Technical data guide for regular temperaments: Difference between revisions
added note about svals |
mNo edit summary |
||
| Line 12: | Line 12: | ||
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)}}. | ||
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 as a whole 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 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. | ||