15/14: Difference between revisions

Line 20:

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

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

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>

== Terminology ==

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 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>[https://~~marsbat~~.~~space~~/pdfs/crystal-growth.pdf ~~Marc Sabat (2008)~~ Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space]</ref>.

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 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'']. Plainsounud Music Edition, 2008.</ref>.

== Approximation ==

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.

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

== References ==

Line 37:

* [[List of superparticular intervals]]

* [[Gallery of just intervals]]

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

* [[1ed15/14|AS15/14]] - its ambitonal sequence

[[Category:Semitone]]

[[Category:Chroma]]

[[Category:Mercurial]]

@@ Line 20: / Line 20: @@
 * [[22/15]] and [[11/7]]
 * [[26/15]] and [[13/7]]
-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>
 == Terminology ==
-/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 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>[https://marsbat.space/pdfs/crystal-growth.pdf Marc Sabat (2008) Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space]</ref>.
+/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 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'']. Plainsounud Music Edition, 2008.</ref>.
 == Approximation ==
-/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.
+/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).
 == References ==
@@ Line 37: / Line 37: @@
 * [[List of superparticular intervals]]
 * [[Gallery of just intervals]]
-* [[15/14ths equal temperament|AS15/14]] - its ambitonal sequence
+* [[1ed15/14|AS15/14]] - its ambitonal sequence
 [[Category:Semitone]]
 [[Category:Chroma]]
 [[Category:Mercurial]]