Trienstonic clan: Difference between revisions

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== Trienstonian ==

This low-accuracy temperament is generated by a fifth, tuned very sharp such that a stack of three reach a ~7/4. [[5edo]] is the tuning that conflates 7/6~9/8 (+2 generator steps) with ~8/7 (-3 generator steps). If you do not care about the intervals of 9 in this temperament, you can tune the fifth sharper for the [[7-odd-limit]], leading to an [[5L 3s|oneirotonic]] scale or otherwise a [[5L 2s|diatonic]] scale with negative small steps. A tuning that prioritizes the 7-odd-limit also tunes the fifth sharper than the [[pentic]] range, instead generating an [[antipentic]] scale. Trienstonian can be considered the [[2.3.7 subgroup|2.3.7]] analog of [[mavila]] temperament, with extremely sharp fifths rather than extremely flat ones.

This low-accuracy temperament is generated by a fifth, tuned very sharp such that a stack of three reach a ~7/4. [[5edo]] is the tuning that conflates 7/6~9/8 (+2 generator steps) with ~8/7 (-3 generator steps). If you do not care about the intervals of 9 in this temperament, you can tune the fifth sharper for the [[7-odd-limit]], leading to an [[5L 3s|oneirotonic]] scale or otherwise a [[5L 2s|diatonic]] scale with negative small steps. A tuning that prioritizes the 7-odd-limit also tunes the fifth sharper than the [[pentic]] range, instead generating an [[antipentic]] scale. Trienstonian can be considered the [[2.3.7 subgroup|2.3.7]] analog of [[mavila]] temperament, with extremely sharp fifths rather than extremely flat ones, being on the other side of 3\5 from [[archy]] fifths, just like how mavila fifths are on the other side of 4\7 from [[meantone]] fifths.

[[Subgroup]]: 2.3.7

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: mapping generators: ~2, ~3

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Members of the clan discussed elsewhere are:

* ''[[~~Wallaby~~]]'' (+35/32) → [[Very low accuracy temperaments #~~Wallaby~~|Very low accuracy temperaments]]

* [[Antonian]] (+10/9 or +15/14) → [[Very low accuracy temperaments #Septimal antonian|Very low accuracy temperaments]]

* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]

* ''[[Sharptone]]'' (+21/20) → [[Meantone family #Sharptone|Meantone family]]

* ''[[Sharpie]]'' (+25/24) → [[Dicot family #Sharpie|Dicot family]]

* ''[[Mite]]'' (+27/25) → [[Bug family #Mite|Bug family]]

* ''[[Wallaby]]'' (+35/32) → [[Very low accuracy temperaments #Wallaby|Very low accuracy temperaments]]

* [[Blackwood]] (+49/48) → [[Limmic temperaments #Blackwood|Limmic temperaments]]

* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]

* ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]]

* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]

* [[Blackwood]] (+49/48) → [[Limmic temperaments #Blackwood|Limmic temperaments]]

Considered below are ~~father, sharptone,~~ uncle, octokaidecal, and parakangaroo.

Considered below are uncle, octokaidecal, and parakangaroo.

~~== Father ==~~

~~{{Main| Father }}~~

~~See [[Father family #Septimal father]].~~

~~== Sharptone ==~~

~~See [[Meantone family #Sharptone]]~~.

== Uncle ==

: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum#~~Uncle_~~(5-limit)~~|Syntonic–diatonic equivalence continuum~~]].''

: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Uncle (5-limit)]].''

Uncle tempers out 256/245, mapping the interval class of 5 to -6 generator steps, as a major 2-step in oneirotonic or a diminished fifth in diatonic.

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== Octokaidecal ==

~~The~~ 5-limit [[~~restriction~~]] ~~of octokaidecal is supersharp, which tempers~~ out [[~~800~~/~~729~~]], the ~~difference between~~ the [[~~27/20~~]] ~~wolf fourth and~~ the [[~~40/27~~]] ~~wolf fifth, splitting the octave into two 27/20~40/27 semioctaves. It generally requires a very sharp fifth, even sharper than 3\5,~~ as ~~a generator. This means that five steps from the~~ [[~~generator sequence #JI scales obtained from guided generator sequences|Zarlino generator sequence~~]] ~~starting with 6/5 are tempered to one and a half octaves. The only reasonable 7-limit extension adds 28/27 and 50/49 to the comma list, taking advantage of the existing semioctave.~~

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Supersharp]].''

