Monzo lattices

From TD 694, Joe Monzo wrote:

> But it's useful for me because it allows me to plug the
> (non-integer) exponents of 2, 3, and 5 into my lattice formula
> and illustrate meantones on my lattices, along with JI pitches
> for reference. Since my lattice formula can already optionally
> include '2' as a factor, I am now able to lattice JI, meantones,

> and ETs all on the same graph, which I think is a good thing. :)
>
> My ultimate objective is to be able to plot *any* and all types
> of tunings on the same lattice, so that any tuning may be seen
> as a subset of the total infinite set of pitches in a way which
> makes clear the rational underpinnings of the harmonic
> relationships. Paul and I both have our doubts about how
> successfully my particular lattice formula does this, but
> so far, for me, it works best.
>

Thanks, Joe for the Woolhouse elaborations. Yes, I do remember this
discussion from the list, and I'm assuming that you just "fine tuned"
the mathematics at the very end between yourself and Paul Erlich. I
enjoyed "scrolling through" that part, anyway... Thanks for the
clarifications as to what was going on...

Now I have a question concerning the Monzo lattices. The lattices that
Paul Erlich was showing me had intervals of the same kind all going in
the same direction as a straight line on the lattices -- the fifths of
the limit 3 horizontally, the thirds of the limit 5 vertically, and the
7th limit going out from the page as the 3rd dimension, correct?? This
pertained for BOTH the rectangualar AND triangular lattices.

However, I notice that the Monzo lattices do not do this, but the 3
limit and 5 limit lines CROSS, and the 7 limit lines are tilted at only
a slight angle from the 5 limit lines...

Is there some advantage to this type of plotting?? I certainly can see
that some very beautiful lattices are made, but are the relationships as
clear as in the "traditional" lattices??

Monz, Paul, help...

___________________ ________ ___ __ __
Joseph Pehrson

> [Joseph Pehrson, TD 702.19]
>
> Now I have a question concerning the Monzo lattices. The
> lattices that Paul Erlich was showing me had intervals of the
> same kind all going in the same direction as a straight line
> on the lattices -- the fifths of the limit 3 horizontally,
> the thirds of the limit 5 vertically, and the 7th limit going
> out from the page as the 3rd dimension, correct?? This
> pertained for BOTH the rectangualar AND triangular lattices.
>
> However, I notice that the Monzo lattices do not do this, but
> the 3 limit and 5 limit lines CROSS, and the 7 limit lines
> are tilted at only a slight angle from the 5 limit lines...
>
> Is there some advantage to this type of plotting?? I certainly
> can see that some very beautiful lattices are made, but are the
> relationships as clear as in the "traditional" lattices??

None of the prime-axes in my lattices ever cross *in theory*.
What you're probably seeing is the result of looking at the
2-dimensional output of my n-dimensional lattices.

You can find the principles upon which I base my lattice formula
in my article on lattices:
http://www.ixpres.com/interval/monzo/lattices/lattices.htm

In brief: from a central reference point, which represents 1/1,
each prime-axis radiates outward at an angle determined by the
'octave'-reduced cents-value of the initial prime harmonic.

For example, the cents value of the 3rd harmonic is:

log(3) * (1200/log(2)) = ~1901.955 cents

To '8ve'-reduce it, we get:

~1901.955 mod 1200 = 701.955 cents

That figure should be readily recognized as the cents-value
of the 3/2 'perfect 5th'.

Next we have to find the degree-value of the angle. Since
there are 360 degrees in a circle, and 1200 cents in the
'octave' (which we can also think of as a circle), we multiply
our cents-value by 360/1200 [= 3/10] to obtain the degrees.
Thus, the angle for the 3-axis is:

~701.955 cents * (3/10) = ~210.6 degrees

So if we imagine a circle (which represents the '8ve')
around each lattice-point, with all prime-axes radiating
outward from each lattice-point at a predetermined angle,
and with some arbitrary point along those circles chosen
to represent 0 = 1200 cents (I used to use 6 o'clock, as in
my Lattice article - now I use 12 o'clock), the 3-axis will
always extend at an angle of ~211 degrees from any lattice-point.

Similarly, the 5-axis will always extend at an angle of

( ( log(5) * (1200/log(2)) ) MOD 1200 ) * (3/10) degrees
= (~2786.314 cents MOD 1200) * (3/10) degrees
= (~386.314 cents) * (3/10) degrees
= ~115.9 degrees

And so on.

The big advantage to using this formula is that one can
represent *any* prime-factor with a *unique* angle, thus
(in theory) plotting any ratio precisely onto a unique
point according to its prime-factorization. This (for me)
is the whole point of using lattices: to illustrate the
unique numerical qualities of each ratio and its unique
relation to the rest of the pitches in the scale/lattice.

