Loading ...
Sorry, an error occurred while loading the content.

2/7 SC in 'septenarian' "qaternarius"C~G~D~A-E-B~F#.....C, was: Re: continuo...

Expand Messages
  • Andreas Sparschuh
    ... Dear Margo, divide alike Zarlino once had done, the SC = 81/80 = (81*7)/(80*7) = 567/560 into 4 arithmetic subparts [2/7 + 2/7 + 2/7] + 1/7 in his manner:
    Message 1 of 143 , Nov 25, 2007
    View Source
    • 0 Attachment
      --- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
      >
      > Then, again, the example of Zarlino
      > and his student Vincenzo Galilei cautions us that a teacher is not
      > necessarily responsible for all the views of her/his pupil --
      > although Galilei, as it happens, expresses a liking for Zarlino's
      > 2/7-comma as the most pleasing keyboard temperament.

      Dear Margo,

      divide alike Zarlino once had done, the

      SC = 81/80 = (81*7)/(80*7) = 567/560

      into 4 arithmetic subparts [2/7 + 2/7 + 2/7] + 1/7 in his manner:

      567/560 = [ (567/565) * (565/563) * (563/561) ] * (561/560) = 81/80

      That yields in Werckmeister's terms of 8 just pure 5ths
      an corresponding 4-fold subdivision of the:

      PC = 3^12/2^19 = (81/80)*(32768/32805) = SC * schisma

      into

      C 565/567 G 563/565 D 561/563 A-E-B (560/561)(32768/32805)
      F#-C#-G#-D#-A#-Bb-F-C

      or 'septenarian' circulating:

      begin=7=Ab-Eb-Bb-F-C~G~D~A-E-B~F#-G#=7=end

      expanded multiplying consecutive the following lines by factor 3

      Ab_-2: 7 cps or Hz
      Eb_1: 21 (> 20.777777777...)
      Bb_2: 63 (> 62.3333333333...)
      F_3: 189 (> 187 = 561/3)
      C_5: 567 (> 565 (> 563 (> 561 = 187*3 )))
      temper down by: (1 200 * ln(565 / 567)) / ln(2) = ~-6.11744117...Cents
      G_6: 1695 (> 1689 (> 1683 = 187*9 ))
      temper down by: (1 200 * ln(563 / 565)) / ln(2) = ~-6.13913427...Cents
      D_8: 5067 (> 5049 = 187*27)
      temper down by: (1 200 * ln(561 / 563)) / ln(2) = ~-6.16098177...Cents
      A_10: 15147 = 187*81
      E_12: 45441 = 187*243
      B_13: 136323 = 187*729...(> B-7: 7/27)
      lower (1200*ln((560/561)*(32768/32805)))/ln(2) = ~-5.04245318...Cents
      (F#_15: 187*2187>) F#_-5: 7/9
      C#_-3: 7/3
      G#_-2: 7 cycle closed enharmonic back to start: Ab_-2: 7 cps or Hz

      That's chromatic in ascending pitch order as absolute frequencies
      when taken modulo 2^n into the middle octave:

      c 283.5 = 567/2 "middle_C"
      # 298.6666666... = 298+2/3 = 128*7/3
      d 316.6875 = 5067/16
      # 336 = 21*16
      e 355.007812 = 45441/128
      f 378 = 189*2
      # 398.2222222... = 398+2/9 = 512*7/9
      g 423.75 = 1695/4
      # 448 = 7*64
      a 473.34375 = 473+11/32 = 15147/32 ~Praetorius high Choir-Thone~
      # 504 = 63*8
      b 532.511719.. = 532+131/256 = 17*11*3^7/2^8
      c'567 "tenor_C"

      for the corresponding lower Cammerthone version
      simply divide each pitch by 9/8 by of an major-tone downwards.

      so that:

      c_4 becomes 252 Hz = (567*4/9)cps and
      a_4 = 1683/4 = 420.75 Hz

      in order to replace the my meanwhile outdated 9.9.99
      first original "squiggle" 420Hz proposal:
      http://www.strukturbildung.de/Andreas.Sparschuh/
      by the above new improved version, the now actual:

      Rational 2/7-SC-"squiggle" interpretation absolute @ a'=420.75cps

      that tempers barely 4 of the dozen 5ths
      just in Werckmeister's famous 8-pure 5ths layout,
      instead fromerly only 4 pure 5ths once in 1999 at
      DA&F#C#G#Eb. Meanwhile, now that turns out in my
      in my ears as suspicious to much near
      inbetween Kellners modern PC^(1/5) schmeme
      or even worser others ahistoric alleged PC^(1/6) claims.

      I.m.h.o:
      As far as i do see the squiggles now:
      There's no reason why JSB should had
      depart from W's original layout
      in whatsoever interpretation for
      C~G~D~AEB~F#...C
      you wants to prefer in yours taste.

      Never the less:

      Try out the rational 2/7-SC variant :

      !septenarianFC_G_D_AEB_Fsharp.scl
      !
      C 565/567 G 563/565 D 561/563 AEB(560/561)(32768/32805)F#C#G#D#A#BbFC
      !
      12
      !
      256/243 ! C# ~1.05349794...
      563/504 ! D ~1.11706349...
      32/27 ! Eb ~1.18518519...
      563/448 ! E ~1.25669643...
      4/3 ! F ~1.33333333...
      1024/729 ! F# ~1.40466392...
      565/378 ! G ~1.49470899...
      128/81 ! G# 1.58024691...
      187/122 ! A ~1.53278689...
      16/9 ! Bb ~1.7777777...
      1683/896 ! H ~1.87834821...
      2/1

      as alternative choice when considering JSB's squiggels.

      Concluding remark;
      Attend that:
      Above Zarlino's arithmetic 2/7-SC division should not be
      confused with its modern irrational approximation:

      (81/80)^(2/7) = ~1.0035556...
      (1 200 * ln((81 / 80)^(2 / 7))) / ln(2) = ~6.14465417...Cents
      with barely tiny deviation but significant
      impact on the representation.

      not to mention the even less useful: PC^(2/7)

      (1 200 * ln(((3^12) / (2^19))^(2 / 7))) / ln(2) = ~6.70286011...Cents

      or for all those,
      that allege that JSB would had matched already back 1722 TUs
      exactly precisely by ear within 15 minutes?

      720TUs/7 = 102+6/7TUs ~102.857143...TUs

      Whoever beliefes such broade claims except Brad?

      Sorry, but:
      Personally i don't need for an other logarithmic unit
      than the traditional Cents of 1200-EDO.

      Anyhow:
      have a lot of fun with my new actual
      arithmetic 2/7-SC "squiggles"
      that fit even matching into Werckmeister's
      C~G~D~A&B~F# pattern.

      sincerely
      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
      View Source
      • 0 Attachment
        --- 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.
      Your message has been successfully submitted and would be delivered to recipients shortly.