Octokaidecal extends trienstonian by tempering out [[50/49]], thus splitting the octave in half. It generates the [[8L 2s]] (taric) mos scale, with tunings on the other side of [[10edo]] as [[2L 8s]] (jaric). Compared to [[pajara]], decatonic thirds ([[8/7]] and [[7/6]]) and fourths ([[6/5]] and [[5/4]]) have their mappings reversed, meaning octokaidecal can be considered what is to pajara as [[mavila]] is to [[meantone]].

~~=== 5-limit~~ (~~supersharp~~) ~~===~~

~~[[Subgroup]]: 2~~.~~3.5~~

[[~~Comma list~~]]~~: 800/729~~

~~{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}~~

~~: mapping generators: ~27/20~~, ~3

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

* [[WE]]~~: ~27/20 = 596.986{{c}}, ~3/2 = 725.434{{c}}~~ (~~~10/9 = 128.448{{c}})~~

: [[~~error map~~]]~~: {{val| -6.029 +17.450 -13.027 }}~~

* [[~~CWE~~]]~~: ~27/20 = 600.000{{c}}, ~3/2 = 726.548{{c}} (~10/9 = 126.548{{c}}~~)

~~: error map: {{val| 0.000 +24.593 -6.670 }}~~

~~{{Optimal ET sequence|legend=1| 8~~, ~~10, 18, 28b }}~~

[[~~Badness~~]] ~~(Sintel): 2~~.88

~~=== 7-limit ===~~

[[Subgroup]]: 2.3.5.7

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== Parakangaroo ==

: ''For the 5-limit version ~~of this temperament~~, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''

This temperament used to be known as ''kangaroo'', but was decanonicalized in 2024 in favor of a more accurate extension. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. [[15edo]] shows us an obvious tuning.

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[[Category:Temperament clans]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Trienstonic clan| ]]

[[Category:Trienstonic clan| ]] 

[[Category:Trienstonic| ]]

[[Category:Rank 2]]

