Mediant hull: Difference between revisions
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== Properties == | == Properties == | ||
The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale ( | The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex. Each face (of dimension ''r'' − 1) of this simplex is the Farey hull of a choice of ''r'' − 1 tunings | ||
In particular, for 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>), each hyperface corresponds to a degenerate version of the scale with ''r'' − 1 step sizes with X<sub>i</sub> = X<sub>i+1</sub>, corresponding to omitting the 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. | |||
== Notation == | == Notation == | ||