Loading ...
Sorry, an error occurred while loading the content.

Unlocking the mysteries of the wedge invariant

Expand Messages
  • Gene W Smith
    This can be defined in multiple ways as cross-products involving intervals and vals, leading to multiple interpretations. We have for instance Interval
    Message 1 of 1 , Nov 27, 2001
    • 0 Attachment
      This can be defined in multiple ways as cross-products involving
      intervals and vals, leading to multiple interpretations. We have for
      instance

      Interval product:

      [u1,u2,u3,u4]^[v1,v2,v3,v4] = [[u2,u3,u4]X[v2,v3,v4],
      [v3,v2,v1]X[u3,u2,u1]]

      These cross-products can be thought of either as involving interval
      classes, or intervals for which one of the prime powers are zero; from
      the latter point of view something perpendicular would be a val which
      mapped one of the primes to zero. We can exclude one of the primes in
      turn, and take the cross product, and compare it to the wedge invariant,
      and see how it can be interpreted in terms of vals.

      If the wedge invariant is [a,b,c,d,e,f], we find we get in this way four
      vals:

      [ 0] [ a] [ b] [ c]
      [ a] [ 0] [ f] [-e]
      [ b] [-f] [ 0] [ d]
      [ c] [ e] [-a] [ 0]

      Similarly, a cross-product of two vals represents an interval in the
      kernel of both, and so from the wedge invariant we get four commas:

      2^f 3^-b 5^a, 3^d 5^e 7^f, 2^d 5^-c 7^b, 2^e 3^c 7^-a

      It seems therefore that going from the wedge invariant to the temperament
      it signifies is not too difficult.

      Here is paultone as an example:

      64/63^50/49 = h12^h22 = [2,-4,-4,2,12,-11]

      We get the four vals:

      [ 0] [ 2] [ -4] [ -4]
      [ 2] [ 0] [-11] [-12]
      [-4] [11] [ 0] [ 2]
      [-4] [12] [ -2] [ 0]

      All these have 64/63 and 50/49 in the kernel.

      We also get the four commas:

      2^-11 3^4 5^2 = 2025/2048
      3^2 5^12 7^-11 = 2197265625/1977326743
      2^2 5^4 7^-4 = (50/49)^2
      2^12 3^-4 7^-2 = (64/63)^2

      All these are commas of the temperament.
    Your message has been successfully submitted and would be delivered to recipients shortly.