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Corrected version of preferred.... Re: absoute-pitch.. on the violin & piano

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  • Andreas Sparschuh
    ... 11-limit septenarian comma versus the SC ... 3^4/5/16 = (11*3^2/7^2/2)*(7^2*3^2/11/5/2^3) i do call the ratio: 441/440
    Message 1 of 63 , May 2, 2008
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      --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

      > from Werckmeister's
      11-limit septenarian comma versus the SC
      > >
      > > 81/80 = (99/98)*(441/440)
      3^4/5/16 = (11*3^2/7^2/2)*(7^2*3^2/11/5/2^3)

      i do call the ratio:
      441/440
      http://www.petersontuners.com/index.cfm?category=85&sub=89
      as "Werckmeister's 11-limit septenarian schisma"
      Scheibler later in the early 19.th century used that interval
      for defineing our todays still actual 440cps standard:
      http://www.1911encyclopedia.org/Musical_Pitch
      > >
      Werckmeister's 11-limit-septenarian-comma becomes when
      > > divided into 3 epimoric subparts:
      > >
      > > 99/98 = (297/296)*(296/295)*(295/294)
      > >
      That 3-fold decompostion of W's-comma can be used for
      > > tempering the 5ths G-D-A-E flattend by the corresponding amounts:
      > >
      > > G 296/297 D 295/296 A 294/295 E
      > >
      > > yielding on the violin empty stings the absolute pitches:
      >
      G2=99Hz lowest violin__G3=198__string
      G3=148 __G4=296__(<297=3*G2)
      A3=221.25 __A4=442.5__ (<444=3*G3)
      highest violin__E5=661.5__string (<663.75=3*A3)

      > > g3: 198 cps = 99*2 := 220*(9/10) a minor-tone below a3=440Hz/2
      > > d4: 296
      > > a4: 442.5 := 885/2
      > > e5: 661.5 := 1323/2
      > >
      or
      When generalized to a dozen 5ths-cirlce on my own old piano:
      > >
      > > C_5: 523Hz (>522 264 132 66 33) 'tenor-C'
      > > G_2: 99 (((> 98 49=7*7 taken from Werckmeister's "septenarius")))
      > > D_4: (297>) 296 (>295 (>294 147=49*3)
      > > A_5: 885 (>882 441=49*9)
      > > E_6: 1323 = 49*27
      > > B_0: (49*81 = 3969>) 3968 ... 496...31 use all 7 'B's on the keys
      > > F#_2: 93
      > > C#_4: 279 a semitone above 'middle-C'
      > > G#_5: 837
      > > Eb_7: 2511
      > > Bb_6: (7533>) 7532 3716 1883
      > > F_4: (5649>) 5648 2824 1412 706 353
      > > C_5: (1059>) 1058 529 = 23^2 cycle returned back to 'tenor-C'
      > >

      Sorry, the old previous meassge contains here some typo-errors:
      please read always 529 instead there formerly faulty 523.
      I had confused that due to a mistake in all to much hurry.
      Simply forget about the wrong numbers:

      and study instead of that
      again the now corrected version:

      !well_Violin2Piano.scl
      !by A.Sparschuh
      temper from violin empty strings G 296/297 D 295/296 A 294/295 E
      12
      ! middle_C 264.5Hz = 529cps/2
      !
      558/529 ! C#
      598/529 ! D
      2511/2116 ! Eb = 627.75/529
      1323/1058 ! E = 661.5/529
      706/529 ! F
      724/529 ! F#
      792/529 ! G
      837/529 ! G#
      885/523 ! A = 442.5Hz*2 absolute a4
      1883/1058 ! Bb = 941.5/529
      992/529 ! B
      2/1
      !
      !
      the relative deviation of the
      5ths corresponds to the following epimoric decomposition

      F 1058:1059 C 528:529 G 296/297 D 295/296 A 294/295 E 3968:3969 B
      B F# C# G# Eb 7532:7533 Bb 5648:5649 F

      into the 8 superparticular subfactorization of the PC=3^12/2^19.

      > Either I don't understand this,
      or if you prefer the same distribution of the PC=~23.46cents
      in logarithmically values as Cents approximation,
      about the amounts:

      F~ -1.635 ~C~ -3.275 ~G~ -5.839 ~D~ -5.859 ~A~ -5.879 ~E~ -0.436 ~B
      B F# C# G# Eb~ -0.2298 ~Bb~ -0.306 ~F

      correspodning to the above 8 epimoric ratios.

