Vals and tuning space: Difference between revisions

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A val "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[Regular_Temperaments|regular temperament]] and thereby define the temperament. A val is written in the form <a1 a2 a3 ... ak|, where the numbers a1 a2 a3 ... are the number of steps along the chain that the first k primes are mapped to. This can be generalized so that a1 a2 a3 ... represent the number of steps any JI basis is mapped to, whereas a JI basis for a [[Just_intonation_subgroups|just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

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|.

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".

If V is a val and M is a monzo of the same rank, then the [http://mathworld.wolfram.com/AngleBracket.html angle bracket] <V|M>, which can also be written V(M), is the result of applying the [http://en.wikipedia.org/wiki/Group_homomorphism homomorphism] V to M. For example, if V = <12 19 28 34| and M = |-5 2 2 -1> then <V|M> equals 12*(-5) + 19*2 + 28*2 - 34 = 0

It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or [[JIP|JIP]], which in weighted coordinates is <1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then <JIP|M>, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = <1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.

The rank-1 [[7-limit|7-limit]] patent [[val|val]] corresponding to [[31edo|31edo]] is <31 49 72 87|. This tells us that 31 steps reaches the 2, approximately 49 the 3, 72 the 5, and 87 the 7. In weighted coordinates, it becomes