Vals and tuning space: Difference between revisions
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<span style="display: block; text-align: right;">[[:de:Val|Deutsch]] - [[ヴァルと音程空間|日本語]]</span> | <span style="display: block; text-align: right;">[[:de:Val|Deutsch]] - [[ヴァルと音程空間|日本語]]</span> | ||
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=Definition= | =Definition= | ||
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Whenever one of the generators of a temperament is a 2/1 the key information is carried by the other vals, assuming octave equivalence (i.e. 3/1=3/2=6/1 etc). Thus the essential character of 5-limit meantone is defined by a single val (the one for the 3/2 generator), written <0 1 4|. | Whenever one of the generators of a temperament is a 2/1 the key information is carried by the other vals, assuming octave equivalence (i.e. 3/1=3/2=6/1 etc). Thus the essential character of 5-limit meantone is defined by a single val (the one for the 3/2 generator), written <0 1 4|. | ||
== | ==Definition for mathematicians== | ||
The p-limit [[Monzos_and_Interval_Space|monzos]] M form a free abelian group, or ℤ-module, of finite rank pi(p), which is the number of primes up to and including p. The [http://planetmath.org/encyclopedia/DualModule.html dual ℤ-module] M* is [http://en.wikipedia.org/wiki/Group_isomorphism isomorphic] to M, but not in a canonical way. Hence it, the group (Z-module) of '''vals''', is also a free abelian group of rank pi(p). Just as monzos are often written as [http://mathworld.wolfram.com/Ket.html kets], vals are typically written as [http://mathworld.wolfram.com/Bra.html bras]. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist [[Yves_Hellegouarch|Yves Hellegouarch]] seems to have been the first to write about them, under the name "degrees". | The p-limit [[Monzos_and_Interval_Space|monzos]] M form a free abelian group, or ℤ-module, of finite rank pi(p), which is the number of primes up to and including p. The [http://planetmath.org/encyclopedia/DualModule.html dual ℤ-module] M* is [http://en.wikipedia.org/wiki/Group_isomorphism isomorphic] to M, but not in a canonical way. Hence it, the group (Z-module) of '''vals''', is also a free abelian group of rank pi(p). Just as monzos are often written as [http://mathworld.wolfram.com/Ket.html kets], vals are typically written as [http://mathworld.wolfram.com/Bra.html bras]. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist [[Yves_Hellegouarch|Yves Hellegouarch]] seems to have been the first to write about them, under the name "degrees". | ||