15/14: Difference between revisions

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{{Infobox Interval

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

| Name = septimal diatonic semitone, aberschismic chromatic semitone, major chromatic semitone

| Color name = ry1, ruyo unison

| Sound = jid_15_14_pluck_adu_dr220.mp3

}}

'''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>.

'''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>.

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:

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

It also arises in higher limits as:

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]]

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

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 ==

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = septimal diatonic semitone, septimal major semitone
+| Name = septimal diatonic semitone, aberschismic chromatic semitone, major chromatic semitone
 | Color name = ry1, ruyo unison
 | 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 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>.
+'''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>.
 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:
@@ Line 21: / Line 21: @@
 * [[7/5]] and [[3/2]]
-It also arises in higher limits as:
+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]]
 * [[26/15]] and [[13/7]]
+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 ==