16ed5/3: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
'''16ed5/3''' | '''16ed5/3''' is the [[Ed5/3|equal division of the just major sixth]] into sixteen parts of 55.2724 [[cent]]s each, corresponding to 21.7106[[edo]]. It is very closely related to the [[Escapade family|escapade temperament]]. It is vaguely equivalent to [[22edo]]. | ||
It very accurately approximates a number of low complexity just intervals, such as: [[4/3]] (<1¢), [[5/4]] (<1¢), [[11/8]] (<2¢), [[11/10]] (<1¢), [[16/15]] (<2¢), and [[25/16]] (<2¢). It also approximates the [[3/2|just fifth]] and [[2/1|octave]] to within 17¢, making it a flexible non-octave scale. Notably, having a period of [[5/3]], the diatonic minor third ([[6/5]]) is the period-reduced diatonic octave. This means both are approximated identically (16¢ sharp). | It very accurately approximates a number of low complexity just intervals, such as: [[4/3]] (<1¢), [[5/4]] (<1¢), [[11/8]] (<2¢), [[11/10]] (<1¢), [[16/15]] (<2¢), and [[25/16]] (<2¢). It also approximates the [[3/2|just fifth]] and [[2/1|octave]] to within 17¢, making it a flexible non-octave scale. Notably, having a period of [[5/3]], the diatonic minor third ([[6/5]]) is the period-reduced diatonic octave. This means both are approximated identically (16¢ sharp). | ||
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== Commas == | == Commas == | ||
Depending on your mapping, 16ed5/3 can be said to temper a number of commas, including the | Depending on your mapping, 16ed5/3 can be said to temper a number of commas, including the [[diaschisma]], the [[marvel comma]], [[64/63|Archytas' comma]], and the [[jubilisma]], all discussed in the temperaments section. In addition, being an even division of the 5/3, it tempers the [[sensamagic comma]], as the half mosoctave is midway between [[9/7]] and [[35/27]]. This is analogous to the tritone in 2n edo systems. The [[keema]] is tempered due to the septimal interpretation of the diatonic sevenths, and the [[mothwellsma]] is tempered by two major mos3rds ([[7/6]]) resulting in an augmented mos5th ([[11/8]]). | ||
== Temperaments == | == Temperaments == | ||
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The diaschisma can also be tempered by taking 5 generators to mean a [[3/2]] ((<sup>4</sup>/<sub>3</sub>)<sup>5</sup>=(<sup>3</sup>/<sub>2</sub>)·(<sup>5</sup>/<sub>3</sub>)<sup>2</sup>), while the marvel comma can also be tempered with a stack of 3 generators, making a [[10/7]] ((<sup>4</sup>/<sub>3</sub>)<sup>3</sup>=(<sup>10</sup>/<sub>7</sub>)·(<sup>5</sup>/<sub>3</sub>)). | The diaschisma can also be tempered by taking 5 generators to mean a [[3/2]] ((<sup>4</sup>/<sub>3</sub>)<sup>5</sup>=(<sup>3</sup>/<sub>2</sub>)·(<sup>5</sup>/<sub>3</sub>)<sup>2</sup>), while the marvel comma can also be tempered with a stack of 3 generators, making a [[10/7]] ((<sup>4</sup>/<sub>3</sub>)<sup>3</sup>=(<sup>10</sup>/<sub>7</sub>)·(<sup>5</sup>/<sub>3</sub>)). | ||
The tempered marvel comma also means that the two large [[ | The tempered marvel comma also means that the two large [[tritone]]s ([[64/45|pental]] and [[10/7|septimal]]) are addressed by the same scale step. The tempered diaschisma, on the other hand, means that both pental tritones are also addressed by the same scale step. | ||
Both of the 7-limit approaches also temper Archytas' comma | Both of the 7-limit approaches also temper Archytas' comma as a result of equating the [[16/9]] with [[7/4]], and the jubilisma ([[50/49]]) due to tritone equivalence. These are relatively large commas, given the step size (about half, and 7/11ths respectively). | ||
This shows the close relationships with [[ | This shows the close relationships with [[srutal]] and [[pajara]] octave temperaments. In 16ed5/3's case, there is a close equivalence to [[22edo]]'s pajara tuning. | ||
As 3 semitones make a period-reduced octave, and it alludes to tritone tempering, [[User:Ayceman|I]] propose the name '''tristone''' for the basic [[Diaschismic family|diaschismic temperament]], based on the 16/15 to 6/5 relationship, as well as the following variants and extensions: | As 3 semitones make a period-reduced octave, and it alludes to tritone tempering, [[User:Ayceman|I]] propose the name '''tristone''' for the basic [[Diaschismic family|diaschismic temperament]], based on the 16/15 to 6/5 relationship, as well as the following variants and extensions: | ||
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[[Vals]]: 9ed5/3, 16ed5/3 | [[Vals]]: 9ed5/3, 16ed5/3 | ||
[[Category:Nonoctave]] | [[Category:Nonoctave]] | ||
[[Category:Edonoi]] | [[Category:Edonoi]] | ||