      > or something is very wrong with the
      > numbers.
      In deed -i have to agree-
      my first data were somewhat out of control.

      Many thanks for making me aware of my blunder,
      that had urgently demanded some bug-fixing.

      > The fifths D-A and A-E are tempered >23 cents,
      not anymore , but now
      all that both 5ths are less tempered than PC^1/4 =~ 5.865 Cents
      compareable to G-A in Werckmeister's#3.

      > and D-F# is
      > tempered by >55 cents.
      that 3rd: D-F# is barely 186/185 ~9.33Cents wide
      but attend the 3rd C-E with barely 2646/2645 ~0.654Cents wider
      than 5/4, hence almost nearly to pure JI.
      >
      > Or is this a joke (since you said, "have a lot of fun")? :-)
      That was never intened as hoax, even in its faulty version.
      so,
      now after that proof reading/checking/correction
      that patched revision is really meant seriously adjusted
      for properly usage.

      Yours Sincerely
      A.S.
    • Andreas Sparschuh
      ... Hi George, ... in deed, even that *.scl-file contained some unfixed bug. ... Many thanks again for that repair. ... when considering more evaluated digits,
      Message 63 of 63 , May 8, 2008
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        --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

        Hi George,
        > > epimoric decomposition
        > >
        > > F 1058:1059 C 528:529 G 296/297 D 295/296 A 294/295 E 3968:3969 B
        > > B F# C# G# Eb 7532:7533 Bb 5648:5649 F
        > >
        > >Cents approximation,
        > > about the amounts:
        >
        > >F~ -1.635 ~C~ -3.275 ~G~ -5.839 ~D~ -5.859 ~A~ -5.879 ~E~ -0.436 ~B
        > > B F# C# G# Eb~ -0.2298 ~Bb~ -0.306 ~F
        > >
        >
        > I believe there are still a few mistakes.
        in deed, even that *.scl-file contained some unfixed bug.

        > From the sizes of the
        > fifths you give, I think that perhaps you meant this:
        >
        Now -as far as i can see- those ratios appearto be correct:
        > 558/529 ! C#
        > 592/529 ! D
        > 2511/2116 ! Eb = 627.75/529
        > 1323/1058 ! E = 661.5/529
        > 706/529 ! F
        > 744/529 ! F#
        > 792/529 ! G
        > 837/529 ! G#
        > 875/523 ! A = 442.5Hz*2 absolute a4
        > 1883/1058 ! Bb = 941.5/529
        > 992/529 ! B
        >
        Many thanks again for that repair.

        > This will result in:
        when considering more evaluated digits,
        as calculated by "Google"s arithmetics, which yields:


        > F~ -1.636 ~C
        (1 200 * ln(1 058 / 1 059)) / ln(2) = ~-1.63555425...

        > ~C~ -3.276 ~G
        (1 200 * ln(528 / 529)) / ln(2) = ~-3.27575131...

        >~G~ -5.839 ~D
        (1 200 * ln(296 / 297)) / ln(2) = ~-5.83890621...
        arises that deviation here due to your's rounding procedere?

        ~D~ -5.784 ~A
        1 200 * ln(295 / 296)) / ln(2) = ~-5.85866566...
        arises that deviation here due to your's rounding procedere?

        ~A~ -5.953 ~E
        (1 200 * ln(294 / 295)) / ln(2) = ~-5.8785593...

        same question as for D~A?
        Or what else could be the reason for the tiny
        discrepancy amounting about tiny 1/10 Cents
        inbetween ours calculations of the relative deviations
        in the tempered 5ths flatnesses?


        ~E~ -0.436 ~B
        (1 200 * ln(3 968 / 3 969)) / ln(2) = ~-0.436243936...


        > B F# C# G# all just pure 5ths

        Eb~ -0.2298 ~Bb
        (1 200 * ln(7 532 / 7 533)) / ln(2) = ~-0.229835254...

        Bb~ -0.306 ~F
        (1 200 * ln(5 648 / 5 649)) / ln(2) = ~-0.306494477...

        at least we both do agree now in all others 5ths except D~A~E.


        What do you think about that well-temperement,
        with an almost JI the C-major chord:

        C:E:G = 4 : 5*(2646/2645) : 6*(529/528)

        ?

        Yours Sincerely
        Andreas
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