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~~

~~| FJS name = A157~~

}}

'''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[~~Wikipedia:Triangular_number|~~triangular number]].

'''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:

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

* [[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]]

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

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

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

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

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

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

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]]

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

~~Although~~ called ~~"septimal~~ diatonic semitone" for its proximity (and conflation in systems such as septimal [[meantone]]) with the classic diatonic semitone [[16/15]], 15/14 is a [[Wikipedia:chromatic semitone|chromatic semitone]] in both [[~~Helmholtz-Ellis~~ notation]] and [[Functional Just System]] ~~because~~ it is [[~~5120~~/~~5103~~]] ~~larger than~~ the ~~apotome~~ [[~~2187~~/~~2048~~]].

== Terminology ==

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>.

== 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.

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).

== 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 [[~~octave~~ complement]]

* [[7/5]] – its [[fifth complement]]

* [[~~Wikipedia: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, 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
-| FJS name = A1<sup>5</sup><sub>7</sub>
 }}
-'''15/14''' is a [[superparticular]] ratio with a numerator which is the fifth [[Wikipedia:Triangular_number|triangular number]].
+{{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:
-It may be found as the interval between many [[7-limit]] ratios, including:
 * [[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]]
 * [[4/3]] and [[10/7]]
 * [[7/5]] and [[3/2]]
-* [[22/15]] and [[11/7]]
 * [[14/9]] and [[5/3]]
 * [[8/5]] and [[12/7]]
+* [[7/4]] and [[15/8]]
+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]]
-* [[7/4]] and [[15/8]]
-Although called "septimal diatonic semitone" for its proximity (and conflation in systems such as septimal [[meantone]]) with the classic diatonic semitone [[16/15]], 15/14 is a [[Wikipedia:chromatic semitone|chromatic semitone]] in both [[Helmholtz-Ellis notation]] and [[Functional Just System]] because it is [[5120/5103]] larger than the apotome [[2187/2048]].
+== Terminology ==
+/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>.
+== 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.
+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]].
+{{todo|complete list}}
 == See also ==
-* [[Gallery of Just Intervals]]
+* [[28/15]] – its [[octave complement]]
-* [[28/15]] its [[octave complement]]
+* [[7/5]] – its [[fifth complement]]
-* [[Wikipedia: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]]