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The '''trienstonic clan''' of temperaments | {{Technical data page}} | ||
The '''trienstonic clan''' of [[rank-2 temperament|rank-2]] [[temperament]]s are low-complexity, high-error temperaments that [[tempering out|temper out]] [[28/27]], the septimal third-tone or trienstonic comma. This equates very different intervals with each other; in particular, [[9/8]] with [[7/6]], [[8/7]] with [[32/27]], and [[4/3]] with [[9/7]]. Trienstonian is close to the edge of what can be sensibly called a temperament at all; in other words, it is an [[exotemperament]]. | |||
== 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. | |||
[[Subgroup]]: 2.3.7 | |||
Subgroup: 2.3.7 | |||
[[Comma list]]: 28/27 | [[Comma list]]: 28/27 | ||
{{Mapping|legend=2| 1 0 -2 | 0 1 3 }} | |||
: mapping generators: ~2, ~3 | |||
{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }} | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1196.254{{c}}, ~3/2 = 719.306{{c}} | |||
: [[error map]]: {{val| -3.746 +13.604 -14.655 }} | |||
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 719.606{{c}} | |||
: error map: {{val| 0.000 +17.651 -10.007 }} | |||
{{Optimal ET sequence|legend=1| 2d, 3d, 5 }} | |||
{{ | |||
[[Badness]] (Sintel): 0.235 | |||
=== Overview to extensions === | |||
Adding 16/15 to 28/27 leads to father, 21/20 gives sharptone, 256/245 gives uncle, and 35/32 gives wallaby. These all use the same generators as trienstonian. | |||
50/49 gives octokaidecal with a semi-octave period. 25/24 gives sharpie; 27/25 gives mite. Those split the generator in two. 1029/1000 gives parakangaroo; 126/125 gives opossum. Those split the generator in three. 128/125 gives inflated with a 1/3-octave period. Finally, 49/48 gives blackwood, with a 1/5-octave period. | |||
Members of the clan discussed elsewhere are: | |||
* ''[[Wallaby]]'' (+35/32) → [[Very low accuracy temperaments #Wallaby|Very low accuracy temperaments]] | |||
* ''[[Sharpie]]'' (+25/24) → [[Dicot family #Sharpie|Dicot family]] | |||
* ''[[Mite]]'' (+27/25) → [[Bug family #Mite|Bug 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. | |||
== Father == | |||
{{Main| Father }} | |||
See [[Father family #Septimal father]]. | |||
[[ | == Sharptone == | ||
See [[Meantone family #Sharptone]]. | |||
== Uncle == | == Uncle == | ||
Subgroup: 2.3.5.7 | : ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum]].'' | ||
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. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 28/27, 256/245 | [[Comma list]]: 28/27, 256/245 | ||
{{Mapping|legend=1| 1 0 12 -2 | 0 1 -6 3 }} | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1190.224{{c}}, ~3/2 = 725.221{{c}} | |||
[[ | : [[error map]]: {{val| -9.776 +13.490 +3.707 -2.939 }} | ||
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 731.394{{c}} | |||
: error map: {{val| 0.000 +29.439 +25.324 +25.355 }} | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]] eigenmonzo | * [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3 | ||
* [[9-odd-limit]] eigenmonzo | * [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5 | ||
{{ | {{Optimal ET sequence|legend=1| 5, 13d, 18, 23bc, 41bbcd }} | ||
[[Badness]]: | [[Badness]] (Sintel): 1.84 | ||
== | == 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. | |||
=== 5-limit (supersharp) === | === 5-limit (supersharp) === | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 800/729 | [[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]]: | [[Badness]] (Sintel): 2.88 | ||
=== 7-limit === | === 7-limit === | ||
Subgroup: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 28/27, 50/49 | [[Comma list]]: 28/27, 50/49 | ||
{{Mapping|legend=1| 2 0 -5 -4 | 0 1 3 3 }} | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~7/5 = 596.984{{c}}, ~3/2 = 725.210{{c}} (~15/14 = 128.226{{c}}) | |||
: [[error map]]: {{val| -6.031 +17.224 -13.699 +0.774 }} | |||
* [[CWE]]: ~7/5 = 600.000{{c}}, ~3/2 = 726.319{{c}} (~15/14 = 126.319{{c}}) | |||
