- View Source
> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@>

Dear Keenan,

> wrote:

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> >

Reordering that 5ths-circle-pitches

> > I really can't understand what this means.

into an arithmetical ascending series

yields chromatical:

> C4 264 Hz middle C

Dividing all that 12 pitches by C=264Hz normalizes

> 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

>

the frequencies it into

dimensinonless scala-file ratios with C=1/1:>

It's a kind of

> ! 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.

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.>

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

> > 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.

Above procedere divides the PC into> > > PC=3^12/2^19=531441/524288= subpartition

Alreday Andreas Werckmeister in his "Musicalische Temperatur 1691"

> > > (297/296)(111/110)(165/164)(123/122)(122/121)(1089/1090)(819/820)

> > > (615/616)(231/232)(87/88)

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. - View Source--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:

> But when you say "The actual 11-limit project ignores 2, 243/242 and

seems to me

> 441/440" and you go about creating a lattice for the result, it

> that would usually be called a 5-limit lattice, isn't that right?

Not really. It isn't like 225/224, where you can look at it as a way

of mashing the 7s down into the 5-limit lattice. Here the horizontal

lattice relationship is 49/40 (7-limit) and the vertical is 10/7

(again, 7-limit.) So it's clearly a 7-limit lattice, but squished down

to a plane of pitch-classes. The deal is, two 49/40s in a row gives

you your 3/2 (by tempering), and a 49/40 times 10/7 is 7/4 (no

tempering) and times 10/7 again is 5/2 (no tempering.) So it boils

down to using 49/40 and 60/49 for the same thing, and using two of

these neutral thirds to reach the fifth.

In terms of this lattice, the fifth is [2,0], and the major third

[1,2]. The determinant of [[2,0], [1,2]] is 4, so only 1/4 of the

lattice consists of the pure 5-limit.

At the> same time the temperament acknowledges the equivalence of this to an

wants

> 11-limit functionality. And as I recall, the concept of temperament

> to remain a little vague about the actual tuning, so you could in this

others,

> example make the 7- and 11-limit intervals exact at the expense of

> or you could make others exact at the expense of 7 and 11. Or you

could do

> some other optimization.

Or you could just use 72-et or 130-et (a division I'm exploring these

days.) If you want to favor the 7-limit a bit, 171-et.