Fokker block: Difference between revisions

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Let's take [[5-limit]] just intation, and pick the [[25/24|just chromatic semitone]] (25/24) and the [[syntonic comma]] (81/80) as our chromas.

The octave equivalent lattice is generated by fifths and just major thirds.

Since <math>25/24 = 2^{-3} \cdot 3^{-1} \cdot 5^2</math>, it has coordinates <math>(-1, 2)</math> in the octave-equivalent lattice.

The syntonic comma is reached by stacking 4 fifths and going down a major third, so it has coordinates <math>(4, -1)</math>.

[[File:Fokker_block_zarlino.png|400px|thumb|none|Fokker block corresponding to the just diatonic scale. The gray grid is the interval lattice, and the black lines show the sublattice generated by the chromas. The fundamental domain is colored in blue.]]

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Let's now use the [[128/125|diesis]] (128/125) and the [[syntonic comma]] (81/80) as our chromas, in 5-limit JI as above.

The diesis is the difference between an octave and three major thirds, so it has coordinates <math>(0, -3)</math>.

~~The syntonic comma is reached by stacking 4 fifths and going down a major third, so it has coordinates <math>(4, -1)</math>~~

[[File:Fokker_block_duodene.png|400px|thumb|none|Duodene as a Fokker block. Note that the fundamental domain is shifted from the origin to obtain the familiar scale.]]

@@ Line 41: / Line 41: @@
 Let's take [[5-limit]] just intation, and pick the [[25/24|just chromatic semitone]] (25/24) and the [[syntonic comma]] (81/80) as our chromas.
 The octave equivalent lattice is generated by fifths and just major thirds.
+Since <math>25/24 = 2^{-3} \cdot 3^{-1} \cdot 5^2</math>, it has coordinates <math>(-1, 2)</math> in the octave-equivalent lattice.
+The syntonic comma is reached by stacking 4 fifths and going down a major third, so it has coordinates <math>(4, -1)</math>.
 [[File:Fokker_block_zarlino.png|400px|thumb|none|Fokker block corresponding to the just diatonic scale. The gray grid is the interval lattice, and the black lines show the sublattice generated by the chromas. The fundamental domain is colored in blue.]]
@@ Line 55: / Line 57: @@
 Let's now use the [[128/125|diesis]] (128/125) and the [[syntonic comma]] (81/80) as our chromas, in 5-limit JI as above.
 The diesis is the difference between an octave and three major thirds, so it has coordinates <math>(0, -3)</math>.
-The syntonic comma is reached by stacking 4 fifths and going down a major third, so it has coordinates <math>(4, -1)</math>
 [[File:Fokker_block_duodene.png|400px|thumb|none|Duodene as a Fokker block. Note that the fundamental domain is shifted from the origin to obtain the familiar scale.]]