Mediant hull: Difference between revisions

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The '''mediant hull''' or '''Farey hull''' of a set of [[equal-step tuning]]s for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by ~~traversing~~ a finite ~~path~~ (resulting in another equal-step tuning) or infinite ~~path~~ (resulting in a tuning with irrational step ratio) on the ~~corresponding [[scale tree]]~~. We write the mediant hull of tunings by putting double bars <code>||</code> between the tunings. For example, <code>5L 2s(7||12)</code>, the mediant hull of 7edo [[diatonic]] and 12edo diatonic, is [[TAMNAMS|soft-of-basic]] diatonic.

The '''mediant hull''' or '''Farey hull''' of a set of [[equal-step tuning]]s for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by taking 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 <code>||</code> between the tunings. For example, <code>5L 2s(7||12)</code>, the mediant hull of 7edo [[5L 2s|diatonic]] and 12edo diatonic, is [[TAMNAMS|soft-of-basic]] diatonic. (Note: Double bars are used for formatting on Discord, so you should use <code>``</code> or escape the double bars.)

Formally, given a 2-step scale structure ''S'' and a set of ~~edo~~ tunings ''T1, ..., Tr'', given by generators ''m1/n1, ..., mr/nr'' for S, the ''mediant hull'' or ''Farey hull'' of ''T1, ..., Tr'' is

Formally, given a 2-step scale structure ''S'' with equave ''E'' and a set of equal-step tunings ''T1, ..., Tr'', given by generators ''m1\n1edE, ..., mr\nredE'' for S, the ''mediant hull'' or ''Farey hull'' of ''T1, ..., Tr'' is

<math>T_1 \mid\mid T_2 \mid\mid \cdots \mid\mid ~~T_n~~ = \operatorname{cl}\bigg\{\dfrac{a_1 m_1 + \cdots + a_r m_r}{a_1 n_1 + \cdots + a_r n_r} : (a_1, ..., a_r) \in \mathbb{Z}^r_{\geq 0} \setminus (0, 0, ..., 0) \bigg\}</math>

<math>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)\mathrm{ed}E} : (a_1, ..., a_r) \in \mathbb{Z}^r_{\geq 0} \setminus (0, 0, ..., 0) \bigg\}</math>

where <math>\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 is obvious.

where <math>\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 ==

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'' − 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 X1 ≥ X2 ≥ ... ≥ Xr), the Farey hull of the degenerate tunings X1:... :Xi:Xi+1:...:Xr = 1:...:1:0:...:0 and X1:...:Xr = 1:...:1, is an (''r'' − 1)-simplex. 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).

== Notation ==

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

Properly, one should write out the generators or step ratios of all the input edo tunings, for example <code>5L 2s (7\12||11\19)</code> or <code>5L 2s ((2:1)||(3:2))</code>. Less carefully, we can also use just the edo numbers: <code>5L 2s (12||19)</code> (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, <code>diasem(7||26||31)</code> 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]].

== See also ==

* [[Mediant]]

[[Category:Terms]]

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-The '''mediant hull''' or '''Farey hull''' of a set of [[equal-step tuning]]s for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by traversing a finite path (resulting in another equal-step tuning) or infinite path (resulting in a tuning with irrational step ratio) on the corresponding [[scale tree]]. We write the mediant hull of tunings by putting double bars <code>||</code> between the tunings. For example, <code>5L 2s(7||12)</code>, the mediant hull of 7edo [[diatonic]] and 12edo diatonic, is [[TAMNAMS|soft-of-basic]] diatonic.
+The '''mediant hull''' or '''Farey hull''' of a set of [[equal-step tuning]]s for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by taking 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 <code>||</code> between the tunings. For example, <code>5L 2s(7||12)</code>, the mediant hull of 7edo [[5L 2s|diatonic]] and 12edo diatonic, is [[TAMNAMS|soft-of-basic]] diatonic. (Note: Double bars are used for formatting on Discord, so you should use <code>``</code> or escape the double bars.)
-Formally, given a 2-step scale structure ''S'' and a set of edo tunings ''T<sub>1</sub>, ..., T<sub>r</sub>'', given by generators ''m<sub>1</sub>/n<sub>1</sub>, ..., m<sub>r</sub>/n<sub>r</sub>'' for S, the ''mediant hull'' or ''Farey hull'' of ''T<sub>1</sub>, ..., T<sub>r</sub>'' is
+Formally, given a 2-step scale structure ''S'' with equave ''E'' and a set of equal-step tunings ''T<sub>1</sub>, ..., T<sub>r</sub>'', given by generators ''m<sub>1</sub>\n<sub>1</sub>edE, ..., m<sub>r</sub>\n<sub>r</sub>edE'' for S, the ''mediant hull'' or ''Farey hull'' of ''T<sub>1</sub>, ..., T<sub>r</sub>'' is
-<math>T_1 \mid\mid T_2 \mid\mid \cdots \mid\mid T_n = \operatorname{cl}\bigg\{\dfrac{a_1 m_1 + \cdots + a_r m_r}{a_1 n_1 + \cdots + a_r n_r} : (a_1, ..., a_r) \in \mathbb{Z}^r_{\geq 0} \setminus (0, 0, ..., 0) \bigg\}</math>
+<math>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)\mathrm{ed}E} : (a_1, ..., a_r) \in \mathbb{Z}^r_{\geq 0} \setminus (0, 0, ..., 0) \bigg\}</math>
-where <math>\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 is obvious.
+where <math>\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 ==
+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; 2) of this simplex is the Farey hull of a choice of ''r'' &minus; 1 of the starting tunings.
+In particular, 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>), 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, is an (''r'' &minus; 1)-simplex. 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).
 == Notation ==
-Properly, one should write out the generators or step ratios of all the input edo tunings, for example <code>5L 2s (7\12||11\19)</code> or <code>5L 2s ((2:1)||(3:2))</code> Less carefully, we can also use just the edo numbers: <code>5L 2s (12||19)</code> (but there is potential for confusion when an edo has multiple tunings of a given scale structure).
+Properly, one should write out the generators or step ratios of all the input edo tunings, for example <code>5L 2s (7\12||11\19)</code> or <code>5L 2s ((2:1)||(3:2))</code>. Less carefully, we can also use just the edo numbers: <code>5L 2s (12||19)</code> (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, <code>diasem(7||26||31)</code> 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]].
+== See also ==
+* [[Mediant]]
 [[Category:Terms]]