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gneralizing Werckmeister's septenarius Newton's 53, was: Re:Buxtehude

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  • Andreas Sparschuh
    ... Dear Mike & all others, for Newton s horogramm in 53: http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif in the
    Message 1 of 16 , Jul 14, 2008
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      --- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
      >
      > Or, if anyone has any recommendations for recordings of classical
      > works done in 53-tet or other temperaments, or even perhaps JI...

      Dear Mike & all others,

      for Newton's "horogramm" in 53:
      http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif

      in the
      http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
      style, i do reccomend the epimoric
      stepwise cycle of 5ths modulo octaves in Bosanquet's notation
      in terms of the corresponding absolute pitches:

      0; c-_-4 = 1 ... c-_4=256Hz unison as general reference to the unit
      1; g-_-2 = 3 ! 5th
      2; d-_-1 = 9 ! major-tone
      3; a-_0 = 27 ! Pythagorean 6th
      4; e-_2 = 81 ! ditone
      5; b-_3 = 243 ! octave:limma
      6; gB_6 = 729 ! tritone
      7; dB_7 = 2,187 ! apotome 4,374 [< 4,375 = 5*a\_5] the 'ragisma'
      8; aB_3 = 205 410 820 1,640 3,280 6,560 (<6,561 = 3^8)
      9; eB_5 = 615
      10; bB_5 = 1,845
      11; f\_7 = 2,767 5,534 (<5,535)
      12; c\_7 = 2,075 4,150 8,300 (<8,301)
      13; g\_4 = 389 778 1,556 3,112 6.224 (<6,225)
      14; d\_6 = 1,167
      15; a\_5 = 875=5*f._3 ; 1,750 3,500 (<3,501) JI 3rds: F. -> A\
      16; e\_7 = (41 82 164 328 656 1,312 2,624<) 2625 = 5*c._3
      17; b\_8 = (123 ... 7.872<) 7875 = 5*g._6
      18; gb_10 = (369 ... 23,616<) 23,625 = 5*d._8 last JI 3rd among 4
      19; db_6 = 1107
      20; ab_4 = 415 830 1,660 3,320 (<3321) neoBaroque tuning-forks
      21; eb_6 = 1,245
      22; bb_6 = 1,867 3,734 (<3,735)
      23; f._3 = 175 350 700 1,400 2,800 5,600(<5601) instead W's "176"
      24; c._5 = 525 tenor_C5 ; middle_C4 = 262.5 Hz
      25; g._6 = 1,575
      26; d._8 = 4,725
      27; a._9 = 14,175
      28; e._11 = 42,525
      29; b._12 = 127,575
      30; f#_12 = (1,495 ... 95,680<) 95,681 ... 382,724 (<382,725)
      31; c#_8 = 4,485
      32; g#_9 = 13,455
      33; d#_11 = 40,365
      34; a#_12 = 121,095
      35; f/_12 = 90,821 181,642 363,284 (<363,285)
      36; c/_13 = 136,231 272,462 (<272,463)
      37; g/_12 = 102,173 204,346 408,692 (<408,693)
      38; d/_13 = 153,259 306,518 (<306,519)
      39; a/_4 = 449 ... 459,776 (<459,777)
      40; e/_6 = 1,347
      41; b/_7 = 4,041
      42; f&_9 = 12,123 := f#/ with '&'='#'*'/' about 6 commata sharper
      43; c&_11 = 36,369
      44; g&_12 = 109,107
      45; d&_11 = 40,915 ... 327,320 (<327,321)
      46; a&_9 = 15,343 ... 122,744 (<122,745)
      47; f+_11 = 46,029 := f//_11 with '+':= '//' double comma elevation
      48; c+_12 = 69,043 138,086 (<138,087)
      49; g+_10 = 25,891 ... 207,128 (<207,129)
      50; d+_12 = 77,673
      51; a+_12 = 116,509 233,018 (<233,019)
      52; ( e+ = f- )_13 = 177,763 349,526 (<349,527) enharmoic change
      53=0'; b+_3 = c-_4 = 256Hz=2^8 ... 2^19=524,288 (<524,299) back home

      that cycle matches almost
      http://en.wikipedia.org/wiki/53_equal_temperament
      it also subdivides the
      "Mercator's Comma. Mercator's Comma is of such small value to begin
      with (~3.615 cents)"
      into the above 23 epimoric subfactors instead of 53 equal ones.
      Attend within that the schisma 32805:32768 inbetween:

