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Thus it is to be expected that no one in the "hard core" just intonation community has bothered to make use of Erv Wilson's limitlessly rich CPS method for generating just scales. And, obviously since composers like myself and Warren Burt and Jonathan Glasier subsist far outside the "hard core" just intonation community of "experts," *naturally* we are the only composers to have used Wilson's scales in any substantial musical way.
This follows from the inevitable and universal law governing all human affairs: to wit--
This is a basic reality of human life known well for thousands of years. Socrates was the first to point out the fact that most "experts" are in fact incompetent, most people in authority don't have a clue, and genuine ability and expertise is in fact vanishingly rare. For imparting this priceless gem of wisdom to the Athenians, Socrates was rewarded in the most obvious possible way: he was tried, convicted of "corrupting Athenian youth," and forced to drink hemlock, after which his corpse was kicked into a pauper's grave. Thus it seems aberrant that no one has yet handed me a cup and told me to drink deep. Perhaps after a few more posts like this...? No doubt.
"One double hemlock, coming up..."
So far, however, offering new ideas and telling the truth in public (two infallible marks of low breeding and a vicious nature) has merely earned me a few surly grumbles from the robe-and-tassel crowd. This is a much milder response that one would expect--no torches, no pitchforks, not even a noose tossed over a tree.
Why, that's like to get a body all encouraged, chilluns.
So it seems apt to conclude this set of posts by discussing some extensions to Erv Wilson's combination product set methods of generating musical scales.
[Permit me in advance to point out that these ideas are not mine; Erv himself was well aware of them long before my first mention of the subject. As always, Erv saw farther than any of us long before any of us imagined looking, and as usual Erv chose to keep his theoretical breakthroughs to himself. However, Erv considers this set of extensions to his CPS methods so obvious and so self-evident that he doesn't consider it a breach of confidence to discuss them. Alas, many of his far more interesting musical advances remain outside the realm of what can be discussed in public, since Erv has forbidden me or John Chalmers from bruiting them about. Perhaps time will change Erv's mind.]
The "standard" Wilson combination product set method of generating musical scales employs a set of mutually prime integers as generators. To get exotic numbers of pitches from the scale, generators which are factors of one another can be used--as we've seen in previous posts in this series.
Example:
Dividing the product of the 4 mutually prime generating integers 3, 5, 7 and 11 by 2 out of every 4 of these generators yields 6 different just intonation ratios, or--in Wilson's usage-- a hexany.
However, there exist *many* flavors of hexanies... and only a few of them are just scales.
Hexanies can be generated from sets of irrational numbers as well as from sets of integers.
In fact Wilson's method can generate an infinite range of scales: For instance, instead of taking 2 out of 4 integers, we could choose 2 out of 4 four infinite continued fraction convergents...or we could choose 2 out of 4 indices of a given prime (which would produce a just intonation hexany but not the standard Wilson hexany), or we could choose 2 out of 4 transcendental numbers.
In fact we could choose just about any recurrence relation or other mathematical operation to produce the raw set of generators from which the n out of m Wilson CPS is then sieved by combinatoric methods:
3*5*7*9/3*5 = 63/32 3*5*7*9/9*7 = 15/8 3*5*7*9/3*7 = 45/32 3*5*7*9/3*9 = 35/32 3*5*7*9/5*7 = 27/16 3*5*7*9/ 5*9 = 21/16but the Wilson CPS process can also give
