Technical data guide for regular temperaments: Difference between revisions
| Line 20: | Line 20: | ||
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 '''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 2<sup>-4</sup> × 3<sup>4</sup> × 5<sup>-1</sup>) to the unison, in other words ''tempering out'' 81/80. | 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 '''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 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, 6561/6400, 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, 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 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. | ||