Yahoo Tuning Groups Ultimate Backup mills-tuning-list RE: Muddy Waters

I have not recieved TD 1122, so I write with some tentativeness . . .

I wrote that in the rectangular lattice philosophy,

>>i.e., a major seventh is just as dissonant/complex as a major
>>sixth.

Marion replied,

>I must be missing something here. I thought a major seventh was
>15:8, and a major sixth was 5:3. The last time I checked the LCM
>of 5:3 was 15 and the LCM of 15:8 was 120.

Note that I did state that I was assuming octave equivalence, by which I
meant exact equivalence in all respects. (If we drop the assumption
altogether, then I would argue that prime or odd limits will not do;
every integer should be a distinct limit representing a certain degree
of harmonic complexity.) Equating the LCM and ROHS ideas as Graham has
done requires that we consider 2 to be a distinct factor. So I should
really be comparing 5:3 with 15:1 directly if I am specifically
addressing the LCM variety of the rectangular lattice philosophy. 15:1
is a very wide interval so the usual roughness we associate with
dissonance does not come into play. But if we could somehow control for
the width of the interval, I claim that 5:3 should be considered more
consonant than 15:1. For example, I think 14:1 is slighlty more
consonant that 15:1, once the unfamiliarity of the former interval is
overcome. But 5:3 is more consonant than any 7-limit interval.

I think a basic argument for the triangular as opposed to rectangular
lattice representations can be founded upon traditional theory.
Traditional theory deems 15:1 and all its octave equivalents to be
dissonant, and 5:3 and all its octave equivalents to be consonant. Aside
from second inversions, traditional theory deems major and minor triads
to be consonant, while all inversions of the triads 1:3:15 and 1:5:15
are considered dissonant. A 2-d triangular lattice therefore seems a far
better fit to traditional theory than a 2-d rectangular lattice.

To address Graham's questions: One way of constructing a lattice with a
useful metric is to make the length of each interval, according to a
"city-block metric," a monotonic function of its (odd) limit. James
Tenney and Graham have both proposed using log(n) as the length of each
step along the axis representing the factor n. The problem I have with
that is that m/n and m*n will both have a metric of log(m)+log(n).
Following traditional theory, Partch, and my own ears, I think that for
any odd number n, the ratios n/m or m/n, where m is an odd number less
than n, and arbitrary powers of 2 are allowed, are more consonant that
ratios involving odd numbers larger than n. The idea of a lattice and an
associated metric is to make the distance associated with a consonant
interval shorter than that associated with a dissonant interval. One way
to make the lattice my beliefs is to use isosceles triangles; the base,
representing 3:1, is given length log(3), while the sides, representing
5:1 and 5:3, are both given length log(5). One could also envision using
scalene triangles and giving 5:3 a longer length than 5:1 if one felt
that 5:1 was more consonant than 5:3. One could even preserve the
appearance of the rectangular lattice and allow a "short-cut" diagonally
through the block for 5:3, but not for 15:1. Then the length of 5:3
would be sqrt(log(3)^2 + log(5)^2), but 15:1 would still be
log(3)+log(2). This sort of representation just barely works because
sqrt(log(3)^2 + log(5)^2)9487 and the smallest compound interval
length is log(9)1972.

In the 5-limit case, the plane is filled by triangles. The right-side-up
triangles are major triads, and the upside-down triangles are minor
triads. In the 7-limit case, space is filled by tetrahedra and
octahedra. The right-side-up tetrahedra are 7-o tetrads, the upside-down
tetrahedra are 7-u tetrads, and the octahedra are Wilson hexanies. In
general, a lattice of mutually prime axes will have the property that
space is filled by all possible CPSs where the set of factors is the set
of axes. One of each of these CPSs fit together like pieces of a puzzle
to form an Euler genus that uses each factor exactly once. Since the
Euler geni fill the space, so do the CPSs.

When the axes are not mutually prime, as is the case when 9-limit ratios
are considered consonant, things get a little stranger since the same
note may appear in more than one place in the lattice. The easiest way
around this is to use the prime lattice and introduce a lot of
"shortcuts" corresponding to 9-limit and other odd ratios, but that
destroys the CPS-filling-Euler-genera-filling-space picture. However,
the usual otonal and utonal hypertriangles, the most compact structures
in these lattices, will fail to be the only saturated consonant entities
if the axes are not prime, so the CPS picture is not so useful in those
cases anyway . . .

Marion, I am using geometrical pictures to express my thoughts and your
LCM idea, as Graham pointed out, is equivalent to a particular
geometrical picture as well, a rectangular one where 2 is a distinct
factor that has its own axis, and the log of the LCM is just the
distance (according to the city-block metric) between the two most
distant points in the chord. In other words, the LCM is a measure of the
longest extent of the ROHS of a chord. Since I feel that a minor triad
is more consonant than a major seventh chord, I don't like this method
and would prefer instead to use the triangular lattice. What replaces
the LCM in that case? If we use the isosceles triangles eluded to above,
the answer is just the Partch limit!

Much more to come . . .

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RE: Muddy Waters

🔗"Paul H. Erlich" <PErlich@...>

🔗mr88cet@texas.net (Gary Morrison)