In my last post, I drew:
>
> ---- --. 21 --- -. 27 --- ---
> / ,/' /. ,/' / / /
> /. ' / /..' / / / /
> 22 --- --- 14 .-- --- 18 --- --. 23
> / / / \. / /. ,/' /
> / / / .\ / /..' / /
> --- --- --. 15 -.-24 .-- ---
> / / / .,/'\ / . / /
> / / /. ' ./ \ / . / /
> ---- --. 25 -- 20 -- 16 --.- ---
> / ,/' / /. / \ . / /
> /. ' / / /. / \./ /
> 26 --- --- --- -.- --- 17 ---
> / / / / . / / /
> / / / / . / / /
> ---- --- --- --.- --- ---
> / / / / . / / /
>/ / / / ./ / /
>---- --- --- --- 19 --- ---
>

For the sake of those who may have had some
trouble following me, I had meant to
include a listing of the harmonics on
the lattice that are already in 5-limit,
i.e., are not bridges, and thus have no
"error". They are:

5-LIMIT NOTE
matrix
HARMONIC 3^x*5^y = ratio
15 | 1 1 | 15/8
16 | 0 0 | 1/1
18 | 2 0 | 9/8
20 | 0 1 | 5/4
24 | 1 0 | 3/2
25 | 0 2 | 25/16
27 | 3 0 | 27/16

The omission of this table explains
the gaps in the table of bridges, and
may also help if you don't understand
the prime-factor matrix notation.

In a recent post I synopsized a book
by Fokker where he makes important use
of the 5--7 bridge (225/224).

Another thing I wanted to mention was
that this same process would work just
fine for a Pythagorean system, since
the distance between the Pythagorean
and Syntonic commas is so small (a schisma).
Pythagorean "bridging" to higher primes
has already been explored a bit in this
forum by Margo Schulter (3--7 bridges) and
also in some of Erv Wilson's writings
(the 3--5, 3--7, and 3--11 bridges).

To my knowledge no one's discussed representing
ratios higher than 11-limit with either
3- or 5-limit, with the single exception
of the 3--19 bridge (513/512). This was
implied as a bridge, altho not actually
discussed as such, as long ago as c. 210 BC
by Eratosthenes, and it plays a prominent role
in my analysis of the 14th-century Marchetto
of Padua.
--------
See:

Erv Wilson, "On the development of intonational
systems by extended linear mapping", Xenharmonikon 3
www.anaphoria.com

Margo Schulter, "Septimal schisma as xenharmonic bridge?"
http://www.ixpres.com/interval/td/schulter/septimal.htm

Joe Monzo, "Speculations on Marchetto of Padua's 'Fifth-Tones'"
http://www.ixpres.com/interval/monzo/marchet.htm

- Monzo


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