Technical data guide for regular temperaments: Difference between revisions

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The ''subgroup'' (or ''domain basis'') 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 [[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]] intervals.

The ''subgroup'' (or ''domain basis'') 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 [[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), aka ''rational'', intervals.

In a subgroup, all intervals must be reachable by stacking (up and down) copies of a few "generating intervals", or ''[[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. 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 (i.e. 2/3 × 2/1 = 4/3).

A subgroup is generally expressed as a list of its generators separated by dots: e.g. 2.3.5 is the subgroup of all intervals consisting of combinations of [[2/1]], [[3/1]] and [[5/1]]. The 2.3.5 subgroup is equivalent to the [[5-limit]], ~~because it contains all~~ [[prime ~~harmonic~~]]s up to 5, ~~but~~ temperament ~~data~~ tables ~~typically~~ prefer the ~~first notation~~.

The generators of the entirety of JI are the infinite set of prime numbers: 2, 3, 5, 7, etc.; therefore the most common type of subgroup of JI uses a subset of primes (or, if 2 is in the subset, equivalently octave-reduced prime harmonics such as 3/2, 5/4, 7/4, etc.) as its generators. A subgroup is generally expressed as a list of its generators separated by dots: e.g. 2.3.5 is the subgroup of all intervals consisting of combinations of [[2/1]], [[3/1]] and [[5/1]]. The 2.3.5 subgroup is equivalent to the [[5-limit]], the subgroup defined by the [[prime harmonics]] up to 5, though for maximum clarity the temperament tables currently prefer spelling out the primes explicitly.

However, it may be reasonable in some cases to include composite numbers in a subgroup: the subgroup 2.7.9.11.15 includes ''some'' intervals that contain 3 and 5 in their factorization (such as 9/7, 15/8, or 5/3 - the last being interpreted as 15/9), but not others (it would not contain an interval like 3/2 or 5/4, since these can't be reached from multiplying and dividing 9 and 15 with primes); or even fractions, like the subgroup 2.3.11.13/5.17 (note that this is interpreted as 2.3.11.(13/5).17), which includes intervals of 13 and intervals of 5, but only when a power of 13 is matched by an equal power of 5 on the other side of the fraction.

=== Comma list ===

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 {{Main|Just intonation subgroup}}
 {{See also|Domain basis}}
-The ''subgroup'' (or ''domain basis'') 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 [[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]] intervals.
+The ''subgroup'' (or ''domain basis'') 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 [[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), aka ''rational'', intervals.
 In a subgroup, all intervals must be reachable by stacking (up and down) copies of a few "generating intervals", or ''[[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. 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 (i.e. 2/3 × 2/1 = 4/3).
-A subgroup is generally expressed as a list of its generators separated by dots: e.g. 2.3.5 is the subgroup of all intervals consisting of combinations of [[2/1]], [[3/1]] and [[5/1]]. The 2.3.5 subgroup is equivalent to the [[5-limit]], because it contains all [[prime harmonic]]s up to 5, but temperament data tables typically prefer the first notation.
+The generators of the entirety of JI are the infinite set of prime numbers: 2, 3, 5, 7, etc.; therefore the most common type of subgroup of JI uses a subset of primes (or, if 2 is in the subset, equivalently octave-reduced prime harmonics such as 3/2, 5/4, 7/4, etc.) as its generators. A subgroup is generally expressed as a list of its generators separated by dots: e.g. 2.3.5 is the subgroup of all intervals consisting of combinations of [[2/1]], [[3/1]] and [[5/1]]. The 2.3.5 subgroup is equivalent to the [[5-limit]], the subgroup defined by the [[prime harmonics]] up to 5, though for maximum clarity the temperament tables currently prefer spelling out the primes explicitly.
+However, it may be reasonable in some cases to include composite numbers in a subgroup: the subgroup 2.7.9.11.15 includes ''some'' intervals that contain 3 and 5 in their factorization (such as 9/7, 15/8, or 5/3 - the last being interpreted as 15/9), but not others (it would not contain an interval like 3/2 or 5/4, since these can't be reached from multiplying and dividing 9 and 15 with primes); or even fractions, like the subgroup 2.3.11.13/5.17 (note that this is interpreted as 2.3.11.(13/5).17), which includes intervals of 13 and intervals of 5, but only when a power of 13 is matched by an equal power of 5 on the other side of the fraction.
 === Comma list ===