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

___________________________________________________________________

You don't need to buy Internet access to use free Internet e-mail.

Get completely free e-mail from Juno at http://www.juno.com/getjuno.html

or call Juno at (800) 654-JUNO [654-5866]

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

___________________________________________________________________

You don't need to buy Internet access to use free Internet e-mail.

Get completely free e-mail from Juno at http://www.juno.com/getjuno.html

or call Juno at (800) 654-JUNO [654-5866]