Technical data guide for regular temperaments: Difference between revisions
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The entirety of JI can be generated by the infinite set of [[prime number]]s {{nowrap|(2, 3, 5, 7, …)}}. In practice, most subgroups are generated by a few primes only (hence the term ''subgroup'', where JI is the larger ''group''). A common kind of subgroups are [[prime limit]]s, which are generated by all prime harmonics up to a certain limit. For example, the [[5-limit]] is generated by all primes up to 5 (i.e. 2, 3 and 5). | The entirety of JI can be generated by the infinite set of [[prime number]]s {{nowrap|(2, 3, 5, 7, …)}}. In practice, most subgroups are generated by a few primes only (hence the term ''subgroup'', where JI is the larger ''group''). A common kind of subgroups are [[prime limit]]s, which are generated by all prime harmonics up to a certain limit. For example, the [[5-limit]] is generated by all primes up to 5 (i.e. 2, 3 and 5). | ||
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 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 2.7.9.11.15 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." | 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." | ||