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

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  • a_sparschuh
    ... ___ B4 492:=123*4 B (61,122)123 instead former-wrong typo : 462 ... The !!! 5th B F# has to be flattend down by the product of (123/122)*(122/121) =
    Message 1 of 44 , May 22, 2006
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      --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
      correction:
      > > 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 492:=123*4 > > B (61,122)123 "instead former-wrong "typo": 462
      !!! F# 363________ >> F# 363(366,183) "366/363=122/121
      > > 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 528:=33*16 > > C 33
      >
      > I can't understand how you get from B to F#.
      The !!! 5th B>F# has to be flattend down by the product of
      (123/122)*(122/121) = 123/121 = 61.5/60.5
      because 122 cancels out in nominator versus denominator.
      I frankly admit:
      Tempering the 5th: B>F# about
      (1 200 * ln(61.5 / 60.5)) / ln(2) = ~28.38...Cents
      flat sounds a bit harsh even in my ears, due to enforcing the:
      http://www.google.de/search?as_q=&num=10&hl=de&btnG=Google-Suche&as_epq=alphorn+fa&as_oq=&as_eq=&lr=lang_en&as_ft=i&as_filetype=&as_qdr=all&as_occt=any&as_dt=i&as_sitesearch=&as_rights=&safe=images
      11/8 alphorn-fa on C>F#.

      > It seems like you're
      > trying to make all the fifths differ from 3/2 by a superparticular
      > ratio,
      Yes, in deed, in imitating Werckmeister's "Septenarian" way,
      See for deeper ananlysis also the later decomposition into prime-factors.

      > but 363/246 differs from 3/2 by 123/121, which is not
      > superparticular.
      but the composite 61.5/60.5:=123/121 satisfies again that proprty,
      if we allow additional half-integral superparticulars as valid too.

      > It's unclear to me what the parentheses and the
      > ordering of the numbers mean.
      The value in parentheses versus the bare without the
      parenthesis indicate the amount of tempering the 5ths.
      Hence:
      The values enclosed inbetween the brackets represent
      only the virtual pitches, that an just-pure 5th step
      (factor 3:2) would have merely thought ,
      instead/versus the real tempered pitch-numbers,
      without any parentheses barely.
      >
      > More importantly, what are you trying to achieve with this
      > procedure?
      Just an circle of a dozen tempered 5ths that includes an
      > > 4:5:6:7 :8 :9 :10:11 :12 chord on the keys
      > > C:E:G:Bb:C':D':E':F#':G'.

      >
      > > > ! sync_beat_11-limit.scl
      > > > !
      > > > synchronous beating 11-limit scale for C4=264Hz or A4=444Hz
      > > > 12
      > > > !
      _____ 545/528 !now corrected instead faulty denominator 524 formerly
      > > > 9/8
      > > > 615/528
      > > > 5/4
      > > > 87/66 ! =(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
      >
      > Shouldn't C# be 545/528 rather than 545/524?
      The denominator 528 is correct, due to C5 528:=33*16,
      hence just another typo error in transferring from
      paper to posting. Sorry! Thanx for yours patience.
      'hope, that now at least my numbers fit accurate.
      >
      >
      > I'm familiar with that, but it seems a little old-fashioned now that
      > we have a solid mathematical theory of regular temperaments.
      Antediluvianic integer arithmtics avoids the faultyness of
      logarithms in the "regular" theory, in order to get rid of
      accumualting logarithmic rounding-errors, that you have inavoidable
      alyways inherent included in modern ET "regular" systems.

      Consider the Advantages of the traditional way:
      1.Everything can executed easily merely by pencil and paper,
      without any need of electronic calculators or even slide rulers.
      2. All 5ths beatings are synchroneous to 1 Hz or Metronome: 60 beats.
      3. You got exaclty all the demanded ratios 4:5:....,11:12 just pure,
      instead merelyarbitray incontrolable numerical approximations
      whatsoever.
      4. The way of calculating represents the procedere in practical tuning
      too.
      Are that enough convincing arguments in order to prefer W's old method?
      > > > > Are the octaves really supposed
      > > > > to be stretched by ...the... instruments.
      > > http://en.wikipedia.org/wiki/Inharmonicity
      >
      > I'm quite familiar with that, but when you gave 55 and 111 as Hertz
      > values for the same pitch class I didn't know what to think.

