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Wedge products and the torsion mess

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  • genewardsmith@juno.com
    Perhaps wedge products are the best way of cleaning this up. If we write 2^a 3^b 5^c 7^d as a e2 + b e3 + c e5 + d e7 we can take wedge products by the
    Message 1 of 4 , Nov 26, 2001
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      Perhaps wedge products are the best way of cleaning this up. If we
      write 2^a 3^b 5^c 7^d as a e2 + b e3 + c e5 + d e7 we can take wedge
      products by the following rule ei^ei = 0, and if i != j, then
      e1^ej = - ej^ei. In the 5-limit case, the wedge product will be, in
      effect, the correspodning val. In the 7-limit case, we get something
      six dimensional, which if we added another interval would give us a
      val. However, it still can be used to test for torsion.

      50/49 = e2+2e5-2e7, 2048/2025 = 11e2+4e3-2e5. Taking the wedge
      product gives us 50/49^2048/2025 =
      4e2^e3 - 24e2^e5 - 8 e3^e5 - 4 e5^e7. This has a common factor of 4.
      On the other hand 50/49^54/63 = -2 e2^e3 - 12 e2^e5 + 5 e2^e7
      + 4 e3^e5 + 2 e3^e7 - 2 e5^e7, with a gcd of 1 for the coefficients.
      All is, therefore, not lost, I think. I'll ponder the question
      further.
    • genewardsmith@juno.com
      ... One way to see what is going on is this: if the wedge product has a common factor, then whatever we pick as another basis interval in order to compute the
      Message 2 of 4 , Nov 26, 2001
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        --- In tuning-math@y..., genewardsmith@j... wrote:

        >I'll ponder the question
        > further.

        One way to see what is going on is this: if the wedge product has a
        common factor, then whatever we pick as another basis interval in
        order to compute the corresponding val will also have a common factor
        when we take determinants, and hence show torsion according to our
        usual test of the gcd of the coefficients of the val. Therefore the
        torsion is already present in the two elements we started with.
        2048/2025 and 50/49 cannot be extended in a non-torsion way to three
        7-limit intervals, in other words, which would be suitable for a
        block. This is the same problem as before, in a more insidious form.
      • Paul Erlich
        ... three ... Well that s a nice clarification. So perhaps it would have been better to focus on scales, rather than linear temperaments, after all!
        Message 3 of 4 , Nov 26, 2001
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          --- In tuning-math@y..., genewardsmith@j... wrote:

          > 2048/2025 and 50/49 cannot be extended in a non-torsion way to
          three
          > 7-limit intervals, in other words, which would be suitable for a
          > block.

          Well that's a nice clarification.

          So perhaps it would have been better to focus on scales, rather than
          linear temperaments, after all!
        • genewardsmith@juno.com
          ... than ... Not really--we will still get uniqueness after booting out the torsion crud, and some of the things I am getting this way it would not have
          Message 4 of 4 , Nov 26, 2001
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            --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

            > So perhaps it would have been better to focus on scales, rather
            than
            > linear temperaments, after all!

            Not really--we will still get uniqueness after booting out the
            torsion crud, and some of the things I am getting this way it would
            not have occured to look at. I still plan on seeing if we are missing
            something we shouldn't by looking at it from the other side also.

            Of course this is one more piece of weirdness it probably would be a
            pain to explain to a non-mathematical readership. :(
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