

I have realised that this subject will allow me to do two things which I have never been able to do before. A new approach to a new field:
1) Think about it. Do not read anything.
2) Write down all ideas, algorithms paradigms etc.... about how you would do it, from your position of ignorance.
3) Stop thinking and read everything which has been written about the subject.
4) Repeat step 2 from a position of knowledge.
Of course rarely do you get to start from a position of true ignorance, which is where I am at the moment. You are usually influenced by knowing something about how the experts do it.
The second thing relates to something I suggested in a scientific context which everyone thought was a terrible idea. Suppose there are some numbers you would like to know but can't get experiementally and you are not clever enough to be able to calculate them. Guess, (estimate sounds better), them all, and guess all the error bars and then proceed as though they were the real data. What does it predict? At least if you ever get the real data then you know how good an estimator you are.
Though it looks like we can't calculate the absolute complexity of a piece we ought to be able to guess the relative complexities of 2 or 3 pieces together. With all these ratios it is like having the real data but not knowing what the units are.
Misusc is right to think that fractals are the key. Complexity and fractals probably can get you to the same place by different methods which have some underlying equivalence though of course high fractal dimensions as in the Mandelbrot set actually come from simple not complex objects, (just a quartic equation), but a coastline is infinitely complex if you go down to the shape of the last grain of sand before the sea, so there is much to understand here.
I once did some research in quantum chemistry and fractals and found out that I could not really understand what the mathematicians were saying and could not get them to say anything other than it might be this or it might be that but I do not know and they could not understand what I was getting at. Several people said there was something really good there but the work has circulated half finished for years without getting into a good journal.
Anyway if anyone to like to guess the relative complexities of a Mozart, Beethoven and Schoenberg piece, (I like Op18 no. 1 as Theo Wyatt used to use it as a benchmark of medium difficulty, and the Chamber Symphony No. 1, do not know which Mozart....) let me know. I am at stage 1).



Awesome!
After reading and thinking about the other post, I went and looked into fractals.
Ok, I am years behind...or should I say centuries!
but, I think there is something here that will be very good to investigate. And funny enough, I did come across Mandlebrot (ha ha).
So, here is where I am at:
q1= q0^2 + q0
q2= q1^2 + q1
q3= q2^2 + q2
q4= q3^2 + q3 ....
I am going to try and learn the math to understand exactly what this is! A combination of steps 1,2 and 3 I guess. It's gonna take some time. But, to be honest, my goal is not just to understand more about Mozart and the boys. I am mostly interested in the idea of their being Mathematics to explain musical quality in terms of the density of complexity compared to a "seed" idea so I can contemplate it when I write my own music.
At present, I am struggling through a new piece in which I feel am directly confronted with this issue. The work I want to compose is based on three thematic concepts, which are all derived in part from a 12 tone row. The row it self is quite interesting  P Db = Db C Bb G F E B D Ab A Eb F#  incredibly symmetrical (so much so that it must have been used before) but completely opposite of Webern. At any rate, the 1st and 3rd themes are in stone  very simple derivations of this material (the kind of treatments highlighting the tonality that would make many new music heads squirm! oh well)  but it is the 2nd theme that is troubling. I want it to be more complex  a more atonal abstraction, yet rhythmically more simple  I know it must be in order to balance the other themes, and the role it will play throughout the work for large scale tension. But I am having a heck of a time working it out (6th try so far), and I am wondering if the above equation might help...maybe...
I know that there isn't any "real" answer. If there was, well, computers would be composing, and we just input variables (today I am going to compose Cs = 0.8240.1711i  a Julia set!), and I think we are a long way from that becoming the standard! But, I am interested in how Mathematics intersects with music  not just in the Xenakis kind of way, which seems quite literal, but in a more generalized way.
(note, I have heard some interesting pieces composed using Macs (improvising interface) and mathematical constructs by Robert Morris)
"it might be this or it might be that but I do not know"  maybe this is enough to expect from mathematics and art....in composing, we are confronted with endless choices...it might be this, it might be that...i do not know for certain, but this will be my choice. I am at least interested in knowing what the this or that is, and take it from there.
Martin, please share any more ideas you have  vague as they might be. Do any of these ideas come into play in your own composing?
back to composing...



