25/24, that is,~~ (55/48)/(25/24) ~~= 11/10. Therefore, since a period plus a gen is 6/5 and a period minus a gen is 11/10, we reach 12/11 at 2 gens.~~

	~~Remarkably, trisedodge admits an extension to the full [[29-limit]] which, except for prime 13, is surprisingly obvious/simple a way to extend the 11-limit representation~~.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	[[Badness]] (Sintel): 5.93		[[Badness]] (Sintel): 5.93

			=== Overview to extensions ===
			The second comma of the comma list defines which 7-limit family member we are looking at. Among these are septimal trisedodge (65d & 80), which adds [[4000/3993]], and coblack (50 & 65), which adds [[126/125]]. Remarkably, septimal trisedodge admits an extension to the full [[29-limit]], which, except for prime 13, is obvious and simple a way to extend the 11-limit representation.

			Temperaments discussed elsewhere include [[15th-octave temperaments #Quindecic\|quindecic]] and [[Stearnsmic clan #Decistearn\|decistearn]]. Considered below are trisedodge and coblack.

			Septimal trisedodge and coblack have the common [[2.3.5.11-subgroup\|2.3.5.11 subgroup]] [[restriction]], called countdown, considered immediately below. In this temperament, the generator can be taken to be ~11/10, reached as a period minus 25/24, that is, (55/48)/(25/24) = 11/10. Therefore, since a period plus a gen is 6/5 and a period minus a gen is 11/10, we reach 12/11 in 2 generator steps.

	=== Countdown ===		=== Countdown ===
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	== Septimal trisedodge ==		== Septimal trisedodge ==
			We can extend trisedodge to the [[17-limit]] by using the sharp tendency of prime 5 to justify tempering out [[256/255]] ({{S\|16}}). Note that prime 3 is also tuned sharp (though less than prime 5) in optimized tunings. We can then extend it to the 19-limit by tempering out [[361/360]] ({{S\|19}}) or equivalently [[400/399]] ({{S\|20}}), whose naturalness becomes much clearer when we consider it in the [[23-limit]], where we equate [[23/19]] with a stack of two [[11/10]]'s, tempering out [[2300/2299]] ([[S-expression\|S20/S22]]), relying on the obvious mapping of [[23/16]] as one period above [[5/4]] so that [[~]][[23/20]] is tuned to 1\5. The mapping of 23 also implies tempering out [[276/275]] (the difference between [[55/48]] and [[23/20]]), which is [[3025/3024]] flat of [[253/252]]. Finally, there is an obvious mapping for [[29/16]] as two periods above [[11/8]] so that [[~]][[29/22]] is tuned to 2\5 and that [[~]][[32/29]] is equated with [[~]][[11/10]], the generator.

			This defines trisedodge as being an unambiguously full [[29-limit]] temperament, with an interesting feature of having possible alternative mappings for primes 7 and 13. Prime 7 can either be mapped the more accurate way as septimal trisedodge does or it can be mapped as in [[#Coblack\|coblack]], while prime 13 can alternatively be found as 8 generators ''up'' instead of down, corresponding to [[#Trisey\|trisey]], though using both of those mappings simultaneously only really makes sense in [[80edo]], which is a reasonable edo tuning for it and happens to correspond to the 80-note generator chain of trisedodge required for finding every prime relative to the same root, though note that [[11/10]] is practically just there so that intervals of 29 require error cancellation of the oversharp 29th harmonic to help justify harmonically

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

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	{{Optimal ET sequence\|legend=1\| 15, 50d, 65d, 80 }}		{{Optimal ET sequence\|legend=1\| 15, 50d, 65d, 80 }}

	Badness (Sintel): 3.48		[[Badness]] (Sintel): 3.48

	=== 11-limit ===		=== 11-limit ===
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	==== 17-limit ====		==== 17-limit ====
	We extend to prime 17 by using the sharp tendency of prime 5 to justify tempering out ([[16/15]])/([[17/16]]) = [[256/255\|S16]]. Note that prime 3 is also tuned sharp (though less than prime 5) in optimized tunings.

	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17

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	==== 19-limit ====		==== 19-limit ====
	We extend to prime 19 by tempering out [[361/360\|361/360 = S19]] or equivalently [[400/399\|400/399 = S20]], whose naturalness becomes much clearer when we consider it in the 23-limit as the result of tempering out ([[23/19]])/([[11/10\|22/20]])<sup>2</sup> = [[2300/2299\|S20/S22]], relying on the surprisingly obvious mapping of [[23/16]] as one period above [[5/4]] so that [[~]][[23/20]] = 1\5.

	Subgroup: 2.3.5.7.11.13.17.19		Subgroup: 2.3.5.7.11.13.17.19

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	==== 23-limit ====		==== 23-limit ====
	As mentioned, prime 23 is found as prime 5 plus a period (up to octave-equivalence). This id done by tempering out ([[55/48]])/([[23/20]]) = [[276/275]] which is [[3025/3024]] flat of [[253/252]]. Curiously, the [[CTE]] and [[CWE]] tunings are almost exactly the same here (different by about a hundredth of a cent).