      ...Gb 2624:2625 Db 3320:3321 Ab Eb 3734:3735 Bb 5600:5601 F...

      gaining JI heptatonics for C-major in

      1. major and minor 3rds:
      [F.] 5:4 [A\] 6:5 [C.] 5:4 [E\] 6:5 [G.] 5:4 [B\] 6:5 [D.]

      2. as scale of whole&semi-tones:
      [C.] 9:8 [D.] 10:9 [E\]16:15[F.] 9:8 [G.] 10:9 [A\] 9:8 [B\]16:15[c.]

      or in commatic ascending order, as in Newton's 1664 drawing:


      !septenarian53well.scl
      Sparschuh's 53 generalization of Werckmeister's septenarius
      53
      2075/2048 ! 1; c\_7 : 2^11
      525/512 ! 2; c._5 : 2^9 ~tenor_C in ET in reference to a4=440Hz
      136231/131072! 3; c\_13 : 2^17
      69043/65536 ! 4; c+_12 : 2^16
      2187/2048 ! 5; dB_7 : 2^11 apotome
      1107/1024 ! 6; db_6 : 2^10
      4485/4096 ! 7; c#_8 : 2^12
      36369/32768 ! 8; c&_11 : 2^15
      9/8 ! 9; d-_3 : 2^3 Pythaogorean major-tone
      1167/1024 !10; d\_6 : 2^10
      4725/4096 !11; d._8 : 2^12
      153259/131072!12; d\_6 : 2^10
      77673/65536 !13; d+_12 : 2^16
      615/512 !14; eB_5 : 2^9 (5:4)*(123:128)
      1245/1024 !15; eb_6 : 2^10 (5:4)*(249:256)
      40365/32768 !16; d#_11 : 2^15 (5:4)*(8073:8192)
      40915/32768 !17; d&_11 : 2^15 (5:4)*(8183:8192)
      81/64 !18; e-_2 : 2^6 Pythagorean ditone
      2625/2048 !19; e\_7 : 2^11 = C.*(5:4) JI 3rds in [C.]->[E\]
      42525/32768 !20; e._11 : 2^15
      1347/1024 !21; e/_6 : 2^10
      174763/131072!22;(e+=f-)_13:2^17 = (4:3)*(524,289:524,288) enharm.ch.
      2767/2048 !23; f\_7 : 2^11
      175/128 !24; f._3 : 2^7 instead of Werckmeister's "176" choice
      90821/65536 !25; f/_12 : 2^16
      46029/32768 !26; f+_11 : 2^15
      729/512 !27; gB_6 : 2^9 tritone
      23625/16384 !28; gb_10 : 2^14 = D.*(5:4) JI 3rds in [D]->[F#\]
      95681/65536 !29; f#_12 : 2^16
      12123/8192 !30; f&_12 : 2^16
      3/2 !31; g-_-2 : 2 the initial 5th step at begin
      389/256 !32; g\_4 : 2^8
      1575/1024 !33; g-_5 : 2^10
      102173/65536 !34; g/_12 : 2^16
      25891/16384 !35; g+_10 : 2^14
      205/128 !36; aB_3 : 2^7
      415/256 !37; ab_4 : 2^8 neoBaroque tuning-fork in 415Hz
      13455/8192 !38; g#_9 : 2^13
      109107/65536 !39; g&_12 : 2^16
      27/16 !40; a-_0 : 2^4 Pythagorean 6th
      875/512 !41; a\_5 : 2^9 = F.*(5:4) JI 3rds [F.]->[A\]
      14175/8192 !42; a._9 : 2^13
      449/256 !43; a/_4 : 2^8
      116509/65536 !44; a+-12 : 2^16
      1845/1024 !45; bB_6 : 2^10
      1867/1024 !46; bb-6 : 2^10
      121095/65536 !47; a#_12 : 2^16
      15343/8192 !48; c&_9 : 2^13
      243/128 !49; b-_3 : 2^7 Pythagorean 7th or octave:limma
      7875/4096 !50; b\-8 : 2^12
      127575/65536 !51; b._12 : 2^16
      4041/2048 !52; b/_7 : 2^11
      2/1 ! 53==0 ; ( b+ = 2*c+ )_3 Helmholtz's enharmonics: B// = C\\
      !
      !
      that yiedls -compared against 53EDO- an intersting
      53-comma key-charcteristics
      http://www.wmich.edu/mus-theo/courses/keys.html
      because all the 3rds range in fine graduation inbetween

