Reply to Graham Breed's Miscellany

>LATTICES
>--------

>It would be more correct to say "Paul Erlich's Triangular
>Lattice Diagram, which is of the type used by Breed". Paul
>was using them before me, definitely extended them to the
>7-limit first, and drew them in ASCII first. He also named
>them "triangular". As I say on my website (and have said
>before on this list) _I_ would have called them hexagonal,
>like I was taught in Condensed Matter Physics.

>I don't know how much, if anything, is original to Paul.
>He can speak for himself on that.

My 7-limit lattice is of course a face-centered cubic lattice, distinct
from the hexagonal lattice which packs space just as efficiently but is
less symmetrical. I came up with the idea on my own, and while at first
it perplexed John Chalmers, he later found it to be quite similar to
what Erv Wilson had done (exactly the same, in fact, except that Wilson
may (or may not) have simply seen the lattice as existing in two
dimensions, while I definitely see the two-dimensional representation as
being a projection of the three-dimensional one). In fact, my triangular
lattices show Wilson's CPS scales as the most symmetrical arrangements
of tones, while cubic lattices show Euler genera as more symmetrical.

>PRIME VS ODD LIMIT
>------------------

>The most paradoxical chord from an odd limit perspective I
>find is 4:6:9. I think of it as extended 3-limit, because
>it's the fifths that give it it's character. It is also
>highly octave specific. It sounds more consonant
>(concordant) than a major triad in root position, but
>transpose it to 8:9:12 and it's clearly more dissonant.

Well, I don't know if it really beats a major triad, but otherwise, I
quite agree. Here's the interesting thing. When I worked out a model for
harmonic entropy, which should also describe critical band roughness if
the partials decrease in amplitude in some specific fashion, I derived
that to a good approximation, the complexity of a just ratio is directly
related to its DENOMINATOR. Later, imposing octave equivalence made me
change this to ODD LIMIT, but I admit that it's possible that octave
equivalence does not really come in to the "objective" dissonance of an
interval.

Now the chord 4:6:9 has three intervals: 3:2, 3:2, and 9:4. The highest
denominator is 4. The chord 4:5:6 has a 3:2, a 5:4, and a 6:5. The
highest denominator is 5. So judging from the intervals alone, you
appear to be right that the 4:6:9 is more consonant. However, I think
the chord as a whole has something that the intervals alone don't
explain, as evidenced by the differing levels of consonance in otonal
and utonal chords. On the whole-chord basis, which has proved
impenetrable to the type of analysis that led me to the denominator rule
for intervals, I think the major triad would win.

The chord 8:9:12 has the intervals 3:2, 4:3, and 9:8. The highest
denominator is 8 -- pretty dissonant!

>There may be ways the ear relates to prime factors. Different
>overtones will be reinforced, and the difference tone pattern
>may change.

Can you give a concrete example?

>The good intervals in equal temperaments don't usually
>constitute anything like an odd limit. It's generally more
>efficient to say how well different primes are approximated.

This may not work if consistency doesn't hold.

>For exactness, state the signed errors, and you can work
>other intervals out from that.

Isn't that what Carl Lumma said? But no, you guys are wrong, and that's
the whole reason for the consistency concept. Wendy Carlos, Yunik and
Swift, and others also seem to have missed out on the importance of
consistency.

>Incidentally, it seems to me that the concept of level-n
>consistency is closely related to the recently defined
>radius of the scale. If you want to use all the notes in
>a radius 2 scale, it helps if it's level-2 consistent as well.

I don't think it helps all that much. I think Paul Hahn would agree with
you, though.

>LUCYTUNING
>----------

>Gary Morrison wrote:

>> The semitruth: LucyTuning definitely can stack up more fifths above a
tonic
>> before approximately closing the circle than either quarter-comma or
third-
>> comma meantone. Third-comma meantone comes "close enough" at 19 fifths,
>> and quarter-comma at 31 fifths. LucyTuning doesn't get there until
about
>> 88 fifths. But this is only a semitruth, because taken in absolutes,
the
>> circle of fifths never closes in ANY typical meantone tuning.

>The most irrational meantone in this respect is Kornerup's
>phi based tuning. Is that right?

I think so. There are more "irrational" tunings with one generator, but
they cannot be considered meantones.

>I think of it as the
>standard melodic meantone.

I've thought about that too, but I don't know. It is awfully close to
the harmonically optimal meantones in my paper.

>GUITAR TUNING
>-------------

>Incidentally, having different notes on different strings is
>a _good_ thing. It means you get more chords than you would
>otherwise.

Huh? It definitely means fewer positions in which to play a given chord.