Just intonation subgroup: Difference between revisions
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A '''just intonation subgroup''' is a {{w|Free abelian group|group}} generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Using subgroups implies a way to organize [[just intonation]] intervals such that they form a lattice. Therefore it is closely related to [[regular temperament theory]]. | A '''just intonation subgroup''' is a {{w|Free abelian group|group}} generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Using subgroups implies a way to organize [[just intonation]] intervals such that they form a lattice. Therefore it is closely related to [[regular temperament theory]]. | ||
Just intonation subgroups can be described by listing their [[generator]]s with | Just intonation subgroups can be described by listing their [[generator]]s with full stops between them; we use said convention below. In standard mathematical notation, let ''c''<sub>1</sub>, ..., ''c''<sub>''r''</sub> be positive reals, and suppose ''v''<sub>''k''</sub> is the musical interval of log<sub>2</sub>(''c''<sub>''k''</sub>) octaves. Then | ||
<math>c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, ..., v_k\}.</math> | <math>c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, ..., v_k\}.</math> | ||