Trisedodge family: Difference between revisions

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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>.

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

== 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

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[[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 ===

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== 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

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Badness (Sintel): 3.48

[[Badness]] (Sintel): 3.48

=== 11-limit ===

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==== 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

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==== 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

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==== 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

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==== 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

@@ Line 3: / Line 3: @@
 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 ==
-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
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 [[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 ===
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 == 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
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 {{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }}
-Badness (Sintel): 3.48
+[[Badness]] (Sintel): 3.48
 === 11-limit ===
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 ==== 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
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 ==== 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
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 ==== 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
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 ==== 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