Mediant hull

The mediant hull or Farey hull of a set of equal-step tunings for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by traversing a finite (resulting in another equal-step tuning) or infinite (resulting in a tuning with irrational step ratio) Farey sum of the starting values. We write the mediant hull of tunings by putting double bars || between the tunings. For example, 5L 2s(7||12), the mediant hull of 7edo diatonic and 12edo diatonic, is soft-of-basic diatonic. (Note: Double bars are used for formatting on Discord, so you should use `` or escape the double bars.)

Formally, given a 2-step scale structure S with equave E and a set of equal-step tunings T₁, ..., T_r, given by generators m₁\n₁edE, ..., m_r\n_redE for S, the mediant hull or Farey hull of T₁, ..., T_r is [math]\displaystyle{ T_1 \mid\mid T_2 \mid\mid \cdots \mid\mid T_r = \operatorname{cl}\bigg\{\dfrac{a_1 m_1 + \cdots + a_r m_r}{(a_1 n_1 + \cdots + a_r n_r)edE} : (a_1, ..., a_r) \in \mathbb{Z}^r_{\geq 0} \setminus (0, 0, ..., 0) \bigg\} }[/math]

where [math]\displaystyle{ \operatorname{cl} }[/math] denotes the topological closure (specifying the generator is enough, given the period, since the period doesn't change upon taking the mediant). The generalization to higher-rank scale structures (requiring more than one non-period generator) is obvious.

Properties

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.

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

Notation

Properly, one should write out the generators or step ratios of all the input edo tunings, for example 5L 2s (7\12||11\19) or 5L 2s ((2:1)||(3:2)). Less carefully, we can also use just the edo numbers: 5L 2s (12||19) (but there is potential for confusion when an edo has multiple tunings of a given scale structure).

This works similarly for rank-3 and higher scales: for example, diasem(7||26||31) specifies the mediant hull of the 7edo (L:m:s = 1:1:0), 26edo (L:m:s = 4:2:1) and 31edo (L:m:s = 5:2:1) tunings for diasem.