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Re: A Rational Well Temperament

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  • Andreas Sparschuh
    ... That s made by construction in 5ths: C = 90 45 G = 3C = 135 D: (3G=405) 404 202 101 A: (3D=303) 302 151 E: (3A=453) 452 226 113 B: (3e=339) 338 169
    Message 1 of 63 , Apr 9 12:54 PM
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      --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

      > > > > I think it is
      > > > > 90:95:101:107:113:120:127:135:143:151:160:169:180

      That's made by construction in 5ths:
      C = 90 45
      G = 3C = 135
      D: (3G=405) > 404 202 101
      A: (3D=303)> 302 151
      E: (3A=453) > 452 226 113
      B: (3e=339) > 338 169
      F# (3B=507) < 508 254 127 attend the broade layperson's dog-5th
      C# (3F#=381) > 380 190 95
      G# (3C#=285) < 286 143 another even worser wide dog-5th
      Eb (3G#=429) > 428 214 107
      Bb (3Eb=321) > 320 160 80 40
      F = 3Bb = 120 60 30 15
      C = 3F = 45 cycle closed


      >... there is the wide fifth 143/95 which is tempered by 286/285,
      > about 2/7 comma.
      that sound none well, but is still only good,
      if you really intend there an 'open' 5th alike in meantonics.
      > > >
      > > > Noting that 285=3*95, I think the smallest base number that
      allows all
      > > > fifths to be tempered less than 321/320 will turn out to be over
      107.
      > > What about ?
      > >
      > > 125:132:140:148:157:167:176:187:198:210:222:235:250
      >
      in 5ths cycle:
      C 125
      G (375) > 374 187
      D (561) > 560 280 140 70
      A 210 105
      E (315) > 314 157
      B (471) > 470 235
      F# (1175) < 1184 592 148 74 37 even worser wide than the above ex.
      C# 222 111
      G# (333) < 334 167 another problematic wide 5th
      D# (501) > 500 250 125

      in order to fix such ugly broade-5th bugs, just consult my:
      http://www.strukturbildung.de/Andreas.Sparschuh/Mainz_1999.jpg
      without such dog-5ths defects,
      as found in some 'esotheric' reinterpretations,
      that do not appear in my 1999 original 'discovery' version.

      > Wait! There's even better one
      > 101:107:113:120:127:135:143:151:160:169:180:191:202
      > No fifth differs more than 1/4 pythagorean comma from just.

      That's expanded:
      C: 101
      G: (303) > 302 151
      D: (453) > 452 226 113
      A: (339) > 338 169
      E: (507) < 508 254 127 at least barely only one broade 5th
      B: (381) > 382 191
      F# (573) > 572 286 143
      C# (429) > 428 214 107
      G# (312) > 320 160 80 40 20 10 5
      Eb 15
      Bb 45
      F: 135
      C: (405) > 404 202 101
      >
      > And with this
      >
      > 131:139:147:156:165:175:185:196:208:220:234:247:262
      > No fifth differs more than 321/320 from just.
      >
      C 131
      G (393) > 392 196 98 49
      D 147
      A (441) > 440 220 110 55
      E 165
      B (495) > 494 247
      F# (741) > 740 370 185 {Proposal in order to get rid of the dog}
      C# (555) < 556 278 139 { here you'd better let 555 unchanged }
      G# (417) > 416 ... 13 {...832 1664 < (1665 = 3*555) }
      Eb 39
      Bb 117
      F: (351) > 350 175 instead W's choice 176=11*2^4
      C: (525) > 524 262 131

      {considering that little change converts your's originally
      'open'-tuning into an
      "well"-temperament in the sense of W. & Bach, without the dog}

      Attend:
      Already Werckmeister used almost about the same ratios in:
      http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
      http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_IV_.28VI.29:_the_Septenarius_tunings

      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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