      8192:6561 ~384Cents (schimatic 3rd) <<<???<<< and 5:4 ~386Cents

      Attend for instance the 3rd [A\] -> [DB]
      that becomes about an
      http://en.wikipedia.org/wiki/Ragisma
      flattend.
      That small interval was also historically also used for coin-making:
      http://de.wikipedia.org/wiki/Karlspfund
      "Bei historischen Längenmaßen liegt der Variationskoeffizient im
      allgemeinen unter 1/500, was eine Genauigkeit von ± 0,2 % bedeutet. So
      gelten bei den Längenmaßen z.B. 1/2400 oder 1/4374, also die 7-glatten
      Ratios 2401 : 2400 und 4375 : 4374, sowie ihre Reziprokwerte nicht als
      eigentliche Ratios, sondern nur als Kommata."

      Yours Sincerely
      A.S.
    • threesixesinarow
      ... mto.93.0.3.lindley7.gif Can someone help these lazy people? http://en.wikipedia.org/wiki/ Talk:53_equal_temperament#Newton.27s_unpublished_manuscripts.3F
      Message 2 of 16 , Jul 14, 2008
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        --- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
        wrote:

        > Newton's "horogramm" in 53:
        >
        >http://mto.societymusictheory.org/issues/mto.93.0.3/
        mto.93.0.3.lindley7.gif

        Can someone help these lazy people?

        http://en.wikipedia.org/wiki/
        Talk:53_equal_temperament#Newton.27s_unpublished_manuscripts.3F
      • Carl Lumma
        ... Why do you call them that? The above is already linked in their discussion. -Carl
        Message 3 of 16 , Jul 14, 2008
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          Clark wrote:

          > > Newton's "horogramm" in 53:
          > >
          > >http://mto.societymusictheory.org/issues/mto.93.0.3/
          > >mto.93.0.3.lindley7.gif
          >
          > Can someone help these lazy people?
          >
          > http://en.wikipedia.org/wiki/
          > Talk:53_equal_temperament#Newton.27s_unpublished_manuscripts.3F

          Why do you call them that? The above is already
          linked in their discussion.

          -Carl
        • threesixesinarow
          ... I know some howlers where historians just relied on a picture or worse instead of reading the accompanying text and at least one of those editors didn t
          Message 4 of 16 , Jul 15, 2008
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            --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
            >
            > Clark wrote:
            >
            > > > Newton's "horogramm" in 53:
            > > >
            > > >http://mto.societymusictheory.org/issues/mto.93.0.3/
            > > >mto.93.0.3.lindley7.gif
            > >
            > > Can someone help these lazy people?
            > >
            > > http://en.wikipedia.org/wiki/
            > > Talk:53_equal_temperament#Newton.27s_unpublished_manuscripts.3F
            >
            > Why do you call them that? The above is already
            > linked in their discussion.
            >
            > -Carl
            >
            I know some howlers where historians just relied on a
            picture or worse instead of reading the accompanying text
            and at least one of those editors didn't bother to look it
            up in Lindley's book.
          • threesixesinarow
            ... http://books.google.com/books?id=x_A5AAAAIAAJ http://www.archive.org/details/onsensationsofto00helmrich
            Message 5 of 16 , Jul 15, 2008
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              --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
              >(I've re-ordered Helmholtz' sensation of tone but it takes
              > months to get here to Australia).

