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

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  • a_sparschuh
    ... Dear Keenan, here comes additional the conversion-step into the middle octave too: C4 264:= 33*8 C 33 G4 396:= 99*4 G 99 D4 297_______ D
    Message 1 of 44 , May 20, 2006
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      > --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@>
      > wrote:
      Dear Keenan,
      here comes additional the conversion-step into the middle octave too:

      C4 264:= 33*8 > > C 33
      G4 396:= 99*4 > > G 99
      D4 297_______ > > D (37,74,148,296)297
      A4 444:=111*4 > > A (55,110)111
      E4 330:=165*2 > > E (41,82,164)165
      B4 462:=123*4 > > B (61,122)123
      F# 363________ >> F# 363(366,183)
      C# 272.5=545/2 >> C# (273,546)545,1090(1089)
      G# 409.5=819/2 >> G# (205,410,820)819
      Eb 307.5=615/2 >> Eb (77,154,308,616)615
      Bb 462:=231*2 > > Bb (29,58,116,232)231
      F4 348:= 87*4 > > F (11,22,44,88)87
      C5 524:=33*16 > > C 33
      > >
      > > I really can't understand what this means.
      Reordering that 5ths-circle-pitches
      into an arithmetical ascending series
      yields chromatical:

      > C4 264 Hz middle C
      > C# 272.5
      > D4 297
      > Eb 307.5
      > E4 330
      > F4 348
      > F# 363
      > G4 396
      > G# 409.5
      > A4 444 Hz, that's 4Hz sharper above standard normal-pitch 440Hz
      > Bb 462
      > B4 492
      > C5 524=264*2=C4*2
      >
      Dividing all that 12 pitches by C=264Hz normalizes
      the frequencies it into
      dimensinonless scala-file ratios with C=1/1:
      >
      > ! sync_beat_11-limit.scl
      > !
      > synchronous beating 11-limit scale for C4=264Hz or A4=444Hz
      > 12
      > !
      > 545/524
      > 9/8
      > 615/524
      > 5/4
      > 348/264 ! =(4/3)(87/88)
      > 11/8
      > 3/2
      > 273/176
      > 37/22 ! =(5/3)(111/110)
      > 7/4
      > 41/22 ! =(15/8)(164/165)
      > 2/1

      > > Why did you round
      > > everything to whole numbers of hertz?
      > There is no rounding here.
      It's a kind of
      http://tonalsoft.com/enc/b/bridging.aspx
      by using the superparticular (epimoric) bridges
      as tempering steps inbetween the 5ths.
      The round brackets indicate the tempering steps in 5hts:
      about how far should the according 5ths be flattened or widened.
      >
      > > Are the octaves really supposed
      > > to be stretched by 10-20 cents?
      > not at all, because.....that turns out individual different from
      > instrument to instrument.
      http://en.wikipedia.org/wiki/Inharmonicity

      Above procedere divides the PC into
      > > > PC=3^12/2^19=531441/524288= subpartition
      > > > (297/296)(111/110)(165/164)(123/122)(122/121)(1089/1090)(819/820)
      > > > (615/616)(231/232)(87/88)

      Alreday Andreas Werckmeister in his "Musicalische Temperatur 1691"
      http://diapason.xentonic.org/ttl/ttl01.html
      used that superparticular-subdivision method successfully
      in his #6, the "Septenarius"-tuning:
      http://launch.groups.yahoo.com/group/tuning/message/63471

      As far as i do know at the moment:
      You are the first man since 315 years,
      that doubts about W's idea:
      > > This math doesn't work out.
      Amazing!?

      Above modified version is merely adapted for producing an pure
      4:5:6:7 :8 :9 :10:11 :12 chord on the keys
      C:E:G:Bb:C':D':E':F#':G'.
      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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