Composition technique with Union Theory

[mikkeljs]

I just came across union theory in Xenakis´s Herma for piano, though I don´t know how exactly he used it. But this theory fits very well for my idea of composition, which I talked about before. I thought that fractals could  describe everything in the physics. But now, union theory, seems as you can simply put everything together.
It works like this:
You have a set A, which is a complexity expressed by numbers, fx A:{2,5,7,9}

Now, my idea is to create 2 new sets B and C, which might be different from eachother and from A, but they must be created from a common system. (this system should be entirely derived "around" A, so that there are no holes!

The first number in B could be 5, and the first number in C could be 9. Then we have to handle those two numbers in the same fashion through the other numbers in A, fx:

B:{5,7,12,14} , where 5+2=7, 5+7=12 and 5+9=14
C:{9,11,14,16} , where 9+2=11, 9+5=14 and 9+7=16

Now we keep doing new processes until we get the same numbers in B and C. If we can create such a common value, it would be, if I´m right, non-algebraic and rather mystical, since the complexity is equal to or higher than degree 5 (the quintic equation, which Abel proved, was unsolvable).

Do you think, that is right all the way through?

[bwv 1080]

The first number in B could be 5, and the first number in C could be 9. Then we have to handle those two numbers in the same fashion through the other numbers in A, fx:

B:{5,7,12,14} , where 5+2=7, 5+7=12 and 5+9=14
C:{9,11,14,16} , where 9+2=11, 9+5=14 and 9+7=16

Yes, various operations on sets are used alot.  I am assuming you are mapping the integers to semitones, so +5 means a transposition of a 4th.  Depending on whether or not you want spatial sets (i.e. fixed octaves) you may want to use mod-12 arithmetic (mod simply means the remainder after division by the mod #) which would keep the sets within an octave (i.e. 5+2=7, 5+7=0 5+9=2 - so the resulting set is [027]

Now we keep doing new processes until we get the same numbers in B and C. If we can create such a common value, it would be, if I´m right, non-algebraic and rather mystical, since the complexity is equal to or higher than degree 5 (the quintic equation, which Abel proved, was unsolvable).

I do not see how that follows.


You might check out this book:
http://www.amazon.com/Basic-Atonal-Theory-John-Rahn/dp/0028731603/ref=sr_1_3?ie=UTF8&s=books&qid=1211645221&sr=1-3

[mikkeljs]

Yes, various operations on sets are used alot.  I am assuming you are mapping the integers to semitones, so +5 means a transposition of a 4th.  Depending on whether or not you want spatial sets (i.e. fixed octaves) you may want to use mod-12 arithmetic (mod simply means the remainder after division by the mod #) which would keep the sets within an octave (i.e. 5+2=7, 5+7=0 5+9=2 - so the resulting set is [027]

Thanks for the advise!  I was only thinking in abstract numbers so far, in order to imagine a simple composition. But I will keep your idea of mod-12 arithmetic in mind.

 

[mikkeljs]

I do not see how that follows.

I just mean, Abel said, that the quintic equation and any other higher equation, can´t be solved algebraically. I thought it´s the same with non-continous expressions, no? Of cause the calculator mashine or a computer program can give you solutions on very complex equations, but that works in another way, that is more primitive and not 100% correct. Fx it draws a generated graph in order to decide the solution for an higher equation, but often you only get a close number and not the exact.

I mean, if something can´t be calculated, then it would be a little miracle, if you found the solution, even if it was easy to find. There is something mystical about that. 


by the way, thanks for the link to the book. Looks really interrestingly and I will put it on top of the wish list for my next birthday! 

[mikkeljs]

I just typed on my calculator, to see how many random combinations you could get from two sets with 5 different numbers each, and the result was 120 (less, if some numbers were common).  The same number of possible solutions for a quintic equation!

So the sets with 5 numbers, really are the minimum complexity, if the music has to be at least as complex as the quintic equation.

Now I´m wondering, if you can describe two combined sets each having 6 numbers each, with a seria of sets with only 5 numbers each. And I found out, that the same possible combination as two combined 6-numbered sets, would be a single operation on a seria of 7 5-numbered sets, where you can choose one and one other of the 6 sets:

one set operation by random combination on
[1,2,3,4,5,6]
[7,8,9,10,11,12] , where all sets involved should be entirely derived          possible combinations 6x5x4x3x2x1

and one set operation by random combination on
[1,2,3,4,5]
[6,7,8,9,10]
[11,12,13,14,15]
[16,17,18,19,20]
[21,22,23,24,25]
[26,27,28,29,30]
[31,32,33,34,35] , where all sets involved should be entirely derived           possible combinations 6x(5x4x3x2x1)     

so does that mean, that you can principally describe everything with sets with only 5 numbers?

[bwv 1080]

I just typed on my calculator, to see how many random combinations you could get from two sets with 5 different numbers each, and the result was 120 (less, if some numbers were common).  The same number of possible solutions for a quintic equation!

Yes but that is trivial - the binomial theorem uses the same combinatorial calculation.

So the sets with 5 numbers, really are the minimum complexity, if the music has to be at least as complex as the quintic equation.

But one has no relation to the other

Now I´m wondering, if you can describe two combined sets each having 6 numbers each, with a seria of sets with only 5 numbers each. And I found out, that the same possible combination as two combined 6-numbered sets, would be a single operation on a seria of 7 5-numbered sets, where you can choose one and one other of the 6 sets:

one set operation by random combination on
[1,2,3,4,5,6]
[7,8,9,10,11,12] , where all sets involved should be entirely derived          possible combinations 6x5x4x3x2x1

and one set operation by random combination on
[1,2,3,4,5]
[6,7,8,9,10]
[11,12,13,14,15]
[16,17,18,19,20]
[21,22,23,24,25]
[26,27,28,29,30]
[31,32,33,34,35] , where all sets involved should be entirely derived           possible combinations 6x(5x4x3x2x1)     

so does that mean, that you can principally describe everything with sets with only 5 numbers?

Again this is trivial, you can apply the same logic to 4 note sets relative to 5, 3 relative to 4 and so forth.

[mikkeljs]

Again this is trivial, you can apply the same logic to 4 note sets relative to 5, 3 relative to 4 and so forth.

Thanks! You helps my thoughts going faster!