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Abstract Preparation Bibliography Main Text
Here are a few links to more or less interesting sites. Note how many scientists are interested in the link between maths and music.
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
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.
Main text (not usually available until after the lecture).