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New septenarius for a'=440 Hz, was Re: Werckmeister's Septinarius temperament

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  • Andreas Sparschuh
    ... right, in order to get rid of the oversharp wolfs in W s original stringlength numbers, alike in his famous #3 the quaternarius has only 4 flattend and 8
    Message 1 of 26 , Apr 13 11:24 AM
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      --- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
      >
      >
      > Comments below!
      >
      > > a'440Hz a220 A110
      > > e'330 e165
      > > b'495
      > > 1485/1484 f#"742 f#'371
      >
      > Instead of 496 and 372. This avoids having a wide e-b fifth and is
      > better for G major and D major.

      right, in order to get rid of the oversharp wolfs in W's original
      stringlength numbers, alike in his famous #3 the 'quaternarius'
      has only 4 flattend and 8 pure 5ths:
      A E B>F# C# G# Eb Bb F C>G>D>A

      >
      > > 1113/1112 c#"556 c#'278 c#139
      > > 417/416 g#208 G#104
      > > eb'312 eb156 Eb78
      > > bb'468 bb234 Bb117
      > > f'351
      > > 1053/1052 c"526 c'263/262 131
      > > g'393/392 196 98 49=7*7
      >
      > I don't think this is so good, you have C-G tempered by 262/263
      > which is about 1/3 comma...
      >
      ...flattend than a pure 5th: 3/2.
      So far about that strongest detuned 5th C>G.
      Consider accordingly the belonging 3rd C>E in the major-chord C>E>G:
      with compareable sharpness:
      e165 e'330 e"660 e'"1330/1315=5*c"263
      shortening by common factor 5 yiels
      a tempering of C>E about 264/265,
      so that the C-major chord consists in

      C(1/1)-E*(265/264)-G*(262/263)
      or
      4:(5*(265/264)):(6*262/263)
      instead barely 4:5:6 without the idea of tempering
      3rds and 5ths in the same range of magintude.

      Many 'experts' do recommend to sharpen the 3rds about the amount in
      amplitude alike the corresponding 5ths become flattened, so that the
      beatings of 3rds and 5ths beat almost the same in reverse directions.
      Skilled organ-builders use that effect in order to demonstrate
      that their robust organs survive even such impressive resonances.
      It appears that already
      http://en.wikipedia.org/wiki/Arnolt_Schlick
      knew that old well-known tuning-trick/method in his instructions.

      > Better to have 315/350 f'350 f175 and 525/524 c''524 c262 131 ... ?
      Good idea, if you intend to stay nearer to W's original version,
      when somehow aiming to approximate "ET" however.

      >
      > Then C-G is pure and
      makes that only sense according the above demands
      when C-E is also pure chosen?

      >
      > >
      Correction of: !septenarius440Hz.scl

      > > 351/263 ! F# [should be 271] for Gene: 371 is the correct pitch.

      > or:
      >
      > !septenarius440Hzmk2.scl
      > !
      > TD's septenarius @ middle c'=262Hz or a'=440Hz
      > !
      > 12
      > !
      > 278/262 ! C# short 138/131
      > 294/262 ! D short 147/131
      > 312/262 ! Eb short 156/131
      > 330/262 ! E short 165/131
      > 350/262 ! F short 175/131
      > 371/262 ! F#
      > 393/262 ! G
      > 416/262 ! G# 208/131
      > 440/262 ! A 220/131
      > 468/262 ! Bb 234/131
      > 495/262 ! B
      > 2/1

      The pure middle c' in reference to a'=440Hz becomes in the just case
      440Hz*3/5 = 264Hz. Hence i do prefer the nearer 263Hz instead
      yours lower 262Hz, which appears just a little bit to flat lowered
      in my personal taste, especially when having just intonation
      in mind or ear instead the virtual "et".
      Most professional and skilled tuners do to keep
      the frequent keys with few accidentials somehow purer than
      the less used keys with many accidentials, so that the
      strange keys got more pythagorean 3rds, by purer
      or even just pure 5ths inbetween them.

      >
      > (cf. 12ET frequencies: 261.6, 277.2, 293.7, 311.1, 329.6, 349.2,
      > 370.0, 392.0, 415.3, 440.0, 466.2, 493.9 ...)
      those irrational numbers are far to complicated for solving the
      problem. Who needs the advanced precision of 4 decimal digits
      in the octave from c' to c"?

      But if you want to approach "et" whatsoever, then 262 would be the
      better choice for converging "et" as the above approximation suggest.