              http://books.google.com/books?id=x_A5AAAAIAAJ
              http://www.archive.org/details/onsensationsofto00helmrich
            • Carl Lumma
              ... We ve discovered that certain Google books that are available in the U.S. are not available down under. However, the second link, to the internet archive,
              Message 6 of 16 , Jul 15, 2008
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                > >(I've re-ordered Helmholtz' sensation of tone but it takes
                > > months to get here to Australia).
                >
                > http://books.google.com/books?id=x_A5AAAAIAAJ
                > http://www.archive.org/details/onsensationsofto00helmrich

                We've discovered that certain Google books that are available
                in the U.S. are not available down under. However, the second
                link, to the internet archive, looks like a winner!

                -Carl
              • Andreas Sparschuh
                ... Dear CACCOLA & all others that want to understand N s draft sketch, here comes some more work for lazy people beyond my earlier interpretations of
                Message 7 of 16 , Jul 16, 2008
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                  --- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@...> wrote:
                  > > Newton's "horogramm" in 53:
                  > >
                  > >http://mto.societymusictheory.org/issues/mto.93.0.3/
                  > mto.93.0.3.lindley7.gif
                  >
                  > Can someone help these lazy people?
                  >
                  > http://en.wikipedia.org/wiki/
                  > Talk:53_equal_temperament#Newton.27s_unpublished_manuscripts.3F
                  >
                  Dear CACCOLA & all others that want to understand N's draft sketch,

                  here comes some more work for "lazy people" beyond
                  my earlier interpretations of Newton's layout:
                  http://launch.groups.yahoo.com/group/tuning/message/72161
                  http://launch.groups.yahoo.com/group/tuning/message/73536
                  Provenace:
                  "
                  Newton's autograph drawing is dated November 1665:
                  Literature reference:
                  The original manuscript is located at:
                  Cambridge Univ.Lib. Signature: Ms.Add.4000,fol.105v
                  "
                  Has anybody in that group access to that original or an copy of that,
                  and could please offer an reprint to us group-members here in that forum?

                  All i know about N's diagramm bases barely
                  on the views of my old colleauge and friend and expert in that field:
                  http://en.wikipedia.org/wiki/Mark_Lindley
                  He refers to N's delineation
                  in his encyclopecic ripely in depth article
                  'tuning & temperature' standard reference article:
                  "Stimmung und Temperatur"
                  within the book:
                  (F. Zaminer, Editor, Geschichte der
                  Musiktheorie, Vol. 6: Hören, Messen und Rechnen in der Frühen Neuzeit
                  (Wissenschaftliche Buchgesellschaft, 1987) pp.205-210
                  Sadly -sorry i'm afraid-
                  in print that's available solely in German :-(
                  How unfortunately!

                  Lindley's scholar article reproduces also an similar
                  even earlier "horogramm"
                  -appearently an forerunning predecessor-
                  made by
                  http://en.wikipedia.org/wiki/Ren%C3%A9_Descartes
                  that appeared in his famous:
                  "1618. Compendium Musicae.
                  A treatise on music theory and the
                  aesthetics of music written for Descartes's
                  early collaborator Isaac Beeckman."
                  http://en.wikipedia.org/wiki/Isaac_Beeckman

                  R. Descartes reports about that encounter when meeting Beekman personally:
                  http://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/descartes.html
                  " Around 1618 I believe, I began to study mathematics once again under
                  the Dutch scientist Isaac Beekman, who I had met one day walking
                  through the streets when he translated a Dutch placard for me that
                  turned out to be a math problem."

