Technical data guide for regular temperaments: Difference between revisions

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As a last note, factorizations are generally abbreviated in the form of a (subgroup) [[monzo]], which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be {{monzo|-5 2 2 -1}} (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup).

=== Mapping and ~~sval~~ mapping ===

=== Mapping and subgroup-val mapping ===

A regular temperament has a structure defined by a set of ''generators'', whose number is equivalent to the ''rank'' of the temperament. Like JI itself, the set of all distinct intervals available to the regular temperament can be created by stacking these generators. Unlike JI, the determination of which intervals are generators is often highly nontrivial given the comma basis or other information.

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For an example, let us look at meanpop, an [[11-limit]] extension of meantone. Its mapping is given by {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}, with the "mapping generators" being ~2 and ~3 (with the tilde used to indicate ''tunings'' of these intervals under the temperament), where each ''vector'' within the mapping indicates the number of each generator in the stack used to reach a prime harmonic. This particular mapping tells us that 2/1 and 3/1 are reached by one of the generators ~2 and ~3 (trivially) each; that 5/1 is reached by 4 times ~3 upward and 4 times ~2 downward; that 7/1 is reached by 10 times ~3 upward and 13 times ~2 downward; and that 11/1 is reached by 24 times ~2 upward and 13 times ~3 downward. Therefore, the 11th harmonic in this temperament is quite complex (even if we regard ~2, the octave, as "free"), especially because it is reached the ''opposite'' way that 3, 5, and 7 are and so ratios of 11 with these other primes are even more complex. Thus intervals of 11 will not appear until quite a long way down the [[chain of fifths]], and only in rather large scales built out of tempered intervals.

In subgroups other than full prime-limits, mappings are sometimes called ~~"sval~~ mappings"; the only distinction here is that the columns of the mapping do not indicate all consecutive primes but only the basis elements of the subgroup. These are distinct from "gencom mappings" with zero entries for primes not included in the subgroup.

In subgroups other than full prime-limits, mappings are sometimes called ''subgroup-val mappings''; the only distinction here is that the columns of the mapping do not indicate all consecutive primes but only the basis elements of the subgroup. These are distinct from ''gencom mappings'' with zero entries for primes not included in the subgroup.

One last note is that mappings may use a slightly different (if equivalent) set of generators from elsewhere in the temperament data: for meanpop, for instance, the ~~"canonical"~~ generator, for which optimal tunings are specified, is in fact ~3/2, rather than ~3. In these cases, the mapping should (but does not always) specify the set of generators used for the ''mapping''.

One last note is that mappings may use a slightly different (if equivalent) set of generators from elsewhere in the temperament data: for meanpop, for instance, the conventional generator, for which optimal tunings are specified, is in fact ~3/2, rather than ~3. In these cases, the mapping should (but does not always) specify the set of generators used for the ''mapping''.

=== Extensions and restrictions ===

@@ Line 28: / Line 28: @@
 As a last note, factorizations are generally abbreviated in the form of a (subgroup) [[monzo]], which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be {{monzo|-5 2 2 -1}} (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup).
-=== Mapping and sval mapping ===
+=== Mapping and subgroup-val mapping ===
-{{Main|Mapping}}
+{{Main| Mapping }}
-{{See also|Smonzos and svals}}
+{{See also| Subgroup monzos and vals }}
 A regular temperament has a structure defined by a set of ''generators'', whose number is equivalent to the ''rank'' of the temperament. Like JI itself, the set of all distinct intervals available to the regular temperament can be created by stacking these generators. Unlike JI, the determination of which intervals are generators is often highly nontrivial given the comma basis or other information.
@@ Line 37: / Line 37: @@
 For an example, let us look at meanpop, an [[11-limit]] extension of meantone. Its mapping is given by {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}, with the "mapping generators" being ~2 and ~3 (with the tilde used to indicate ''tunings'' of these intervals under the temperament), where each ''vector'' within the mapping indicates the number of each generator in the stack used to reach a prime harmonic. This particular mapping tells us that 2/1 and 3/1 are reached by one of the generators ~2 and ~3 (trivially) each; that 5/1 is reached by 4 times ~3 upward and 4 times ~2 downward; that 7/1 is reached by 10 times ~3 upward and 13 times ~2 downward; and that 11/1 is reached by 24 times ~2 upward and 13 times ~3 downward. Therefore, the 11th harmonic in this temperament is quite complex (even if we regard ~2, the octave, as "free"), especially because it is reached the ''opposite'' way that 3, 5, and 7 are and so ratios of 11 with these other primes are even more complex. Thus intervals of 11 will not appear until quite a long way down the [[chain of fifths]], and only in rather large scales built out of tempered intervals.
-In subgroups other than full prime-limits, mappings are sometimes called "sval mappings"; the only distinction here is that the columns of the mapping do not indicate all consecutive primes but only the basis elements of the subgroup. These are distinct from "gencom mappings" with zero entries for primes not included in the subgroup.
+In subgroups other than full prime-limits, mappings are sometimes called ''subgroup-val mappings''; the only distinction here is that the columns of the mapping do not indicate all consecutive primes but only the basis elements of the subgroup. These are distinct from ''gencom mappings'' with zero entries for primes not included in the subgroup.
-One last note is that mappings may use a slightly different (if equivalent) set of generators from elsewhere in the temperament data: for meanpop, for instance, the "canonical" generator, for which optimal tunings are specified, is in fact ~3/2, rather than ~3. In these cases, the mapping should (but does not always) specify the set of generators used for the ''mapping''.
+One last note is that mappings may use a slightly different (if equivalent) set of generators from elsewhere in the temperament data: for meanpop, for instance, the conventional generator, for which optimal tunings are specified, is in fact ~3/2, rather than ~3. In these cases, the mapping should (but does not always) specify the set of generators used for the ''mapping''.
 === Extensions and restrictions ===