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[[Badness]] (Sintel): 1.43
[[Badness]] (Sintel): 1.43


Music:
; Music
* ''[https://www.youtube.com/watch?v=vdjhC9i5KF4 Four Short Experiments in Octave Stretched 42edo (Dec 2024)]'' by [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=vdjhC9i5KF4 ''Four Short Experiments in Octave Stretched 42edo''] (2024) by [[Budjarn Lambeth]]


=== 11-limit ===
=== 11-limit ===
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== Marfifths ==
== Marfifths ==
The ''marfifths'' temperament (19 & 140) tempers out the [[hemimage comma]], 10976/10935. It splits the interval of a major thirteenth (~10/3) into three marvelous fifth ([[112/75]]) intervals, and uses it for a generator.
Named by [[Xenllium]] in 2021, marfifths tempers out the 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 19 & 140 }} temperament. It is generated by a marvel fourth of [[75/56]] (or a marvel fifth of [[112/75]]), three of which minus an octave make the hanson generator of ~6/5. Its [[ploidacot]] is zeta-18-cot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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=== Diatessic ===
=== Diatessic ===
The ''diatessic'' temperament (121 & 140) is closely related to the '''diatess tuning''' (generator: 505.727281 cents).
Diatessic may be described as {{nowrap| 121 & 140 }}} and is closely related to the Diatess tuning (generator: 505.727281 cents).


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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=== Marf ===
=== Marf ===
The ''marf'' temperament (19 & 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.
Marf may be described as {{nowrap| 19 & 121 }}. It has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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== Marthirds ==
== Marthirds ==
The ''marthirds'' temperament (19 & 193) tempers out the breeze comma (laquadru-atruyo comma), [[2460375/2458624]]. It splits the interval of minor tenth (~12/5) into four marvelous major third ([[56/45]]) intervals, and uses it for a generator.
Named by [[Xenllium]] in 2021, marthirds tempers out 2460375/2458624, the [[breeze comma]], and may be described as the {{nowrap| 19 & 193 }} temperament. It is generated by a marvel-comma-flat classical major third, [[56/45]], four of which minus an octave make the hanson generator of [[6/5]]. Its [[ploidacot]] is zeta-24-cot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Main| Sqrtphi }}
{{Main| Sqrtphi }}


The just value of sqrt (φ) is 416.545 cents.  
Sqrtphi tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 49 & 72 }} temperament. The just value of sqrt(φ) is 416.545 cents, and this temperament gives a close approximation of it.
 
Note that in the data below, the generator is given as its [[octave complement]], which stands in for [[~]][[11/7]] from the [[11-limit]] onwards. Five generators octave reduced make the hanson generator of ~[[6/5]]. The [[ploidacot]] for this temperament is 19-sheared 30-cot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== Quartkeenlig ==
== Quartkeenlig ==
Quartkeenlig uses a generator in the 11-limit that is 33/32~36/35 tempered together, and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with 7/6, keenanisma, 385/384, tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching|stretched 23edo]].
Named by [[Eliora]] in 2022, quartkeenlig uses a generator that is a quartertone of [[33/32]][[~]][[36/35]] tempered together in the [[11-limit]], and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with [[7/6]], keenanisma, [[385/384]], tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). As six quartertones make the hanson generator of ~[[6/5]], its [[ploidacot]] is alpha-36-cot. It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching|stretched 23edo]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== Subgroup extensions ==
== Subgroup extensions ==
=== Kleismic (2.3.5.13) a.k.a. cata ===
=== Kleismic (2.3.5.13) a.k.a. cata ===
Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])([[625/624]]) and 325/324 = (625/624)([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324|S10/S12 = S25*S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]}. For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].
Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]]), [[676/675|S13/S15 = S26]]}. For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].


Subgroup: 2.3.5.13
Subgroup: 2.3.5.13
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