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Rewrite to better address the names (displacing "septimal major semitone", not attested anywhere)
 
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{{Infobox Interval
{{Infobox Interval
| Name = septimal diatonic semitone, septimal major semitone
| Name = septimal diatonic semitone, aberschismic chromatic semitone, major chromatic semitone
| Color name = ry1, ruyo unison
| Color name = ry1, ruyo unison
| Sound = jid_15_14_pluck_adu_dr220.mp3
| Sound = jid_15_14_pluck_adu_dr220.mp3
}}
}}
{{Wikipedia|Septimal diatonic semitone}}
{{Wikipedia|Septimal diatonic semitone}}
'''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[triangular number]]. Because it contains exactly one of each prime up to 7, it appears as the interval between many simple [[7-limit]] ratios, including:
'''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:
* [[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]]
* [[4/3]] and [[10/7]]
* [[7/5]] and [[3/2]]
* [[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 the difference between:  
* [[14/13]] and [[15/13]]
* [[14/13]] and [[15/13]]
* [[14/11]] and [[15/11]]
* [[14/11]] and [[15/11]]
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* [[26/15]] and [[13/7]]
* [[26/15]] and [[13/7]]


== Theory ==
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]]).  
15/14 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 ''[[Wikipedia:chromatic semitone|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>.
 
15/14 is the difference between certain [[interval qualities]] of thirds and sixths: between (classic) minor and supermajor, and between subminor and (classic) major. These are the pairs of intervals separated by 15/14:
* 7/6 and 5/4 (otonal thirds)
* 6/5 and 9/7 (utonal thirds)
* 14/9 and 5/3 (otonal sixths)
* 8/5 and 12/7 (utonal sixths)
[[36/35]] separates the otonal pairs from the utonal pairs; 4/3 separates the thirds from the sixths.
 
They can also be organized into pairs of intervals differing by 36/35:
* 7/6 and 6/5 (minor thirds)
* 5/4 and 9/7 (major thirds)
* 14/9 and 8/5 (minor sixths)
* 5/3 and 12/7 (major sixths)
15/14 separates the minor pairs from the major pairs; 4/3 separates the thirds from the sixths.
 
36/35 separates 15/14 from [[25/24]] on one side and [[54/49]] on the other.


== Approximation ==
== Approximation ==
15/14 is very accurately approximated by [[10edo]] (1\10) and all [[linus]] temperaments. The [[linus comma]], 5.6¢, is the amount by which a stack of ten 15/14's falls short of the octave.
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).
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).
{{Interval edo approximation|max_edo=131|15/14}}
 
{{Interval edo approximation|max edo=131|15/14}}


== Temperaments ==
== Temperaments ==
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