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 [[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'' and a set of equal-step tunings ''T1, ..., Tr'', given by generators ''m1\n1, ..., mr\nr'' 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>

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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 [[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'' 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>, ..., 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
 <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>