Technical data guide for regular temperaments: Difference between revisions
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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. | 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. | 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. Composites or fractions treated as primes in this context are often called "formal primes" or "basis elements." | ||
=== Comma list === | === Comma list === | ||