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Music, however, involves something more than numbers & words. Music involves a phenomenon known as "sound."
This annoying disturbance of the air is viewed with disgust in many quarters, particularly the music theory department of universities like UCSD.
(We in the southern california microtonal group call it "SCUD," because the Music Dept. of the University of California at San Diego is a weapon of intellectual mass destruction-- the students remain standing, but their imaginations have been destroyed.)
However, those who subscribe to this tuning forum presumably recognize that what music actually SOUNDS like has some bearing on the subject. And so, here are some practical tips for using Wilson CPSs in actual compositional settings.
Mind you, these aren't offered as "the way to do it." Instead, they're simply a few strategies which have proven useful tried & tested Wilson CPS compositions. As you work with Wilsons CPS scales you'll no doubt find other strategies which prove equally (or more) useful. As always, you must decide for yourself and ignore the pronouncements of "experts."
First:
Substituting a single different prime which is still mutually prime to the rest of the generators of the CPS produces a scale that's slightly different in a few pitches, yet which tends to sound similar to the original CPS.
For example, the 3,6 [1,3,5,7,11,13] eikosany and the 3,6 [1,3,5,7,11,17] eikosanies sound very similar. And most of their pitches are the same.
This offers a simple and straightforward method of "transferring" out of a Wilson CPS into another entirely different Wilson CPS.
By programming 8 of these scales into my TX802 (which enjoys the cachet of 8 different tuning tables), I can effortlessly and transparently move between Wilson CPS scales.
Suppose, for example, we want to move
from 3,6 [1,3,5,7,11,13] to
3,6, [3,5,7,11,13,17]. This can
be done either by stellating the
ekosany (a complex process and hard
to conceptualize), or you could simply
move from the "touching points" of the
3,6 [1,3,5,7,11,13] eik into the
"touching points" of the 3,6, [1,3,5,7,13,17]
eik into the "touching points" of the
3,6 [1,3,5,11,13,7] eik into the
common scale degrees (AKA "touching
points" in ratio space) of the
3,6, [1,5,5,11,13,17] into the
common scale degrees of the
3,6 [3,5,7,11,13,17].
See how easy that was?
The overall modulatory process introduces some fascinating new chords, or "facets" as Erv Wilson calls 'em. This usage arises from Erv's visualization of the eik in ratio space: the triangular "facets" are in fact chords and the total number of facets is equal to the total number of different 3-note chords.
The process described above is akin to taking various vertex-edge paths along these facets through a stellated eikosany to another eikosany embedded within the stellation. Again, this is something that Erv visualizes easily and intutively but which the rest of us might find easier to conceptualize as a movement between sets of pitches (it certainly is for me!).
Again, this is a process known well to Erv and long since discussed and dealt with in his (unpublished) writings.
A process which Erv has not to my knowledge dealt with, however, is the combination of non-12 equal temperaments with Wilson CPSs. More properly, the issue at hand is how to move most adroitly from a Wilson CPS which uses just pitches to any given equal temperament and back again.
It turns out that it isn't that hard. As we've seen, the Wilson CPS exhibits a just 3/2 fifth ONLY in those cases where a 1,3 is present (or the equivalent: 3,9 will do as well) in the generating set of integers.
However, we've also seen that "spiking up" one of the generators to a large integer will "knock askew" all the members of the CPS. Thus, to move from a Wilson CPS with just pitches into, say, the 19-tone equal-tempered scale, we need to move from a wilson CPS, say, of 2,4, [1,3,5,7] to one of 2,4, [1,3,5,X] where X is a prime number much larger than 7. By guessing around and playing with numbers, it's possible to find some large prime that knocks the 3/2 term of the 2,4 [1,3,5,7] hexany out of kilter so that it's now about 7 cents flat. Then, voila! All we need to do is modulate from 2,4, [1,3,5,7] to 2,4, [1,3,5,X] and from there (via the flattened 3/2) to the 7-cent-flat fifth of the 19- tone equal tempered scale. We can "transfer" back (Ivor Darreg used the term "transfer" to indicate musical movement between different tuning systems in the course of a composition) from 19-TET to the 2,4 [1,3,5,X] hexany just as easily, and from there to 2,4, [1,3,5,7].
Now, it's equally obvious that we can just as easily move from a Wilson CPS to, say, 15-TET, and once in 15-TET, we could easily move to 12-TET or 9-TET, which both share 3 common tones with 15-TET...and this opens the door to "transfer" between *many* different types of scales.
Lastly, it's worth pointing out that the Wilson CPS scales can just as well be viewed as timbres. In particular, taking subsets of the very large CPSs and arranging them as timbres works particularly well, since the very-slightly-offset pitches produce a warm complex of beats in the resulting additive synthesis sounds. Thus, while it's nominally true that very large CPSs don't really exhibit an acoustically useful gamut of pitches across their full range where scale generation is concerned, the very large Wilson CPS are definitely superbly useful for building additive synthesis timbres.
The opens up the possibility of composing
timbres by extending a given CPS.
Thus, the 4,8 [1,3,5,7,9,11,13,17] heb
might use timbres from the
5, 10 252-any [1,3,5,7,9,11,13,17,19,23].
The ear will hear a connection, since
many different 4,8 [1,3,5,7,9,11,13,17] hebs
are embedded within the
5, 10 252-any [1,3,5,7,9,11,13,17,19,23],
but the connection will remain audible
at a low level...and with the added
benefit of considerable "warmth" from
the very slightly offset partials of
many of the 252-any's pitches in an
additive timbre, giving sounds thus
constructed a lovely chorus effect not
unlike that of string choir.
The next and final post discusses a few of the more obvious extensions of the basic Wilson CPS method of generating scales.
--mclaren
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