Technical data guide for regular temperaments: Difference between revisions
m Update todo |
|||
| Line 24: | Line 24: | ||
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. | ||
However, if a tuning tempers out multiple independent commas, the situation gets more complicated, for the set of tempered | However, if a tuning tempers out multiple independent commas, the situation gets more complicated, for the set of intervals that are tempered out in fact forms a lattice ''generated'' by more than one generator (in other words, a nontrivial subgroup of JI), and the choice of which specific intervals to consider generators (which in this context are ''basis commas'') is not always obvious. For instance, septimal meantone tempers out the intervals {{nowrap|[[126/125]] {{=}} 2 × 3<sup>2</sup> × 5<sup>−3</sup> × 7}}; {{nowrap|[[225/224]] {{=}} 2<sup>−5</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 7<sup>−1</sup>}}, and 81/80, but {{nowrap|81/80 {{=}} (126/125) × (225/224)}}, and therefore these three commas are ''not all independent''–but all of them are useful, in that all three define prominent ''temperament families'' (collections of regular temperaments that share a tempered comma in common): 81/80 defines [[meantone]], 126/125 defines [[starling]], and 225/224 defines [[marvel]]. Various methods exist for choosing which commas are selected to be basis commas, which are associated with the technique of [[matrix echelon forms]]; in the case of septimal meantone, the basis commas are chosen to be 81/80 and 126/125 at the price of obscuring the fact that it also tempers out 225/224. | ||
As a last note, factorizations are generally abbreviated in the form of a (subgroup) [[monzo]], which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be {{monzo|-5 2 2 -1}} (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup). | As a last note, factorizations are generally abbreviated in the form of a (subgroup) [[monzo]], which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be {{monzo|-5 2 2 -1}} (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup). | ||