	Subgroup: 2.3.5.7.11.13.17.19.23		Subgroup: 2.3.5.7.11.13.17.19.23

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	==== 29-limit ====		==== 29-limit ====
	There's a surprisingly obvious mapping of [[29/16]] as two periods above [[11/8]] so that [[~]][[29/22]] = 2\5 and meaning equating [[~]][[32/29]] with [[~]][[11/10]], the generator. This defines trisedodge as being an unambiguously full 29-limit temperament, with an interesting feature of having two possible mappings of prime 7 and 13; prime 7 can either be mapped the more accurate way as septimal trisedodge does or it can be mapped as it is approximated in [[5edo]], while prime 13 can alternately be found as 8 generators ''up'' instead of down, corresponding to [[#Trisey]], though using both of those mappings simultaneously only really makes sense in [[80edo]], which is a reasonable edo tuning for it and happens to correspond to the 80-note MOS of trisedodge required for finding every prime relative to the same root, though note that [[11/10]] is practically just there so that intervals of 29 require error cancellation of the oversharp 29th harmonic to help justify harmonically

	Subgroup: 2.3.5.7.11.13.17.19.23.29		Subgroup: 2.3.5.7.11.13.17.19.23.29

FloraC: Switch to Sintel's badness, WE & CWE tunings

2026-09-05T11:53:17Z

Switch to Sintel's badness, WE & CWE tunings

Show changes

FloraC: Units & misc. cleanup

2026-09-05T10:58:20Z

Units & misc. cleanup

Show changes

FloraC: Recategorize

2026-08-14T12:24:33Z

Recategorize

← Older revision		Revision as of 12:24, 14 August 2026
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	== References ==		== References ==

	~~[[Category:Temperament families]]~~
	[[Category:Trisedodge family\| ]] <!-- main article -->		[[Category:Trisedodge family\| ]] <!-- main article -->
	[[Category:Trisedodge\| ]] <!-- key article -->		[[Category:Trisedodge\| ]] <!-- key article -->
	[[Category:~~Rank~~ 2]]		[[Category:Temperament families]]
			[[Category:Catalogs of rank-2 temperaments]]

FloraC: Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

2025-11-27T12:18:33Z

Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

← Older revision		Revision as of 12:18, 27 November 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Trisedodge family\| ]] <!-- main article -->		[[Category:Trisedodge family\| ]] <!-- main article -->
	[[Category:Trisedodge\| ]] <!-- key article -->		[[Category:Trisedodge\| ]] <!-- key article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

FloraC: Septimal blackwood is now canon

2025-09-11T15:14:06Z

Septimal blackwood is now canon

← Older revision		Revision as of 15:14, 11 September 2025
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	{{See also\| Cloudy clan #Coblack }}		{{See also\| Cloudy clan #Coblack }}

	In addition to 126/125, the coblack temperament tempers out the [[cloudy comma]], 16807/16384, which is the amount by which five septimal supermajor seconds ([[8/7]]) fall short of an octave. Coblack was also named by Petr Pařízek, who considered it a counterpart of [[~~blacksmith~~]]<ref name="naming"/>.		In addition to 126/125, the coblack temperament tempers out the [[cloudy comma]], 16807/16384, which is the amount by which five septimal supermajor seconds ([[8/7]]) fall short of an octave. Coblack was also named by Petr Pařízek, who considered it a counterpart of [[blackwood]]<ref name="naming"/>.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	Badness: 0.045070		Badness: 0.045070

	== ~~Notes~~ ==		== References ==

	[[Category:Temperament families]]		[[Category:Temperament families]]

Lériendil: Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

2025-06-24T00:40:44Z

Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:40, 24 June 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Trisedodge family\| ]] <!-- main article -->		[[Category:Trisedodge family\| ]] <!-- main article -->
	[[Category:Trisedodge\| ]] <!-- key article -->		[[Category:Trisedodge\| ]] <!-- key article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

VectorGraphics: wedgie

2025-06-12T04:16:20Z

wedgie

← Older revision		Revision as of 04:16, 12 June 2025
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	{{Mapping\|legend=1\| 5 1 7 21 \| 0 3 2 -3 }}		{{Mapping\|legend=1\| 5 1 7 21 \| 0 3 2 -3 }}

	~~Wedgie: {{multival\| 15 10 -15 -19 -66 -63}}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	{{Mapping\|legend=1\| 5 1 7 14 \| 0 3 2 0 }}		{{Mapping\|legend=1\| 5 1 7 14 \| 0 3 2 0 }}

	~~{{Multival\|legend=1\| 15 10 0 -19 -42 -28 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:

FloraC: Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

2025-06-06T16:38:19Z

Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

← Older revision		Revision as of 16:38, 6 June 2025
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	Badness: ?		Badness: ?

	Badness (~~Dirichlet~~): 1.609		Badness (Sintel): 1.609

	==== 19-limit ====		==== 19-limit ====
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	Badness: ?		Badness: ?

	Badness (~~Dirichlet~~): 1.542		Badness (Sintel): 1.542

	==== 23-limit ====		==== 23-limit ====
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	Badness: ?		Badness: ?

	Badness (~~Dirichlet~~): 1.463		Badness (Sintel): 1.463

	==== 29-limit ====		==== 29-limit ====
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	Badness: ?		Badness: ?

	Badness (~~Dirichlet~~): 1.380		Badness (Sintel): 1.380

	==== Trisey ====		==== Trisey ====

FloraC: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"

2025-04-13T13:43:18Z

Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"

← Older revision		Revision as of 13:43, 13 April 2025
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	{{Technical data page}}~~<br><br>~~		{{Technical data page}}
	The '''trisedodge family''' tempers out the [[trisedodge comma]], 30958682112/30517578125 = {{monzo\| 19 10 -15 }}.		The '''trisedodge family''' tempers out the [[trisedodge comma]], 30958682112/30517578125 = {{monzo\| 19 10 -15 }}.