                  Probably D. refers in his advanced blueprint
                  -written already at age of 22 years-
                  to the questionable treatment of the subject by an precursor:
                  http://en.wikipedia.org/wiki/Robert_Fludd
                  's outlines:
                  http://www.celestialmonochord.org/log/images/celestial_monochord.jpg
                  or if you prefer the same pic in somewhat higher resolution:
                  http://www.imaginatorium.org/books/monochd.gif
                  Attend God's divine hand out of the clouds that tunes the string.
                  http://www.bach1722.com/
                  'Il temperamento di Dio' == ???"God's temperament"???
                  including an Italian foro di propagare:
                  http://www.edumus.com/forum/read.php?21,54338,page=1
                  Parlare a vuoto di un agrumento putativo inprofessionale:
                  Che palle!
                  .......
                  ......
                  .....
                  ....
                  ...
                  ..
                  .

                  But better let's retrun back at that point of no return
                  -away from that nonsensial layman's flubdub-
                  toggeling reverse to our's historically real serious D&N:

                  Beyond Fludd's heptatonics
                  Descartes explains there in his compedium en detail
                  -alike in Newton's later refinement too-
                  how to obtain the 53-commata scale from shifting
                  hexachords by 4ths(4:3) from hard(durum) to soft(molle) modes
                  by changeing the keys: G-C-F in refernce to
                  the incomlete C-major scale,
                  because in coeval hexachords the pitch-class of 'B' is left out there.

                  http://en.wikipedia.org/wiki/Hexachord

                  In order to understand N's conceptual design
                  read the 5 concentric circles from inside to outside as scales over:
                  outest = 44Bb(ut,) > 21F(ut) > 53=0C(Ut) > 31G(UT) > 9(UT') = innerst

                  so that Descartes's C-major 'scala-naturalis' is located in the
                  middle at the center of N's 5 concentric circles:
                  That old gamut (GAMMA-UT) is labeld in today's modern concept as:

                  1. extended tonic durum-hexachord C-major-Mixolydian 7-tone scale:
                  consisting of 2
                  http://en.wikipedia.org/wiki/Tetrachord
                  s and an major-tone 9:8 intbetween both of them:
                  Yilding an C-major
                  http://en.wikipedia.org/wiki/Tonic_%28music%29
                  scale with legenda:

                  C=0Ut=1:1 +8 8Re=10:9 +9 17Mi=5:4 +5 22Fa=4:3 first-tetrachord22Fa4:3
                  22Fa=4:3 +9 31Sol=3:2 major-tone9commata inbetween both tetrachords
                  31Sol=3:2 +8 39La +5 44--16:9 +9 (53=0)Ut'=2:1 second-tetrachord

                  On the one hand:
                  Step from that one cirle nearer to the main focal point,
                  that corresponds to an transition an 5th upwards into the:
                  http://en.wikipedia.org/wiki/Dominant_%28music%29
                  2. dominant hard(sharp)-hexachord G-major-Mixolydian 7-tone scale
                  with legenda:
                  G=31UT=3:2 +8 39RE=5:3 +9 48MI=15:8 +5 53=0FA=1:1 first-tetrachord
                  53=0FA=1:1 +9 9SOL=9:8 major-tone 9:8 consists of 9 commata steps
                  9SOL=9:8 +8 17LA=5:4 +5 22--=4:3 +9 31UT second-tetrachord

                  On the other hand:
                  By stepping in reverse direction by
                  one 4th (4:3) from the middle-C-major cicle
                  into outwards direction we yield an change of key into the:
                  http://en.wikipedia.org/wiki/Subdominant
                  3. subdominant moll(soft)-hexachord F-major-Mixolydian 7-tone scale:
                  Legenda:
                  F=22ut=4:3 +8 30re40:27 +9 39mi=5:3 +5 44fa=16:9 first-tetrachord
                  44fa=16:9 +9 53=0sol=1:1 major-tone 9:8 of 9 commatas
                  0sol1:1 +8 8la=10:9 +5 13--=50:27 +9 22ut4:3

