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Bach-squiggle from Werckmeister's Septenarius, wasRe: half-diminished septima...

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  • Andreas Sparschuh
    ... Just that subdivsion of the SC=81/80 into =(225/224)(126/125) 2 superparticular factors happens also too @ notes !280! D. 28/25 =(10/9)(126/125) =
    Message 1 of 24 , Mar 2, 2007
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      --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
      wrote:

      > You (it was: Tom Dent) are raising 16/9 by 225/224 to get 25/14,...
      Just that subdivsion of the SC=81/80 into =(225/224)(126/125)
      2 superparticular factors happens also too @ notes

      !280! D. 28/25 =(10/9)(126/125) = (225/224)(9/8)
      !420! A. 42/25 = (5/3)(126/125) = (225/224)(27/16) see down

      in my first 1999 Bach-squiggle interpretation as an cycle of a dozen
      partially temperated 5ths:

      Subdivision of the PC:=3^12/2^19 into 8 superparticular factors:
      http://www.strukturbildung.de/Andreas.Sparschuh/

      A. 105,210; !420! cps (or Hz) @ begin
      E. 157; _314_ ;/(315:=105*3)
      B. 235; _470_ ;/(471:=147*3)
      F# 11,22,44,88,176; _352_ ;704/(705:=235*3)
      C# 33:=11*3
      G# 99:=33*3
      Eb _297_ :=99*3
      Bb ; _445_ ,890/(981:=297*3)
      F. ; _333.5_ ;667,1334/(1335:=445*3)
      C. 125; _250_ ;500,1000,2000/(2001:=667*3)
      G. 187; _374_ ;/(375:=125*3)
      D. 35,70,140; !280! ;560/(561:=187*3)
      A. 105:=35*3 cycle terminates @ same pitch as already on start

      short:
      A 314/315 E 470/471 B 704/705 F# C# G# Eb 890/891 Bb 1334/1335 F
      2000/2001 C 374/375 G 560/561 D A

      Consiting in a decomposition of the PC into the 8-fold product:
      PC = 3^12/2^19 = 531441/524288 =

      (315/314)(471/470)(705/704)(891/890)(1335/1334)(2001/2001)(375/374)(560/561)

      That's recombined in ascending pitch-order in
      _abs_ frequency/pitch-name/relative ratio/deviation from 3&5 limits
      ..................................................................
      _250_ C. __1/1__ = middle 'C' has absolute 250 cps or Hz
      _264_ C# 132/125 = (256/243)(8019/8000) = (99/100)(16/15)
      !280! D. _28/25_ = (10/9)(126/125) = (225/224)(9/8) see initial rem.
      _297_ Eb 297/250 = (32/27)(8019/8000) = (99/100)(6/5)
      _314
      _333.5F. 667/500 = (4/3)(2001/2000) tiny sharp 4th
      _352_ F# 176/125 = (10/7)(616/625) = (176/175)(7/5) ~sept. tritone
      _374_ G. 187/125 = (374/375)(3/2) flat 5th
      _396_ G# 198/125 = (128/81)(3969/4000) = (49/50)(8/5 octaved down 3rd)
      !420! A. _42/25_ = (5/3)(126/125) = (225/224)(27/16) syntonic vs. pyth
      _445_ Bb _89/50_ = (16/9)(801/800) = (89/90)(9/5)
      _470_ B. _47/25_ = (15/8)(376/375) = (6016/6075)(243/128)
      _500_ c' __2/1

      attend also @ Bb: SC:=81/80=(801/800)(89/90)

      !sparschuh1999.scl
      !
      Sparschuh's 1999 interpetation of J.S. Bach's 1722 WTC squiggles
      12
      !
      ! 1/1 ! 1.000
      132/125 ! 1.056
      28/25 ! 1.12
      297/250 ! 1.188
      157/125 ! 1.256
      667/500 ! 1.334
      176/125 ! 1.408
      187/125 ! 1.496
      198/125 ! 1.584
      42/25 ! 1.68
      89/50 ! 1.78
      47/25 ! 1.88
      2/1 ! 2.000


      that was inspired by Andreas Werckmeisters squiggle on p.91 of
      his "Musicalische Temperatur" Quedlinburg 1691
      and his "Septenarius"-tuning Chap. XXVII pp.71-74
      http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html


      C. 393/392; _196_ ;98,49 := 7^2 initialization
      G. 525/524,264; _131_ :=393/3
      D. 351/350; _175_ :=525/3 original questionable 176 corrected to 175
      A. _117_ :=351/3
      E. _156_ ;78,39:=117/3
      H. 417/416,208; _104_ ;52,26,13:=39/3
      F# 279/278; _139_ :=417/3
      C# _186_ ;93:=279/3
      G# 495/496,248; _124_ ;62,31:=93/3
      D# _165_ :=495/3
      B. 441/440,220; _110_ ;55:=165/3
      F _147_ :=441/3
      C 49:=147/3 back home @ 7*7=49

      PC decomposition:
      C 392/393 G 524/525 D 350/351 A E H 416/417 F# 278/279 C# G# 496/495
      D# B 441/440 F C

      Yielding the recombined string-lengnts in descending counting-order:

      196 C. __1/1__
      186 C# _98/93_ = (256/243)(3969/3968) = (245/248)(16/15)
      175 D. 196/175 = (10/9)(882/875) = (1568/1575)(9/8)
      165 D# 196/165 = (32/27)(441/440) = (98/99)(6/5) superparticular
      156 E. _49/39_ = (5/4)(196/195) = (3136/3159)(81/64)
      147 F. __4/3__
      139 F# 196/139 = (7/5)(140/139) = (686/695)(10/7) tritone
      131 G. 196/131 = (416/417)(3/2) tiny flat 5th
      124 G# _49/31_ = (128/81)(3969/3968) =(245/248)(8/5)
      117 A. 196/117 = (5/3)(196/195) = (3136/3159)(27/16)
      110 B. _98/55_ = (16/9)(441/440) = (98/99)(9/5) superparticular
      104 H. _49/26_ = (15/8)(196/195) = (3136/3159)(243/128)
      098 c' __2/1__

      have a lot of fun with that
    • Tom Dent
      ... I m not sure what mapping means in this context. If we take the unit to be 1/3 comma then a pure third is C E_3 now the 5-limit just tritone is C F#_3
      Message 2 of 24 , Mar 2, 2007
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        --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
        >
        > Hiya Tom,
        >
        > What's the mapping for this temperament? Can you show
        > how a dominant 7th chord would be notated?
        >
        > -Carl
        >

        I'm not sure what 'mapping' means in this context. If we take the unit
        to be 1/3 comma then a pure third is

        C E_3

        now the 5-limit 'just' tritone is

        C F#_3

        but the septimal tritone is

        C F#_4

        i.e. one 'unit' lower.

        The 7/4 is then

        Ab^3 F#_4 or equivalently Ab F#_7

        The 7/6 is

        F G#_7

        the 9/7 is for example

        F#_3 Bb^4

        So a dom 7 to my way of thinking is

        C E_3 G Bb^1

        an aug 6 would be

        C E_3 G A#_7

        etc. etc.

        If you are prepared to forget the schisma (a la 159- or 171- equal),
        then Bb=A#_3, therefore the aug 6 might be rewritten as a 'harmonic 7th'

        C E_3 G Bb_4

        which corresponds to the approximation that 64/63 is 4/3 of 81/80.

        Hope that is clear enough...
        ~~~T~~~
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