lattice wrap-up

[Lumma, TD 125:]
> 49/32 isn't more complex than 25/16 because it
> took two 7-limit rungs and 25/16 took two 5-limit ones.

You really think not? 49 isn't more complex than 25?
It isn't because 7 is more complex than 5?

> What am I not getting? The triangular lattice can
> represent multiple factors on a single rung,
> the rectangular lattice cannot. This is true no
> matter what is going on with the 2's, as far as
> I can see. For example, 7/5 is one rung in a
> triangular lattice whereas it is two rungs on a
> rectangular one.

I don't know if this has any bearing on what you're asking,
but thought it should be pointed out.

Inasmuch as all the different lattices and tonal systems
we have been considering with this topic are based on
an implicit assumption of a 1/1, they are all Monophonic
(in the Partchian sense), and therefore *ALL* ratios
are in one sense theoretically one rung away from 1/1.

-monz

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[Erlich:]
> 7:4 and 5:4 are each three rungs in the rectangular
> lattice, but they still come out a litle simpler
> than 7:5 since the rungs along the 2-axis are so short.

> Ah, I see where you're coming from... and this should
> make calculating things easy by use of prime factorization.

Does this mean you guys are coming around to rectangular?
It sounds like we're all beginning to meet.

In regard to Paul's statement that "the rungs along the
2-axis are so short", see the cover of Xenharmonikon 6,
Erv Wilson's logarithmic harmonic spiral diagram. This
is truly one of Erv's most amazing visual models.

I was staring at it the other day while also reading Erlich's
long (and excellent!) posting on/quoting of Terhardt,
and thinking about some of the things I've posted here
recently about the role of tonalness in calculating or
representing tri-and-higher-adic harmony, and I thought
it useful to imagine a prime-lattice of the type I use,
with a harmonic spiral appearing for each local 1/1
as it occurs in the music. The lattice would combine
the rectangular/triangular prime structure with, theoretically,
a harmonic spiral associated with each lattice-point.

[Keenan:]
> I have to agree that for purposes of predicting
> dyadic dissonance, prime factorisation is pointless.

Just so that we're clear on this, I agree with this too.

I'm just not that interested in dyads - I'm trying to
formulate theories about larger musical structures.

Dave, your post is a great summary on musical complexity!

> 200:250:300 Hz
> interval complexity (period)
> 4:5 4/200Hz = 20ms
> 5:6 5/250Hz = 20ms
> 2:3 2/200Hz = 10ms
>
> 200:240:300 Hz
> interval complexity (period)
> 4:5 4/240Hz = 16.67ms
> 5:6 5/200Hz = 25ms
> 2:3 2/200Hz = 10ms

Now, this is something worth quoting!

I thought it worthwhile to look at the ratios of the
complexity periods within these chords - that seems
to explain better than anything why the otonal chord
sounds so much more "consonant" (I'm almost afraid to
use the damn word now!) than the utonal.

It was also interesting to see how these ratios were
reflected in the ratios of the intervals themselves.

The otonal chord presents no need of real calculating:
the ratios of its complexity periods are 1:2, 1:1, and 1:2.
Interestingly, these ratios do not reflect at all on
the complexity of the actual numbers of the interval ratios.

The utonal chord presents a scenario which is, however,
much more . . . complex :)

There's a 3:2 relationship between the complexity periods
of the 4:5 and 5:6 intervals - the 5s, one a numerator and
the other a denominator, cancel, and numerator 6 and
denominator 4 are reduced in half.

There's a 5:3 ratio between the complexity periods of
the 4:5 and 2:3 - these numbers are in both numerators.

And there's a 5:2 ratio between the complexity periods
of 5:6 and 2:3 - these numbers are in both denominators.

Would the scientists please explain what this means?
if anything?

[Keenan:]
> Now this is the hard part; taking account of
> tolerance in chords.

I suggest that everyone be very cautious about disallowing
the relevance of prime-factorization when they embark on this
task. I'm convinced that it plays a major part when
considering pitch-sets of more-than-two distinct notes.

-monzo
http://www.ixpres.com/interval/monzo/homepage.html
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