totient(3)*totient(5)*totient(7)*totient(9)/totient(3)*totient(5) totient(3)*totient(5)*totient(7)*totient(9)/totient(3)*totient(7) totient(3)*totient(5)*totient(7)*totient(9)/totient(5)*totient(7) totient(3)*totient(5)*totient(7)*totient(9)/totient(3)*totient(9) totient(3)*totient(5)*totient(7)*totient(9)/totient(9)*totient(7) totient(3)*totient(5)*totient(7)*totient(9)/totient(5)*totient(9)(the enumeration of the pitches of this scale is left as an exercise for the enterprising xenharmonist) and the Wilson CPS process can also give
3!*5!*7!*9!/7!*9! = 1.40625 3!*5!*7!*9!/3!*5! = 1.70331 3!*5!*7!*9!/3!*7! = 1.29776 3!*5!*7!*9!/3!*9! = 1.1535644 3!*5!*7!*9!/5!*7! = 1.038208 3!*5!*7!*9!/5!*9! = 1.8457031but the Wilson CPS process can also give
ln(3!)*ln(5!)*ln(7!)*ln(9!)/ln(3!)*ln(5!) = 1.7052757 ln(3!)*ln(5!)*ln(7!)*ln(9!)/ln(3!)*ln(7!) = 0.9152701 = 1.8305 ln(3!)*ln(5!)*ln(7!)*ln(9!)/ln(3!)*ln(9!) = 1.2754418 ln(3!)*ln(5!)*ln(7!)*ln(9!)/ln(5!)*ln(7!) = 1.433612226 ln(3!)*ln(5!)*ln(7!)*ln(9!)/ln(5!)*ln(9!) = 1.90937824 ln(3!)*ln(5!)*ln(7!)*ln(9!)/ln(7!)*ln(9!) = 1.0722542and the Wilson CPS process can also give
[3,3,3,3...]*[5,5,5,5...]*[7,7,7,7...]*[9,9,9,9...]/[3,3,3,3...]*[5,5,5,5...] = 7.1400551*9.10977268 = 1.01631685 [3,3,3,3...]*[5,5,5,5...]*[7,7,7,7...]*[9,9,9,9...]/[3,3,3,3...]*[7,7,7,7...] = 5.1925826*9.10977268 =1.47822647 [3,3,3,3...]*[5,5,5,5...]*[7,7,7,7...]*[9,9,9,9..]/[3,3,3,3...]*[9,9,9,9...] = 5.1925826*7.1400551 = 1.15860393 [3,3,3,3...]*[5,5,5,5...]*[7,7,7,7...]*[9,9,9,9...]/[5,5,5,5...]*[9,9,9,9...] = 3.3027775*7.1400551 = 1.47387583 [3,3,3,3...]*[5,5,5,5...]*[7,7,7,7...]*[9,9,9,9...]/[7,7,7,7...]*[9,9,9,9...] = 3.3027775*5.1925826 = 1.07187156 [3,3,3,3...]*[5,5,5,5...]*[7,7,7,7...]*[9,9,9,9...]/[5,5,5,5...]*[7,7,7,7...] = 3.3027775*9.10977268 =1.88047218Where the notation [1,1,1....] indicates the infinite continued fraction
Clearly there are an unlimited number of other "flavors" of CPS scales, since there are an unlimited number of different types of operations which might be performed on the set of initial generators.
Moreover, the generators could easily be themselves algebraic or transcendental irrationals: for instance, the infinite continued fraciton CPS could be altered by taking
This brings up a signal point:
All forms of musical scale generation involve essentially a 2-step process--
Nota bene: This entire section is lifted bodily from my 1990 article "General Methods Of Generating Musical Scales," published in Xenharmonikon 13. It remains interesting that Your Humble Corresondent is still the only person on the planet to use a full range of the possible Wilson CPS scales, rather than the drastically and unnecessarily limited subset which are mistakenly thought to describe the full range of Erv Wilson's infinitely-extensible method of generating scales.
Just as Harry Partch could not be awarded a doctorate for his work while he was alive--but now that he's dead, far lesser intellects can be awarded doctorates for making minor commentaries on Partch's work--in like manner, the so-called "new music" establishment cannot recognize or utilize any of Erv Wilson's breakthroughs since Wilson is still alive.
Once Erv is dead, of course, doctorates will blossom like crabcrass in a garden...PhD after PhD awarded to pony-tailed po-mo wannabes for neglible footnotes to Wilson's basic work. Regardless, those of you with imagination and get-up- and-go and some interest in actually making music might want to start exploring some *truly* interesting extensions of Wilson's work in a compositional context. (Say, oh, perhaps 4 or 5 out of the 350+ subscribers to this forum?)
As the final and most bizarre demonstration of the properties of extended Wilson CPS scales, observe the tuning which falls out of the 3,8 [1.22222, 2.33333, 3.444444, 4.55555, 5.666666, 6.777777, 7.888888, 8.99999].
According to Manuel Op de Coul's SCALA program, this Wilson CPS produces the 9-tone equal-tempered scale.
It is left as an exercise to the enterprising xenharmonist to determine why and how.
(Nota Bene: this was a joke. Naturally no one realizes. Manuel's SCALA suffers from a bug. If you enter non-integer factors in the CPS, SCALA goes berserk and outputs junk--as in the above case. Manuel has never fixed this bug.)
--mclaren
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