                  So far N's scribble agrees in the 3 innerst circles fully with
                  Descartes's elaborated original scheme.
                  But beyond D's hexachords in barely the kernel keys
                  F > C > G
                  N. delivers additional
                  Bb > (F > C > G) > D
                  in extending D's range from the
                  double-subdomiant=Bb>(subdom.=F > tonic=C > dom.=G)>double-dom.=D
                  by 2 additional outer scales.
                  N. also extends by the note '--' the classial Hexachords
                  into to the mixolydian scale, with distances in commatas:
                  C +8 D\ +9 E\ +5 F +9 G +8 A\ +5 Bb +9 C'
                  instead of the todays common usual 'Ionian'-C-major scale:
                  C +9 D +8 E\ +5 F +9 G +8 A\ +9 B\ +5 C'

                  N's outest circle exterior corresponds analogous to the
                  double-subdominant-flattend Bb-major mixolydian heptatonic scale:
                  Legenda:
                  Bb=44ut, 52re, 8mi, 13fa, 22sol, 30=la, 35--, 44ut,=Bb

                  And finally:
                  respectively inside the interiorst cycle represents
                  double-dominant-sharpend D-major mixolydian hepatonics:
                  Legenda:
                  D=9UT' 17RE' 26MI' 31FA' 40SOL' 48LA' 53=0--' 9UT'=D
                  in comleting all 5 cases.

                  hope that helps to illustrate and understand
                  N's division of the octave into 53.
                  when transferred into modern terminology.
                  from traditionally medivial hexachordian terminology
                  over N's personal Bb-F-C-G-D "Mixolydian" concept
                  over of Rameau's "Ionian" major 'triad' F-C-G

                  Rameau's terms for an change of the actual key:

                  C=tonique 1:1
                  with its 2 transpositons:
                  F=sous-dominant by an 4th 4:3, labeled by an b-sign(molle)in the score
                  C=super-dominant by an 5th 3:2, corresponds an #-sign(durum)in score

                  resulting finally in the point of:
                  http://en.wikipedia.org/wiki/Hugo_Riemann
                  's
                  http://en.wikipedia.org/wiki/Tonnetz


                  when applied to the good-old song,
                  http://en.wikipedia.org/wiki/Ut_queant_laxis
                  then N's drawing represents simply
                  the transpositions of that chant through
                  the pentard of keys:
                  outside = Bb > F > C > G > D = inside

                  For HIP experts:
                  Attend N. starts in his original considerations at GAMMA-UT = 0 == 53
                  http://tede.ibict.br/tde_busca/processaArquivo.php?codArquivo=354
                  http://jrma.oxfordjournals.org/cgi/reprint/115/2/145.pdf
                  (Warning: That both do deliver exhausting depletive informations!)

                  That can be easier expressed in todays modern terms by the
                  simplifications over the last few centuries:
                  http://en.wikipedia.org/wiki/Solf%C3%A8ge

                  Here ends my historically excursus,
                  into the conservative N's old mediaevel world,
                  his Baroque coevals considered his way of treating
                  the subject as an antediluvian-reactionar-fossil
                  relict of the middle-age, but still worth to read
                  for the purpose of inspiration of 53 refinements...

                  ...just let me conclude in quoteing an prince of poets laureats:
                  J.W.Goethe
                  http://www.zitate-online.de/literaturzitate/allgemein/16754/was-du-ererbt-von-deinen-vaetern-hast-erwirb.html
                  „Was du ererbt von Deinen Vätern hast, erwirb es, um es zu besitzen."
                  Translation:
                  "What you have (once) inherit from Yours (late) fathers,
                  retrieve it (yourself again anew), in order to possess it
                  (henceforward as yours own personal belongings)."

