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 | 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 | 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 | <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 X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ 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'' − 1)-simplex. 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 == | ||
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]]. | 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]] | [[Category:Terms]] | ||
Latest revision as of 11:30, 9 February 2023
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 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 || 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, ..., Tr, given by generators m1\n1edE, ..., mr\nredE for S, the mediant hull or Farey hull of T1, ..., Tr 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)\mathrm{ed}E} : (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
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 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.