: error map: {{val| 0.000 +24.364 -7.358 +10.130 }} | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]] | * [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
{{ | {{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.930 | ||
=== 11-limit === | === 11-limit === | ||
Line 113: | Line 120: | ||
Comma list: 28/27, 50/49, 55/54 | Comma list: 28/27, 50/49, 55/54 | ||
Mapping: | Mapping: {{mapping| 2 0 -5 -4 7 | 0 1 3 3 0 }} | ||
Optimal tunings: | |||
* | * WE: ~7/5 = 595.139{{c}}, ~3/2 = 726.397{{c}} (~15/14 = 131.258{{c}}) | ||
* | * CWE: ~7/5 = 600.000{{c}}, ~3/2 = 729.485{{c}} (~15/14 = 129.485{{c}}) | ||
{{ | {{Optimal ET sequence|legend=0| 8d, 10, 18e }} | ||
Badness (Sintel): 1.00 | |||
== | == Parakangaroo == | ||
: ''For the 5-limit version of this temperament, 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. | |||
[[Subgroup]]: 2.3.5.7 | |||
Subgroup | |||
[[Comma list]]: 28/27, 1029/1000 | [[Comma list]]: 28/27, 1029/1000 | ||
{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }} | |||
: mapping generators: ~2, ~10/7 | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 596.984{{c}}, ~10/7 = 638.135{{c}} | |||
: [[error map]]: {{val| -2.883 +12.450 +3.685 -19.845 }} | |||
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/7 = 639.302{{c}} | |||
: error map: {{val| 0.000 +15.952 +6.710 -15.104 }} | |||
{{ | {{Optimal ET sequence|legend=1| 2cd, …, 13cd, 15 }} | ||
[[Badness]]: | [[Badness]] (Sintel): 1.97 | ||
=== 11-limit === | === 11-limit === | ||
Line 197: | Line 158: | ||
Comma list: 28/27, 77/75, 245/242 | Comma list: 28/27, 77/75, 245/242 | ||
Mapping: | Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1196.971{{c}}, ~10/7 = 638.230{{c}} | |||
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.480{{c}} | |||
{{Optimal ET sequence|legend=0| 15 }} | |||
Badness: | Badness (Sintel): 1.43 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 28/27, 40/39, | Comma list: 28/27, 40/39, 66/65, 147/143 | ||
Mapping: | Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1194.720{{c}}, ~10/7 = 637.413{{c}} | |||
* CWE: ~2 = 1200.000{{c}}, ~10/7 = 639.609{{c}} | |||
{{Optimal ET sequence|legend=0| 15 }} | |||
Badness: | Badness (Sintel): 1.35 | ||
[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||
[[Category:Trienstonic clan| ]] <!-- | [[Category:Pages with mostly numerical content]] | ||
[[Category:Trienstonic]] | [[Category:Trienstonic clan| ]] <!-- Main article --> | ||
[[Category:Trienstonic| ]] <!-- Key article --> | |||
[[Category:Rank 2]] | [[Category:Rank 2]] |
Latest revision as of 11:45, 16 July 2025
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The trienstonic clan of rank-2 temperaments are low-complexity, high-error temperaments that temper out 28/27, the septimal third-tone or trienstonic comma. This equates very different intervals with each other; in particular, 9/8 with 7/6, 8/7 with 32/27, and 4/3 with 9/7. Trienstonian is close to the edge of what can be sensibly called a temperament at all; in other words, it is an exotemperament.
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 oneirotonic scale or otherwise a diatonic scale with negative small steps.
Subgroup: 2.3.7
Comma list: 28/27
Sval mapping: [⟨1 0 -2], ⟨0 1 3]]
- mapping generators: ~2, ~3
Gencom mapping: [⟨1 0 0 -2], ⟨0 1 0 3]]
- WE: ~2 = 1196.254 ¢, ~3/2 = 719.306 ¢
- error map: ⟨-3.746 +13.604 -14.655]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 719.606 ¢
- error map: ⟨0.000 +17.651 -10.007]
Optimal ET sequence: 2d, 3d, 5
Badness (Sintel): 0.235
Overview to extensions
Adding 16/15 to 28/27 leads to father, 21/20 gives sharptone, 256/245 gives uncle, and 35/32 gives wallaby. These all use the same generators as trienstonian.
50/49 gives octokaidecal with a semi-octave period. 25/24 gives sharpie; 27/25 gives mite. Those split the generator in two. 1029/1000 gives parakangaroo; 126/125 gives opossum. Those split the generator in three. 128/125 gives inflated with a 1/3-octave period. Finally, 49/48 gives blackwood, with a 1/5-octave period.
