Technical data guide for regular temperaments: Difference between revisions
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A subgroup is generally expressed as a list of its generators separated by dots. For example, "2.3.5" denotes the aforementioned 5-limit. Primes are not required to be consecutive; [[2.3.7 subgroup|2.3.7]] is an equally valid subgroup. A shorthand exists where full ''p''-limits within an extended subgroup are denoted by L''p'', e.g. the 2.3.5.7.11.17.29.31 subgroup can be written as "L11.17.29.31"; however this notation is not common and therefore remains discouraged for clarity. | A subgroup is generally expressed as a list of its generators separated by dots. For example, "2.3.5" denotes the aforementioned 5-limit. Primes are not required to be consecutive; [[2.3.7 subgroup|2.3.7]] is an equally valid subgroup. A shorthand exists where full ''p''-limits within an extended subgroup are denoted by L''p'', e.g. the 2.3.5.7.11.17.29.31 subgroup can be written as "L11.17.29.31"; however this notation is not common and therefore remains discouraged for clarity. | ||
However, it may be reasonable in some cases to include composite numbers in a subgroup: the subgroup 2.9.15.7.11 (note that these are, by convention, sorted in order of prime limit rather than numerical order) 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. Fractions may be included as well, 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 2.9.15.7.11 (note that these are, by convention, sorted in order of prime limit rather than numerical order) 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. Fractions may be included as well, 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; or the subgroup 2.5/3.7/3.11/3, which additionally includes intervals like 7/5 and 11/5 but not intervals like 7/4 or 11/8. Composites or fractions treated as primes in this context are often called "formal primes" or "basis elements." | ||
=== Comma list === | === Comma list === | ||