I say 'in theory' because in practice, representing lattices
of >3 dimensions on a 2-dimensional surface, there will always
be some type of overlapping. But by using software to draw
the lattices on a computer screen (which I'm in the process
of developing) one can zoom in to a magnification and/or
rotate the diagram in 3-d space so that overlapping points
or lines can be clearly separated.

Because we are limited to 3 dimensions (or 4, if we include
time by the use of animation) in perceiving spatial relationships,
there is no single way to represent >3-D structures that is
better than any other.

The method used by Paul and others (including me) to draw
lattices here on this List is simply a bow to the limitations
of ASCII representation. The conventions used here, which
you describe in the quote above, are the clearest way to
represent a 2- or 3-D lattice in email. For most *musical*
purposes, this is sufficient, as the vast majority of rational-
based music and music-theory is 1-, 2-, or 3-dimensional
(the best examples being Pythagorean, JI, and 7-limit,
respectively).

But since I'm interested in the history of tuning theory,
I wanted to be able to view *any* rational tuning as a
subset of the 'total lattice', thus the need for unique angles.
If an ancient or Renaissance theorist used ratios with such
factors as 19, 31, or 499, I need to be able to represent
those uniquely.

For some criticism levelled against my lattice formula,
primarly by Paul, see the List archives from about a year ago.
His main complaint is that my lattice formula plots ratios
in a way that seems to indicate a level of 'complexity'
that is the opposite of what is numerically and empirically
true, so that, for instance, 15/8 and 6/5 are exactly the
same distance from 1/1, when it seems that 6/5 should be
much closer.

But for my purposes (so far), the advantages of my formula
far outweigh these kinds of disadvantages.

--------

An extremely interesting footnote:

I've been ransacking the libraries on information about the
Sumerians lately, because I think I've 'cracked the code' for
some Babylonian mathematical problems that mathematicians and
Assyriologists have so far been unable to decipher, and which
seem to me to be information handed down from the days of
the Sumerian hegemony.

The problems make use of a 'coefficient' called 18 IM.LA.
No one has been able to figure out what the Sumerian logogram
'IM.LA' refers to.

But the formulas always produce 2/3 (of a string-length, is my
guess). So the problems in question appear to me to be a formula
for calculating the (so-called) Pythagorean scale, at *least*
1000 years before Pythagoras, and perhaps another millenium
or two before that, if the Sumerians had figured out the formula
(as seems to be indicated by the Babylonian use of the Sumerian
symbols in the formulae).

What I haven't been able to figure out is the role that the
'coefficient' plays in the formula.

The Babylonians used a sexagesimal (= base-60) numbering
system, so '18' really means 18/60 = 3/10. AHA!

As one can see from my answer to Joseph above, the ratio of
degrees to cents happens to provide exactly the same coefficient
of 3/10.

So perhaps this formula actually concerns not a Pythagorean
scale, but rather some type of cents-calculation?

Hmmmm..............
Could temperament actually be a Babylonian (or Sumerian) invention,
from c. 4000 years ago?....

-monz

Joseph L. Monzo San Diego monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
| 'I had broken thru the lattice barrier...' |
| -Erv Wilson |
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--- In tuning@egroups.com, Joe Monzo <MONZ@J...> wrote:

> The method used by Paul and others (including me) to draw
> lattices here on this List is simply a bow to the limitations
> of ASCII representation.

Not at all -- although the limitations are annoying, I draw the
lattices here with the same lines and directions as I would use on
paper.

> The conventions used here, which
> you describe in the quote above, are the clearest way to
> represent a 2- or 3-D lattice in email. For most *musical*
> purposes, this is sufficient, as the vast majority of rational-
> based music and music-theory is 1-, 2-, or 3-dimensional
> (the best examples being Pythagorean, JI, and 7-limit,
> respectively).

right.
>
> For some criticism levelled against my lattice formula,
> primarly by Paul, see the List archives from about a year ago.
> His main complaint is that my lattice formula plots ratios
> in a way that seems to indicate a level of 'complexity'
> that is the opposite of what is numerically and empirically
> true, so that, for instance, 15/8 and 6/5 are exactly the
> same distance from 1/1, when it seems that 6/5 should be
> much closer.

Actually, Monz's lattice puts 15/8 closer to 1/1 and 6/5 farther from
1/1. To partially remedy these sorts of problems, I suggested to Monz
that he map all the positive powers of primes to angles between 0
degrees and 180 degrees, instead of between 0 degrees and 360
degrees.
This would also remedy another problem that Monz has previously
acknowledged with his lattices, that there is no immediate way to
distinguish whether a line at angle of x degrees is meant to
represent
a prime interval of about 10x/3 cents, or the inverse of a prime
interval of about 10x/3 plus or minus 600 cents.

When I suggested this multi-purpose remedy to Monz, he said that
someone else had suggested it to him right before me. Apparently,
though, Monz has developed too strong a familiarity with his original
formulation to switch mid-stream (Monz?) . . .