      >
      > Can one use the nice ratio 63/50 = 1.26 to build a 'septenarian' >near-
      > equal tuning?
      I.m.o: the nearer one draws to approach "et",
      the most frequently used 3rds get to much worse detuned

      Luckily nobody can tune irrational intervals in practice,
      so that the worsest case: "et" remains barely a theoretically fiction,
      excluded from real implementation on a real sounding instrument.
      Simply try out how well can you reproduce by yours ears:
      on the one hand:
      a pure 5th ratio 3:2=1.5 ~702cents
      and on the other hand:
      sqrt(2) = 600 Cent "et"-tritous, that's geometrically interpreted:
      http://mathworld.wolfram.com/PythagorassConstant.html

      Experimental result:
      There is no psychoacusitcally evidence for departening
      the ratio of a 5th 3:2 for the benefit of sqrt(2) ET-tritone.
      Quoting Herrmann Helmholtz: "The ear prefers simple ratios."


      > eg Eb-G = 150:189 ...
      > via Eb150 Bb225 (675) F337 (1011) C505 (504) G378=189

      But by that procedure one does also loose to much of the 'Baroque'
      key-characteristics.
      http://www.societymusictheory.org/mto/issues/mto.95.1.4/mto.95.1.4.code.html
      http://de.wikipedia.org/wiki/Tonartencharakter
      >
      > then continue:
      > ... (567) D283 (849) A424=212 (636) E635 (634) B951 (2853/2848)
      > F#356=178 C#267 (801) G#400 Eb300
      >
      > seems to work nicely at late Baroque pitch levels - only three pure
      > fifths between Eb-Bb, G#-Eb, F#-C#.
      >
      hmm, is it still apt to call that 'baroque'-style?
      As far as i understood late "Baroque" post-meantone instructions:
      There i found a tendency to keep the 5ths inbetween the accidentials
      F#-C#-G#-Eb-Bb
      purer more pure (within the upper black keys on the piano)
      than in the ordinary F-C-G-D-A-E-B: that got generally more tempering.

      In extreme form i do start from my prototype model.
      The procedure consists in a chain of 11 almost pure 5ths,
      that contains the JI pitches, but also schismic Pythagorean-enharmonics:

      Here the chain
      F-A-C-E-G-B-D
      are all 3 pure major 4:5:6 chords
      in exact beatless just proportions:

      A 440. 220 110 55
      E 165
      B 495.
      F# 1485
      4455 C# 4454 2227
      6681 G# 6680 3340 1670 835
      2505 Eb 2504 1252 626 313.
      Bb 939
      2817 F 2816 1408 704 352. 176 88 44 22 11
      C 33
      G 99
      D 297. / 296 148 74 37
      111 A 110 55

      A E B F#~C#~G#~Eb Bb~F C G D~~~~~~~~~~~~~~~~~~~~~~~~~~~~A

      The schisma 32805/32768=5*3^8/2^15=
      (4455/4454)(6681/6680)(2505/2504)(2817/2816)
      is tempered out by the subdivsion in to that
      product of 4 superparticular factors.
      Respectively the
      SC=81/80=(297/296)(111/110) into 2 parts at one @ D>A 40:27

      Rearranging same pitches in ascending order yields:

      C' 264Hz middle-C
      C# 278.375
      D' 297
      Eb 313
      E' 330
      F' 352
      F# 371.25
      G' 396
      G# 417.5
      A' 440Hz reference pitch
      Bb 469.5
      B' 495
      C" 528

      so far about the 11 other frquencies that i percieve instantly
      also in mind immediatley when hearing a 440Hz tuning-fork by ear,
      or simpy when imagening that pitch-levels enwraped when reading
      musical scores in any fitting tuning.

      schismatic_just440Hz.scl
      !
      sparschuh's-schisma-subdivision(4455/4454)(6681/6680)(2505/2504)(2817/2816)
      !
      2227/2112 ! C#
      9/8 ! D
      313/264 ! Eb
      5/4 ! E
      4/3 ! F
      45/32 ! F#
      3/2 ! G
      835/528 ! G#
      5/3 ! A=440Hz
      313/176 ! Bb
      15/8 ! B
      2/1

      But, how about that almost similar alternative one at the moment on my
      piano?

