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Re: desire for meantone with an 11-limit interval on piano

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  • a_sparschuh
    ... multiply all ratios by factor 264=11*3*2*2 in order to obtain integral absolute pitch-frequencies ... C4 264 from 1/1 middle C=264Hz C# 280 35/33 (D4 297
    Message 1 of 44 , May 26, 2006
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      --- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
      multiply all ratios by factor 264=11*3*2*2 in order to
      obtain integral absolute pitch-frequencies
      > > ! bihexany.scl
      > > Hole around [0, 1/2, 1/2, 1/2]
      > > 12
      > > !
      C4 264 from 1/1 middle C=264Hz
      C# 280 > 35/33
      (D4 297 > 9/8) inserted instead Cb=B# for smoothening the 5ths
      Eb 308 > 7/6
      E4 330 > 5/4
      F4 336 > 14/11
      F# 360 > 15/11
      G4 396 > 3/2
      G# 420 > 35/22
      A4 440 > 5/3 !normal pitch 440Hz
      Bb 462 > 7/4
      B4 480 > 20/11
      (Cb=B# 504 > 21/11)
      C5 528 > 2

      or arranged in a cycle of a dozen partial tempered 5ths

      C 33,..,264 begin, starting @ middle-C
      G 99:=33*3,198,396
      D (37,74,148,296)297:=99*3
      A 55,110(111:=37*3),220,440 ! (111:110)*(297:296)=81:80 the SC
      E (41,82,164)165:=55*3,330
      B 15,30,60,,(123:=41*3)120,240,480 ! (165:164)*(41:40)=33:32
      F# 45:=15*3,..,360
      C# 35,70,(135:=45*3)140,280 ! 140:135=28:27
      G# (13,26,52,104)105:=35*3,210,420
      Eb (39:=13*3,78)77,154,308
      !(105:104)*(77:78)=2695:2704=299.4444.../300.4444...
      Bb (7,...,228)231:=77*3,462 ! 231:228=77:76
      F (11,22)21:=7*3,42,84,168,336
      C 33:=11*3 cycle closed, ready done.

      i.m.o: heavy harsh tempering, but it works.
      A.S.
    • Gene Ward Smith
      ... seems to me ... Not really. It isn t like 225/224, where you can look at it as a way of mashing the 7s down into the 5-limit lattice. Here the horizontal
      Message 44 of 44 , May 27, 2006
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        --- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:

        > But when you say "The actual 11-limit project ignores 2, 243/242 and
        > 441/440" and you go about creating a lattice for the result, it
        seems to me
        > that would usually be called a 5-limit lattice, isn't that right?

        Not really. It isn't like 225/224, where you can look at it as a way
        of mashing the 7s down into the 5-limit lattice. Here the horizontal
        lattice relationship is 49/40 (7-limit) and the vertical is 10/7
        (again, 7-limit.) So it's clearly a 7-limit lattice, but squished down
        to a plane of pitch-classes. The deal is, two 49/40s in a row gives
        you your 3/2 (by tempering), and a 49/40 times 10/7 is 7/4 (no
        tempering) and times 10/7 again is 5/2 (no tempering.) So it boils
        down to using 49/40 and 60/49 for the same thing, and using two of
        these neutral thirds to reach the fifth.

        In terms of this lattice, the fifth is [2,0], and the major third
        [1,2]. The determinant of [[2,0], [1,2]] is 4, so only 1/4 of the
        lattice consists of the pure 5-limit.

        At the
        > same time the temperament acknowledges the equivalence of this to an
        > 11-limit functionality. And as I recall, the concept of temperament
        wants
        > to remain a little vague about the actual tuning, so you could in this
        > example make the 7- and 11-limit intervals exact at the expense of
        others,
        > or you could make others exact at the expense of 7 and 11. Or you
        could do
        > some other optimization.

        Or you could just use 72-et or 130-et (a division I'm exploring these
        days.) If you want to favor the 7-limit a bit, 171-et.
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