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Proposal for Mietke's 1719-1784 lost "Bach" harpsichord Re:compl.19 17 2 WT

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  • Andreas Sparschuh
    ... Hi Carl, Tom & all others, yours ... 19,17,3 approachs work similar as my older attempts in: http://launch.groups.yahoo.com/group/clavichord/message/8737 .
    Message 1 of 31 , Oct 6, 2008
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      --- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
      > > Here's a tally of the 19-limit consonances.
      > >
      > > 19/18 1
      > > 18/17 4
      > > 17/16 2
      > > 19/17 0
      > > 9/8 2
      > > 19/16 5
      > > 24/19 2
      > > 4/3 4
      > > 17/12 2
      > > ---------
      > > 22
      > (a,b,c,b,a,b,1,1,b,1,1,b) where a=6137/6144, b=512/513, c=4617/4624.

      Hi Carl, Tom & all others,

      yours ... 19,17,3 approachs work similar as my older attempts in:
      http://launch.groups.yahoo.com/group/clavichord/message/8737
      .
      That proposal intends to offer an arguable
      http://en.wikipedia.org/wiki/Historically_informed_performance
      tuning on modern replicas of
      http://en.wikipedia.org/wiki/Michael_Mietke
      "Clavessin"
      "He delivered a harpsichord to the court at Cöthen in 1719 on the
      recommendation of Johann Sebastian Bach, which was probably the
      instrument for which Bach composed Brandenburg concerto no.5 as a
      show-piece."
      The scholary literature offers some further details
      about Bachs's journey to Berlin for buying there the instrument.
      Even the bill is still preserved.
      Dated: Cöthen, March 1st 1719,
      It lists $130 expenses altogehter.
      Reprint in:
      Bach-Dokumente, Kassel 1969, Vol.II p.73-74, #95
      Translation into english by A.Mendel, Bach reader p.431,490

      But when Bach moved in 1723 to Leipzig
      that piano remained back in Coethen at least until 1784,
      recorded among the court's inventory:
      "Specifickation derer Fürstlichen Instrumenten in der
      Musikalienkammer"
      lists it on March, 8th, 1784 as "defect".

      The problem:
      Today, nobody knows any more how Bach had tuned that instrument.

      But HIPerformers need for Mietke-replicas
      adequate modern so called "Bach"-tunings,
      preferably in the coeval "Cammerthone"
      pitch of a4=(~405+-5)cps
      as known from Mietke's others instruments too.

      Hence a modern "Bach"-tuning should contain that specification,
      in order to represent the state of the art in the research of the
      last ~10 years:

      New proposal for absolute pitches located at the frequencies:

      243 middle_c'
      256
      272 d'
      304
      304 e'
      312 f'
      342
      364 g'
      392
      406 a' cps or Hz
      432
      456 b'
      486 c"

      as again obtained from Bach's 1722(1723?) WTC autograph,
      when reading the 'squiggle' pattern layout as an instruction
      how to temper 5ths the corresponding to the "squiggles"
      http://en.wikipedia.org/wiki/Well_Tempered_Clavier

      Here an older photo of JSB's "decorative-ornaments" that i do prefer:
      http://www.strukturbildung.de/Andreas.Sparschuh/Bach_Handschrift.jpg

      That doodle at the top can be interpreted symbolic as:

      Concise: start~2~2~2~1-1-1~3-3-3-3-3-3=end

      when understood as an cylce 12 times 5ths

      Or when labeld with note-names:
      start=C~2G~2D~2A~2E-1B-1F#-1C#~3G#-3D#-3A#-3F-3C=end

