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understanding George's: (was, Re:..) Rational Well Temperament

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  • Andreas Sparschuh
    ... Hi George, Tom & all others, ... Did you mean by that following corresponding absolute pitches, or which other frequncies had you concrete in mind when
    Message 1 of 63 , Apr 23 8:58 AM
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      --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
      Hi George, Tom & all others,
      >
      > ! Dent-YN-RWT.scl
      > !
      > Tom Dent's Young-Neidhardt well-temperament (rationalized by G. Secor)
      > 12
      > !
      > 560/531 ! C#
      > 2643/2360 ! D
      > 70/59 ! Eb
      > 74/59 ! E
      > 315/236 ! F
      > 1329/944 ! F#
      > 883/590 ! G
      > 280/177 ! G#
      > 890/531 ! A
      > 105/59 ! A#
      > 887/472 ! B
      > 2/1
      >
      Did you mean by that following corresponding absolute pitches,
      or which other frequncies had you concrete in mind when
      creating that particular one by the factor 59?

      C: 531 := 59*9
      C# 560 280 140 70 35 = 7*5
      D: 594.675 := 264.3*9/4
      Eb 630 315 = 7*5*9
      E: 666 333 = 37*9
      F: 708 354 177 = 59*3
      F# 747.625 = 1329*9/16
      G: 794.7 = 88.3*9
      G# 840 = 240*3
      A: 890 445Hz ? probably yours intend A4=445 cps ?
      Bb 945 = 105*9
      B: 997.875 = 887*9/8

      Hence i do agree with your's observation:
      > The brats aren't nearly as simple as with the harmonic temperaments
      > we've been looking at lately, but OTOH the fifths are much more
      > consistent in size (all tempered <4 cents).
      >
      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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