User:Arseniiv/Isodifferential subdivision: Difference between revisions

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'''Isodifferential''' or '''linear subdivision'''{{idiosyncratic}} of an interval <math>s</math> into <math>d</math> parts is an [[Delta-rational chord#Isodifferential chord|isodifferential chord]] of <math>d+1</math> notes, spanning <math>s</math> —that is, a chord <math>a_0 : a_1 : a_2 : \ldots : a_{d-1} : a_d</math> with <math>\frac{a_d}{a_0} = s</math>, where frequency differences between consecutive notes are the same: <math>a_1 - a_0 = a_2 - a_1 = \ldots = a_d - a_{d-1}</math>. If the interval <math>s</math> is rational, this chord can be represented with integer <math>a_k</math> values.

'''Isodifferential''' or '''linear subdivision'''{{idiosyncratic}} of an interval <math>s</math> into <math>d</math> parts is an [[Delta-rational chord#Isodifferential chord|isodifferential chord]] of <math>d+1</math> notes, spanning <math>s</math> —that is, a chord <math>a_0 : a_1 : a_2 : \ldots : a_{d-1} : a_d</math> with <math>\frac{a_d}{a_0} = s</math>, where frequency differences between consecutive notes are the same: <math>a_1 - a_0 = a_2 - a_1 = \ldots = a_d - a_{d-1}</math>. If the interval <math>s</math> is rational, this (then ''isoharmonic'') chord can be represented with integer <math>a_k</math> values.

Isodifferential subdivision of <math>s</math> into two intervals <math>a, h</math> is already quite notable: <math>a</math> is the arithmetic mean of <math>s</math> and 1, the unison —this is just by definition chosen above,— and <math>h</math> is the harmonic mean of <math>s</math> and 1.

Dividing a superparticular interval in this way gives two superparticular intervals, which gives rise to "the Archytas's pyramid", a binary tree

Dividing a [[superparticular]] interval in this way gives two superparticular intervals, which gives rise to "the Archytas's pyramid", a binary tree

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which generates every superparticular interval by linearly dividing an octave into 2, 4, 8 parts and so on.

which generates every superparticular interval by linearly dividing an [[octave]] into 2, 4, 8 parts and so on.

One can also successively divide a tritave into 3 parts, generating a ternary tree with every interval <math>\tfrac{n+2}n</math> that isn't also a superparticular (so, with <math>n</math> odd).

One can also successively divide a [[tritave]] into 3 parts, generating a ternary tree with every interval <math>\tfrac{n+2}n</math> that isn't also a superparticular (so, with <math>n</math> odd).

== Properties ==

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-'''Isodifferential''' or '''linear subdivision'''{{idiosyncratic}} of an interval <math>s</math> into <math>d</math> parts is an [[Delta-rational chord#Isodifferential chord|isodifferential chord]] of <math>d+1</math> notes, spanning <math>s</math> —that is, a chord <math>a_0 : a_1 : a_2 : \ldots : a_{d-1} : a_d</math> with <math>\frac{a_d}{a_0} = s</math>, where frequency differences between consecutive notes are the same: <math>a_1 - a_0 = a_2 - a_1 = \ldots = a_d - a_{d-1}</math>. If the interval <math>s</math> is rational, this chord can be represented with integer <math>a_k</math> values.
+'''Isodifferential''' or '''linear subdivision'''{{idiosyncratic}} of an interval <math>s</math> into <math>d</math> parts is an [[Delta-rational chord#Isodifferential chord|isodifferential chord]] of <math>d+1</math> notes, spanning <math>s</math> —that is, a chord <math>a_0 : a_1 : a_2 : \ldots : a_{d-1} : a_d</math> with <math>\frac{a_d}{a_0} = s</math>, where frequency differences between consecutive notes are the same: <math>a_1 - a_0 = a_2 - a_1 = \ldots = a_d - a_{d-1}</math>. If the interval <math>s</math> is rational, this (then ''isoharmonic'') chord can be represented with integer <math>a_k</math> values.
 Isodifferential subdivision of <math>s</math> into two intervals <math>a, h</math> is already quite notable: <math>a</math> is the arithmetic mean of <math>s</math> and 1, the unison —this is just by definition chosen above,— and <math>h</math> is the harmonic mean of <math>s</math> and 1.
-Dividing a superparticular interval in this way gives two superparticular intervals, which gives rise to "the Archytas's pyramid", a binary tree
+Dividing a [[superparticular]] interval in this way gives two superparticular intervals, which gives rise to "the Archytas's pyramid", a binary tree
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-which generates every superparticular interval by linearly dividing an octave into 2, 4, 8 parts and so on.
+which generates every superparticular interval by linearly dividing an [[octave]] into 2, 4, 8 parts and so on.
-One can also successively divide a tritave into 3 parts, generating a ternary tree with every interval <math>\tfrac{n+2}n</math> that isn't also a superparticular (so, with <math>n</math> odd).
+One can also successively divide a [[tritave]] into 3 parts, generating a ternary tree with every interval <math>\tfrac{n+2}n</math> that isn't also a superparticular (so, with <math>n</math> odd).
 == Properties ==