Mediant hull: Difference between revisions

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Todo; rank 3 example

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'' − 1) 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, for the set of all possible tunings of a scale structure (with step sizes X1 ≥ X2 ≥ ... ≥ Xr), each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with Xi = Xi+1 (resp. Xr = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X1:... :Xi:Xi+1:...:Xr = 1:...:1:0:...:0 (resp. X1:...:Xr = 1:...:1).

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 Todo; rank 3 example
-The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' &minus; 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' &minus; 1) of this simplex is the Farey hull of a choice of ''r'' &minus; 1 of the starting tunings.
+The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' &minus; 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' &minus; 2) of this simplex is the Farey hull of a choice of ''r'' &minus; 1 of the starting tunings.
 In particular, for the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> &ge; X<sub>2</sub> &ge; ... &ge; X<sub>r</sub>), each hyperface corresponds to the degenerate version of the scale (with ''r'' &minus; 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).