> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@>
> wrote:
Dear Keenan,
here comes additional the conversion-step into the middle octave too:

C4 264:= 33*8 > > C 33
G4 396:= 99*4 > > G 99
D4 297_______ > > D (37,74,148,296)297
A4 444:=111*4 > > A (55,110)111
E4 330:=165*2 > > E (41,82,164)165
B4 462:=123*4 > > B (61,122)123
F# 363________ >> F# 363(366,183)
C# 272.5=545/2 >> C# (273,546)545,1090(1089)
G# 409.5=819/2 >> G# (205,410,820)819
Eb 307.5=615/2 >> Eb (77,154,308,616)615
Bb 462:=231*2 > > Bb (29,58,116,232)231
F4 348:= 87*4 > > F (11,22,44,88)87
C5 524:=33*16 > > C 33
> >
> > I really can't understand what this means.
Reordering that 5ths-circle-pitches
into an arithmetical ascending series
yields chromatical:

> C4 264 Hz middle C
> C# 272.5
> D4 297
> Eb 307.5
> E4 330
> F4 348
> F# 363
> G4 396
> G# 409.5
> A4 444 Hz, that's 4Hz sharper above standard normal-pitch 440Hz
> Bb 462
> B4 492
> C5 524=264*2=C4*2
>
Dividing all that 12 pitches by C=264Hz normalizes
the frequencies it into
dimensinonless scala-file ratios with C=1/1:
>
> ! sync_beat_11-limit.scl
> !
> synchronous beating 11-limit scale for C4=264Hz or A4=444Hz
> 12
> !
> 545/524
> 9/8
> 615/524
> 5/4
> 348/264 ! =(4/3)(87/88)
> 11/8
> 3/2
> 273/176
> 37/22 ! =(5/3)(111/110)
> 7/4
> 41/22 ! =(15/8)(164/165)
> 2/1

> > Why did you round
> > everything to whole numbers of hertz?
> There is no rounding here.
It's a kind of
http://tonalsoft.com/enc/b/bridging.aspx
by using the superparticular (epimoric) bridges
as tempering steps inbetween the 5ths.
The round brackets indicate the tempering steps in 5hts:
about how far should the according 5ths be flattened or widened.
>
> > Are the octaves really supposed
> > to be stretched by 10-20 cents?
> not at all, because.....that turns out individual different from
> instrument to instrument.
http://en.wikipedia.org/wiki/Inharmonicity

Above procedere divides the PC into
> > > PC=3^12/2^19=531441/524288= subpartition
> > > (297/296)(111/110)(165/164)(123/122)(122/121)(1089/1090)(819/820)
> > > (615/616)(231/232)(87/88)

Alreday Andreas Werckmeister in his "Musicalische Temperatur 1691"
http://diapason.xentonic.org/ttl/ttl01.html
used that superparticular-subdivision method successfully
in his #6, the "Septenarius"-tuning:
http://launch.groups.yahoo.com/group/tuning/message/63471

As far as i do know at the moment:
You are the first man since 315 years,
that doubts about W's idea:
> > This math doesn't work out.
Amazing!?

Above modified version is merely adapted for producing an pure
4:5:6:7 :8 :9 :10:11 :12 chord on the keys
C:E:G:Bb:C':D':E':F#':G'.
A.S.