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Re: [tuning] harmonic/subharmonic

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  • Peter Mulkers
    ... And... ... hmm... Let s check this out. So, are these formula s right? ... Given: To Calculate: f1 : f2 : f3 - 1/F1 : 1/F2 : 1/F3 F1=
    Message 1 of 11 , Jun 1, 2000
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      Daniel wolf (01.06.2000) wrote:
      > ... any collection of tones selected from a harmonic
      > series can also be notated as subharmonic and any
      > selection from a subharmonic series can also be
      > notated as harmonic ...

      And...
      Kraig Grady (27.05.2000) wrote:
      > The 4 5 6 is easily found by taking the 2 out of 3 set.
      > For instance 4x5 4x6 5x6, being 20 24 30 (10 12 15)


      hmm... Let's check this out.
      So, are these formula's right?
      ---------------------------------------------------
      Given: To Calculate:
      f1 : f2 : f3 -> 1/F1 : 1/F2 : 1/F3


      F1= f2*f3
      ______________________
      GCD(f1*f2,f1*f3,f2*f3)

      Excel Notation: = (f2*f3)/(GCD(f1*f2,f1*f3,f2*f3)


      F2= f1*f3
      ______________________
      GCD(f1*f2,f1*f3,f2*f3)

      Excel Notation: = (f1*f3)/(GCD(f1*f2,f1*f3,f2*f3)


      F3= f1*f2
      ______________________
      GCD(f1*f2,f1*f3,f2*f3)

      Excel Notation: = (f1*f2)/(GCD(f1*f2,f1*f3,f2*f3)

      ----------------------------------------------------


      f1*F1 = f2*F2 = f3*F3 =
      The distance between harmonic fundamental and subharmonic
      fundamental (WOW: between greatest common subharmonic and
      least common harmonic) for multiads
      is equivalent to complexity n*d for diads?




      Peter Mulkers
      Nijverheidsstraat 80
      B-2840 Rumst
      Belgium
      P.Mulkers@...
    • Paul H. Erlich
      Peter Mulkers wrote, ... That s right, and many (such as Marion on this list) have come up with this before. However, it s not a great measure of dissonance
      Message 2 of 11 , Jun 1, 2000
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        Peter Mulkers wrote,

        >f1*F1 = f2*F2 = f3*F3 =
        >The distance between harmonic fundamental and subharmonic
        >fundamental (WOW: between greatest common subharmonic and
        >least common harmonic) for multiads
        >is equivalent to complexity n*d for diads?

        That's right, and many (such as Marion on this list) have come up with this
        before. However, it's not a great measure of dissonance for multiads; e.g.,
        the 4:5:6:7:9 and 1/(4:5:6:7:9) chords that Daniel Wolf recently posted
        clearly differ in their level of dissonance, but have the same distance
        between harmonic fundamental and guide tone (guide tone is Fokker's term for
        what you call "subharmonic fundamental").
      • Kraig Grady
        Paul! Such things are basic properties behind all the CPS structures. 85% of all post on this list are things that have been said before, even before this list
        Message 3 of 11 , Jun 1, 2000
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          Paul!
              Such things are basic properties behind all the CPS structures. 85% of all post on this list are things that have been said before, even before this list existed!

          "Paul H. Erlich" wrote:

           

          That's right, and many (such as Marion on this list) have come up with this
          before.


          -- Kraig Grady
          North American Embassy of Anaphoria island
          www.anaphoria.com

        • Paul H. Erlich
          Yes, I think the guide tone divided by fundamental was a measure used as early as Euler, if not earlier.
          Message 4 of 11 , Jun 1, 2000
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            Yes, I think the guide tone divided by fundamental was a measure used as
            early as Euler, if not earlier.
          • Kraig Grady
            Paul! Yes! but it was unfortunate that he failed to include 1 in his elements which caused him to overlook the hexany when it was already contained inside one
            Message 5 of 11 , Jun 1, 2000
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              Paul!
                  Yes! but it was unfortunate that he failed to include 1 in his elements which caused him to overlook the hexany when it was already contained inside one of his structures. By omitting the 1 as a true element of a set, i am not sure how he would have seen the formula I quoted unless he saw the subharmonic version in the 3 5 7 where 1 would not be nessicery.

