Mediant hull: Difference between revisions
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The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 2) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings. | The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 2) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings. | ||
In particular, the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), is the Farey hull of the degenerate tunings X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 and X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1. | In particular, the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), is the Farey hull of the degenerate tunings X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 and X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1. Each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1). | ||
== Notation == | == Notation == | ||