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1/3 SC Werckmeister3 interpretation Re: Werckmeister IV variant: 1/3-SC fifths?

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  • Andreas Sparschuh
    ... it is also possible to read Werckmeister s #3 pattern C~G~D~A E B~F#...C in 1/3 SC terms: C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C# G# D# Bb F
    Message 1 of 143 , Mar 28, 2008
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      --- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
      >
      >.... it occurred to me that if the
      > "comma" may be either Pythagorean or syntonic, with the schisma
      > regarded as not so important, then why not 1/3-syntonic comma
      > tempering for the narrow and wide fifths alike?

      > ! werckmeisterIV_variant.scl
      > !
      > Werckmeister IV with 1/3 syntonic comma temperings
      > 12
      > !
      > 85.00995
      > 196.74124
      > 32/27
      > 393.48248
      > 4/3
      > 45/32
      > 694.78624
      > 785.01123
      > 891.52748
      > 1003.25876
      > 15/8
      > 2/1
      >
      >
      > ! WerckmeisterIV_variant_c.scl
      > !
      > Werckmeister IV variation, 1/3-SC, all intervals in cents
      > 12
      > !
      > 85.00995
      > 196.74124
      > 294.13500
      > 393.48248
      > 498.04500
      > 590.22372
      > 694.78624
      > 785.01123
      > 891.52748
      > 1003.25876
      > 1088.26871
      > 2/1
      >
      > The 1/3-comma variation seems
      > to fit this model -- at least if, like Costeley (1570) and Salinas
      > (1577), we are ready to accept fifths tempered by this great a
      > quantity, as in a regular 1/3-comma meantone or 19-EDO. Zarlino (1571)
      > found 1/3-comma temperament "languid," ....

      it is also possible to read Werckmeister's #3 pattern

      C~G~D~A E B~F#...C

      in 1/3 SC terms:

      C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C# G# D# Bb F C

      as refinement of his JI tuning presented in his book:
      "Musicae mathematicae hodegus curiosus"
      FFM 1687: p.71: a'=400cps
      extracted from his "Nat�rlich" (natural) scale,
      there defined in absolute pitch-frequencies:

      c" 480 cps
      (db 512)
      c# 500
      d" 540
      d# 562.5
      eb 576
      e" 600
      f" 640
      f# 675
      g" 720
      g# 750
      ab 768
      a" 800 overtaken from Mersenne's reference-tone a'=400Hz
      b" 864
      h" 900
      c"'960

      The W3 pattern can be understood as
      modification of layout pattern,
      in absolute terms,
      as cycle of partially tempered 5hts:

      Db 1 unison, implicit contained in his absolute "hodegus" tuning
      Ab 3
      Eb 9
      Bb 27
      F 81 (>80+2/3 (>80+1/3 (80 40 20 10 5)))
      C 243 (>242 (>241 (>240 120 60 30 15)))
      G (729 >) 726 (>723 (>720 360 180 90 45))
      D 2169 (>2160 1080 540 270 135)
      A 405 compare to Chr. Hygens(1629-95) Amsterdam determination:~407 Hz
      E 1215
      B 3645
      F# (10935=32805/3 >) 32768/3 ... 1/3
      C# 1 returend back unison again

      that's relative in chromatically ascending order as Scala-file:

      !Werckmeister3_one3rd_SC_variant.scl
      !
      Werckmeister's famous C~G~D-A-E-B~F#...C pattern as 1/3 SC + schisma
      !C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C#=Db Ab Eb Bb F C
      !
      256/243 ! Db=C# enharmonics @ absolute Mersenne's 256cps unison
      241/216 ! D
      32/27 ! Eb
      5/4 ! E
      4/3 ! F
      1024/729 ! F#
      121/81 ! G = (11/9)^2 = (3/2)*(243/242)
      128/81 ! Ab
      5/3 ! A
      16/9 ! Bb
      15/8 ! B (german H)
      2/1

      attend:
      That one contains more pure intervals than other interpretations.

      if you have some better ratios for W3 -even nearer to JI?-,
      please let me know about that.

