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W's 1/11th division of the SC into epimoric subfactors:wasRe: Werckmeister 1697

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  • Andreas Sparschuh
    ... Dear Tom, alike Zarlino & others bevor him, W. divided the SC=80:81 regulary into: 2 parts: (160/161)(161:162) alike Kirnberger 3 inbetween D-A-E 3 parts:
    Message 1 of 3 , Jul 9, 2008
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      --- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> translated:
      >...I may also let the fifths be, as described in the
      > monochord, some pure and some tempered, which also goes quite well. I
      > also wanted to give another such small division [Bruch] so that the
      > fifths are all tempered a little one with another on the monochord;
      > [I] also wanted to depict the partition but because such divisions are
      > tiresome to construct with the compass and the ingratitude much too
      > great, so I have misgivings to write any more about it; primarily
      > because it requires [a] publisher and expense. But since sensation
      > [and] reason are the judges, so [one] has from my monochord or
      > ``sensus'' so much information, that he can well find his feet and
      > manage for himself, and there is the self-evident proof in it, to see
      > and hear what is certain; > ... More when I get round to it!
      >
      Dear Tom,
      alike Zarlino & others bevor him,
      W. divided the SC=80:81 regulary into:

      2 parts: (160/161)(161:162) alike Kirnberger 3 inbetween D-A-E
      3 parts: (242:243)(242:241)(241:240) alike Stanhope inbetw: G-D-A-E
      http://groenewald-berlin.de/text/text_T041.html
      Groenwald confuses in his reprenstation PC versus SC.
      see: Lindley's "Stimmung & Temeratur"
      or
      http://launch.groups.yahoo.com/group/tuning/message/75760
      "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
      "
      http://launch.groups.yahoo.com/group/tuning/message/75449
      "http://diapason.xentonic.org/ttl/ttl01.html
      on p.37, Chap. XVII
      when considering some arithmetical subdivisions of
      81:81 for tempering:
      "Wenn ein Comma in zwey Theile getheilt wird /
      so stehen in kleinesten Zahlen 162. 161. 160.
      In drey Theile sind die kleinsten Termini 243. 242. 241. 240.
      So es in vier Theile gemachet; stehen die kleinsten Termini 324. 323
      322. 322. 320. Die äussersten sind das comma...."
      tr:
      'If a comma is divided into 2 parts,
      then arise in the smallest numbers 162. 161. 160.
      In 3 parts the smallest termini become 243. 242. 241 240.
      when made into 4 parts, the smallest Termini get 324. 323. 322. 320.
      the outer ones represent the comma...'"

      http://harpsichords.pbwiki.com/f/Kirn_1871.html
      "Oder wenn man von C nach e 80 : 81 in vier Quinten vertheilen will,
      kann es folgender Art geschehen:

      C-G 216 : 323 temperirte Quinte = 2/3 - 1/324
      216 : 324 reine Quinte
      -------------------------------------
      G-d 215 1/3 : 322 temperirte Quinte = 2/3 - 1/323
      215 1/3 : 323 reine Quinte
      -------------------------------------
      A-e 214 2/3 : 321 temperirte Quinte = 2/3 - 1/322
      214 2/3 : 322 reine Quinte
      -------------------------------------
      D-A 214 : 320 temperirte Quinte = 2/3 - 1/321
      214 : 321 reine Quinte
      -------------------------------------


      Analogous
      in generalizing W's concept of SC subdivisions Neidhardt obtained:
      11 an corresponding epimoric subfactors:
      (880:881)(881:882)(882:883)*...*(890:891)
      so that:
      Werckmeister11/Kirnberger11/Neidhardt11's approx. of 12 ET yiels:

      F# 32805:32768 C# 890:891 G# 889:890 Eb 888:889 Bb...
      ...A 882:883 E 881:882 B 880:881 f#

      That makes in modern 20th century TUs = PC(1/720) units:

      F#
      schisma:
      (720TU*ln(32768/32805) / ln((3^12) / (2^19)) = ~-59.9607138...TUs
      C#
      (720TU * ln(890 / 891)) / ln((3^12) / (2^19)) = ~-59.665901...TUs
      G#
      (720TU * ln(889 / 890)) / ln((3^12) / (2^19)) = ~-59.732979...TUs
      Eb
      (720TU * ln(888 / 889)) / ln((3^12) / (2^19)) = ~-59.8002081...TUs
      Bb
      .
      .
      .

      .
      .
      A
      (720TU * ln(882 / 883)) / ln((3^12) / (2^19)) = ~-60.2067818...TUs
      E
      (720TU * ln(881 / 882)) / ln((3^12) / (2^19)) = ~-60.2750822...TUs
      B
      (720TU * ln(880 / 881)) / ln((3^12) / (2^19)) = ~-60.3435377...TUs

      Espeically for those in that group here that do claim to be able to
      tune 12-EDO stepwise in that precision,
      or even an alleged "Squiggle"-reinterpretation
      in exactly 60TUs steps, without sligthest error in deviation.

      I.m.h.o:
      It's an modern absurd oversimplification
      to impute that Bach had only tempered barely in 60TU steps.
      Completely ahsitorically!

      Yours Sincerely
      A.S.
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