I will have a look at the symmetry and possibilities of your series Scott. Will be back on that later.
I have not applied any complexity ideas yet. The last two years I have mostly spent the winter on preparing original pieces and transcriptions of all sorts of stuff to get material for several summer schools / musical house parties which I am involved with running. The original music I write for this is in a mid 20thC accessable style as I think any extreme techniques would turn people off and they would go and play Jenkins instead, (even though Jenkins is marvellous not what I want...).
However within these quite traditional pieces there are mathematical concepts, which nobody has yet noticed. Some of these might be new and immediately useful to people out there....
An old one is to use the Fibonacci series to generate a small climax. This is as old as Bartok's Music for Strings Percussion and Celesta, see the celesta solo in the slow movement.
http://en.wikipedia.org/wiki/Fibonacci_number
This series piles up very rapidly and so if you use it for counting notes in time the music gets very black after a few bars........ (There are probably some series using complex numbers which go round in circles or spirals, these could be good.)
Another concept is to extend the baroque idea of diminution / augmentation into the pitch and fractional time domain. One way of modifying a theme is to compress it, even down into microtones. According to Bartok this is done naturally in some Serbian and Croatian folk music. I have never gone into microtiones yet, but only because of practicalities. Expansion works better than I expected. Augmentation by stretching whole phrases by 10 or 15 percent works well and can be done by several notational tricks as well as just at the quarter note level. Of course all these can mixed and the thematic material cut up and some compressed, some expanded in time / pitch domain. (What about instrumental colour domain too? I have not done that one.) Anyway ther are a few ideas I have applied.



I also thought last night that there are no applications of complex numbers in structural complexity, though of course spectralists are indirectly dealing with complex numbers all the time as they occur so naturally in any description of frequencies. But I was wrong. Frequency at the multisecond level and repetition can use complex numbers.
Just take the number 1/root(2) + i/root(2) and keep multiplying it by itself.
Firstly you get i, then 1/root(2) + i/root(2). Then 1 then
1/root(2)  i/root(2). Then i. Then 1/root(2)  i/root(2). Then 1. Then after the next multiplication you are back to where you started. Plot it on an Argand Diagram
http://people.bath.ac.uk/pgc20/argand.html
You have gone round in a circle. More complex maths allows you to move forward and backward in time, round in spirals etc. but using a simple(?) elegant description. Something to think about there.
It is the magic of the square root of minus one which allows this repeated application of the same number not to either explode upwards or go down to nothing.



Have you heard of the software "Open Music"?
it allows you to work with all sorts of mathematics, while giving you an immediate musical equivalent.
very interesting,up to a certain point.
and it saves you an enormous amount of time



I have heard of it but forgot about it when I could not download it from IRCAM. I will put that in my bibliography for stage 3, the list of things I must not read until I have already written..... Thanks Nicholas.



There is a couple of things I forgot to say ealier, the first of which must be well known to minimalists: that is the use of duplets e.g. septuplets and quintuplets in 6/8 time, (6 12 and 24 have plenty of factors), to expand and compress time. I have used this a bit.
The other thing concerning time is stretto. Difficult in fugue because if you work back from the stretto to the beginning the fugue will probably be rather stereotyped, if you don't a strict stretto will not harmonise.......
Frescobaldi is a master of strettos for winding up the tension at the end of a piece. I wondered about an inverse Fibonacci series to generate a stretto but it would go entry, entry, two more entries, faster, much faster, then whallop as all the voices piled in..... Could be exciting.....



Nicolas,
Nice to hear form you!
Care to share any results (even if just words...) regarding experiments with Open Music? Or in other composers?
What were the Mathematics used? What did they apply to in the composition?