      On A2,3,4 only 111,222,444 got tuned in practice.
      in contrast remain (55,110,220,440 in the brackets)
      merely virtual meant, without got tuned real in practice:
      A (55,110)111 merely 111 matters to represent the significant pitch
      E 165:=55*3
      hence the 5th A>E amounts =165/111. It becomes 111/110
      (1 200 * ln(111 / 110)) / ln(2) = ~15.67....Cents
      flattend down, than if it would be just pure
      3/2 = 165/110 =, because =(165/111)(111/110).

      In general:
      Subtracting any arbitray argument N the difference (-1)
      is algebraic equivalent to an multiplication of N times (N-1)/N.
      Proof: N * ((N-1)/N) = N-1. q.e.d. done by shortening.

      >Forget it.
      Why?
      But W's old method yields exact the desired result,
      instead merely approximating the true ratios by
      irrational-act numbers of the "regular" ET theory.
      The "regular" ET theory excludes inherently, due to of beeing
      resticted only to irrational-numbers, to obtain the correct ratios of the
      4:5:6:7...:11:12 chord in an finte amount of numerical steps,
      directly correct, neither on the paper nor on the machine,
      and must hence refused as inferior,
      in applying Occams-razor!
      >
      > > 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)
      insert here: (545/546) becaue i forgot that factor.
      > > > >(819/820)
      > > > > > (615/616)(231/232)(87/88)
      > Um,... I just pointed out that that
      > mathematical equation is false.
      ok, let's factorize the 11 terms out into prime decomposition:
      __297/296__ __11*3^3/37*2^3
      __111/110__ ____37*3/55*2
      __165/164__ ____55*3/41*2^2
      __123/122__ ____41*3/61*2
      __122/121__ ___61*2/11^2
      _1089/1090_ 11^2*3^2/545*2
      __545/546__ _____545/273*2 "that one went lost last time, sorry
      __819/820__ ___273*3/205*2^2
      __615/616__ ___205*3/77*2^3
      __231/232__ ____77*3/29*8
      ___87/88___ ____29*3/11*2^3
      That results in
      total over all : 3^12/2^19,
      the collective product over all 11 superparticulares.
      Factors not equal to powers of 3 or 2 do cancel out each others in
      nominator versus denominator, so that just merely the PC=3^12/2^19
      remains:
      Simply add the exponents of the 3s and respectively of the 2s.
      That yields two sums in the powers of the 3s: =12,
      respectively in the 2s: =-19,
      makes concluding final: 3^12/2^19=531441/528244. q.e.d.

      > Probably because you left out 545/546.
      That remark looks already alike,
      you got meanwhile be able to understand
      a little more about comprehending the concept.
      >
      > That's quite clear to me, but what about the other notes?
      they are chosen in above way inbetween the given specified ratios,
      in order to interpolate the cycle of 12 5hts,
      as smooth as possible,
      under the predetermined restrictions like:
      > > pure
      > > 4:5:6:7 :8 :9 :10:11 :12 chord on the keys
      > > C:E:G:Bb:C':D':E':F#':G'.

      Summary in other words:
      The values in parenthesises serve merely as
      auxiliary-variables, working for the intermediate
      5ths-tempering calculation steps inbetween,
      but become dispensable for yielding the final result,
      hence they do appear carried along in brackets merely virtual,
      in order to indicate the priority of the intended 12 bare
      pitch-frequencies against the assumed pure ones,
      in order to compensate by/on the way the PC,
      by its subdivision into superparticular factors.

      Is that to grasp really so difficult?
      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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