The Logic Off Topic list

Hendrik, you didn't read Mandelbrot's book by any chance did you?

>The nice coloured pictures that everyone has come to know as fractals 
>are nothing but maps of this behaviour.  .... Do the 
>values become huge?  Colour it red.  Do they move towards zero? 
>Colour it black.  Etc.


        The method of creation of these picture is simple enough, but what 
principle 
        causes these iterative processes to be self-similar? I always 
wondered that.
        (There are all manner of other algorhitms by the way - for example, 
on the 
        complex plain (where many fractals live), picking a point and seeing 
which
        of the 4 roots of z^4 -1 that point converges to using Newton's 
Method.) Its 
        interesting that there is a connection of some type between geometry 
and "algebra".


>straight lines, circles, squares.  Looking at just a single leaf from 
>a tree it's painfully obvious how inadequate circles and lines are to 
>explain its shape. It was therefore thought that the "laws of nature" 
>would be extremely complex since the shapes of nature are so complex.
...
>So apparently achieving "natural complexity" _was_ possible without 
>the need for terribly complex formulas.  In fact, the formulas used 
>were so simple, that nowadays you could even let a 15-year old with a 
>computer make fractals.  


        But the "natural" patterns like coastlines, clouds, etc.. have a 
significant amount 
        of randnomness in them - not simple mathematical formulas.


>Wouldn't it be possible, using similar algorithms, to 
>create complex sounds?  Up to a point the answer is probably: yes. 
>You can create "fractal waveforms" and use those instead of plain saw 

        There are at least two levels of fractacality - the wave form of a 
single pitch
         and the actual composition of tones. In either case, the method 
fails to be 
         truly fractal because there are small finite number of levels of 
self-similarity.


>That's not amazing, I think: music in general is highly structured 
>and non-chaotic.  Fractals, by their very nature, _are_ chaotic, and 


       I think you are mixing the concepts of randomness and order. 
       Many fractals are highly ordered - for example, the Koch curve
       (an equilaterial triangle where the sides are lengthened by 4/3 each
        iteration and you end up with a 'fuzzy' boundary, with an infinite
        perimeter, but finite area, and fractal dimension of ln(4)/ln(3)).

>so don't lend themselves that well to "automated composition".  All 
>imo of course.

        This is how this topic came up originally - a totally randomly created
         series of tone would be garbage - but if there were some 
self-similarity
         like the way a chorus had a similar melody to the verse line but 
transposed
         speed up 2x, with added embellishment notes - but similar in some 
way -
         then there must be some way of composing music that sounds good. In 
the 
         original article 20 years ago, the writer took samples of various 
types of 
         music, and processed it "fractally" and generated original music 
that sounded
         authentic in some way (but no hit songs! Unless you consider Britney 
Spears
         and the rest to be fractal processed madonna songs or whatever).


>Think about it: how would you translate the "graph" of a coastline 
>into music?  Would it be interesting music?  The human eye is rather 
>good at seeing structure on different scales simultaneously -- you 
>can see structures that are just a few millimeters big and at the 
>same time see structure on the 2-meter-canvas scale.  

        Thats just similarity, not self-similarity. You can recognize the 
        same musical motifs played high and fast or low and slow in 
        music from Beethoven's 5th to Queen's Night at the Opera.


>I think the average human ear is less capable in that respect. 


    Maybe or maybe not. Isn't there something of a universal agreement
    on the beauty of certain melodies ... from Beethoven's Fur Elise or
    Tchaikosky's Swan Lake, to "Yesterday", "Eleanor Rigby", "Satisfaction"
    "Billy Jean", .... these things have enjoyed popularity because there is 
      something beautiful about their patterns of notes. And this is where my 
      interest is. If many (but not all) people can appreciate one melody as
      beautiful but another as rubbish, there must be some inherent rules
      or reasons for this.


>Still that doesn't mean fractal principles are totally unsuited for 
>making music.  Even in music there's an amount of chaos present -- 
>chorusses repeat but are not exact copies, each performance of a 
>piece is slightly different, etc, etc.  Plenty of "structured chaos" 
>there.  Maybe we just have to find the right perspective before being 
>able to apply fractals in music in a meaningful way.

       I think you are talking about some "random" variation of a very 
controlled thing
     - actual "chaos" is more the reverse - some unexpected order in a 
seemingly 
      chaotic situation.