15/14: Difference between revisions

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{{Infobox Interval

~~| Ratio = 15/14~~

| Name = septimal diatonic semitone, septimal major semitone

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

~~| Cents = 119.44281~~

| Name = septimal diatonic semitone, ~~ ~~septimal major semitone

| Color name = ry1, ruyo unison

~~| FJS name = A157~~

| Sound = jid_15_14_pluck_adu_dr220.mp3

}}

'''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[triangular number]]. It ~~may be found~~ as the interval between many [[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]]

* [[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:

* [[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]]

In combination with [[19/17]] it forms a good approximation of [[golden meantone]]. The untempered combination of five 19/17 and two 15/14 leads to an interval that is sharp to an octave by the [[mercurial comma]]: ~~<code>(~~(19/17)^5 * (15/14)^2 = (2~~/1)~~ / (mercurial comma))</~~code>~~

== 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).

== ~~Terminology~~ ==

== Temperaments ==

~~15/14 is traditionally called a ''diatonic semitone'', perhaps for its proximity (and conflation in systems such as septimal~~ [[~~meantone~~]]~~) with the classic diatonic semitone~~ [[~~16/15~~]]~~. However, 15/14 is~~ a ~~''[[Wikipedia:chromatic semitone|chromatic semitone]]'' in both~~ [[~~Helmholtz-Ellis notation~~]] ~~and [[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>[https://marsbat.space/pdfs/crystal-growth.pdf Marc Sabat (2008) Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space]</ref>.

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

* [[Septidiasemi]]

* [[Subsedia]]

~~== Approximation ==~~

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

15/14 ~~is very accurately approximated by~~ [[~~10edo|10EDO~~]] (1\10) ~~and all linus temperaments. The [[15/14ths equal temperament|linus comma]], 5.6¢, is the amount by which a stack of ten 15/14's falls short of the octave.~~

* [[Tertiosec]] (1\3)

~~== References ==~~

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

~~<references/>~~

== See also ==

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* [[List of superparticular intervals]]

* [[Gallery of just intervals]]

* [[15/14ths equal temperament|AS15/14]] - its ambitonal sequence

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

== References ==

[[Category:Semitone]]

[[Category:Chroma]]

~~[[Category:Superparticular]]~~

[[Category:Mercurial]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Ratio = 15/14
+| Name = septimal diatonic semitone, septimal major semitone
-| Monzo = -1 1 1 -1
-| Cents = 119.44281
-| Name = septimal diatonic semitone, <br>septimal major semitone
 | Color name = ry1, ruyo unison
-| FJS name = A1<sup>5</sup><sub>7</sub>
 | Sound = jid_15_14_pluck_adu_dr220.mp3
 }}
 {{Wikipedia|Septimal diatonic semitone}}
-'''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[triangular number]]. It may be found as the interval between many [[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]]
-* [[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:
+* [[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]]
-In combination with [[19/17]] it forms a good approximation of [[golden meantone]]. The untempered combination of five 19/17 and two 15/14 leads to an interval that is sharp to an octave by the [[mercurial comma]]: <code>((19/17)^5 * (15/14)^2 = (2/1) / (mercurial comma))</code>
+== 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}}
-== Terminology ==
+== Temperaments ==
-/14 is traditionally called a ''diatonic semitone'', perhaps for its proximity (and conflation in systems such as septimal [[meantone]]) with the classic diatonic semitone [[16/15]]. However, 15/14 is a ''[[Wikipedia:chromatic semitone|chromatic semitone]]'' in both [[Helmholtz-Ellis notation]] and [[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>[https://marsbat.space/pdfs/crystal-growth.pdf Marc Sabat (2008) Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space]</ref>.
+The following [[linear temperament]]s are [[generate]]d by a [[~]]15/14:
+* [[Septidiasemi]]
+* [[Subsedia]]
-== Approximation ==
+In addition, this [[fractional-octave temperament]] is generated by a ~15/14:
-/14 is very accurately approximated by [[10edo|10EDO]] (1\10) and all linus temperaments. The [[15/14ths equal temperament|linus comma]], 5.6¢, is the amount by which a stack of ten 15/14's falls short of the octave.
+* [[Tertiosec]] (1\3)
-== References ==
+Several [[10th-octave temperaments]] treat ~15/14 as the period, including [[decoid]] and [[linus]].
-<references/>
+{{todo|complete list}}
 == See also ==
@@ Line 39: / Line 50: @@
 * [[List of superparticular intervals]]
 * [[Gallery of just intervals]]
-* [[15/14ths equal temperament|AS15/14]] - its ambitonal sequence
-[[Category:7-limit]]
+== References ==
+<references/>
 [[Category:Semitone]]
 [[Category:Chroma]]
-[[Category:Superparticular]]
 [[Category:Mercurial]]