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Here are a few links to more or less interesting sites. Note how many scientists are interested in the link between maths and music.

- Maths Basics for Computer Music
- www.math.niu.edu/~rusin/papers/uses-math/music/ - Mathematics and Music
*Here are some collections of information about the interplay between music and mathematics. I have collected a bibliography of such items as they float past me in cyberspace, and I went hunting for references in the reviewing journal Mathematical Reviews for mentions of music. In addition I am interested in a few other specific topics.* - http://www.math.uga.edu/~djb/html/math-music.html - Mathematics and Music Course
- www.shef.ac.uk/misc/personal/mup99np/res/mathmus.htm:
*Musical innovation never just happens. It is always part of a history. The way in which it uses this history can vary from adopting principles to rejecting the ideas. Thus said I started a journey through time to find the roots of fractal music.* - plus.maths.org:
*Self-similar syncopations: Fibonacci, L-systems, limericks and ragtime by Kevin Jones* - The art of numbers:
*It's not unheard of for mathematicians and artists to look at each other with suspicion, and yet numerical relationships underly much form in art and nature. Indeed, many mathematicians are driven by a strongly aesthetic sense of creativity: "There is no permanent place in the world for ugly mathematics", as the mathematician G.H. Hardy once said.* - Fibonacci Numbers and the Golden Ratio

- Wondrous Numbers
- Pi
- Mixmaster
- Game of Life
- Euclid's Algorithm:
The highest common factor of two numbers: take two numbers, say 3654 and 1365:

3654/1365 gives remainder 924

1365/924 gives remainder 441

924/441 gives remainder 42

441/42 gives remainder 21

42/21 gives remainder 0 - The formula represents the pairs (a,b) of natural numbers uniquely.
- Jacques Laskarr's calculations
- My seminar/lecture on Musical Number Theory
- My seminar/lecture on Numbers

Many people have attempted to make links between mathematics and music. Notably, figures such as Pythagorus attempted to make links between the positions of celestial bodies and music, calling the result the 'music of the spheres'. Later, numerology was considered important - in the Ars Nova period (the fourteenth century), music was notated according to its pulse relationship with the number three, representing the Christian Trinity - the most 'perfect' time was considered to be three groups of three notes each - what we might well notate as 9/8. This was also represented as a circle. The 'C' which we interpret as 4/4 is in fact a broken circle indicating 'imperfect' time.

Since then there have been other relationships made, principally between the ability of mathematics to understand and develop 'patterns' and music ability to make use of the same. The development of serial music in the first part of the twentieth century, while not really 'mathematical', is an attempt to make use of the patterns inherent in musical lines. In particular, the composer Webern was one of the first to make almost exclusive use of these patterns in his short and concentrated music virtually devoid of traditional melody and harmony.

Rhythm too has been investigated for mathematical potential. Clearly, many rhythms can be expressed in terms of their (relative) durational values - that is, so many semiquavers, followed by so many, etc., etc. Composers such as Messiaen (for instance the piano and 'cello parts of the first movement of *Quator pour la Fin du Temps*) and Harrison Birtwistle have taken up an idea from the fourteenth century - a line may have a talea - a fixed melody, and a color - a fixed durational sequence. These are of different lengths and so when combined, the two will 'revolve' - not quite repeating. Stockhausen's *Kontrapunkte* uses serial methods with rhythm.

In all these case, mathematics as such is rarely used - number, or patterns of numbers are. This also makes sense because of the nature of sound - a varying pattern of waves or air-pressure variations. Sounds constructed without such patterns are commonly described as noise.

MIDI clearly presents itself as a highly appropriate arena for the auditioning of these processes. As many of the basic musical parameters are defined in clear and simple numerical terms, these parameters can be controlled and manipulated by mathematical processes. Indeed, the resulting patterns can often be easily heard, (although they often can't, too). There are a few things you have to look out for. Most MIDI parameters are based around a range of 0-127, 0-255, 0-15, and others. Of course, many, if not most mathematical patterns involve numbers substantially beyond these ranges, or only move between, for instance, 0 and 1. Common functions that have been used for these purposes are fractals and attractors. Included in the sample programme *Formulator* are implementations of the function to test for a number's 'wondrousness', and of the astronomical *Mixmaster* function.

*Formulator*, (see below), currently only deals with general MIDI, and fairly restricted general MIDI at that. However, exactly the same principals could be applied to any system exclusive messages. Using software that details an instrument's sysex messages (for instance the *Sysex* programme included with the Mabry MIDI controls), you can find the codes that edit a part of a sound's timbre, and then use the above (or any) data to manipulate that. The beauty of numbers is exactly that - the same data can be used for a multitude of purposes. Remember that in this case, you'd still need to play a note (using general MIDI in order to hear it's effect.

What is knowledge? Is it the same to 'know' how to play an instrument as to 'know' how to multiply 8 by 7? How much 'knowledge' do we have when we are born?

When some of us calculate mathematical problems, we do it 'automatically' - we 'know' that 8 multiplied by 7 equals 56. We 'know' this because - we've done it before, we've used a 'method' that we 'know' works, that we used a reliable calculator, etc. We very rarely imagine, or even notate, a group of eight oranges, say, and then six more similar groups, and count each individual orange to come up with the solution.

Consider.

Prepare.

- New Scientist #2294