      A 440. 220 110
      330 E 329.
      B 987
      2961 F# 2960 1480 740 370. 185
      C# 555
      1665 G# 1664 832 416. 208 104 52 26 13
      Eb 39
      Bb 117
      F 351.
      1053 C 1052 526 263.
      789 G 788 394. 197
      591 D 590 285./284 147
      441 A 440. (or 3*285=885 A 880 440.)

      with strongest tempering @ D>A: 885/880=(285/284)(441/440)=177/176
      but still less than SC^(1/2) ~161/160 or ~162/161,
      hence rather tolerable than
      the ancient Erlangen-monochord or Kirnberger#1,
      that charge a full SC on D>A alike the above 'schismatic_just.scl'.

      So far my reccomendation for those who prefer to stay nearer at JI
      than to the i.m.o. over-detuned "ET", that i do meanwhile consider as
      outdated intuneable fiction.

      sparschuh_gothic_style440Hz.scl
      !
      12
      !
      555/526 ! C# 277.5 Hz
      285/263 ! D
      312/263 ! Eb
      329/263 ! E
      351/263 ! F
      370/263 ! F#
      394/263 ! G
      416/263 ! G#
      440/263 ! A reference-pitch 440Hz
      468/263 ! Bb
      987/526 ! B 493.5 Hz
      2/1

      on the keys

      +-----------
      | C 263 middle-C
      +--|277.5=C#
      | D 285
      +--|312=Eb
      | E 329
      +-----------
      | F 351
      +--|370=F#
      | G 394
      +--|416=G#
      | A 440
      +--|468=Bb
      | B 493.5
      +-----------
      | C'526
      &ct.

      If you dont't like any of that, it's up to you to create yours own
      personal version, according yours private preferences.

      A.S.
    • Andreas Sparschuh
      ... agreed, hence i do return to W s original 131. ... also right, hence so the resulting ratios get even more simple: The calculations benefit from that by
      Message 2 of 26 , Apr 17 11:49 AM
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        --- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
        >
        > > 1113/1112 c#"556 c#'278 c#139
        > > 417/416 g#208 G#104
        > > eb'312 eb156 Eb78
        > > bb'468 bb234 Bb117
        > > f'351
        > > 1053/1052 c"526 c'263/262 131
        > > g'393/392 196 98 49=7*7
        >
        > I don't think this is so good, you have C-G tempered by 262/263
        > which
        > is about 1/3 comma...
        agreed, hence i do return to W's original 131.
        >
        > Better to have 315/350 f'350 f175 and 525/524 c''524 c262 131 ... ?
        >
        also right, hence so the resulting ratios get even more simple:
        The calculations benefit from that by less computational0 overhead.

        Follow the classical way:
        start traditional @ pitch-class GAMMA=G alike in:

        http://www.celestialmonochord.org/log/images/celestial_monochord.jpg

        GG 49 := 7*7 (GAMMA-ut, the empty string in the picture)
        D 147
        3D441 > a'440 a220 A110 AA55Hz=the AA-string of a double-bass
        e 165
        3e495 > b'494 b247
        3b741 > f#"740 f#'370 f#185
        c# "555
        3c#"1665 > g#"'1664 g#"832 g#'416 g#208 G#104 GG#52 GGG#26 GGGG#13
        EEb 39
        Bb 117
        3Bb351 > f'350 f175
        3f525 > c"524 c'263 c131
        3c393 > g'392 g196 G98 GG49=7^2 returned

        !septenarius_GG49Hz.scl
        sparschuh's version @ middle-c'=262Hz or a'=440Hz
        12
        !absolute pitches relativ to c=131 Hz
        555/524 ! c# 138.75 Hz
        147/131 ! d
        156/131 ! eb
        165/131 ! e
        175/131 ! f
        185/131 ! f#
        196/131 ! g
        208/131 ! g#
        220/131 ! a 440Hz/2
        234/131 ! bb
        247/131 ! b
        2/1


        That results on my old piano in the first/lowest octave:

        AAA 27.5 Hz lowest pitch, on the first white key on the left side
        BBBb29.25 next upper black key
        BBB 30.875 http://en.wikipedia.org/wiki/Double_bass "at~30.87 hertz"..
        CC_ 32.875
        CC# 34.6875 := c"555/16
        DD_ 36.75
        EEb 39 := GGGG#13Hz*3
        EE_ 41.25 ..."E1 (on standard four-string basses) at ~41.20 Hz
        FF_ 43.75
        FF# 46.25
        GG_ 49 := 7*7 Werckmeister's/Scheibler's initial septimal choice
        GG# 52 = GGGG#13Hz*4
        AA_ 55 = 440Hz/8 ; 3 octaves below Scheibler's choice

        http://mmd.foxtail.com/Tech/jorgensen.html
        #133: "Johann Heinrich Scheibler's metronome method of 1836"

        http://www.41hz.com
        "41 Hz is the frequency of the low E string on a double bass or an
        electric bass." if it has none additional 5th string
        for midi(B0)=BBB 30.875 Hz an 2nd above AAA 27.5 Hz,
        the lowest A on the piano, without attending or even careing
        "string-imharmonicty"
        http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN0000760000S1000S22000004&idtype=cvips&gifs=yes