      Or even when expanded into full detailed pitches:
      start=C 243 := 3^5 obtained from the last 5 ternary 'squiggles'
      ~2~ G 91 182 364 728 (<729 :=3^6)
      ~2~ D 68 136 272 (<273 := 3*G)
      ~2~ A 203 (<204 := 3*68)
      ~1- E 19 38 76 152 304 608 (<609 := 3*A) ; 3 single 'squiggles'
      -1- B 57
      -1~ F# 171
      ~3- C# 1...512 (<513 := 3*F#) ; the concluding 5 triply 'squiggles'
      -3- G# 3
      -3- D# 9 := 3^2
      -3- A# 27 := 3^3
      -3- F 81 := 3^4
      -3. C=end 243 := 3^5

      That's in today's modern terms of the:
      http://www.xs4all.nl/~huygensf/scala/scl_format.html

      !Sparschuh_proposal_Mietke.scl
      !absolute pitches from the middle_c' on:
      c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456
      !
      12
      !
      256/243 ! ~90.22...(cents)
      272/243 ! ~195.81... D
      32/27 ! ~294.13...
      304/243 ! ~387.74... E (5/4)*(1216/1215)
      4/3 ! ~498.04... F
      38/27 ! ~591.65...
      364/243 ! ~699.58... G (3/2)*(728/729)
      128/81 ! ~792.18...
      406/243 ! ~888.63... A4=406cps ~ Coeval Berlin, Coethen Cammer-thone
      16/9 ! ~996.09...
      152/81 ! ~1089.69... B
      2/1

      Attend,
      that tuning contains some consecutive partials
      out of the overtone-series inbetween the 4 semitones:

      16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1

      in reference to:
      http://en.wikipedia.org/wiki/Johann_Gottfried_Walther
      specification in his:
      "He wrote a handbook for the young Duke with the title Praecepta der
      musicalischen Composition, 1708. It remained handwritten until Peter
      Benary's edition (Leipzig, 1955)."

      in german:
      http://de.wikipedia.org/wiki/Johann_Gottfried_Walther
      's
      "Praecepta der musicalischen Composition, Weimar 1708;
      Neu hrsg. von Peter Benary in:
      Jenaer Beiträge zur Musikforschung. Band 2,
      Breitkopf & Härtel, Leipzig 1955 "

      bye
      A.S.
    • Carl Lumma
      ... Heh, not quite. (P P P b P P a P P b c a) also has 68 intervals. And despite having one 32/27 and one non-17-limit tritone, its pattern of M3s is more
      Message 31 of 31 , Oct 26, 2008
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        I wrote:

        > With 7 pure 5ths, the number is 68:
        > (p p b p p a p p b p c a) a=(-3 1 1) b=(-2 0 -1) c=(3 -2 0)
        >
        > Each of these scales uniquely hits the maximum number of
        > 19-limit consonances in its category.

        Heh, not quite. (P P P b P P a P P b c a) also has 68
        intervals. And despite having one 32/27 and one non-17-limit
        tritone, its pattern of M3s is more equal, and its key-color
        pattern is more sinusoidal on the circle of fifths.

        !
        (P P P b P P a P P b c a) a=(-3 1 1) b=(-2 0 -1) c=(3 -2 0)
        12
        !
        19/18
        323/288
        19/16
        64/51
        4/3
        24/17
        3/2
        19/12
        57/34
        16/9
        32/17
        2/1
        !

        As for 4 different nonpure 5ths... with 5 pure 5ths and
        searching to radius 5, we get 19-limit intervals = 54:
        (p p b c p p a p b c d a)
        a=(-3 1 1) b=(-1 -1 -3) c=(0 1 2) d=(3 -2 0)

        With 6 pure 5ths and radius 5, we get 60:
        (p p p b p p p b c b d a)
        a=(-3 1 1) b=(-2 0 -1) c=(0 1 2) d=(3 -2 0)

        With 7 pure 5ths and radius 9, we get 64:
        (p p b p p a p p b p c d)
        a=(-3 1 1) b=(-2 0 -1) c=(-2 3 6) d=(2 -4 -5)

        With this last, I think I broke Scala. 309237645312/206806264579
        shows up in cents even though it's rational in the .scl file.

        -Carl
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