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

multiply all ratios by factor 264=11*3*2*2 in order to

obtain integral absolute pitch-frequencies

> > ! bihexany.scl

> > Hole around [0, 1/2, 1/2, 1/2]

> > 12

> > !

C4 264 from 1/1 middle C=264Hz

C# 280 > 35/33

(D4 297 > 9/8) inserted instead Cb=B# for smoothening the 5ths

Eb 308 > 7/6

E4 330 > 5/4

F4 336 > 14/11

F# 360 > 15/11

G4 396 > 3/2

G# 420 > 35/22

A4 440 > 5/3 !normal pitch 440Hz

Bb 462 > 7/4

B4 480 > 20/11

(Cb=B# 504 > 21/11)

C5 528 > 2

or arranged in a cycle of a dozen partial tempered 5ths

C 33,..,264 begin, starting @ middle-C

G 99:=33*3,198,396

D (37,74,148,296)297:=99*3

A 55,110(111:=37*3),220,440 ! (111:110)*(297:296)=81:80 the SC

E (41,82,164)165:=55*3,330

B 15,30,60,,(123:=41*3)120,240,480 ! (165:164)*(41:40)=33:32

F# 45:=15*3,..,360

C# 35,70,(135:=45*3)140,280 ! 140:135=28:27

G# (13,26,52,104)105:=35*3,210,420

Eb (39:=13*3,78)77,154,308

!(105:104)*(77:78)=2695:2704=299.4444.../300.4444...

Bb (7,...,228)231:=77*3,462 ! 231:228=77:76

F (11,22)21:=7*3,42,84,168,336

C 33:=11*3 cycle closed, ready done.

i.m.o: heavy harsh tempering, but it works.

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.