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Mattheson & Tuerck on Werckmeister's : C~G~D~A-E-B~F#.....C..

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  • Andreas Sparschuh
    ... I overtook my current point of view about the wrong PC^(1/4) re-interpretation from: Johann Mattheson s Grosse Generalbass Schule Big thourough basso
    Message 1 of 143 , Nov 27, 2007
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      --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
      >
      > --- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@>
      > wrote:
      > > on my proposal:
      > > C
      > > temper down:(1 200 * ln(565 / 567)) / ln(2) = ~-6.11744117...Cents
      > > G
      > > temper down:(1 200 * ln(563 / 565)) / ln(2) = ~-6.13913427...Cents
      > > D
      > > temper down:(1 200 * ln(561 / 563)) / ln(2) = ~-6.16098177...Cents
      > > A
      > > E
      > > B
      > >lower(1200*ln((560/561)*(32768/32805)))/ln(2)=~-5.04245318...Cents
      > > F#
      > (...)
      >
      > Please state your present position clearly.
      I overtook my current point of view about the
      wrong PC^(1/4) re-interpretation from:

      Johann Mattheson's
      "Grosse Generalbass Schule"
      'Big thourough basso contiuo school'
      2nd. Ed.
      Hamburg 1731,
      facsimile reprint: Hildesheim 1968
      on p. 164, footnote a;
      "Seit 1691 stehet dieselbe scala...
      ...vor Werckmeisters Buch von der Temeratur...
      ...daß sie nun bey nahe 40. Jahren nichts daran zu tadeln gefunden
      hat. Jetzo aber,..,wirft man sie alsbald in den logarthmischen
      Schmeltz-Tiegel, wendet sie zu einem Gebrauch an, dazu sie keines
      Falles verfertiget worden..."

      tr:

      '...about W's temperature...
      ...now there had nothing to citicize about that for almost 40 years.
      But now: it had been be thrown in at the deep end of the
      logarithmic melting-crucible, abused in a way,
      that was never intended at the time of its invention..."


      Similar excludes
      http://de.wikipedia.org/wiki/Daniel_Gottlob_T%C3%BCrk
      PC^(1/4) for Werckmeister by giving and discussing
      the following approximative stringlengths in his:

      http://dispatch.opac.d-nb.de/DB=4.1/LNG=DU/LRSET=1/SET=1/SID=a43c2c1f-0/TTL=1/SHW?FRST=4
      ISBN: 3-910019-14-5
      pp. 477-482

      C 8192
      # 7776
      D 7331
      # 6912
      E 6540
      F 6144
      # 5832
      G 5480
      # 5184
      A 4905
      # 4608
      C 4096

      also given in the edition
      http://diapason.xentonic.org/ttl/ttl01.html
      on p.40.

      that's in 5ths:

      C 2048/2055 G 21920/21993 D 14662/14715 A-E-B 2180/2187 F#C#G#D#A#FC
      or
      C ~-5.91cent ~G~ 5.76cent ~D~ 6.25cent ~AEB~ 5.55cent F#...C

      even Kellner disagrees also with PC^(1/4) too:

      http://members.pcug.org.au/~apurdam/tempers.html

      pseudo- "Werckmeister III"
      0
      90
      192
      294
      390
      498
      588
      696
      792
      888
      996
      1092
      1200

      vs:
      Werckmeister(Kellner)
      0
      90
      195
      294
      389
      498
      588
      697
      792
      892
      996
      1091
      1200

      or:
      C 697 G 698 D 697 AEB 697 F#...C


      Does anybody here in that group knows K's original ratios of that?

      http://launch.groups.yahoo.com/group/tuning/message/27024
      "By COMMA Werckmeister meant
      in this context just any MICRO-INTERVAL!!
      There exits no neutral comma with Werckmeister, but he utilizes
      MICRO-INTERVALS of various sizes, whereever necessary."

      > I want to be sure I
      > understand this.
      read:
      Mark Lindley (in his 'Stimmung &Temperatur') on Zarlino's 2/7-SC:
      facsimle on p.166

      a 567
      c 566
      f 565
      g 564
      h 563
      i 562
      k 561
      c 560


      > Are you asserting that you now believe Bach's
      > drawing at the top of the WTC title page is meaningless as to
      > tuning,
      Not at all,
      as long as JSB's 11-eyes become interpreted as:
      C~G~D~AEB~F#...C

      > and that Bach simply slammed Werckmeister's most famous layout onto
      > his harpsichords
      right!

      but certainly not
      > more or less sloppily
      > i.e. with the tempered 5ths logarithmically equal
      that can be excluded.


      and B-F# noticeably less tempered than the
      > other three)?
      alike Tuerck did so too.
      >
      > Or are you saying something else?
      C 2/7sc G 2/7sc D 2/7sc AEB 2/sc+schisma F#...C

      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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