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val

[Joe Monzo]

A val is a list of numbers of the form < a b c ... ] , where a,b,c... are integers. It is a covector on the tonespace lattice (i.e., a linear functional on the vectors in the lattice). It is a specific dimensionality referring to the 1-dimensional case of the more general term multival.

One important function of certain vals is that they give the prime-mapping, in number of scale degrees, of a tuning system, whether a just-intonation periodicity-block or a temperament. The numbers are listed according to the prime series:

A mapping to any interval in the scale can be found by multiplying the numbers a,b,c... in the val by their respective exponents x,y,z... in the interval's monzo [ x y, z ... > and adding them -- the sum is the mapping in scale degrees of that interval.

For example:

The 2,3,5-val for 12-edo (which is also a breed) is < 12 19 28 ] . This says that 12-edo:

  • maps the 2:1 to 12 degrees of the 12-edo scale,
  • maps 3:1 to 19 degrees, and
  • maps 5:1 to 28 degrees.

Octave-equivalence means that all notes separated by a 2:1 ratio are considered equivalent -- thus, all mappings in 12-edo can be reduced mod-12 because 12 is the number of 12edo scale degrees spanning the equivalence-interval, therefore:

  • the 2:1 ratio and all its octave-equivalents is mapped to 12 mod 12 = 0 degrees, and thus is the equivalence-interval,
  • the 3:2 ratio and all its octave-equivalents is mapped to 19 mod 12 = 7 degrees, and
  • the 5:4 and its equivalents is mapped to 28 mod 12 = 4 degrees.

This can be seen clearly in calculating the mapping by multiplying the val with the monzo and adding the products, thus:

  • The 2,3,5-monzo of 3:2 is [ -1 1, 0 > . To find the 12-edo mapping: 
      < 12  19  28 ]  =  val (breed) of 12edo
* [ -1   1,  0 >  =  2,3,5-monzo of 3:2
----------------
   -12  19   0

   -12 + 19 + 0  =  7

so 7 degrees of 12edo maps 3:2 , or  2(7/12) ~= 3/2
  • The 2,3,5-monzo of 5:4 is [ -2 0, 1 > . To find the 12edo mapping: 
      < 12  19  28 ]  =  val (breed) of 12edo
* [ -2   0,  1 >  =  2,3,5-monzo of 5:4
----------------
   -24   0  28

   -24 + 0 + 28  =  4

so 4 degrees of 12edo maps 5:4 , or 2(4/12) ~= 5/4

A specific type of val which applies to equal-temperaments is a breed.

. . . . . . . . .
[Gene Ward Smith, Yahoo tuning, message 55115 (Fri Jul 30, 2004 12:46 pm)]

[The val] <1 *| gives the exponent of 2; this is the "2-adic valuation" of number theory; <0 1 *| is the 3-adic valuation and so forth. Example:

<0 1 4| gives the number of meantone fifths needed to get to 3 and 5 in meantone, and <1 1 0| the corresponding number of octaves; in other words as a pair this is the <octave|,<generator| mapping for 5-limit meantone.

Equal temperaments are associated to a single val, linear temperaments to a pair of vals, and so forth.

. . . . . . . . .
[Gene Ward Smith, Yahoo tuning-math, message 2569 (Thu Jan 10, 2002 11:11 pm)]
val

A map from a rational tone group to the integers, which respects multiplication.

If h is a val, then:

h(a*b) = h(a) + h(b);

h(1) = 0; and

h(1/a) = -h(a).

If we write the rational number "a" as a = 2^e2 * 3^e3 * ... * p^ep [that is, if we prime-factor it], we may denote it by a row vector [e2, e3, ...., ep]. In that case we denote a val by a column vector of integers of the same dimension.

In the language of abstract algebra, a val is a homomorphism from the tone group to the integers.

. . . . . . . . .
[Gene Ward Smith, Intervals and Vals]

Given the p-limit group Np of intervals, there is a non-canonically isomorphic dual group Vp of vals.

A val is a homomorphism of Np to the integers Z. Just as an interval may be regarded as a Z-linear combination of basis elements representing the prime numbers, a val may be regarded as a Z-linear combination of a dual basis, consisting of the p-adic valuations.

For a given prime p, the corresponding p-adic valuation vp gives the p-exponent of an interval q

So, for instance:

v2(5/4) = -2
v3(5/4) = 0
v5(5/4) = 1

If an interval is written as a row vector of integers, a val is simply a column vector of integers.

. . . . . . . . .

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