15/14: Difference between revisions

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{{Infobox Interval

~~| JI glyph =~~

| Name = septimal diatonic semitone, aberschismic chromatic semitone, major chromatic 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~~]] ~~ratio~~ with a ~~numerator which~~ is the ~~fifth~~ [~~http~~://en.~~wikipedia~~.org/~~wiki~~/~~Triangular_number triangular number]~~.

'''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>.

~~It may be found as the interval between many [[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]]

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

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

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

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

* [[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 the difference between:

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

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

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

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

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

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

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

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)<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 ==

* [[~~Gallery of Just Intervals~~]]

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

* [[28/15]] its ~~inverse interval~~

* [[7/5]] – its [[fifth complement]]

* [~~http://en.wikipedia.org/wiki~~/~~Septimal_diatonic_semitone Septimal diatonic semitone - Wikipedia]~~

* [[List of superparticular intervals]]

* [[Gallery of just intervals]]

== References ==

~~[[Category:7-limit]]~~

~~[[Category:Interval]]~~

[[Category:Semitone]]

[[Category:~~Superparticular~~]]

[[Category:Chroma]]

[[Category:~~todo:expand~~]]

[[Category:Mercurial]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| JI glyph =
+| Name = septimal diatonic semitone, aberschismic chromatic semitone, major chromatic 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]] ratio with a numerator which is the fifth [http://en.wikipedia.org/wiki/Triangular_number triangular number].
+{{Wikipedia|Septimal diatonic semitone}}
+'''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>.
-It may be found as the interval between many [[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]]
-* [[14/13]] and [[15/13]]
 * [[7/6]] and [[5/4]]
 * [[6/5]] and [[9/7]]
-* [[14/11]] and [[15/11]]
+* [[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 the difference between:
+* [[14/13]] and [[15/13]]
+* [[14/11]] and [[15/11]]
 * [[22/15]] and [[11/7]]
-* [[14/9]] and [[5/3]]
-* [[8/5]] and [[12/7]]
 * [[26/15]] and [[13/7]]
-* [[7/4]] and [[15/8]]
+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 ==
+/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 ==
-* [[Gallery of Just Intervals]]
+* [[28/15]] – its [[octave complement]]
-* [[28/15]] its inverse interval
+* [[7/5]] – its [[fifth complement]]
-* [http://en.wikipedia.org/wiki/Septimal_diatonic_semitone Septimal diatonic semitone - Wikipedia]
+* [[List of superparticular intervals]]
+* [[Gallery of just intervals]]
+== References ==
+<references/>
-[[Category:7-limit]]
-[[Category:Interval]]
 [[Category:Semitone]]
-[[Category:Superparticular]]
+[[Category:Chroma]]
-[[Category:todo:expand]]
+[[Category:Mercurial]]