        Consider the 3rds qualities in violins empty sting G-D-A-E block order:

        1: G > B > Eb> G
        2: D > F#> Bb> D
        3: A > C#> F > A
        4: E > G#> C > E

        1: G > B > Eb > G.
        absolute analysis:
        GG 49.
        5*49 = 245 < b'247 < 248 124 62 31
        5*31 = 155 < eb156 Eb78 EEb39
        5*39 = 195 < g196 G98 GG7*7.
        relative diesis 128/125 subpartition:
        G 123.5/122.5 B 96/95 Eb 196/195 G
        G ~14.1 cents B ~18.1 Eb ~ 8.86c G

        2: D > F# > Bb > D.
        abs:
        d147. < 148 74 37
        5*37 = f#185 < 186 93
        5*31*3=3*155 < 156*3 78*3 39*3 = Bb117
        5*39*3=3*195 < 196*3 98*3 49*3 = d147.
        rel:
        D 148/147 F# (1/9+85)/(84+1/9) Bb 196/195 D
        D ~ 11.4c F# ~ 20.5 cents Bb ~ 8.86cents D

        3: A > C# > F > A.
        abs:
        AA55. A110 < 111 = 37*3
        5*111= c#"555 = 5*111 < 112*5 56*5 28*5 14*5 7*5
        5*35 = f175 < 176 88 44 22 11
        5*11 = AA55.
        rel:
        A 111/110 C# 112/111 F 176/175 A ; with all 3 factors superparticular
        A ~ 15.7c C# ~ 15.5c F ~ 9.86c A

        4: E > G#> C > E.
        abs:
        e165. < 166 83
        5*83 = 415 < g#'416 g#208 G#104 GG#52 GGG#26 GGGG#13 ยงยง
        5*13 = 65 130 < c131 < 132 66 33
        5*33 = e165.
        rel:
        E (6/7+118)/(117+6/7) G# 131/130 C 132/131 E
        E ~ 14.6 cents ~ G# ~ 13.3 cents C ~ 13.2c E
        ยงยง GGGG# 13 Hz has negative "midi"-index G#_-1,
        which midi-keyboard supports negative key indices?


        Summary:
        3rds martix in "Cents"
        5ths in top>down order
        3rds in left>right direction respectively:

        1: G 14.1 B_ 18.1 Eb 8.86 G
        2: D 11.4 F# 20.5 Bb 8.86 D
        3: A 15.7 C# 15.5 F_ 9.86 A
        4: E 14.6 G# 13.3 C_ 13.2 E

        Conversely "ET" detunes all 3rds about the same amount:
        (128/125)^(1/3) = ~13.7Cents or ~127/126,
        Attend that: the "septenarius" fits therefore better than ET to
        horns and trumpets in Eb,Bb & F, with inherent natural
        3rds Eb>G, Bb>D, & F>A that turn out less than 10Cents out of tune
        in the septenarius case.

        > (cf. 12ET frequencies: 261.6, 277.2, 293.7, 311.1, 329.6, 349.2,
        > 370.0, 392.0, 415.3, 440.0, 466.2, 493.9 ...)

        In the "ET" case, it is difficult to resolve the
        septenarian root-factors above alike:
        11,13,31,37 & 83 below the well known 3,5 & 7 limits.
        >
        > Can one use the nice ratio 63/50 = 1.26 to build a 'septenarian' >near-
        > equal tuning?
        on the one hand is:
        (63/50)(4/5)=126/125
        but but on the other side
        the diesis 128/125=(128/127)(127/126)(126/125)
        contains 3 factors.
        hence:
        (63/50)^3 = 2.000376... > 2/1
        overstretched octave
        or as superparticular ratio:
        ((63/50)^3)/2= 250047/250000=(7/47+5320)/(5319+7/47)
        ~1/2 per mille
        but a better approximation of the octave delivers 127/126
        the factor in the middle:
        ((5 / 4) * (127 / 126))^3 = ~1.99999802............




