15/14: Difference between revisions

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{{Infobox Interval

~~| JI glyph =~~

| Name = septimal diatonic semitone, septimal major semitone

~~| Ratio = 15/14~~

| Color name = ry1, ruyo unison

~~| Monzo = -1 1 1 -1~~

~~| Cents = 119.44281~~

| Name = septimal diatonic semitone

| Sound = jid_15_14_pluck_adu_dr220.mp3

~~| Color name = ry1, ruyo semitone~~

}}

15/14 is a [[~~superparticular|~~superparticular]] ratio with a numerator which is the fifth [~~http~~://en.~~wikipedia~~.org/~~wiki~~/~~Triangular_number triangular number~~].

'''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>.

~~It may be found~~ as the interval between many [[~~7-limit|~~7-limit]] ratios, ~~including~~:

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]]

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

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

* [[7/4]] and [[15/8]]

* [[9/5]] and [[27/14]]

~~<ul><li>16/15 and 8/7</li><li>14/13 and 15/13</li><li>7/6 and 5/4</li><li>6/5 and 9/7</li><li>14/11~~ and ~~15/11</li><li>~~4/3 and 10/7~~</li><li>~~7/5 and 3/2~~</li><li>22/15 and 11/7</li><li>14/9 and 5/3</li><li>8/5 and 12/7</li><li>26/15 and 13/7</li><li>7/4 and 15/8</li></ul>~~

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]]

[~~http:~~//en.~~wikipedia~~.~~org~~/~~wiki~~/~~Septimal_diatonic_semitone http~~://en.~~wikipedia~~.~~org~~/~~wiki~~/~~Septimal_diatonic_semitone~~]

It also arises in higher limits as:

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

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

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

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

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

== 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]].

== See also ==

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

* [[7/5]] – its [[fifth complement]]

* [[List of superparticular intervals]]

* [[Gallery of just intervals]]

== References ==

[[Category:Semitone]]

[[Category:Chroma]]

[[Category:Mercurial]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| JI glyph =
+| Name = septimal diatonic semitone, septimal major semitone
-| Ratio = 15/14
+| Color name = ry1, ruyo unison
-| Monzo = -1 1 1 -1
-| Cents = 119.44281
-| Name = septimal diatonic semitone
 | Sound = jid_15_14_pluck_adu_dr220.mp3
-| Color name = ry1, ruyo semitone
 }}
-/14 is a [[superparticular|superparticular]] ratio with a numerator which is the fifth [http://en.wikipedia.org/wiki/Triangular_number triangular number].
+{{Wikipedia|Septimal diatonic semitone}}
+'''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>.
-It may be found as the interval between many [[7-limit|7-limit]] ratios, including:
+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]]
+* [[14/9]] and [[5/3]]
+* [[8/5]] and [[12/7]]
+* [[7/4]] and [[15/8]]
+* [[9/5]] and [[27/14]]
-<ul><li>16/15 and 8/7</li><li>14/13 and 15/13</li><li>7/6 and 5/4</li><li>6/5 and 9/7</li><li>14/11 and 15/11</li><li>4/3 and 10/7</li><li>7/5 and 3/2</li><li>22/15 and 11/7</li><li>14/9 and 5/3</li><li>8/5 and 12/7</li><li>26/15 and 13/7</li><li>7/4 and 15/8</li></ul>
+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]]
-[http://en.wikipedia.org/wiki/Septimal_diatonic_semitone http://en.wikipedia.org/wiki/Septimal_diatonic_semitone]
+It also arises in higher limits as:
+* [[14/13]] and [[15/13]]
+* [[14/11]] and [[15/11]]
+* [[22/15]] and [[11/7]]
+* [[26/15]] and [[13/7]]
+== Approximation ==
+/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}}
+== 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 ==
+* [[28/15]] – its [[octave complement]]
+* [[7/5]] – its [[fifth complement]]
+* [[List of superparticular intervals]]
+* [[Gallery of just intervals]]
+== References ==
+<references/>
+[[Category:Semitone]]
+[[Category:Chroma]]
+[[Category:Mercurial]]