                  Quest:
                  Which native english speaker in that group here can offer
                  for us an more appropriate and accurate translation,
                  that is more apt, than my humble attempt?

                  With sorry for beeing so much verbose
                  Yours Sincerely
                  A.S.
                • threesixesinarow
                  ... wrote: http://mto.societymusictheory.org/issues/mto.93.0.3/ mto.93.0.3.lindley7.gif http://en.wikipedia.org/wiki/
                  Message 8 of 16 , Jul 17, 2008
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                    --- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
                    wrote:

                    http://mto.societymusictheory.org/issues/mto.93.0.3/
                    mto.93.0.3.lindley7.gif

                    http://en.wikipedia.org/wiki/
                    Talk:53_equal_temperament#Newton.27s_unpublished_manuscripts.3F

                    >
                    > [Lindley] refers to N's delineation
                    > in his encyclopecic ripely in depth article
                    > 'tuning & temperature' standard reference article:
                    > "Stimmung und Temperatur"
                    > within the book:
                    > (F. Zaminer, Editor, Geschichte der
                    > Musiktheorie, Vol. 6: Hören, Messen und Rechnen in der
                    > Frühen Neuzeit (Wissenschaftliche Buchgesellschaft,
                    > 1987) pp.205-210

                    Do you think you can amend the statement on Wikipedia
                    so it more accurately reflects how he refers to Newton's
                    treatment of 53 equal in this article?

                    > Lindley's scholar article reproduces also an similar
                    > even earlier "horogramm"
                    > -appearently an forerunning predecessor-
                    > made by [René Descartes]
                    > that appeared in his famous:
                    > "1618. Compendium Musicae.

                    Neat, and there's more than one diagram.

                    Musicae compendium / Renati Des Cartes (1695)
                    http://gallica.bnf.fr/notice?N=FRBNF37240052

                    Excellent compendium of musick with necessary and
                    judicious animadversions there upon : by a person of
                    honour / Renatus Descartes (1653)
                    http://gallica.bnf.fr/notice?N=FRBNF37240054

                    Clark
                  • Andreas Sparschuh
                    ... Dear Clark and all others that seek a deeper understanding of Newton, sorry, i m afraid, all i know about N s53 stems barely from one source alone: Mark
                    Message 9 of 16 , Jul 17, 2008
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                      --- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@...> wrote:
                      >
                      > Do you think you can amend the statement on Wikipedia
                      > so it more accurately reflects how he refers to Newton's
                      > treatment of 53 equal in this article?
                      >
                      Dear Clark and all others that seek a deeper understanding of Newton,

                      sorry, i'm afraid,
                      all i know about N's53 stems barely from one source alone:

                      Mark Lindley critizises Newton for an supposed neglect of the
                      schismatic 5*3^8:2^15 subdivision,
                      which consequently would result in an double allocation
                      for the 3rd at step 17 out of 53 into:

                      http://en.wikipedia.org/wiki/Interval_(music)
                      "A schismic major third is a schisma different from a just major
                      third, eight fifths down and five octaves up, F♭ in C."

                      L. assumes that N. had no idea of discerning properly inbetween:
                      1.) 2^13:3^8 = 8192:6561 ~384Cents an Pythagoren diminshed 4th
                      2.) 5:4 ~386Cents JI 3rd
                      in presuming that N. wasn't clear aware of the schismatic concept:
                      http://en.wikipedia.org/wiki/Schismatic_temperament
                      "Mark Lindley and Ronald Turner-Smith argue that schismatic tuning was
                      briefly in use during the late medieval period."
                      Fully agreed.

                      For reassessing the uncertainty about N's view,
                      i simply need more historically checkable facts for verification,
                      that i do suspect in N's unpublished autograph.

                      For an apt review i would have to study N's own concrete ratios
                      myself:
                      How do they fit to N's drawing?

