15/14: Difference between revisions

'''15/14''' is an interval in [[7-limit]] [[just intonation]] measuring about 119.4 [[cent]]s, traditionally known as the '''septimal diatonic semitone''' for its proximity (and conflation in [[marvel]] tunings such as septimal [[meantone]]) with the classical diatonic semitone [[16/15]]. However, it functions as a ''[[chromatic semitone]]'', as is supported by [[Sagittal notation]], [[Helmholtz–Ellis notation]] and the [[Functional Just System]], viewed as the [[Pythagorean apotome]] altered by an [[aberschisma]]. This gives rise to the more precise name '''aberschismic chromatic semitone''', or as [[Marc Sabat]] has taken to call it, the '''major chromatic 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:

* [[8/5]] and [[12/7]]

* [[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]]

* [[14/13]] and [[15/13]]

* [[14/11]] and [[15/11]]

* [[22/15]] and [[11/7]]

* [[26/15]] and [[13/7]]

 

Finally, since it is a [[superparticular ratio]] with a numerator which is the fifth [[triangular number]], it is a [[triangle-particular]] ratio with factorization ([[25/24]])⋅([[36/35]]).  

 

== Approximation ==

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)⁵ × (15/14)² = 2 / (mercurial comma).

 

{{Interval edo approximation|max edo=131|15/14}}

 

== Temperaments ==

The following [[linear temperament]]s are [[generate]]d by a [[~]]15/14:

 

In addition, this [[fractional-octave temperament]] is generated by a ~15/14:

 

Several [[10th-octave temperaments]] treat ~15/14 as the period, including [[decoid]] and [[linus]].

 

* [[28/15]] – its [[octave complement]]

* [[7/5]] – its [[fifth complement]]

 

== References ==

<references/>