Subgroup basis matrix: Difference between revisions
m Mike Battaglia moved page Subgroup Mapping Matrices (V-maps) to Subgroup Basis Matrices: Clarify terminology; "subgroup mapping" sounds like a mapping matrix *on* a subgroup |
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A [[Temperament_Mapping_Matrices_(M-maps)|temperament mapping matrix]], or M-map, is a Z-module homomorphism (aka abelian group homomorphism) '''T''': J → K from the free Z-module (abelian group) J of JI ratios to a new free Z-module K, where K then comes to represent tempered intervals, that is to say, intervals of an [[Abstract_regular_temperament|abstract regular temperament]]. We can also consider Z-module homomorphisms '''S:''' J* → L*, where J* is the Z-module of linear functionals (elements of Hom(J, Z)) on J, and where we map directly from J* to another Z-module of linear functionals L*; this Z-module is unrelated to K above. A bit of analysis will reveal that these homomorphisms restrict vals to [[Smonzos_and_Svals|svals]] on a certain subgroup, and that the Z-module L which the elements of L* act on are [[Smonzos_and_Svals|smonzos]]. Hence, since these new homomorphisms can also be represented by integer matrices, we will call such matrices '''subgroup mapping matrices''', or "val-maps" or '''V-maps''' when context demands they be distinguished from their temperamental counterparts, the [[Temperament_Mapping_Matrices_(M-maps)|M-maps]]. | |||
If we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix V is said to be a mapping matrix for a subgroup G of a JI module J if and only if the column module of V spans G and if V is of full column rank. Note that, unlike with M-maps, we drop the restriction that G must be saturated, so that we specifically allow for subgroups with prime powers and the like. | |||
The column module of any subgroup mapping matrix is the submodule of J corresponding to the subgroup G. The row module of any subgroup mapping matrix V is the module of [[Smonzos_and_Svals|svals]] which take coefficients representing, in order, the mappings for the intervals specified by the columns of V. Note that, much like with M-maps, there is not a unique mapping matrix for any subgroup: any matrix V of full-column rank which has columns that form a basis for G will also send vals to svals on that subgroup, but the coefficients of the svals will change to reflect the basis of V. | |||
Of note is that, much like temperament homomorphisms, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos. For any V-map V and associated subgroup G defined by the columns of V, the kernel of V consists of those vals tempering out G. These vals have the property that, for any val k in the kernel and any other val v, (k+v)∙V = k∙V + v∙V = 0 + v∙V = v∙V. In other words, any two vals differing by an element in the left null module will restrict to the same sval. Rather than saying that these null vals are "tempered out," we instead say that they are '''restricted away''', as their subgroup restriction under V is the zero sval. | Of note is that, much like temperament homomorphisms, these new subgroup homomorphisms also have a kernel, but this kernel is now a subspace of vals rather than monzos. For any V-map V and associated subgroup G defined by the columns of V, the kernel of V consists of those vals tempering out G. These vals have the property that, for any val k in the kernel and any other val v, (k+v)∙V = k∙V + v∙V = 0 + v∙V = v∙V. In other words, any two vals differing by an element in the left null module will restrict to the same sval. Rather than saying that these null vals are "tempered out," we instead say that they are '''restricted away''', as their subgroup restriction under V is the zero sval. | ||