Technical data guide for regular temperaments: Difference between revisions
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=== Comma list === | === Comma list === | ||
{{Main|Comma basis}} | {{Main|Comma basis}} | ||
An abstract regular temperament can be thought of as a family of ''valuations'' (''tunings'' of the temperament) of the primes in its subgroup that satisfy certain equations; if we bold the numbers to make it clear that we are speaking of them as abstract variables, an example of such an equation would be {{nowrap|'''3'''<sup>4</sup> {{=}} '''2'''<sup>4</sup> × '''5'''}}. Each of these equations corresponds to setting a JI interval to be equal to the unison ([[1/1]]); the equation specified here sets [[81/80]] (with factorization | An abstract regular temperament can be thought of as a family of ''valuations'' (''tunings'' of the temperament) of the primes in its subgroup that satisfy certain equations; if we bold the numbers to make it clear that we are speaking of them as abstract variables, an example of such an equation would be {{nowrap|'''3'''<sup>4</sup> {{=}} '''2'''<sup>4</sup> × '''5'''}}. Each of these equations corresponds to setting a JI interval to be equal to the unison ([[1/1]]); the equation specified here sets [[81/80]] (with factorization {{nowrap|2<sup>−4</sup> × 3<sup>4</sup> × 5<sup>−1</sup>}}) to the unison, in other words ''tempering out'' 81/80. | ||
The ''comma basis'' is a list of such intervals–''commas''–that are tempered out by the temperament, thereby restricting the set of possible tunings. If a tuning only tempers out one comma, then the only intervals within the subgroup that are set to the unison are that comma, and its positive and negative powers (in the case above, 80/81, {{nowrap|6561/6400 {{=}} (81/80)<sup>2</sup>}}, etc.); therefore there is only one logical choice for "which comma" you claim is tempered, that being the simplest of these powers greater than the unison. Rank-2 temperaments in subgroups including 3 primes, such as the [[5-limit]], only temper out one comma, for instance. | The ''comma basis'' is a list of such intervals–''commas''–that are tempered out by the temperament, thereby restricting the set of possible tunings. If a tuning only tempers out one comma, then the only intervals within the subgroup that are set to the unison are that comma, and its positive and negative powers (in the case above, 80/81, {{nowrap|6561/6400 {{=}} (81/80)<sup>2</sup>}}, etc.); therefore there is only one logical choice for "which comma" you claim is tempered, that being the simplest of these powers greater than the unison. Rank-2 temperaments in subgroups including 3 primes, such as the [[5-limit]], only temper out one comma, for instance. | ||