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almost just anothers...was: Re: A Rational Well Temperaments

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  • Andreas Sparschuh
    ... Hi George, Tom & all others, ... 313 E 315 334 F 335 352 F# 354 375 G 376 396 G# 398 418 A 421 cps or Hz 445.5 Bb 447 469.5 B 472 501 C
    Message 1 of 63 , Apr 18 9:54 AM
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      > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
      >
      Hi George, Tom & all others,
      ...just compare my actual refined 'squiggle' versus your similar...
      >...old suggestion:

      313 > E 315
      334 > F 335
      352 > F# 354
      375 > G 376
      396 > G# 398
      418 > A 421 cps or Hz
      445.5 > Bb 447
      469.5 > B 472
      501 > C 502
      528 > C# 531
      560 > D 563
      594 > Eb 597

      !squiggle418cps.scl
      12
      Sparschuh's 2008 Cammerton a4=418Hz 'squiggle'
      176/167 ! C#
      560/501 ! D
      198/167 ! Eb
      626/501 ! E = (5/4)*(2504/2505)
      4/3 ! F
      704/501 ! F#
      250/167 ! G = (3/2)*(500/501)
      264/167 ! G#
      839/501 ! A
      297/167 ! Bb
      313/167 ! B
      2/1

      obtained from tempering some 5ths sharper than pure by the amounts:
      C 500/501 G 374/375 D 224/225 A 209/201 E 626/627 B 2816/2817 F#...
      ...F# - C# - G# - Eb - Bb 2672/2673 F - C.
      >

      > George's proposal:
      > 364, 384, 408, 432, 456, 486, 512, 545, 576, 610, 648, 683, 728:
      >
      > ! WTPB-24c.scl
      > !
      > George Secor's 24-triad proportional-beating well-temperament (24c)
      > 12
      > !
      > 96/91
      > 102/91
      > 108/91
      > 114/91
      > 243/182
      > 128/91
      > 545/364
      > 144/91
      > 305/182
      > 162/91
      > 683/364
      > 2/1
      >
      that sounds really fine,
      works similar on 91=13*7 alike my HIP:
      http://en.wikipedia.org/wiki/Historically_informed_performance
      Contius reconstruction:

      F: 3^6 = 729 ( > 728 364 182 91 = 13*7 )
      C: 3*91 = 273
      G: (3*273 = 819 1638 >) 1637 ( > 1632 816 408 204 102 51)
      D: (51*3 = 153 306 612 >) 611 ( >608 304 152 76 38 19)
      A: 19*3 = 57
      E: (57*3 = 171 342 684 >) 683
      B: (3*683 = 2049 >) 2048 ... 1=3^0
      F# 3
      C# 9 = 3^2
      G# 27 = 3^3
      Eb 81 = 3^4
      Bb 243 = 3^5
      F: 729 = 3^6

      all 5ths inbetween the accidentials or nominals are just 3/2 pure.

      !ContiusHIP.scl
      12
      HIP reconstruction of Contius organ-tuning in Halle by A. Sparschuh
      192/91 ! C#
      611/546 ! D
      108/91 ! Eb
      683/546 ! E = (5/4)*(1366/1365)
      243/182 ! F
      128/91 ! F#
      1637/1092 ! G = (3/2)*(1637/1638)
      144/91 ! G#
      456/273 ! A Choir-tone 456cps or Hz
      162/91 ! Bb
      512/273 ! B
      2/1
      !

      Which one of that ones would you prefer or even improve?

      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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