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This 50 message thread spans 4 pages:  < <   1  [2]  3   4  > >  
  Re: Algebraic complexity in music  MartinY at 19:25 on 23 June 2009

Just as I was thinking my thoughts were theoretical and useless my public conversation with Nicholas has allowed me to make a firm prediction. One listens to a new piece, the experience mostly uses auditory parts of the brain. One listens to it repeatedly and what is going on changes, using much more of the visual system and whatever parts of the brain are used in geometrical and algebraic processing. If this is done with a subject's head inside a MRI scanner there should be a clear change in the pictures as the piece becomes more familiar. Has anybody's university done this?

Useless FACT realised on the 23rd...... All the compositions which have been written and all the compositions which WILL be written can be stored in one real, algebraic not transcendental number (look up Boreal numbers). This is a fact but does it have any concievable use. I am a bit sorry music is not transcendental.

However all possible performances is a different matter, is it?

  Re: Algebraic complexity in music  Misuc at 10:36 on 25 June 2009

I know next to nothing about maths.
What I found exhilarating in my reading about fractal geometry is the way 'simple' iteration could lead to 'complex', 'chaotic' and even 'unpredictable'.

I have also read and not understood Xenakis.

I have also bought 'Open Music', but have no tutorial or decent music software or hardware either. So I don't even know where to begin.

But all of this misses the point. These may be the subject of interesting experiments to develop the imagination, but imagination is the key. And the point about imagination - i.e. what is learnable about it - is that it works dialectically not according to formal logic, even fractal.

I once had to write 60 pages about Haydn's 'Farewell' Symphony. Even allowing for a fair amount of guff and gossip this left quite a bit for 'analysis'. I found all kinds of obvious and hidden connections and links - networks of motivic and harmonic relationships etc. and didn't even begin to scratch the surface of what, I suppose, could be a useful algorithmic or fractal description. Such a description could be very interesting and revealing to those who could understand it. But it would only be a description i.e. after the event. You can be sure that Haydn never studied fractals.

His music is still both deeper and more complex than that of Xenakis (a composer who great interests me). This is partly because the 'language' he used (now extinct)lends itself more readily to layers of dialectical thinking that reach further and deeper than that of Xenakis. In the end Xenakis and the sort of methods we are discussing can only present notes and ideas 'as they are' and as they have been invented or discovered in one head.

Try it if you like, but this is a severe limitation. Listen again to the Jenkins piece we were talking about to see the difference. Every semiquaver refers back to ideas that have emerged from the heads of millions and to ideas that are not simply 'as they are' but 'as they might become', as 'they are no more' and 'as they never quite were'. I am aware that I am not making myself very clear. Maybe one of you will find an algorithm which will be able to notate my idea more clearly than I can possibly think it...?

  Re: Algebraic complexity in music  MartinY at 15:35 on 25 June 2009

Sorry Scott - have not looked at your series yet.

Just realised reading a coffee table book that I have had my big book of Escher drawings / paintings on my bookshelf for 15 years without realising, other than peripherally, it could have some relevance to music. I am going to get the book out tonight. I can see already on Google that there are several 'compositions' (and other things) related to Escher.

  Re: Algebraic complexity in music  MartinY at 20:22 on 25 June 2009

I read Xenakis Formalize Music years ago, thought I understood it, and typed out the Fortran program in the book. I have now lost it but I suppose Open Music can do what it could do. If the old program could write a .sib file it might still be useful. The great thing about formal randomness is you can't guess it. Try and guess a heads and tails sequence and see who many of the 36 (is that right?) tests of randomness it passes. It never does because humans do too many head / tail swops.

I gather Formalize Music is regarded as too hard to teach to music post-graduates in the UK but I suppose there will be somewhere where it is used. I will get back to Haydn and series as soon as I can.Misuc's comments about the Farewell Symphony sound very interesting.............