              "Paul H. Erlich" wrote:

              Yes, I think the guide tone divided by fundamental was a measure used as
              early as Euler, if not earlier.

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              -- Kraig Grady
              North American Embassy of Anaphoria island
              www.anaphoria.com

            • Kraig Grady
              Paul! I believe his formula was a proof/example of the one I posted to him! ... -- Kraig Grady North American Embassy of Anaphoria island www.anaphoria.com
              Message 6 of 11 , Jun 1, 2000
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                Paul!
                    I believe his formula was a proof/example of the one I posted to him!

                "Paul H. Erlich" wrote:

                Kraig,

                We're on completely different wavelengths here. I don't know which formula
                you're referring to. I was just referring to the formula Peter Mulkers
                brought up.

                In my opinion, Euler not seeing the hexany was simply an issue of Euler
                using the rectangular lattice vs. Wilson using the triangular lattice.

                -Paul

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                -- Kraig Grady
                North American Embassy of Anaphoria island
                www.anaphoria.com

              • Paul H. Erlich
                Kraig, We re on completely different wavelengths here. I don t know which formula you re referring to. I was just referring to the formula Peter Mulkers
                Message 7 of 11 , Jun 1, 2000
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                  Kraig,

                  We're on completely different wavelengths here. I don't know which formula
                  you're referring to. I was just referring to the formula Peter Mulkers
                  brought up.

                  In my opinion, Euler not seeing the hexany was simply an issue of Euler
                  using the rectangular lattice vs. Wilson using the triangular lattice.

                  -Paul
                • Paul H. Erlich
                  Kraig, Well, if it s the same, my comment stands. I d even go further. The formula in itself is extremely favorable to Euler s rectangular orientation, since
                  Message 8 of 11 , Jun 1, 2000
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                    Kraig,

                    Well, if it's the same, my comment stands. I'd even go further. The formula
                    in itself is extremely favorable to Euler's rectangular orientation, since
                    the formula gives a major seventh chord the same complexity as a major triad
                    or minor triad. However, on the triangular lattice, on which the hexany
                    emerges as a very compact region, the major seventh's true dissonance is
                    better represented

                    -Paul
                  • Kraig Grady
                    Paul! It is my understanding that the Euler developed the generalized lattice. 5s running vertically, 3s horizontally and the seven diagonally. It was this
                    Message 9 of 11 , Jun 1, 2000
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                      Paul!
                          It is my understanding that the Euler developed the generalized lattice. 5s running vertically, 3s horizontally and the seven diagonally. It was this very lattice that Wilson first spotted the hexany which was visible because of the factors. The hexany lattice came afterwards. Wilson development of Lattices is in most cases after the fact of the structure he is illustrating. The lattice is developed to illustrate what he wants the viewer to see or what aspect he wants to bring out. Dallesandro shows many different kind of lattices he has developed.

                      "Paul H. Erlich" wrote:

                       
                      In my opinion, Euler not seeing the hexany was simply an issue of Euler
                      using the rectangular lattice vs. Wilson using the triangular lattice.

                      -Paul
                       
                       


                      -- Kraig Grady
                      North American Embassy of Anaphoria island
                      www.anaphoria.com

                    • Paul H. Erlich
                      OK Kraig -- points well taken. I just note that in my view, what is truly special about the hexany is the number of 7-limit consonant intervals it contains
                      Message 10 of 11 , Jun 1, 2000
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                        OK Kraig -- points well taken. I just note that in my view, what is truly
                        special about the hexany is the number of 7-limit consonant intervals it
                        contains (12) for the number of notes it contains (6). This feature comes
                        out most clearly on the 3-d triangular lattice, otherwise known as the
                        octahedral-tetrahedral (or oct-tet) lattice, which shows each consonance as
                        a connecting line.
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