Just intonation subgroup: Difference between revisions
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i so want to remove "formal prime" here but it's actually used on the discord :sob: |
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A '''just intonation subgroup''' is a | A '''just intonation subgroup''' is a {{w|Free abelian group|group}} generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Using subgroups implies a way to organize [[just intonation]] intervals such that they form a lattice. Therefore it is closely related to [[regular temperament theory]]. | ||
Just intonation subgroups can be described by listing their [[generator]]s with full stops between them; we use said convention below. In standard mathematical notation, let ''c''<sub>1</sub>, ..., ''c''<sub>''r''</sub> be positive reals, and suppose ''v''<sub>''k''</sub> is the musical interval of log<sub>2</sub>(''c''<sub>''k''</sub>) octaves. Then | |||
<math>c_1.c_2.\cdots.c_r := \operatorname{span}_\mathbb{Z} \{v_1, ..., v_k\}.</math> | |||
There are three categories of subgroups: | There are three categories of subgroups: | ||
* Prime subgroups (e.g. 2.3.7) contain only primes | * '''Prime subgroups''' (e.g. 2.3.7) contain only primes | ||
* Composite subgroups (e.g. 2.5 | * '''Composite subgroups''' (e.g. 2.9.5) contain composite numbers and perhaps prime numbers too | ||
* | * '''Fractional subgroups''' (e.g. 2.3.7/5) contain fractional numbers and perhaps prime and/or composite numbers too | ||
For composite and | For composite and fractional subgroups, not all combinations of numbers are mathematically valid [[basis|bases]] for subgroups. For example, 2.3.9 has a redundant generator 9, and both 2.3.15 and 2.3.5/3 can be simplified to 2.3.5. | ||
A prime subgroup that | 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 [[Harmonic limit|''p''-limit JI]]. 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 '''basis element''', '''structural prime''', 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. | |||
== Normalization == | |||
A canonical naming system for just intonation subgroups is to give a [[Normal lists #Normal interval lists|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. | |||
== Index == | |||
{{See also| Wikipedia: Index of a subgroup }} | |||
Intuitively speaking, the '''index''' measures the relative size of the subgroup within another subgroup, which is usually the ''p''-limit. | |||
== 7-limit subgroups == | Subgroups in the strict sense come in two flavors: finite index and infinite index. 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 [[subgroup basis matrix]], whose columns are the [[monzo]]s of the generators. | ||
== Generalization == | |||
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). | |||
== List of selected subgroups == | |||
=== 7-limit subgroups === | |||
{{See also| 2.3.7 subgroup }} | |||
; 2.3.7: | ; 2.3.7: | ||
| Line 54: | Line 66: | ||
; 2.9/5.9/7: | ; 2.9/5.9/7: | ||
* {{EDOs|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }} | * {{EDOs|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }} | ||
* ''Terrain temperament'' subgroup, see [[ | * ''Terrain temperament'' subgroup, see [[Subgroup temperaments #Terrain]] | ||
; 3.5.7: | |||
* Does not have octaves, commonly used for non-octave [[EDT]]s | |||
== 11-limit subgroups == | === 11-limit subgroups === | ||
{{See also| 2.3.7.11 subgroup }} | |||
{{See also| Alpharabian tuning }} | |||
; 2.3.11: | ; 2.3.11: | ||
| Line 73: | Line 90: | ||
; 2.3.7.11: | ; 2.3.7.11: | ||
* {{EDOs|legend=1| 9, 17, 26, 31, 41, 46, 63, 72, 135 }} | * {{EDOs|legend=1| 9, 17, 26, 31, 41, 46, 63, 72, 135 }} | ||
* The [[ | * The [[Gamelismic clan#Radon|Radon temperament]] subgroup, generated by the Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1] | ||
* See: [[Gallery of 2.3.7.11 Subgroup Scales]] | * See: [[Gallery of 2.3.7.11 Subgroup Scales]] | ||
| Line 81: | Line 98: | ||
; 2.5/3.7/3.11/3: | ; 2.5/3.7/3.11/3: | ||
* {{EDOs|legend=1| 33, 41, 49, 57, 106, 204, 253 }} | * {{EDOs|legend=1| 33, 41, 49, 57, 106, 204, 253 }} | ||
* The [[ | * The [[Subgroup temperaments#Indium|Indium temperament]] subgroup. | ||
== 13-limit subgroups == | === 13-limit subgroups === | ||
; 2.3.13: | ; 2.3.13: | ||
| Line 91: | Line 108: | ||
; 2.3.5.13: | ; 2.3.5.13: | ||
* {{EDOs|legend=1| 15, 19, 34, 53, 87, 130, 140, 246, 270 }} | * {{EDOs|legend=1| 15, 19, 34, 53, 87, 130, 140, 246, 270 }} | ||
* The [[ | * The [[Kleismic family#Cata|Cata]], [[The Archipelago#Trinidad|Trinidad]] and [[The Archipelago#Parizekmic|Parizekmic]] temperaments subgroup. | ||
; 2.3.7.13: | ; 2.3.7.13: | ||
| Line 101: | Line 118: | ||
; 2.5.7.13: | ; 2.5.7.13: | ||
* {{EDOs|legend=1| 7, 10, 17, 27, 37, 84, 121, 400 }} | * {{EDOs|legend=1| 7, 10, 17, 27, 37, 84, 121, 400 }} | ||
* The [[ | * The [[No-threes subgroup temperaments#Huntington|Huntington temperament]] subgroup. | ||
; 2.5.7.11.13: | ; 2.5.7.11.13: | ||
* {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }} | * {{EDOs|legend=1| 6, 7, 13, 19, 25, 31, 37 }} | ||
* The [[ | * The [[Hemimean clan#Roulette|Roulette temperament]] subgroup | ||
; 2.3.13/5: | ; 2.3.13/5: | ||
| Line 113: | Line 130: | ||
; 2.3.11/5.13/5: | ; 2.3.11/5.13/5: | ||
* {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }} | * {{EDOs|legend=1| 5, 9, 14, 19, 24, 29 }} | ||
* The [[ | * The [[Subgroup temperaments#Bridgetown|Bridgetown temperament]] subgroup. | ||
; 2.3.11/7.13/7: | ; 2.3.11/7.13/7: | ||
* {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }} | * {{EDOs|legend=1| 5, 7, 12, 17, 29, 46, 75, 196, 271 }} | ||
* The [[ | * The [[Subgroup temperaments#Pepperoni|Pepperoni temperament]] subgroup. | ||
; 2.7/5.11/5.13/5: | ; 2.7/5.11/5.13/5: | ||
* {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }} | * {{EDOs|legend=1| 5, 8, 21, 29, 37, 66, 169, 235 }} | ||
* The [[ | * The [[Subgroup temperaments#Tridec|Tridec temperament]] subgroup. | ||
=== Higher-limit subgroups === | |||
* [[2.11.13.17.19 subgroup]] | |||
* [[2.17/13.19/13 subgroup]] | |||
== See also == | |||
* [[Subgroup basis matrices]] – a formal discussion on matrix representations of subgroup bases | |||
== Notes == | |||
[[Category:Subgroup| ]] <!-- main article --> | |||
[[Category:Just intonation]] | [[Category:Just intonation]] | ||