                      In order to stay honest and fair against N.
                      Without such an verification in reference to N's original
                      i do hesitate to repeat again L's complains
                      of finding fault in N's considerations,
                      that i want accept preliminary only tentative with reservation,
                      barely with a tiny grain of salt.

                      But unfortunately L. presents there no concrete numerical-values
                      in his (supposeable overly hypercritically?) evaluation:

                      At the moment my situation consists still
                      in a gap in my knowledge about N's real data:
                      Simply i know to less about it.

                      All i can say about it hitherto is precious few:
                      From L. discussion arises the question,
                      if N. actually refers to 53-EDO at all,
                      or if he had something complete rational in his mind,
                      when he penned down his 5 intersecting mixolydian hexachords?

                      There remain some open questions:

                      Weather meant N. really Holder's 53-EDO?:

                      The schisma of ~2Cents refers to an finer resolution than
                      the more coarse 2^(1:53) ~22.6415...Cents
                      http://en.wikipedia.org/wiki/Holdrian_comma

                      I really don't know if N. overtook the 2^(1:53) concept from
                      http://en.wikipedia.org/wiki/William_Holder
                      or vice versa just or was it just the other way around?

                      Whom of that invented 53-EDO?

                      Hence, please understand my caution about careless
                      overhasty jumps into premature rash half-cocked conclusions:
                      Without having read N's original myself,
                      i'm not able to decide whether M. Lindley's objections
                      against N's representation is justified or not?

                      But if Mark says so, his review has to be taken serious.

                      My conclusion:
                      All i can report at the moment:
                      My lacking personal insight into N's autograph and my respcet
                      for his genius forbids me to invent disputable hypothesises about
                      his work:
                      http://en.wikipedia.org/wiki/Hypotheses_non_fingo
                      >
                      > > Ren� Descartes, 1618. Compendium Musicae.
                      >
                      > Neat, and there's more than one diagram.

                      > Musicae compendium / Renati Des Cartes (1695)
                      > http://gallica.bnf.fr/notice?N=FRBNF37240052
                      > http://gallica.bnf.fr/notice?N=FRBNF37240054

                      Lindley's copy originates exactly from the horogramm there in that.
                      >
                      But
                      http://en.wikipedia.org/wiki/Edmond_de_Coussemaker
                      's
                      "Scriptores de musica medii aevi (4 delen) (1864-1876)"
                      contains appearently an other one even older
                      more archaic-looking "horogramm" forerunner model of D's & N's.

                      in staying tuned what N. really wrote or may-be even not
                      time will show that
                      Yours Sincerely
                      A.S.
                    • Andreas Sparschuh
                      - --- In tuning@yahoogroups.com, threesixesinarow wrote: Hi all lovers of old slogans, probably ... referred to D s & N s
                      Message 10 of 16 , Jul 17, 2008
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                        - --- In tuning@yahoogroups.com, "threesixesinarow" <CACCOLA@> wrote:
                        Hi all lovers of old slogans,

                        probably
                        > J.W.Goethe
                        referred to D's & N's
                        http://en.wikipedia.org/wiki/Theory_of_Colours
                        in reference to n's quote:
                        http://en.wikipedia.org/wiki/Standing_on_the_shoulders_of_giants
                        "Pigmaei gigantum humeris impositi plusquam ipsi gigantes vident"
                        'If I have seen a little further it is by standing on the shoulders of
                        Giants.'
                        http://www.aerospaceweb.org/question/history/q0162b.shtml

                        That sounds in G's version:
                        http://www.zitate-online.de/literaturzitate/allgemein/16754/was-du-ererbt-von-deinen-vaetern-hast-erwirb.html
                        "Was du ererbt von Deinen Vaetern hast, erwirb es, um es zu besitzen."
                        'Retrieve yours fahters inheritance,
                        for possessing that as internalized own property'

                        G's dictum can i.m.h.o. also be referred well to D's & N's tunings
                        A.S.
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