11/13 limit bridges

I have been investigating a lot of(usually 12-tone)JI tunings lately,looking for ways to elegantly incorporate 7-limit harmonic bridges.Mainly being,two 15/8s equalling a 7/4 and 8 4/3s equals a 5/4(the later known as the schismic third approximation).But what good examples are there of 11 and 13 limit low number tonal bridges?I hope you can give good examples of some,other than 105/104 which is the only one i�ve explored so far.

-=-=-=-=-=-=-
MATS �LJARE
http://www.angelfire.com/mo/oljare
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Mats wrote,

>Mainly being,two
>15/8s equalling a 7/4

Aka the 225:224 unison vector, Fokker's favorite . . .

>But what good examples are there of 11 and 13
>limit low number tonal bridges?I hope you can give good examples of
>some,other than 105/104 which is the only one i´ve explored so far.

We've discussed quite a few 11-limit ones on this list -- for example, on
Graham's page http://x31eq.com/lattice.htm you'll see

121:120
and
243:243

On http://www.ixpres.com/interval/td/erlich/partchpblock.htm you'll see me
show that Partch's scale is essentially the periodicity block of the four
unison vectors

100:99
245:243,
441:440,
and
896:891

Back in November, I created some 13-limit periodicity blocks for Justin
White based on the five unison vectors

100:99,
105:104,
196:195,
275:273,
and
385:384

There's many more . . . one could easily write a computer program to find
the ones that satisfy a given set of maximum bounds on the size of the
numbers, the size of the prime factors, and the size of the intervals . . .

--- In tuning@y..., "Mats Öljare" <oljare@h...> wrote:
> I have been investigating a lot of(usually 12-tone)JI tunings
lately,looking
> for ways to elegantly incorporate 7-limit harmonic bridges.Mainly
being,two
> 15/8s equalling a 7/4 and 8 4/3s equals a 5/4(the later known as the
> schismic third approximation).But what good examples are there of 11
and 13
> limit low number tonal bridges?I hope you can give good examples of
> some,other than 105/104 which is the only one i´ve explored so far.

Since you are using the 224:225 septimal kleisma, you should look at
the 11-limit 384:385 = 3 x 2^7 : 5 x 7 x 11. i.e. stack a major third,
a subminor seventh and a super fourth and you get a perfect twelfth.

These two "bridges" work together beautifully. You will find some of
my "microtemperaments" that distribute them, in the archives.

Regards,
-- Dave Keenan

Mats �ljare wrote,

<<I hope you can give good examples of some,other than 105/104 which
is the only one i�ve explored so far.>>

I've used this rather odd 7-tone scale as a sort of bridged/tempered
3-7-11 lydian hybrid...

,1/1.
.' | `.
32/21-------8/7----+---12/7------9/7------27/14
|
7/5

The way I see it this scale makes interesting and selective use of the
441/440 and the 540/539 and allows for the 243/242 to disappear as
well...

1/1 8/7 9/7 7/5 32/21 12/7 27/14 2/1
1/1 9/8 11*9 4/3 3/2 27/16 7/4 2/1
1/1 12*11 32/27 4/3 3/2 14/9 16/9 2/1
1/1 12~11 11~9 11*8 10/7 18~11 11~6 2/1
1/1 9/8 81/64 21/16 3/2 27/16 11*6 2/1
1/1 9/8 7/6 4/3 3/2 18*11 16/9 2/1
1/1 28/27 32/27 4/3 16*11 128/81 16/9 2/1

(The ratios separated by the asterisks are the theoretically bridged
intervals and those separated by the tilde are the theoretically
tempered ratios.)

--Dan Stearns

>100:99
>245:243,
>441:440,
>and
>896:891
>
>Back in November, I created some 13-limit periodicity blocks for Justin
>White based on the five unison vectors
>
>100:99,
>105:104,
>196:195,
>275:273,
>and
>385:384

These are great!I�ll see what i can do with these...Thanks!

-=-=-=-=-=-=-
MATS �LJARE
http://www.angelfire.com/mo/oljare
_________________________________________________________________________
Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com.

Paul H. Erlich wrote:

> We've discussed quite a few 11-limit ones on this list -- for
example, on
> Graham's page http://x31eq.com/lattice.htm you'll see
>
> 121:120
> and
> 243:243

I considered 243:243, but decided it didn't really simplify anything.
Although it can be approximated with a high degree of accuracy. For
most purposes, 243:242 is preferable.

For the 13-limit, 144:143 may be useful. That's between the two
neutral sixths 13:8 and 18:11. An alternative is 352:351, between
13:8 and the extended 11-limit neutral 6th of 44/27 (=(4/3)*(11/9).
Using both bridges also gives you a 243:242.

With only the 352:351, a neutral triad will still contain only
13-limit intervals.

Note: a law of electronic discussions states that this post will
contain a numerical error.

Graham

--- In tuning@y..., graham@m... wrote:

/tuning/topicId_18467.html#18486
>
> I considered 243:243, but decided it didn't really simplify
anything. Although it can be approximated with a high degree of
accuracy.

That's pretty funny....

_____ _____ _____ _
Joseph Pehrson

Graham wrote,

>I considered 243:243, but decided it didn't really simplify anything.
> Although it can be approximated with a high degree of accuracy. For
>most purposes, 243:242 is preferable.

ha ha . . . that was a joke, right?