Technical data guide for regular temperaments: Difference between revisions
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== Structure properties == | == Structure properties == | ||
=== Subgroup | === Subgroup === | ||
{{Main|Just intonation subgroup}} | {{Main|Just intonation subgroup}} | ||
{{See also|Domain basis}} | {{See also|Domain basis}} | ||
The ''subgroup'' (or ''domain | |||
The ''subgroup'' (or ''domain'') of a regular temperament is the set of all [[interval]]s which are considered to be approximated by the temperament. For example, it is common to consider that the [[frequency ratio]] of [[3/2]] is approximated by [[12edo|12-tone equal temperament]], therefore 3/2 would be included in this set, but other intervals like [[11/8]] could be excluded. Most of the time, a subgroup exclusively contains [[just intonation]] (JI) intervals. | |||
In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called ''[[Periods and generators|generator]]s''. Continuing the previous example, if [[3/2]] is taken as a generator of the subgroup, then [[9/4]] is also included in the subgroup {{nowrap|(3/2 × 3/2 {{=}} 9/4)}}, and so on. If [[2/1]] is added to the list of subgroup generators, then intervals like [[4/3]] can be reached by combining a 3/2 down with a 2/1 up {{nowrap|(2/3 × 2/1 {{=}} 4/3)}}. | In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called ''[[Periods and generators|generator]]s''. Continuing the previous example, if [[3/2]] is taken as a generator of the subgroup, then [[9/4]] is also included in the subgroup {{nowrap|(3/2 × 3/2 {{=}} 9/4)}}, and so on. If [[2/1]] is added to the list of subgroup generators, then intervals like [[4/3]] can be reached by combining a 3/2 down with a 2/1 up {{nowrap|(2/3 × 2/1 {{=}} 4/3)}}. | ||
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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 {{nowrap|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 {{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. | ||
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{{Main| Mapping }} | {{Main| Mapping }} | ||
{{See also| Subgroup monzos and vals }} | {{See also| Subgroup monzos and vals }} | ||
A regular temperament has a structure defined by a set of ''generators'', whose number is equivalent to the ''rank'' of the temperament. Like JI itself, the set of all distinct intervals available to the regular temperament can be created by stacking these generators. Unlike JI, the determination of which intervals are generators is often highly nontrivial given the comma basis or other information. | A regular temperament has a structure defined by a set of ''generators'', whose number is equivalent to the ''rank'' of the temperament. Like JI itself, the set of all distinct intervals available to the regular temperament can be created by stacking these generators. Unlike JI, the determination of which intervals are generators is often highly nontrivial given the comma basis or other information. | ||