--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> J.Smith asked:
> > I'd like to know what all the fuss is about....??
The history behind is documented in:
http://en.wikipedia.org/wiki/Well-Tempered_Clavier
there under the topic: " What tuning did Bach intend?"
>
> the circle of fifths:
>
> C 0.000
> G 698.045
> D 196.090
> A 894.135
> E 392.180
> B 1094.135
> F# 596.090
> C# 98.045
> G#/Ab 798.045
> Eb 298.045
> Bb 998.045
> F 501.955
>
critial review in:
http://em.oxfordjournals.org/cgi/content/full/34/4/613?
maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&fulltext=ortgies&searchid=1&
FIRSTINDEX=0&resourcetype=HWCIT


> So the fifths tempered narrow by 1/6 of a comma are 698.045 cents,
and
> the fifths tempered narrow by 1/12 of a comma are 700.000 cents.
The
> single fifth Bb-F is actually tempered wide by 1/12 of a comma,
which
> works out to be 703.910 cents.
most experts consider L's modern 1/12 PC step-wide as to coarse
especially for J.S. Bach, hence historically inacceptable.

>
it's barely an approximative reinterpretation of the
much finer desigend original discovery version of 1999:
http://www.strukturbildung.de/Andreas.Sparschuh/
Contains a copy of the genuine publication in german.

that
reads JSB's string of 11 loops: ?-111000222222-?
as an advise of an tuning-script

start-x1-x2-x3-y1-y2-y3-z1-z2-z3-z4-z5-end cycle of 5hts

beginning and terminating @:
A x1 E x2 B x3 F# y1 C# y2 G# y3 Eb z1 Bb z2 F z3 C z4 G z5 D _ A

Concrete tuned in absolute pitch frequencies on the corresponding
keys:

A 420 cps 210 105 start
x1 = 314/315
E (315>) 314 157
x2 = 470/471
B (471>) 470 235
x3 = 704/705
F# (705>) 704 352 176 88 44 22 11
y1 pure 5th
C# 33
y2 pure 5th
G# 99
y3 pure 5th
Eb 297 148.5 the 5 triple squiggles mean half intergal beatings
z1 = 890/891
Bb (445.5>) 445 222.5
z2 = 1334/1335
F (667.5>) 667 333.5
z3 = 2000/2001
C (1000.5>) 1000 500 250 125 62.5
z4 = 374/375
G (187.5>) 187 93.5
z5 = 560/561
D(280.5>) 280 140 70 35
conclude with an pure 5th back
A 105 returning to the initial start

yielding a subdivision of the
PC = x1*x2*x3*z1*z2*z3*z4*z5

that's in ascending order:

C4 250 cps middle-C frequency
C# 264 132 66 33
D4 280
Eb 297
E4 314
F4 333.5
F# 352 176 88 44 22 11
G4 374 187
G# 396 198 99
A4 420 coeval Leipzig Cammerthon pitch
Bb 445
B4 470
C5 500

Meanwhile there's also an improved version in todays 440Hz norm,
that i do prefer acoustically on my piano,
because i do consider barely half integral pitch-frequencies
in the middle-octave as to coarsely rough-textured alike
L's 1/12 PC reinterpretation, JSB's "squiggles" deserve i.m.o.
a even keener resolution more smooth in finesse of the resolution
for subtle ears, as Bach once had:
My proposal sounds:

a: 440 Hz
start = 1319/1320
e: (3*a = 1320>) 1319 659.5 339.75
x1 = 3569/3957
h: (3*e = 989.25>) 989 494.5 247.5
x2 = 2966/2967
f#=Gb (3*h = 741.75>) 741.5 370.75
x3 = 4448/4449
c#=Db (3*f# = 1112.25>) 1112 556 278 139
Ab = 3*Db = 417 ; triple pure 5ths
Eb = 3*Ab = 1251 625.5 312.75
Bb = 3*Eb = 938.25 469.125 (>469)
z1 = 3752/3753
F: 3*469 = 1407 703.5 351.75 175.875
z2 = 4220/4221
C: (3*F = 527.625>) 527.5 263.75 131.875
z3 = 3164/3165
G: (3*C = 395.675>) 395.5 197.75 98.875
z4 = 181.5/182.5
D: (3*G = 296.625>) 295 (>294 147)
z5 = 176/177
a: (441>) 440 Hz

That's chromatically in ascending order:

C4 263.75 middle-C
c# 278
D4 295
Eb 312.75
e4 339.75
F4 351.75
f# 370.75
G4 395.5
g# 417
a4 440 cps
Bb 469.125
h4 494.5
C5 527.5

or on a 4" organ stop in the discant as integral beating version:

a 440 cps
e (1320>)1319
h (3957>) 3956 1978 989
f# (2967>) 2968 1483
c# (4449>) 4448 2224 1112 556 278 139
g# 417
Eb 1251
Bb 3753 (>3752 1876 938 469)
F 1407
C (4221>) 4220 2110 1055
G (3165>) 3164 1582 791 (>790 395)
D (591>)590 295 (>294 147)
a (885>) 880 440 cps


!sparschuhJSBloops440Hz.scl
!
Sparschuh's 2007 interpretation of J.S. Bach's WTC loops @ 440 cps
!
12
!
! 1055 = (263.75 middle C) *4
!
1112/1055 ! c#
1180/1055 ! D
1251/1055 ! Eb
1319/1055 ! e
1407/1055 ! F
1483/1055 ! f#
1582/1055 ! G
1668/1055 ! g#
1760/1055 ! a with 1760 := 440 * 4
3753/2110 ! Bb
1978/1055 ! h
2/1
!
!

have a lot of fun in trying out that actual refined version

Any comments or suggestions on that recent relase?

A.S.