lattice metrics

[Paul Erlich, TD 158.9]
> ... you're proposing a metric -- we've had disagreements over
> whether it should be called Euclidean, Cartesian, or what --
> which connects two notes with a straight line and uses the
> distance of the line as a measure of complexity.
> As we discussed before, in your lattice, that gives 15:8
> considerably lower complexity than 6:5, which doesn't seem
> right to me.

???

On my lattice 15:8 and 6:5 have exactly the same complexity,
in terms of a direct vector lattice metric: both have
one rung on each of the 3- and 5- axes. [*]

This doesn't change your argument, because your point is that
15:8 should have greater complexity than 6:5, neither lower *nor*
equal. I can agree with that.

[Erlich]
> You were saying that your lattice makes composites appear
> simpler than their odd or prime represenatations would suggest.
> That doesn't seem to be true.

Hold on - don't get it twisted.
This is what I originally said:

[me, monz, TD 155.10]
> In my lattice formula the prime-factoring
> itself makes composite factors such as 9 and 15 shorter
> distances than their odd/integer values would indicate,
> and to me this is the whole point of using prime instead
> of odd factoring.

My lattices *are* a prime representation, so there's no way
that composites could appear simpler than their prime
representations, unless mine have strange wormholes too.

They do, however, make composites appear simpler than their
*odd* limits, which I think represents acoustical/musical
reality better than an odd-limit. That was the whole reason,
about 10 years ago, that I decided prime-factoring was
useful and/or important in music.

(It turned out to be true later, when I 'invented' my lattices,
that it was even more useful for representing large harmonic
systems with maximal simplicity, IMO.)

Here's what I was getting at:

[Erlich, TD 156.14]
> 1st axis: steps of 3:2 have length log(3)
>
> 2nd axis: steps of 5:4 have length log(5)
> the angle between the first and second axes is determined by the
> condition that steps of 5:3 have length log(5)
>
> 3rd axis: steps of 7:4 have length log(7)
> the angle between the the first and third axes is determined by
> the condition that steps of 7:5 have length log (7)
> the angle between the second and third axes is determined by the
> condition that steps of 7:6 have length log (7)

This isn't logical to me. You're using the *limit* prime
of each ratio to determine the vector, whereas my lattice
formula *adds the complexity for each prime* found as a factor.

So in your formula, it doesn't matter whether 7 is combined
with 2, 3, or 5 - they all have a metric based on 7.

In my formula, 7:4, 7:6, and 7:5 are all different, each metric
longer than the one before it, based on all their prime factors
(numbers inside the table are the exponents of the prime factors
on the top line):

2 3 5 7
7:4 = | -2 0 0 1|
7:6 = | -1 -1 0 1|
7:5 = | 0 0 -1 1|

This reflects well my feeling that 7:4 is less complex than both
7:6 and 7:5, that 7:6 is more complex than 7:4 but less complex
than 7:5, and 7:5 more complex than both 7:4 and 7:6. (Of course,
in this simple example an integer-limit would do the same.)

So on your lattice:

[Erlich]
> the distance corresponding to 9:5 is log(5)+log(3)=log(15).
> However, 9:5 is more consonant than 11:5, which unfortunately
> has a shorter length of log(11).

This is not the case on my lattice.

On a prime lattice (like mine) the 9:5 ratio is 3 rungs away
from 1:1 : 2 positive rungs along the 3-axis, and 1 negative
rung along the 5-axis.

The way my formula works, the 3^2 vector (with
a length of 2 * 3) gives the same length (= complexity)
that a '6' would have on an integer-limit lattice, or half-way
between 5 and 7 on an odd-limit lattice, which uses this type of
straight-value length measurement (as opposed to using logs).
Then the 5^-1 vector adds more complexity to it, which I think
is appropriate, as 9:5 *is* more complex than 9:4, or 9:8.

The metric (i.e., direct connection) length for 9:5 is
~8.12, longer than that for 7:4 [= 7.0], which
agrees with my, and I think most people's, perceptions.
And the metric for 11:5 is 8.6, a bit longer than 9:5's,
which is what you expressed a desire to acheive.

The problem I have with this formula is that 16:9 has
a vector length of only 6.0, which is *less* than that
for 7:4, which doesn't agree with my, or most people's,
perceptions.

With my metric, the complementary ratios always have exactly
the same value.

Here are some frequently used ratios (and some others I've
been interested in lately), ranked in order by my lattice metric
(rounded to the nearest 1/10th):

1/1 0.0
2/1 2.0
3/2 4/3 3.0
5/4 8/5 5.0
16/15 15/8 5.6
9/8 16/9 6.0
6/5 5/3 6.0
7/4 8/7 7.0
7/6 12/7 7.1
45/32 64/45 7.5
11/9 18/11 8.0
10/9 9/5 8.1
14/9 9/7 8.4
11/10 20/11 8.6
27/16 32/27 9.0
11/6 12/11 9.1
13/9 18/13 9.4
25/16 32/25 10.0
40/27 27/20 10.6
13/12 24/13 10.9
11/8 16/11 11.0
225/224 448/225 11.2
7/5 10/7 12.0
81/64 160/81 12.0
13/8 16/13 13.0
99/80 160/99 14.4

There are some problems with this. For example, 5:3 should
certainly have a shorter metric than 15:8, altho the difference
is really not that great. 16:9 should be longer than 7:4,
with that difference of 1 unit possibly being significant, and
it most definitely should have a longer metric than 5:3,
which it equals here.

40:27, as I remarked recently, sounds far more complex, and
dissonant, than any of the ratios listed after it here: 24:13 is
another neutral 'major 7th' that I like a lot, 11:8 and 13:8
are both commonly used by many JI composers in stable chords
in a way that 40:27 would never be appropriate. So there's
certainly room for improvement in this formula.

But overall, I feel that it does a pretty decent job of
representing the complexity and/or sonance of intervals
from the perspective of viewing the entire harmonic complex,
that is, not just observing them as individual dyads, but
placing them in relation to each other with a view towards
accomodating their varied harmonic uses, and therefore,
the different lattice metrics that result from different
combinations of the same ratios.

Let's look at those 7-limit ratios again, shall we?,
and reproduce that table with my metric values added:

2 3 5 7 Monzo lattice metric
7:4 = | -2 0 0 1| 7.0
7:6 = | -1 -1 0 1| 7.1
7:5 = | 0 0 -1 1| 12.0

The metric for 7:5 may be a little high (but maybe not),
but I think this is basically a really good representation
of the relative complexities of these three ratios, not
considering any particular musical context.

And all these calculations were done on an 'octave'-equivalent
lattice. I believe that to accurately compare them to your
values I need to make them 'octave'-specific, correct?
I'll make a new table reflecting that in a future posting.

Feedback appreciated - especially from others besides Paul.
I already know I can count on his opinion < {;^) >
but I'd like some others as well.

[*]
As can be seen from the table, which I calculated many
hours after writing this sentence, Paul Erlich is correct
that my metric does indeed give a shorter distance for 15:8
than for 6:5, as he stated. I don't know if I'd call 5.6
a 'considerably lower complexity than 6:5' in relation to
the latter's value of 6.0, but as I said, even if they were
equal, his point is still valid.

Many thanks to Paul for the correspondence-crash-course in
trigonometry that he gave me via email yesterday. Now
hopefully I can follow this complexity thread; at least,
I can now see how my own work fits into the discussion.

-monz

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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