Yahoo Tuning Groups Ultimate Backup tuning lattice wrap up/ Erlich

Erlich [TD 130:]

> Lattices are useful for visualizing the harmonic relationships
> between notes, for composition algorithms utilizing those
> relationships, etc. etc.

Amen.

> Perhaps we should break this up into
> a "Hahn" version (where the rungs are all equal) and an
> "Erlich" version (where the rungs have length log (odd limit)).

For purposes of helping to follow the thread,
can you state those two algorithms briefly, in stepwise fashion?

> any musical interval which is n^2/m^2
> is most simply described either as is or as n/m * n/m,
> and a lattice metric should represent the length of an interval
> as the sum of the lengths of the intervals needed to construct it.
> If n^2/m^2 were not twice the length of n/m,
> that would add many redundant points to the lattice,
> and the goal of the lattice is to present a clear visualization
> of the relationships between tones, etc. etc. . . .

I don't mean to make things more confusing,
but a version of the lattice which correctly represents
the bridges would have to portray this redundancy.

> Joe Monzo wrote:
>
> > Does this mean you guys are coming around to rectangular?
> > It sounds like we're all beginning to meet.
>
> For octave-specific lattices: rectangular.
> For octave-equivalent lattices: triangular.
>
> Joe, your lattices are octave-equivalent but "rectangular"
> (actually, you use parallelograms, but that's equivalent).
>
> So we haven't met yet :)

I've already stated that
I see value in portraying the triangular connections,
but that to add them to my lattices
would make them visually confusing.

On lattices of some smaller systems
(such as pieces I write on my Rational Guitar),
triangulation looks clear on the lattice, and I use it.

This instrument gives me only harmonic ratios over 1/1.
The entire pitch set has the proportions
8:12:14:18:20:21:22:23:24:25:26:27:28:30:34:36:38.
That's fairly small, and I can draw triangular connections
on the lattice of it.

My latest piece for this guitar uses 3/2 (i.e., 12)
as the modal tonic,
with much use of the 7/6 and 13/12 (triangular) intervals.
(BTW, sequencing it was a cinch with my micro.CAL program)

Where I differ with the rest of you is that
I don't really see the need to recognize composite-odd-factors
as a separate rung on the lattice, at least for tri[+]ads,
as I further reduce to prime-factor.

I have acknowledged that odd-factors are important
in defining interval (= dyad) metrics.
But I believe that for more abundant sonorities,
prime factorization simplifies the picture.

The problem I have is, well . . . hang on . . .

> > 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.
>
> Is this on the net?

Last time I looked it wasnt yet.
Kraig says he's working on it.
When it does show up, it'll be in _The Wilson Archives_ at:

www.anaphoria.com

It's nowhere near as dazzling as some of
Erv's later more angular lattices, but
it summarizes all the important harmonic information at one shot,
excepting prime-factorization (if that *is* important).
Altho, it's easily modified to prime just by
ignoring all the composite rungs.

> >> 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.
>
> Again, Joe, it's _relative_ primality that is of concern here,
> not primality itself.

What exactly is the difference between
relative primality and primality itself?

The only thing I've been saying all along
about the importance of primes in music
is that they allow us to view the musical pitch material
in its ultimately reduced and simplified form.

(and also that I think the lowest primes
carry quasi-unique audible information
- I say quasi because of the effect of bridges)

This reduction implies to me
a blurring of the distinction over relativity.

Primality itself provides the underlying framework
of the whole musical structure,
thru which differing perspectives may be taken
and by which the relative distinctions can therefore be seen.

Getting back to where I was going before,
the problem I have over prime-vs.-odd is:
how to account for the harmonic spiral without separate odd rungs
because harmonics do seem to imply an odd-limit.

However, my recent findings suggest that
triangular connections are just as important
in chord-formation and -recogition
as harmonic connections.
(more on this in my next post)

> Joe Monzo wrote,
>
> > 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.
>
> Partch himself only considered ratios within the 11-odd-limit to
> be one rung away from 1/1 -- accounting for 29 of his pitches.
> The other 14 were explicitly built on other ratios and considered
> inharmonic with respect to 1/1.

Yes, this is true - thanks for expanding what I said.
The 14 new additional intervals outside the 11-odd-limit diamond
were added mainly by extension of the powers of 3
to all boundaries of the lattice, symmetrically in both directions.

This did make them inharmonic with 1/1, altho
they are closely related because they are basically
!*triangular*! connections to the bounding ratios.

Also, Partch used more or less than 43 tones at times,
so he was inconsistent with regard to, or at least
did not restrict himself to, this particular system.

But my point was that
at least for a substantial number of pitches under consideration
in any given system,
there is *in this one sense* a direct connection to 1/1,
and *that should be taken into account in the formulae*
when considering tri[+]ads.

As I hinted a few paragraphs ago,
I think more direct connections among lattice-points
of *various different exponents and primes*
is a reasonable model of at least some examples of
harmonic/melodic practice. As I promised,
there will be more on this in my next post.

> Partch's one-footed bride
> (which is designed to be octave-equivalent,
> except for the one "foot")
> implies that he considered the consonance of ratios to decrease
> as their odd limit increased up to 11,
> and beyond that he considered all ratios equally dissonant.
>
> The first part of that points to
> my version of the Hahn-Lumma-Erlich algorithm,
> with declared odd limit of 11,
> implying a 5-dimensional lattice (3, 5, 7, 9, 11) with redundancies
> or a 4-dimensional lattice (3, 5, 7, 11) with wormholes.

OK - I understand the redundancies of the odd lattice,
but what are the wormholes
in this particular example of the prime lattice?

I've been thinking of the wormholes
in terms of the xenharmonic bridges between primes.
It sounds like you mean something different by it.

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

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lattice wrap up/ Erlich

🔗Joe Monzo <joe_monzo@xxxxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>