15/14: Difference between revisions
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{{Wikipedia|Septimal diatonic semitone}} | {{Wikipedia|Septimal diatonic semitone}} | ||
'''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[triangular number]]. It | '''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[triangular number]]. It is traditionally called a ''diatonic semitone'', perhaps for its proximity (and conflation in systems such as septimal [[meantone]] and [[marvel]]) with the classic diatonic semitone [[16/15]]. However, 15/14 is a ''[[chromatic semitone]]'' in both [[Helmholtz–Ellis notation]] and the [[Functional Just System]], viewed as the apotome [[2187/2048]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''major chromatic semitone'' in the same material where [[21/20]] is also named as the minor diatonic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>. | ||
Because it contains exactly one of each prime up to 7, it appears as the interval between many simple [[7-limit]] ratios. In particular, it is the difference between certain [[interval qualities]] of seconds, thirds, sixths, and sevenths: between classical minor and supermajor, and between subminor and classical major. These are the pairs of intervals separated by 15/14: | |||
* [[28/27]] and [[10/9]] | |||
* [[16/15]] and [[8/7]] | * [[16/15]] and [[8/7]] | ||
* [[7/6]] and [[5/4]] | * [[7/6]] and [[5/4]] | ||
* [[6/5]] and [[9/7]] | * [[6/5]] and [[9/7]] | ||
* [[14/9]] and [[5/3]] | * [[14/9]] and [[5/3]] | ||
* [[8/5]] and [[12/7]] | * [[8/5]] and [[12/7]] | ||
* [[7/4]] and [[15/8]] | * [[7/4]] and [[15/8]] | ||
* [[9/5]] and [[27/14]] | |||
In addition, it separates the perfect fourth from the larger septimal tritone, and the perfect fifth from the smaller septimal tritone: | |||
* [[4/3]] and [[10/7]] | |||
* [[7/5]] and [[3/2]] | |||
It also arises in higher limits as: | It also arises in higher limits as: | ||
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* [[26/15]] and [[13/7]] | * [[26/15]] and [[13/7]] | ||
== | == Approximation == | ||
15/14 is | 15/14 is very accurately approximated by [[10edo]] (1\10) and all [[linus]] temperaments. The [[linus comma]], 5.6{{c}}, is the amount by which a stack of ten 15/14's falls short of the octave. | ||
== | In combination with [[19/17]] it forms a good approximation of [[golden meantone]]. The untempered combination of five 19/17's and two 15/14's leads to an interval that is sharp to an octave by the [[mercurial comma]]: (19/17)<sup>5</sup> × (15/14)<sup>2</sup> = 2 / (mercurial comma). | ||
15/14 | |||
{{Interval edo approximation|max edo=131|15/14}} | |||
== Temperaments == | |||
The following [[linear temperament]]s are [[generate]]d by a [[~]]15/14: | |||
* [[Septidiasemi]] | |||
* [[Subsedia]] | |||
In addition, this [[fractional-octave temperament]] is generated by a ~15/14: | |||
* [[Tertiosec]] (1\3) | |||
Several [[10th-octave temperaments]] treat ~15/14 as the period, including [[decoid]] and [[linus]]. | |||
{{todo|complete list}} | |||
== See also == | == See also == | ||
* [[28/15]] – its [[octave complement]] | * [[28/15]] – its [[octave complement]] | ||
* [[7/5]] – its [[fifth complement]] | * [[7/5]] – its [[fifth complement]] | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||