  Re: Algebraic complexity in music  Nicolas Tzortzis at 22:24 on 25 June 2009

I think that the problem many times with mathematics in music is that people tend to focus on the mathematics part,and not on the music.
Since most musicians are not mathematicians,we do have this sense of awe when we look at some weird formulas trying to understand what they exactly mean.
For me what is interesting in Xenakis' approach is the Music.the math is just his way of organizing and of thinking it through.It's what gives him the idea,the starting point.Besides,there are a lot of mistakes in his calculations,and if you ask mathematicians,the math in there is not that great.the music is much more impressive than the math.
If we focus too much on the formulas and forget about the music,then I think we're looking at it from the wrong end.

  Re: Algebraic complexity in music  MartinY at 08:30 on 26 June 2009

Yes - I agree that the maths in Formalize Music is not very profound though as you say if you are good musician you can still get a great piece out of lousy maths, just as you can make a great piece based on totally bogus philosophy.

The thing is, if you had better maths and more of it, would you get any further. Maybe, maybe not. I imagine mathematicians talking to musicians can involve even less information transfer than mathematicians talking to chemists. Mathematicians used to be able to talk to physicists but the subjects have simultaneously advanced and dumbed down such that even that is difficult now apart from at the very top institutions.

  Re: Algebraic complexity in music  Misuc at 09:29 on 26 June 2009

I just want to strongly agree with Nicholas here. Except that it's not just the music as such which counts.

I find Xenakis' music very variable. Some of it tremendously exciting and dynamic and some of it a bit unimaginatiive. What works best for me is precisely the overtly 'mathematical' bits: the exuberant build-up and transfiguration of layer upon layer of clusters of sounds - a sort of ever-shifting aural architecture.

The mathematical conceptions behind all this are perhaps complex. The idea of plotting graph shapes on to a music score and imagining that this has any relationship to a musical shape is ridiculously infantile and inappropriate. It depends, for instance, on the fact that in the score 1st violin parts are written above 2nd violins etc. - change the order and that nice vector isn't a vector any more.

And yet the effect can be exhilarating like almost no other music. [Perhaps this is where good people get trapped by poor maths]

This is not, in my opinion, because of any particular qualities of the 'music' qua music, which is not X's strong point. Nor is it the mathematics, for the reasons I have given. But there is something that X imagined: a certain sound architectronics - incompletely and imperfectly realised - which gives these pieces their life. They have something which transcends music - as do many musical masterpieces. To be music music has to be more (or less) than music.


The bit in brackets "[perhaps this is.....]" should have been at the very end after "....than music"

  Re: Algebraic complexity in music  Misuc at 23:37 on 01 July 2009

I guess what I'm saying is that music is a sort of branch of mathematics, which cannot be translated exactly into another.

Or in the way that some people call mathematics a meta-music, couldn't we say that music is a meta-maths. It is a way of organising patterns of sound-vibrations in a more precise way than can be measured in numbers without a great deal of artificiality.

Some composers, like Bach, occasionally tried to adapt their complex musical ideas to the restrictions of a much cruder elementary maths. But not even in Bach's case did he quite succeed in marrying the two - and they remain odd exceptions to the general rule.

Music has its own logical systems and rules of thought which do not submit to numerological analysis and cannot [at present] be recognised by any computer- or human-generated algorithm. These contain concepts such as balance and stress, playing off contour against harmonic, motivic motion etc. the continual act of summing up, foreshortening and telescoping ideas - transforming a development over time into an instant 'object' etc. etc. Such principles have never been codified either in mathematical form nor in the formal study of music - [musicology]- but the notes we have of how composers in the past taught their art show that these things were ever-present in composers' minds.

  Re: Algebraic complexity in music  MartinY at 08:36 on 02 July 2009

I am now nearly ready to write a lot of things down. I have already found that my initial preconceptions were wrong. Complexity is more of a time dependent feature like dynamics in some pieces, (vis the beginning of Beethoven's 9th Symphony). Also you can look at pieces by Josquin and see some are incredibly simple and others complex but somehow both are very good. I am sure this applies to many contemporary composers, even minimalists!

I think the most useful thing I have suggested is guessing the relative complexity of different passages, rather than imputing them into a computer and calculating, to get numbers to study.

