Just intonation subgroup: Difference between revisions
I no longer endorse the term "formal prime" Tags: Mobile edit Mobile web edit |
Cmloegcmluin (talk | contribs) prefer "basis element" to "formal prime", per recommendation of original coiner of "formal prime" |
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A prime subgroup that does not omit any primes < ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called 5-limit JI, 7-limit JI, etc. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories. | A prime subgroup that does not omit any primes < ''p'' (e.g. 2.3.5, 2.3.5.7, 2.3.5.7.11, etc. but not 2.3.7 or 3.5.7) is simply called 5-limit JI, 7-limit JI, etc. It is customary of just intonation subgroups to refer only to prime subgroups that do omit such primes, as well as the other two categories. | ||
The following terminology has been proposed for streamlining pedagogy: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[…].''d'', call any member of this set a '''formal prime'''. The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. | The following terminology has been proposed for streamlining pedagogy: Given a subgroup written as generated by a fixed (non-redundant) set: ''a''.''b''.''c''.[…].''d'', call any member of this set a '''[[basis]] element''', or '''formal prime'''.<ref>The meaning of "formal" this term is using is "of external form or structure, rather than nature or content", which is to say that a formal prime is not necessarily ''actually'' a prime, but we treat them as if they were. The original coiner of this term, inthar, has recommended its disuse, in favor of the mathematically accurate and generic "basis element", or possibly something else which indicates the co-uniqueness of the elements.</ref> For example, if the group is written 2.5/3.7/3, the basis elements are 2, 5/3 and 7/3. | ||
Subgroups in the strict sense come in two flavors: finite [[Wikipedia: Index of a subgroup|index]] and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire ''p''-limit group. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full [[3-limit]] (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full [[7-limit]] group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the matrix whose rows are the [[monzo]]s of the generators. | Subgroups in the strict sense come in two flavors: finite [[Wikipedia: Index of a subgroup|index]] and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire ''p''-limit group. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full [[3-limit]] (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full [[7-limit]] group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the matrix whose rows are the [[monzo]]s of the generators. | ||
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A canonical naming system for just intonation subgroups is to give a [[Normal lists #Normal interval list|normal interval list]] for the generators of the group, which will also show the [[Wikipedia: Rank of an abelian group|rank]] of the group by the number of generators in the list (the [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is sometimes desirable, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization). Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. Just intonation subgroups can be described by listing their generators with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below. | A canonical naming system for just intonation subgroups is to give a [[Normal lists #Normal interval list|normal interval list]] for the generators of the group, which will also show the [[Wikipedia: Rank of an abelian group|rank]] of the group by the number of generators in the list (the [[Hermite normal form]] should be used here, not the [[canonical form]], because in the case of subgroups, [[enfactoring]] is sometimes desirable, such as in the subgroup 2.9.7 which should not be reduced to 2.3.7 by subgroup canonicalization). Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. Just intonation subgroups can be described by listing their generators with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below. | ||
Non-JI intervals can also be used as | Non-JI intervals can also be used as basis elements, when the subgroup in question contains non-JI intervals. For example, 2.sqrt(3/2) (sometimes written 2.2ed3/2) is the group generated by 2/1 and 350.978 cents, the square root of 3/2 (a neutral third which is exactly one half of 3/2). This is closely related to the [[3L 4s]] mos tuning with neutral third generator sqrt(3/2). | ||
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