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 dots between them. In standard mathematical notation, let ''p''<sub>1</sub>, ..., ''p''<sub>''r''</sub> be distinct primes, and let ''a''<sub>''k''</sub> | Just intonation subgroups can be described by listing their [[generator]]s with dots between them. In standard mathematical notation, let ''p''<sub>1</sub>, ..., ''p''<sub>''r''</sub> be distinct primes, and let ''a''<sub>''k''</sub> be the musical interval of log<sub>2</sub>(''p''<sub>''k''</sub>) octaves. Then | ||
<math>p_1.p_2.\cdots.p_r := \operatorname{span}_\mathbb{Z} \{a_1, ..., a_k\}.</math> | <math>p_1.p_2.\cdots.p_r := \operatorname{span}_\mathbb{Z} \{a_1, ..., a_k\}.</math> | ||