Trienstonic clan: Difference between revisions
- CTE & POTE tunings |
m Update |
||
| (13 intermediate revisions by 4 users not shown) | |||
| Line 2: | Line 2: | ||
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]]. | 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. 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 | [[Subgroup]]: 2.3.7 | ||
| Line 18: | Line 11: | ||
{{Mapping|legend=2| 1 0 -2 | 0 1 3 }} | {{Mapping|legend=2| 1 0 -2 | 0 1 3 }} | ||
{{Mapping|legend=3| 1 0 0 -2 | 0 1 0 3 }} | |||
: mapping generators: ~2, ~3 | : mapping generators: ~2, ~3 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
| Line 32: | Line 24: | ||
[[Badness]] (Sintel): 0.235 | [[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: | |||
* [[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) → [[Blackwood family #Blackwood|Blackwood family]] | |||
* ''[[Opossum]]'' (+126/125) → [[Porcupine family #Opossum|Porcupine family]] | |||
* ''[[Inflated]]'' (+128/125) → [[Augmented family #Inflated|Augmented family]] | |||
Considered below are uncle, octokaidecal, and parakangaroo. | |||
== Uncle == | == Uncle == | ||
: ''For the 5-limit version, see [[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. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 61: | Line 69: | ||
== Octokaidecal == | == Octokaidecal == | ||
: ''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]]. | |||
[[ | |||
[[ | |||
[[ | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 118: | Line 107: | ||
== Parakangaroo == | == Parakangaroo == | ||
: ''For the 5-limit version | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kangaroo]].'' | ||
This temperament used to be known as | 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]]: 2.3.5.7 | ||
| Line 171: | Line 160: | ||
[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||
[[Category:Trienstonic clan| ]] <!-- main article --> | |||
[[Category:Trienstonic clan| ]] <!-- | |||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||