No-fives subgroup temperaments: Difference between revisions
→Temperaments with a 2.3.11 gene: rework the intro to alphaxenean; + betaxenean |
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Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament. | Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament. | ||
The name ''io'' was coined by | The name ''io'' was coined by {{u|CompactStar}} in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma. | ||
[[Subgroup]]: 2.3.11 | [[Subgroup]]: 2.3.11 | ||
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=== Alphaxenean === | === Alphaxenean === | ||
Alphaxenean tempers out the [[Alpharabian comma]] and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It | Alphaxenean tempers out the [[Alpharabian comma]], and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It has a [[semi-octave]] period, and the generator can be taken to be the aforementioned undecimal quartertone or that plus a period, which represents ~[[16/11]]. The interval class of a perfect fifth is found by two generator steps and a period. | ||
[[Subgroup]]: 2.3.11 | [[Subgroup]]: 2.3.11 | ||
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=== Octatonic === | === Octatonic === | ||
[[12/11]] is very close to | [[12/11]] is very close to one step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave. | ||
[[Subgroup]]: 2.3.11 | [[Subgroup]]: 2.3.11 | ||
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[[Badness]] (Sintel): 0.515 | [[Badness]] (Sintel): 0.515 | ||
=== Betaxenean === | |||
Betaxenean tempers out the [[Betarabian comma]]. It splits a [[3/1|perfect twelfth]] into two halves of [[512/297]]. | |||
[[Subgroup]]: 2.3.11 | |||
[[Comma list]]: 131769/131072 | |||
{{Mapping|legend=2| 1 0 9 | 0 2 -7 }} | |||
: mapping generators: ~2, ~512/297 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.5931{{c}}, ~512/297 = 950.6872{{c}} | |||
: [[error map]]: {{val| +0.593 -0.581 -0.791 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~512/297 = 950.1972{{c}} | |||
: error map: {{val| 0.000 -1.561 -2.699 }} | |||
{{Optimal ET sequence|legend=1| 5, 14e, 19, 24, 125, 149, 173, 197e, 221e, 466bee, 687beee }} | |||
[[Badness]] (Sintel): 0.709 | |||
=== Infraug === | === Infraug === | ||