I was looking at a lattice diagram
I'd made of the 5-limit xenharmonic
bridges (Fokker's "unison vectors),
when I discovered some interesting
patterns in the progression of harmonics
as represented by the bridges.

I'll use a lattice with the following
convention - rotated from the perspective
usually seen 'round these parts, to conform
to my lattice design (and to fit on the
screen), and without the "triangular" 6:5
and 5:3 vectors:

-- 3^+1 --
/ / /
/ / /
5^+1 -- 1:1 -- 5^-1
/ / /
/ / /
--- 3^-1 --


The patterns I saw were in the harmonic
series from 14 to 28. Doubtless there
are many more to be found. Here's a
diagram showing the 5-limit end of
the bridges for the harmonic series 14-28
(1/1 = 16):

---- --- 21 --- -- 27 --- ---
/ / / / / / /
/ / / / / / /
22 --- --- 14 --- --- 18 --- --- 23
/ / / / / / /
/ / / / / / /
--- --- --- 15 ---24 --- ---
/ / / / / / /
/ / / / / / /
---- -- 25 -- 20 -- 16 --- ---
/ / / / / / /
/ / / / / / /
26 --- --- --- --- --- 17 ---
/ / / / / / /
/ / / / / / /
---- --- --- --- --- ---
/ / / / / / /
/ / / / / / /
---- --- --- --- 19 --- ---


(I call these 5-limit notes bridge-ends,
because the bridge itself is the ratio,
or unison vector, which actually connects
these 5-limit ends with the note on the
other end, the prime target)

I saw 3 different types of intervallic
connections between these sequential bridge-ends:

3^x * 5^y
16/15 = |-1 -1 | : 14-15-16-17
135/128 = | 3 1 | : 17-18 and 19-20-21
25/24 = |-1 2 | : 21-22, 23-24-25-26 and 27-28

This is largely self-evident: i.e., all
three of these intervals are different
5-limit "semitones", which is roughly
the size of the intervals between all the
harmonics in this particular series.
Still, I thought it was interesting
to see the patterns on the lattice, so
I drew the connections on the diagram
(the best I could in ASCII):

---- --. 21 --- -. 27 --- ---
/ ,/' /. ,/' / / /
/. ' / /..' / / / /
22 --- --- 14 .-- --- 18 --- --. 23
/ / / \. / /. ,/' /
/ / / .\ / /..' / /
--- --- --. 15 -.-24 .-- ---
/ / / .,/'\ / . / /
/ / /. ' ./ \ / . / /
---- --. 25 -- 20 -- 16 --.- ---
/ ,/' / /. / \ . / /
/. ' / / /. / \./ /
26 --- --- --- -.- --- 17 ---
/ / / / . / / /
/ / / / . / / /
---- --- --- --.- --- ---
/ / / / . / / /
/ / / / ./ / /
---- --- --- --- 19 --- ---


The 5-limit bridges to the higher primes
are of the following size (or "error"):

5-LIMIT BRIDGE-END
PRIME CENTS matrix
TARGET ERROR 3^x*5^y = ratio
7 + 7.7 | 2 2 | 225/128
11 - 2.2 | 2 4 | 5625/4096
13 + 2.8 |-1 4 | 625/384
17 + 6.8 |-1 -1 | 16/15
19 - 3.4 |-3 0 | 32/27
23 + 3.0 | 2 -2 | 36/25

(didn't bother to go higher than 23).

---------
Some further thoughts on how this may
apply historically:

We've been discussing here recently the
5--7 bridge (225/224). From the above
table, it's clear that the size of this
bridge is the largest among 5-limit
prime-bridges up to 23.

I (and many others) have stated in the
past that popular acceptance of the "gestalt"
of both 17/16 and 19/16 has ocurred most
likely because of the nearness of 12-Eq
notes to these ratios (errors of only
-5.0 and +2.5 cents, respectively).

The 5--17 and 5--19 bridges are only
slightly larger than their 12-Eq
counterparts, +6.8 and -3.4 cents
respectively. Perhaps 17 and 19 were
already pondered as harmonic resources
long before 12-Eq.

Indeed, the 5--11 bridge is even smaller
than the 12eq--19 bridge, and the 5--13 and
5--23 bridges are smaller than all the rest.

So it wouldn't surprise me at all if
composers in the past, working or thinking
in a large 5-limit JI, or perhaps even a
meantone approximation of it, have represented
ratios up to a 23-limit with the smaller bridges
available in the 49-note (7x7) Euler Genera
depicted in my diagram above.

Of course, it's also possible to reduce the
size of the 5--7 bridge further by choosing a
different 5-limit ratio which is closer in
pitch to 7/4, for example, either of these:

3^ 5^ CENTS ERROR
|-1 -5 | 966.5 -2.3
| 7 -4 | 968.4 -0.4

but that would make the 5-limit matrix
larger. The bridges I have shown are
the closest to 1/1 in 5-limit coordinate
distance that are available.

- Monzo
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