  Re: Algebraic complexity in music  MartinY at 08:55 on 08 August 2009

I have not forgotten this subject and I will return to serious work on it when the next summer school is over..... But I got a new idea to throw out, from my comment about silly orchestration.

If a theme and its inversion are the same (only one additional memory bit is needed to say whether it is up or down)..... can you have an inversion of an orchestration?

Suppose you have the most conventional orchestration implied by a closed score. Opposite is the stupidest orchestration you can think of. Are these a theme and an inversion? Are instruments with lots of overtones the inversion of instruments with pure tones? Where do instruments with strong 12th overtones come into the scheme of things? I suppose composers inclined towards spectralism will have opinions about this and maybe have theory papers on it.......

  Re: Algebraic complexity in music  Misuc at 11:08 on 08 August 2009

Behind all this discussion lies a very basic misunderstanding of what music is.

If I now write: noissucsid siht lla dniheb, I will not have conveyed the idea of the retrograde of my first phrase. I will have just written a load of gibberish.

This is just one reason why e.g. Allen Forte's theory about 'the structure of atonal music' is so inappropriate. It leaves out of account what the music is really doing.

For example, the chord going upwards F-B-E can be heard (and is, by Jazz musicians) as a sort of G7 dominant to C, implying an unheard G, G# or D Flat underneath. Upside down, the chord has no such implication.

A better example: in what way would the inversion of Baa-baa blacksheep be 'equivalent' to the original? If you played Beethoven's 5th backwards or upside down, what would be left of the conception? And what is the inversion of a dream?

Music is made out of notes. But the notes that make up a piece of music are not the music. Notes are just instruction for what to do at a certain moment relative to other moments. Time only goes one way: a particular pattern of higher and lower frequencies and their overtones etc. has a certain effect on an attuned listener's ears and whatever lies between them [i.e. the accumulated degree of cultural awareness, experience of life, musical sensitivity (ability to be knocked off-balance) and intelligence etc.] The inversion or reverse of these patterns does not reproduce or invert the effect or meaning of the passage.

But all this does not mean that the use of these symmetries must always be meaningless. There can, of course, be occasions when they come in handy. In tonal music, there are many simultaneous contradictory forces at work.Exact symmetries are always broken through the need to play off one kind of symmetry against the demands of the tonal structure: another kind of near symmetry. In most cases there are thousands of different hidden or broken symmetries and semi-symmetries of all kinds at different rates, going on within a piece, which gives it a rich pond-life. It is very rare to find a piece which foregoes all this complexity in favour of a rigid, limited 'mathematical structure, and pieces which do do not necessarily deserve any special awe at the clever way they are put together.

  Re: Algebraic complexity in music  MartinY at 08:21 on 09 August 2009

noissucsid siht lla dniheb.... This is both right and wrong, I think... Now there are scientific papers which prove most humans cannot detect musical retrogades but can detect musical inversions. At one time I read them but have forgotten. Time only goes one way in speech, music and real life. Retrogades are detected by transforming the listening into some sort of meta language, (like musical notation for some listeners), and reading it using more of the visual system of the brain. The person in the street will not do this nor should he....

However, much musical theory says retrogades and inversions are the same and in information theory they are, as information differentiated by 1 bit. This is not the same as their meaning being the same as that comes from the interaction with an observer, (just like in quantum mechanics but not really anything like that...). The reason I follow wrong theory is that it has often lead to good practical results. There are many examples in science where the correct theory is practically useless and the slightly wrong theory has all the practical applications. Even though I know retrogades cannot be detected I still use them in serial music and have written mirror pieces as have also Bartok, Mozart, JSB and Joe Bloggs. If up were the same as down we would have problems with gravity.

As Misuc has pointed out Beethoven's fifth does not invert or retrogade sensibly. Misuc and I have both made comments about the undesirability of writing music for its paper based structure.

Now back to JSB, when he writes in the Art of Fugue he has specially selected / constructed themes which are good tunes both ways up, and not all themes are as we know. The crab canons in the Offering also have some element of backward selection in their themes.

