Primes, Lattices, Monzo Planetary Graph

Paul Erlich wrote:
> Joe Monzo wrote,

>> Paul Erlich, did you ever consider the fact
>> that the relationships between ratios which
>> are so elegantly portrayed in your triangular
>> lattice diagrams are the result of nothing other
>> than prime factorization?

> The assertion in that question makes no sense to me,
> but Paul Hahn generously replied,

>> This doesn't mean all that much--I'm still struggling
>> with the 7-limit, but if I were to work in 9-limit or
>> higher, I'd probably prefer to diagram the 9s with a
>> separate axis, and have one of the unison vectors
>> be (2 0 0 -1), to reflect my odd-limit orientation.
>> Of course this would be much easier if I were a
>> five-dimensional being who could buy 4-dimensional
>> graph paper.

> I'd have to agree with Paul H., but wonder if there
> is something to Joe's question that he wasn't
> expressing as well as he would have liked?

First comment: I've many times wished that we humans could
work with more than 3 dimensions when up against the problem
of visualizing and representing more than 3 prime dimensions
in music. Fortunately, 7-limit systems _can_ be represented
in three dimensions.

Response to Paul Erlich:
------------------------

I thought my original question was pretty straightforward,
but I never mind elaborating with the help of visual aids.

Picking a particular set of pitches arbitrarily, and
Going back to Tuning Digest 1315, here's the 7-Limit
"Tonality Diamond" in Graham Breed's Triangular Lattice
Diagram, which is of the type used by Erlich:

7-Limit Tonality Diamond Lattice, � la Graham Breed
--------------------------------------------------

5/3-------5/4
/ \ / \
/ \10/7 / \
/ 7/6 \ / 7/4 \
/ \ / \
4/3-------1/1-------3/2
\ / \ /
\ 8/7 / \12/7 /
\ / 7/5 \ /
\ / \ /
8/5-------6/5


Graham leaves his 7-factor-ratios unconnected by
any lines to the 3- and 5-limit ratios. If I replace
the ratios with the prime-factor notation that I use,
and connect the 7-ratios in a manner similar to others,
I get:

Breed Lattice modified by Monzo
-------------------------------

Legend:
- = 3/2, 4/3
/ = 5/4, 8/5
\ = 6/5, 5/3

5^1*3^-1 ---------------- 5^1
/ \ / \
/ \ 5^1*7^-1 / \
/ \ / \ / \
/ \ / \ / \
/ \/ \ / \
/ 7^1*3^-1 -/\-------\/---- 7^1 \
/ \ / \ /\ / \
3^-1 -------\-/---- n^0 --\-/ -------- 3^1
\ / / \ \ /
\ 7^-1 -\--/--------\/-3^1*7^-1 /
\ \/ /\ /
\ /\ / \ /
\ / \ / \ /
\ / 7^1*5^-1 \ /
\ / \ /
5^-1 --------------- 3^1*5^-1

The connecting-lines reveal the major and minor triads,
of which there are four each in this tuning. However,
the 7-ratios still remain unconnected to the 3- and 5-
limit ones.

( Interesting side note inspired by thoughts on
( Schoenberg, who's been discussed here a lot lately:
( in the latter part of his life he strongly reaffirmed
( his faith in Judaism and wrote some pieces on Jewish
( themes, and all his life he apparently had a superstitious
( belief in numerology. If he had seen this diagram, he
( could have been intrigued by the idea of splitting the
( 12-tone row into 2 hexachords, one using the 7-ratio pitches
( connected in the center of the scheme in the form of a Star of
( David to represent a Jewish theme, and the other hexachord
( using the 3- and 5-limit pitches on the perimeter to represent
( a different theme. The difference in prime-limit between
( the two hexachords would harmonically delimit the two sets
( aurally as strongly as they are visually in the diagram.
( Anyone want to collaborate on a piece?)

If I take this scale and redraw it onto my lattice diagram,
I use a different type of line to connect each prime.

same Tonality Diamond, Lattice � la Monzo
-----------------------------------------

Here, 6/5 doesn't figure as an interval that needs to be
connected directly because it's not a prime axis.

Legend:
- = 3^1, 3^-1 (= 3/2, 4/3)
\ = 5^1, 5^-1 (= 5/4, 8/5)
/ = 7^1, 7^-1 (= 7/4, 8/7)



_ - 5^1
5^1*3^-1 - / _\ - 7^1
\ 3^-1*7^1 -/ \ / \
\ / 5^1*7^-1 \ / \ _ - 3^1
\/ _\ - n^0 - \ / \
3^-1 - \ / \ 5^-1*7^1 / \
\ / \ _ / - 3^1*7^-1 \
7^-1 - \/ _ - 3^1*5^-1
5^-1 -


I was merely pointing out to Paul that prime factorization
figures in his own visual representations of pitch resources.
If one is prepared to argue that we don't hear prime qualities
in intervals, how can one find a diagram useful which is
nothing other than the _visual representation of those qualities_?

(I should also note here that all of the diagrams I have
seen which were drawn by Erlich himself were representing
pitch-classes in 22-Eq temperament and the ratios they
_implied_, and not the just ratios themselves. Does this
have any bearing on my response here?)

Re: Paul Hahn's reply,
----------------------

I think I understand pretty well what you're describing with
your

> ...diagram the 9s with a separate axis, and have one of
> the unison vectors be (2 0 0 -1)...

but I'd love to _see_ what this really looks like from
your point of view -- try posting a diagram.

In my endless search for the "(most) perfect" notation,
I've decided in the end to use a variety of representations
simultaneously. In addition to the my lattice diagrams,
I also use musical staff notation with the prime-factors
as accidentals on each note, I also graph the pitches on
a 12-eq frequency graph, and I also use what I call a
Planetary Graph. This last was my solution to the problem
mentioned above of representing more than 3 dimensions.


Explanation of my Planetary Graph
=================================

Unfortunately, it's impossible for me to draw this within the
limited confines of ASCII text. Hopefully, I can get a
website up to illustrate it. (My lattice graphs look better
with real straight lines too) Anyway, here's a description:

1/1 is represented as the "sun" of the graph, and each
increasingly large prime factor is a successive orbit
around this sun. The circumference of the orbit represents
the "pitch-height" of the ratio as a pitch-class within
the octave -- as pitch-height goes higher around the circle,
it eventually returns to 12 o'clock at the octave. A line
is drawn from the center to the point on the circumference
which represents the pitch-height. The prime factorization
of the ratio is revealed by little boxes above or below the
prime-orbit, each box representing an increase of 1 in the
positive and negative exponents respectively.

I have found this to be the most compact and efficient way
of describing very complex just-intonation systems. At a
glance, one can see the relative position of the pitch-classes
in frequency as well as how all of them relate by prime
factorization. To me, these are the two musically important
qualities that ratios possess which I like to see revealed
immediately.

One of the most interesting things about the Planetary Graph
is that it relates back to the ancient Greek and Medieval
European ideas about the "Music of the Spheres", the harmonic
principles which were felt to underly everything in the
universe, from our earthly music to the orbits of the planets
and stars.

Once again, it is a visual representation I thought I invented,
until research for my book showed that the ideas of both
representing the octave with a circle and representing ratios
with radii had been used around 1600 by Lippius, Descartes,
and other theorists of the period. It was useful for showing
how ratios and their complements related to each other, and
thus for illustrating the then-new concept of inversion in chords.
The only new aspect I brought to it was the prime-factor "orbits".

Joseph L. Monzo
monz@juno.com


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