        > eg Eb-G = 150:189 ...
        > via Eb150 Bb225 (675) F337 (1011) C505 (504) G378=189
        >
        > then continue:
        > ... (567) D283 (849) A424=212 (636) E635 (634) B951 (2853/2848)
        > F#356=178 C#267 (801) G#400 Eb300
        >
        > seems to work nicely at late Baroque pitch levels - only three pure
        > fifths between Eb-Bb, G#-Eb, F#-C#.
        >
        that is expanded:
        A 424 212
        3A636 > E635 > 634 317
        B951 > 950 475
        3*475=1425 > F#1424 712 356 178 89
        C# 267
        801 > G# 800 400 200 100 50 25 Werckmeister's tief-Cammerthon 400Hz
        Eb 75
        Bb 225
        675 > F 674 337
        1011 > C 1050 505 > 504 252 126 63
        G 189
        567 > D 566 283
        849 > A 848 424

        recombining that 5ths-circle in ascending ordered pitches yields:
        C 252.5 Hz middle_C
        C#267
        D 283
        Eb300
        E 317.5 or better 317?
        F 337
        F#356
        G 378
        G#400 Beekman's, Descartes's, Mersenne's & Sauveur's standard-pitch
        A 424
        Bb450
        B 475.5 or better 475?
        C'505

        T.D. remarked already in his numbers inbetween:

        >B951 (2853/2848)
        > F#356

        that there appears an unsatisfactory irregular gap:
        of 2853/2848 = 570.6/569.6 = (951/950)(1425/1424)
        induced by the choice of
        E 635 > 634 317 instead
        635 > E 634 317
        That results in the above none-integral superparticular ratio bug.

        Hence i do suggest to replace
        by the tiny changes
        1: E 635-->>>634
        2: B 951-->>>950
        in order to fix the bug.

        so that now all 5th-tempering steps become integral superparticular
        ratios, without any exception:

        A 424 212 106
        3A=318 E 317 instead formerly 317.5
        3E=951 B 950 475 instead formerly 951 475.5
        3B=1425 F# 1424 712 356 178 89
        &ct.
        the rest of the circle remains unchanged.
        Is that ok?


        Analysis of the:
        3rds sharpness, -how much wider than 5/4-
        per diesis subpartition into superparticular factors,
        so that the product of 3 tempered 5ths results an octave
        in each of the 4 blocks:

        1: G > B > Eb > G.
        abs:
        G378. 189 > 190 95
        5*95 = B 475 < 480 240 120 60 30 15
        5*15 = Eb75
        5*75 = 3*125 < 126*3 = G378.
        rel. 2^7/5^3=128/125=
        G 190/189 B 160/159 Eb 126/125 G remember (63/50)(4/5)=126/125
        ?or formerly in the original version:
        ?G378. 189 > 190 95
        ?5*95 = 475 950 < B951 < 960 480 240 120 60 30 15
        ?......
        ?rel. 2^7/5^3=128/125=
        ?G (190/189)(951/950) B 320/317=(2/3+106)/(105+2/3) Eb 126/125 G
        That appears i.m.o. much more complicated than my suggested change.

        2: D > F# > Bb > D.
        abs:
        D283. < 284 142 71
        5* 71 = 355 < F#356 178 89 < 90 45
        5* 45 = Bb225 < 226 113
        5*113 = 565 < D556 283.
        rel. 128/125=
        D (284/283)(356/355) F# 90/89 Bb (226/225)(556/565) D

        3: A > C# > F > A.
        abs:
        A424. 212 106 53
        5* 53 = 265 < C#267 < 268 134 67
        5* 67 = 335 < F 337 < 338 169
        5*169 = 845 < A 848 424.
        rel: 128/125=
        A 133.5/123.5 C# (268/267)(168.5/166.5) F >>>
        >>> F (338/337)((2/3+282)/(281+2/3)) A

        4: E > G# > C > E.
        E 317. < 320 160 80 40 20
        5* 20 = G#100 < 101
        5*101 = C 505 < 506 253
        5*253 = 1265 < E 1268 634 317.
        rel: 128/125=
        E (2/3+106)/(105+2/3) G# 101/100 C (506/505)/(422.666.../421.666...) E
        ?or formerly
        ?E 635? > 640 320 .....
        ?....&ct. alike above...
        ?....
        ?5*253 =1265 < 1270 635?

        try to find out similar improvements in order to reduce the ratios
        to less complicated proportions

        have a lot of fun in whatever tuning you do prefer
        A.S.
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