Technical data guide for regular temperaments: Difference between revisions
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Therefore, it is useful to specify a way to "translate" between JI intervals and stacks of temperament generators—for example, to know how a given JI interval is retuned under the temperament (and a specific tuning of the generators). The ''mapping'' provides this way, by specifying how each of the ''prime harmonics'' (subgroup generators) are equated to a stack of generators of the temperament. | Therefore, it is useful to specify a way to "translate" between JI intervals and stacks of temperament generators—for example, to know how a given JI interval is retuned under the temperament (and a specific tuning of the generators). The ''mapping'' provides this way, by specifying how each of the ''prime harmonics'' (subgroup generators) are equated to a stack of generators of the temperament. | ||
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"), and intervals of 11 will not appear until quite a long way down the [[circle of fifths]], and only in rather large scales built out of tempered intervals. | 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 [[circle of fifths]], and only in rather large scales built out of tempered intervals. | ||
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 "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''. | ||