Technical data guide for regular temperaments: Difference between revisions
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However, if a tuning tempers out multiple independent commas, the situation gets more complicated, for the set of tempered intervals 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 [[126/125]] = 2 × 3<sup>2</sup> × 5<sup>-3</sup> × 7; [[225/224]] = 2<sup>-5</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 7<sup>-1</sup>, and 81/80, but 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. | However, if a tuning tempers out multiple independent commas, the situation gets more complicated, for the set of tempered intervals 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 [[126/125]] = 2 × 3<sup>2</sup> × 5<sup>-3</sup> × 7; [[225/224]] = 2<sup>-5</sup> × 3<sup>2</sup> × 5<sup>2</sup> × 7<sup>-1</sup>, and 81/80, but 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}}. | 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). | ||
=== Mapping and sval mapping === | === Mapping and sval mapping === | ||