Mediant hull: Difference between revisions

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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_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>

<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 (requiring more than one non-period generator) is obvious.

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 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_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>
+<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 (requiring more than one non-period generator) is obvious.