lattice metric

[me, monz, TD 159.15]
> my lattice formula *adds the complexity for each prime* found
> as a factor.

I wanted to add something to this which I think is very important:
because my formula puts positive and negative exponents on
opposite sides of 1:1 (other lattices do too), and because
my lengths are all different, the metric ends up being different
depending on whether a factor has a positive or a negative
exponent.

This means that it is weighting factors differently
depending on whether they appear in the numerator or the
denominators of the ratio, which is something that was part
of the complexity discussion.

Admittedly, it's not doing it right in the case of 6:5 vs. 15:8,
but I thought I'd point out that it's doing it.

-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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[Paul Erlich, TD 160.6]
> Yes but you were just advocating not this city-block metric
> but a Euclidean metric (in which you would have to use the
> Pythagorean theorem or its non-right-triangle variant, the
> law of cosines). Look at your latest posts -- you're a slippery
> fellow!

I'm not being slippery - I'm just trying different lattice
options :)

The bottom line with me is, I designed my lattices as prime-
factor because I felt that that was the simplest way of
portraying the vast amount of harmonic information in a
large JI system.

More and more as time goes on, however, I find that there
is much value in assigning the Euclidean metric to different
ratios on the lattice (what I've been calling 'direct
lattice connections' lately).

Also, I inadvertently posted the first half of that message
hours before it was finished, and didn't realize it.

> In a full musical context, composites will most likely come into
> play more than higher primes. But that is by means of chains of
> connections, which the lattice can already display quite nicely.
> I prefer the lattice to be constructed devoid of any
> considerations of musical context, since those can always vary.
> Without context, I see (and more importantly, hear) no reason to
> consider composite simpler than primes with similar-sized numbers.

Ok, I think basically that makes sense and I can almost agree
with it. But it was you yourself who just complained recently
about 9:5 having a longer metric than 11:5. So it seems that
really we're both looking for a lattice formula that would create
a metric that's in between what we're already getting, your
results being too large on one end and mine being too small
on the other.

I'd like to respond in-depth to the rest of your reply,
but I won't have the time for the next couple of weeks because
I'm hard at work on my Walt Whitman piece for the AFMM.
So I'll be scarce on the List for the rest of May.

-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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I wrote,

>> In a full musical context, composites will most likely come into
>> play more than higher primes. But that is by means of chains of
>> connections, which the lattice can already display quite nicely.
>> I prefer the lattice to be constructed devoid of any
>> considerations of musical context, since those can always vary.
>> Without context, I see (and more importantly, hear) no reason to
>> consider composite simpler than primes with similar-sized numbers.

Joe Monzo wrote,

>Ok, I think basically that makes sense and I can almost agree
>with it. But it was you yourself who just complained recently
>about 9:5 having a longer metric than 11:5.

That's a totally different issue.

>So it seems that
>really we're both looking for a lattice formula that would create
>a metric that's in between what we're already getting, your
>results being too large on one end and mine being too small
>on the other.

That's only true in a few select cases. Besides, with either wormholes or
composite-odd axes, the metric on my lattice is exactly as I want it to be.