Mediant hull: Difference between revisions
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The '''mediant hull''' or '''Farey hull''' of a set of [[edo]] 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 number of mediants of the starting edo tunings, or as a limit of a sequence of finite mediants. 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 [[edo]] 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 number of mediants of the starting edo tunings, or as a limit of a sequence of finite mediants. 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. | ||
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 ''Farey hull'' of ''T<sub>1</sub>, ..., T<sub>r</sub>'' is | 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 | ||
<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_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> | ||