Hi Scott
in Open Music you can find lots of things,for instance:
all the Xenakis' techniques (nonoctaviantic scales, rhythms,sieves, groups, probabilistic theories etc)
Spectral techniques (chord interpolation based on algorithms, spectral compression/dilatation)
many of Ferneyhough's techniques for manipulating intervals and rhythms.
I'm no mathematician to know exactly what the math part of the whole thing is, but it is all based on algorithms,graphic representations of equations etc.
It's a very interesting tool, because you get the chance to experiment a lot with the numbers and you can immediately hear what the musical result would be.It saves you a great amount of time.there is the danger that one could get too into it,but that's up to the composer



"Xenakis' techniques (nonoctaviantic scales, rhythms,sieves, groups, probabilistic theories etc)"
I understand the first three, but it is the probabilistic theories that I feel in the dark with (at least on a specific scale). Anyone?
I am a huge fan of using these other concepts in my own writing, though. Especially non octave transposing scales. I started using this to achieve a sound that seemed reminiscent of Messiaen in my head, but I didn't want to just go and copy his chords.
So, I devised a simple mode built on the cycle of 5ths (1/2,1,1/2,1/2,1 semitones within each 5th). Then, I just built chords on "4ths" using this mode. Worked like a charm. And It pops up again and again in my music, but, different modes with different transpositions  often derived from other material in the piece.
"Spectral techniques (chord interpolation based on algorithms, spectral compression/dilatation)"
This is a field I know little about, but would very much like to learn more. Sometimes some very satisfying results (but not always!). Although, back in my electroacoustic daze, I would take the sounds of tam tams and transpose them down a couple of octaves, thus revealing higher overtones, and then composing around them. This was fun.
To be honest, I'm scared of using a computer in this way to try things out (open music that is). I would need to have a much deeper understanding in my head before I let a computer do the work for me.



I have a feeling that what I am thinking about is going to turn out to be in the category interesting but not very useful wheras generating musical things, any old things, things that have no relevance, things that you do not understand, etc. and seeing if they work as music is obviously useful.



My personal experience so far with Open Music has been very useful,because it helped me acquire this "deeper understanding" that Scott wrote about.
If you get the software and spend some time with it,then you become acquainted with the concepts,the techniques and how they work.
I think it works the other way round:first I let the computer do the work for me,then I understand it,and then I don't need the computer any more.
Because the truth is that,once you've gotten used to it and you understand how every algorithm works, and how changing the parameters influences the results,then you can easily do it on your own,because you know what you're looking for,what harmonies you search for in your head.
and the other truth is that,if you're familiar with Open Music,you can tell a piece that has been composed with Open Music.and trust me,that is non a compliment for the composer.
If the composer does not interfere with the outcome and just takes it as it comes out from the software,the music is very often missing "something".



Yes I can see that ... I would not think of taking actual open music generated stuff as music, more the concepts which come out of the concepts and the concepts which have come out of the concepts which came out of the concepts etc.. Music must exist in the brain not the computer to work.... And you have to relate all this back to some cultural background.
This brings me on to questions about time in music. Time is not a moment, it contains somehow what you have heard before. And yet the complexity of a piece does not seem to increase as you listen to more of it..... Entropy in music?



I'd have to disagree with you on that last part.
I think it largely depends on the piece.
There are pieces that,the more you listen to them,the more you discover,the more you understand the connections that exist in the music,the deeper you can penetrate the music.
Good music has ,I think,that quality.I don't know if I could call that "increasing complexity",but I think that,the more levels of listening a piece has to offer,the more interesting it becomes with time,so ,in a way, the complexity grows with every listening.
I'd have to say that clarity,understanding more of what's going on, is linked with perceived complexity.The more into the music one goes,the more one sees what that music is about.
if I understand correctly what "complexity" is.



I agree with you. I was meaning just one listening, not repeated listening....... The piece does not seem to get harder to listen to the further into it you get though the interrealationships which are happening are increasing bar by bar, just as the amount of information in history seems to increase as time goes onwards. Conventional theory says information is being destroyed somewhere else, (black holes, white dwarves?).
UK TV adverts say the world is in smart / dumb balance, every clever decision is countered by a dumb one. Almost and attractive idea, though I can't remember what it was advertising so it might not have worked.
As you listen to a piece repeatedly the preexisting complexity reveals itself to you.


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