15/14: Difference between revisions

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'''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 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]]

* [[7/6]] and [[5/4]]

* [[6/5]] and [[9/7]]

* [[4/3]] and [[10/7]]

* [[7/5]] and [[3/2]]

* [[14/9]] and [[5/3]]

* [[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]]

It also arises in higher limits as:

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

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

~~== Theory ==~~

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 ==

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).

== Temperaments ==

@@ Line 5: / Line 5: @@
 }}
 {{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 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]]
 * [[7/6]] and [[5/4]]
 * [[6/5]] and [[9/7]]
-* [[4/3]] and [[10/7]]
-* [[7/5]] and [[3/2]]
 * [[14/9]] and [[5/3]]
 * [[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]]
 It also arises in higher limits as:
@@ Line 20: / Line 26: @@
 * [[22/15]] and [[11/7]]
 * [[26/15]] and [[13/7]]
-== Theory ==
-/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>.
-/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)
-/14 separates the minor pairs from the major pairs; 4/3 separates the thirds from the sixths.
-/35 separates 15/14 from [[25/24]] on one side and [[54/49]] on the other.
 == Approximation ==
-/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.
+/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).
-{{Interval edo approximation|max_edo=131|15/14}}
+{{Interval edo approximation|max edo=131|15/14}}
 == Temperaments ==