Subgroup basis matrix: Difference between revisions

<span style="background-color: #ffffff;">A [[Temperament_Mapping_Matrices_(M-maps)|temperament mapping matrix]], or M-map, is a Z-module homomorphism (aka abelian group homomorphism) </span>'''<span style="background-color: #ffffff;">T</span>'''<span style="background-color: #ffffff;">: 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]].</span>

[[Category:Subgroup]]

[[Category:Mapping]]