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 == Trienstonian ==
-This low-accuracy temperament is generated by a fifth, tuned very sharp such that a stack of three reach a ~7/4. [[5edo]] is the tuning that conflates 7/6~9/8 (+2 generator steps) with ~8/7 (-3 generator steps). If you do not care about the intervals of 9 in this temperament, you can tune the fifth sharper for the [[7-odd-limit]], leading to an [[5L 3s|oneirotonic]] scale or otherwise a [[5L 2s|diatonic]] scale with negative small steps. A tuning that prioritizes the 7-odd-limit also tunes the fifth sharper than the [[pentic]] range, instead generating an [[antipentic]] scale. Trienstonian can be considered the [[2.3.7 subgroup|2.3.7]] analog of [[mavila]] temperament, with extremely sharp fifths rather than extremely flat ones.
+This low-accuracy temperament is generated by a fifth, tuned very sharp such that a stack of three reach a ~7/4. [[5edo]] is the tuning that conflates 7/6~9/8 (+2 generator steps) with ~8/7 (-3 generator steps). If you do not care about the intervals of 9 in this temperament, you can tune the fifth sharper for the [[7-odd-limit]], leading to an [[5L 3s|oneirotonic]] scale or otherwise a [[5L 2s|diatonic]] scale with negative small steps. A tuning that prioritizes the 7-odd-limit also tunes the fifth sharper than the [[pentic]] range, instead generating an [[antipentic]] scale. Trienstonian can be considered the [[2.3.7 subgroup|2.3.7]] analog of [[mavila]] temperament, with extremely sharp fifths rather than extremely flat ones, being on the other side of 3\5 from [[archy]] fifths, just like how mavila fifths are on the other side of 4\7 from [[meantone]] fifths.
 [[Subgroup]]: 2.3.7
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 {{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }}
 : mapping generators: ~2, ~3
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 Members of the clan discussed elsewhere are:
-* ''[[Wallaby]]'' (+35/32) → [[Very low accuracy temperaments #Wallaby|Very low accuracy temperaments]]
+* [[Antonian]] (+10/9 or +15/14) → [[Very low accuracy temperaments #Septimal antonian|Very low accuracy temperaments]]
+* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
+* ''[[Sharptone]]'' (+21/20) → [[Meantone family #Sharptone|Meantone family]]
 * ''[[Sharpie]]'' (+25/24) → [[Dicot family #Sharpie|Dicot family]]
 * ''[[Mite]]'' (+27/25) → [[Bug family #Mite|Bug family]]
+* ''[[Wallaby]]'' (+35/32) → [[Very low accuracy temperaments #Wallaby|Very low accuracy temperaments]]
+* [[Blackwood]] (+49/48) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
+* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]
 * ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]]
-* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]]
-* [[Blackwood]] (+49/48) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
-Considered below are father, sharptone, uncle, octokaidecal, and parakangaroo.
+Considered below are uncle, octokaidecal, and parakangaroo.
-== Father ==
-{{Main| Father }}
-See [[Father family #Septimal father]].
-== Sharptone ==
-See [[Meantone family #Sharptone]].
 == Uncle ==
-: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum#Uncle_(5-limit)|Syntonic–diatonic equivalence continuum]].''
+: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Uncle (5-limit)]].''
 Uncle tempers out 256/245, mapping the interval class of 5 to -6 generator steps, as a major 2-step in oneirotonic or a diminished fifth in diatonic.
@@ Line 74: / Line 69: @@
 == Octokaidecal ==
-The 5-limit [[restriction]] of octokaidecal is supersharp, which tempers out [[800/729]], the difference between the [[27/20]] wolf fourth and the [[40/27]] wolf fifth, splitting the octave into two 27/20~40/27 semioctaves. It generally requires a very sharp fifth, even sharper than 3\5, as a generator. This means that five steps from the [[generator sequence #JI scales obtained from guided generator sequences|Zarlino generator sequence]] starting with 6/5 are tempered to one and a half octaves. The only reasonable 7-limit extension adds 28/27 and 50/49 to the comma list, taking advantage of the existing semioctave.
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Supersharp]].''
+Octokaidecal extends trienstonian by tempering out [[50/49]], thus splitting the octave in half. It generates the [[8L 2s]] (taric) mos scale, with tunings on the other side of [[10edo]] as [[2L 8s]] (jaric). Compared to [[pajara]], decatonic thirds ([[8/7]] and [[7/6]]) and fourths ([[6/5]] and [[5/4]]) have their mappings reversed, meaning octokaidecal can be considered what is to pajara as [[mavila]] is to [[meantone]].
-=== 5-limit (supersharp) ===
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 800/729
-{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}
-: mapping generators: ~27/20, ~3
-[[Optimal tuning]]s:
-* [[WE]]: ~27/20 = 596.986{{c}}, ~3/2 = 725.434{{c}} (~10/9 = 128.448{{c}})
-: [[error map]]: {{val| -6.029 +17.450 -13.027 }}
-* [[CWE]]: ~27/20 = 600.000{{c}}, ~3/2 = 726.548{{c}} (~10/9 = 126.548{{c}})
-: error map: {{val| 0.000 +24.593 -6.670 }}
-{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
-[[Badness]] (Sintel): 2.88
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
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 == Parakangaroo ==
-: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].''
 This temperament used to be known as ''kangaroo'', but was decanonicalized in 2024 in favor of a more accurate extension. It splits the perfect twelfth into three generators of ~10/7; its ploidacot is alpha-tricot. [[15edo]] shows us an obvious tuning.
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 [[Category:Temperament clans]]
-[[Category:Pages with mostly numerical content]]
+[[Category:Trienstonic clan| ]] <!-- main article -->
-[[Category:Trienstonic clan| ]] <!-- Main article -->
-[[Category:Trienstonic| ]] <!-- Key article -->
 [[Category:Rank 2]]