Members of the clan discussed elsewhere are:
- Wallaby (+35/32) → Very low accuracy temperaments
- Sharpie (+25/24) → Dicot family
- Mite (+27/25) → Bug family
- Inflated (+128/125) → Augmented family
- Opossum (+126/125) → Porcupine family
- Blackwood (+49/48) → Limmic temperaments
Considered below are father, sharptone, uncle, octokaidecal, and parakangaroo.
Father
See Father family #Septimal father.
Sharptone
See Meantone family #Sharptone.
Uncle
- For the 5-limit version, see Syntonic–diatonic equivalence continuum.
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.
Subgroup: 2.3.5.7
Comma list: 28/27, 256/245
Mapping: [⟨1 0 12 -2], ⟨0 1 -6 3]]
- WE: ~2 = 1190.224 ¢, ~3/2 = 725.221 ¢
- error map: ⟨-9.776 +13.490 +3.707 -2.939]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 731.394 ¢
- error map: ⟨0.000 +29.439 +25.324 +25.355]
- 7-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3
- 9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5
Optimal ET sequence: 5, 13d, 18, 23bc, 41bbcd
Badness (Sintel): 1.84
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 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.
5-limit (supersharp)
Subgroup: 2.3.5
Comma list: 800/729
Mapping: [⟨2 0 -5], ⟨0 1 3]]
- mapping generators: ~27/20, ~3
- WE: ~27/20 = 596.986 ¢, ~3/2 = 725.434 ¢ (~10/9 = 128.448 ¢)
- error map: ⟨-6.029 +17.450 -13.027]
- CWE: ~27/20 = 600.000 ¢, ~3/2 = 726.548 ¢ (~10/9 = 126.548 ¢)
- error map: ⟨0.000 +24.593 -6.670]
Optimal ET sequence: 8, 10, 18, 28b
Badness (Sintel): 2.88
7-limit
Subgroup: 2.3.5.7
Comma list: 28/27, 50/49
Mapping: [⟨2 0 -5 -4], ⟨0 1 3 3]]
- WE: ~7/5 = 596.984 ¢, ~3/2 = 725.210 ¢ (~15/14 = 128.226 ¢)
- error map: ⟨-6.031 +17.224 -13.699 +0.774]
- CWE: ~7/5 = 600.000 ¢, ~3/2 = 726.319 ¢ (~15/14 = 126.319 ¢)
- error map: ⟨0.000 +24.364 -7.358 +10.130]
Optimal ET sequence: 8d, 10, 18, 28b
Badness (Sintel): 0.930
11-limit
Subgroup: 2.3.5.7.11
Comma list: 28/27, 50/49, 55/54
Mapping: [⟨2 0 -5 -4 7], ⟨0 1 3 3 0]]
Optimal tunings:
- WE: ~7/5 = 595.139 ¢, ~3/2 = 726.397 ¢ (~15/14 = 131.258 ¢)
- CWE: ~7/5 = 600.000 ¢, ~3/2 = 729.485 ¢ (~15/14 = 129.485 ¢)
Optimal ET sequence: 8d, 10, 18e
Badness (Sintel): 1.00
Parakangaroo
- For the 5-limit version of this temperament, 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.
Subgroup: 2.3.5.7
Comma list: 28/27, 1029/1000
Mapping: [⟨1 0 -3 -2], ⟨0 3 10 9]]
- mapping generators: ~2, ~10/7
- WE: ~2 = 596.984 ¢, ~10/7 = 638.135 ¢
- error map: ⟨-2.883 +12.450 +3.685 -19.845]
- CWE: ~2 = 1200.000 ¢, ~10/7 = 639.302 ¢
- error map: ⟨0.000 +15.952 +6.710 -15.104]
Optimal ET sequence: 2cd, …, 13cd, 15
Badness (Sintel): 1.97
11-limit
Subgroup: 2.3.5.7.11
Comma list: 28/27, 77/75, 245/242
Mapping: [⟨1 0 -3 -2 -4], ⟨0 3 10 9 14]]
Optimal tunings:
- WE: ~2 = 1196.971 ¢, ~10/7 = 638.230 ¢
- CWE: ~2 = 1200.000 ¢, ~10/7 = 639.480 ¢
Optimal ET sequence: 15
Badness (Sintel): 1.43
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 40/39, 66/65, 147/143
Mapping: [⟨1 0 -3 -2 -4 0], ⟨0 3 10 9 14 7]]
Optimal tunings:
- WE: ~2 = 1194.720 ¢, ~10/7 = 637.413 ¢
- CWE: ~2 = 1200.000 ¢, ~10/7 = 639.609 ¢
Optimal ET sequence: 15
Badness (Sintel): 1.35