      A.S.
    • Andreas Sparschuh
      ... it is also possible to read Werckmeister s #3 pattern C~G~D~A E B~F#...C in 1/3 SC terms: C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C# G# D# Bb F
      Message 143 of 143 , Mar 28, 2008
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        --- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
        >
        >.... it occurred to me that if the
        > "comma" may be either Pythagorean or syntonic, with the schisma
        > regarded as not so important, then why not 1/3-syntonic comma
        > tempering for the narrow and wide fifths alike?

        > ! werckmeisterIV_variant.scl
        > !
        > Werckmeister IV with 1/3 syntonic comma temperings
        > 12
        > !
        > 85.00995
        > 196.74124
        > 32/27
        > 393.48248
        > 4/3
        > 45/32
        > 694.78624
        > 785.01123
        > 891.52748
        > 1003.25876
        > 15/8
        > 2/1
        >
        >
        > ! WerckmeisterIV_variant_c.scl
        > !
        > Werckmeister IV variation, 1/3-SC, all intervals in cents
        > 12
        > !
        > 85.00995
        > 196.74124
        > 294.13500
        > 393.48248
        > 498.04500
        > 590.22372
        > 694.78624
        > 785.01123
        > 891.52748
        > 1003.25876
        > 1088.26871
        > 2/1
        >
        > The 1/3-comma variation seems
        > to fit this model -- at least if, like Costeley (1570) and Salinas
        > (1577), we are ready to accept fifths tempered by this great a
        > quantity, as in a regular 1/3-comma meantone or 19-EDO. Zarlino (1571)
        > found 1/3-comma temperament "languid," ....

        it is also possible to read Werckmeister's #3 pattern

        C~G~D~A E B~F#...C

        in 1/3 SC terms:

        C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C# G# D# Bb F C

        as refinement of his JI tuning presented in his book:
        "Musicae mathematicae hodegus curiosus"
        FFM 1687: p.71: a'=400cps
        extracted from his "Nat�rlich" (natural) scale,
        there defined in absolute pitch-frequencies:

        c" 480 cps
        (db 512)
        c# 500
        d" 540
        d# 562.5
        eb 576
        e" 600
        f" 640
        f# 675
        g" 720
        g# 750
        ab 768
        a" 800 overtaken from Mersenne's reference-tone a'=400Hz
        b" 864
        h" 900
        c"'960

        The W3 pattern can be understood as
        modification of layout pattern,
        in absolute terms,
        as cycle of partially tempered 5hts:

        Db 1 unison, implicit contained in his absolute "hodegus" tuning
        Ab 3
        Eb 9
        Bb 27
        F 81 (>80+2/3 (>80+1/3 (80 40 20 10 5)))
        C 243 (>242 (>241 (>240 120 60 30 15)))
        G (729 >) 726 (>723 (>720 360 180 90 45))
        D 2169 (>2160 1080 540 270 135)
        A 405 compare to Chr. Hygens(1629-95) Amsterdam determination:~407 Hz
        E 1215
        B 3645
        F# (10935=32805/3 >) 32768/3 ... 1/3
        C# 1 returend back unison again

        that's relative in chromatically ascending order as Scala-file:

        !Werckmeister3_one3rd_SC_variant.scl
        !
        Werckmeister's famous C~G~D-A-E-B~F#...C pattern as 1/3 SC + schisma
        !C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C#=Db Ab Eb Bb F C
        !
        256/243 ! Db=C# enharmonics @ absolute Mersenne's 256cps unison
        241/216 ! D
        32/27 ! Eb
        5/4 ! E
        4/3 ! F
        1024/729 ! F#
        121/81 ! G = (11/9)^2 = (3/2)*(243/242)
        128/81 ! Ab
        5/3 ! A
        16/9 ! Bb
        15/8 ! B (german H)
        2/1

        attend:
        That one contains more pure intervals than other interpretations.

        if you have some better ratios for W3 -even nearer to JI?-,
        please let me know about that.

        A.S.
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