Gencom: Difference between revisions

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The reason for putting the generators together with the commas is that notating the gencom as a list of [[monzo]]s allows it to be treated as a [[subgroup basis matrix]]: the group of intervals generated by the gencom is the same no matter how we place the semicolon, and so is the matrix. When this group is a [[harmonic limit|full ''p''-limit group]], as in the example above, the matrix is a {{w|unimodular matrix}}. Inverting and transposing it gives a matrix whose rows are [[val]]s; if ''r'' is the [[rank]] of the temperament, then the first ''r'' rows are the [[mapping|mapping matrix]] corresponding to the generator transversal. More interesting is the case where the gencom generates a [[Just intonation subgroups|JI subgroup]] of some ''p''-limit. In all cases the transpose of the [[pseudoinverse]] of the matrix of monzos gives a matrix of vals whose first ''r'' rows we call the '''gencom mapping''', and which in its entirety we call the '''extended gencom mapping'''. The extended gencom mapping is only a unimodular matrix, and the inversion ordinary matrix inversion, in the case of the full ''p''-limit. However in all cases the transpose pseudoinverse of the gencom matrix is the extended gencom mapping, and the transpose pseudoinverse of the extended mapping is the gencom matrix.

The rows of the extended gencom mapping are in general fractional vals, meaning the coefficients are allowed to be rational numbers. When applied to elements of the subgroup generated by the gencom, these fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom~~; the gencom mapping applied to a subgroup basis matrix is a matrix of generator steps in terms of [[smonzos and svals|subgroup vals (svals)]], which is exactly the sval mapping~~. However, the converse is not the case: if the gencom mapping returns integer values, it does not mean the interval must belong to the gencom subgroup.

The rows of the extended gencom mapping are in general fractional vals, meaning the coefficients are allowed to be rational numbers. When applied to elements of the subgroup generated by the gencom, these fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom. However, the converse is not the case: if the gencom mapping returns integer values, it does not mean the interval must belong to the gencom subgroup.

In other words, the gencom mapping takes ordinary monzos just as the [[subgroup monzos and vals|subgroup-val]] mapping takes subgroup monzos:

$$ V \cdot \vec m = V_G \cdot \vec m_G $$

where ''V'' and ''V''''G'' are gencom and subgroup-val mappings of the same temperament in subgroup ''G'', and '''m''' and '''m'''''G'' are ordinary and subgroup monzos of the same interval in ''G'', respectively.

It follows that the gencom mapping applied to the subgroup basis matrix of the temperament is a matrix of generator steps in terms of subgroup vals, which is exactly the subgroup-val mapping. If ''S'' is the subgroup basis matrix, then

$$ VS = V_G $$

The extended gencom mapping can also be used to determine if an interval ''q'' is in the group of the temperament. Suppose [''c''1 ''c''2 … ''c''n] is a gencom and [''v''1 ''v''2 … ''v''''n''] is the corresponding extended mapping. Then each of ''v''1 (''q''), ''v''2 (''q'') … ''v''''n'' (''q'') must be an integer, and moreover we must have ''q'' = ''c''1^''v''1 (''q'') · ''c''2^''v''2 (''q'') · … · ''c''''n''^''v''''n'' (''q''). This provides sufficient conditions as well as necessary ones.

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 The reason for putting the generators together with the commas is that notating the gencom as a list of [[monzo]]s allows it to be treated as a [[subgroup basis matrix]]: the group of intervals generated by the gencom is the same no matter how we place the semicolon, and so is the matrix. When this group is a [[harmonic limit|full ''p''-limit group]], as in the example above, the matrix is a {{w|unimodular matrix}}. Inverting and transposing it gives a matrix whose rows are [[val]]s; if ''r'' is the [[rank]] of the temperament, then the first ''r'' rows are the [[mapping|mapping matrix]] corresponding to the generator transversal. More interesting is the case where the gencom generates a [[Just intonation subgroups|JI subgroup]] of some ''p''-limit. In all cases the transpose of the [[pseudoinverse]] of the matrix of monzos gives a matrix of vals whose first ''r'' rows we call the '''gencom mapping''', and which in its entirety we call the '''extended gencom mapping'''. The extended gencom mapping is only a unimodular matrix, and the inversion ordinary matrix inversion, in the case of the full ''p''-limit. However in all cases the transpose pseudoinverse of the gencom matrix is the extended gencom mapping, and the transpose pseudoinverse of the extended mapping is the gencom matrix.
-The rows of the extended gencom mapping are in general fractional vals, meaning the coefficients are allowed to be rational numbers. When applied to elements of the subgroup generated by the gencom, these fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom; the gencom mapping applied to a subgroup basis matrix is a matrix of generator steps in terms of [[smonzos and svals|subgroup vals (svals)]], which is exactly the sval mapping. However, the converse is not the case: if the gencom mapping returns integer values, it does not mean the interval must belong to the gencom subgroup.
+The rows of the extended gencom mapping are in general fractional vals, meaning the coefficients are allowed to be rational numbers. When applied to elements of the subgroup generated by the gencom, these fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom. However, the converse is not the case: if the gencom mapping returns integer values, it does not mean the interval must belong to the gencom subgroup.
+In other words, the gencom mapping takes ordinary monzos just as the [[subgroup monzos and vals|subgroup-val]] mapping takes subgroup monzos:
+$$ V \cdot \vec m = V_G \cdot \vec m_G $$
+where ''V'' and ''V''<sub>''G''</sub> are gencom and subgroup-val mappings of the same temperament in subgroup ''G'', and '''m''' and '''m'''<sub>''G''</sub> are ordinary and subgroup monzos of the same interval in ''G'', respectively.
+It follows that the gencom mapping applied to the subgroup basis matrix of the temperament is a matrix of generator steps in terms of subgroup vals, which is exactly the subgroup-val mapping. If ''S'' is the subgroup basis matrix, then
+$$ VS = V_G $$
 The extended gencom mapping can also be used to determine if an interval ''q'' is in the group of the temperament. Suppose [''c''<sub>1</sub> ''c''<sub>2</sub> … ''c''<sub>n</sub>] is a gencom and [''v''<sub>1</sub> ''v''<sub>2</sub> … ''v''<sub>''n''</sub>] is the corresponding extended mapping. Then each of ''v''<sub>1</sub> (''q''), ''v''<sub>2</sub> (''q'') … ''v''<sub>''n''</sub> (''q'') must be an integer, and moreover we must have ''q'' = ''c''<sub>1</sub>^''v''<sub>1</sub> (''q'') · ''c''<sub>2</sub>^''v''<sub>2</sub> (''q'') · … · ''c''<sub>''n''</sub>^''v''<sub>''n''</sub> (''q''). This provides sufficient conditions as well as necessary ones.