Now back to the Pines of Rome (and JSB), just as it would be terribly boring if we were all the same, being around pompous bigots can be fun and being with little Master / Miss perfect a pain in the ...... Some of the kindest people I have known have been pompous bigots, which does not square well with politically correctness. Now when Frederick the Great and Carl Phillip Emanuel tried to write the world's worst tune as a challenge for JSB by some miracle the tune was so bad that it cames out the other side as excellent, and does invert perfectly.

Speaking of pieces which are so bad that they are good: Grand Pianola Music is on the proms on Monday....

  Re: Algebraic complexity in music  Misuc at 13:03 on 09 August 2009

Yes, of course. It's not difficult to recognise inversions. A classic example of a piece of music that is built up almost completely geometrically out of a pattern of transpositions and inversions is the Bach 1st 2-part invention. The moment when this recognition becomes conscious is one of those "aha!" moments that make study and teaching so much fun.

I have often used this as an example,and to model how regular and irregular symmetries and broken symmetries can make interesting structures out of the most boring material (like buildings out of bricks) I have folded sheets of paper in different ways and cut shapes out of them. I am always excited by the results more than are my students. Still....

Draw the pattern of the Bach 1st 2-part invention and the structure is visible. Perhaps Webern worked in this way, I don't know. It is a fascinating part of music.

But don't let's get carried away. It's a particular trick. No more. Baa baa Black Sheep has a more interesting geometry (well maybe not quite, but you know what I mean. Doing retrograde canons and all that kind of stuff is just a sort of show-off trick. Does anybody believe that the more arcane canons in the Art of Fugue actually work?

Of course in considering all these t has us up against a fundamental question which absorbs neurologists, psychologists, philosophers, artists and others.

My son, who studies these things, recently gave me a book by philosopher, Alva NoŽ.

Here is an extract from the online 'blurb'

"Perception is not something that happens to us, or in us," writes Alva NoŽ. "It is something we do." In Action in Perception, NoŽ argues that perception and perceptual consciousness depend on capacities for action and thought -- that perception is a kind of thoughtful activity. Touch, not vision, should be our model for perception. Perception is not a process in the brain, but a kind of skillful activity of the body as a whole. We enact our perceptual experience.

Bach is so-to-speak giving us an example of what NoŽ is saying: testing out the limits of what we can perceive, understand and hence find beautiful......


typographical error

the paragraph which starts: "Of course in considering..." should go on "....these things, Bach has brought us....."

  Re: Algebraic complexity in music  MartinY at 17:09 on 09 August 2009

I keep going on about useless but correct mathematics which does not quite apply...... but my failure to suggest a sensible model of what is an incredibly complex system, the hearing of a piece of music, raises some interesting points about the modelling of time in science.

Normally time is fudged by converting time to frequency and then all the apparatus of Fourier Analysis, Orthogonal Polynomials etc. etc. can be applied but you have lost any sense of time moving forward in the way we readily appreciate in real life. When you treat time and the 3 dimensions equally you get relativity which has its own set of complex (in every sense of the word) mathematics which we will not go into.

Just consider a simple situation, we do two things A and B. If A . B is not the same as B . A i.e. it matters what order you do the two operations in we have a simple model of time. When we convert this to linear algebra we say the two matrices do not commute. These kind of problems in algebra are much more difficult than ones where the operators do commute. This means ones first task is often to get rid of the non-commuting operators in the model and come up with a problem where the linear algebra is commutative and easy. Something is lost in the process. If your retrogrades and inversions are regarded as being equal to the forward theme you have something akin to the simple commuting operator case.

I am going to have a further think about this though I doubt it will reveal anything useful one never knows.

I suppose Dr. Atomic is an example of relativity in modern music though I have not heard even the symphony from it yet......

  Re: Algebraic complexity in music  Misuc at 19:27 on 09 August 2009

....talking of which, does anybody know the marvellous opera: Einstein by the great Paul Dessau (Brecht's favourite misuccomposer]

...and by the way,back to the subject, has anybody come across this modest little programme for testing out some of these patterns? Just a little bit more and it would be really useful for serious composers. As it is it is still fun.


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