a performance dialogue

Thursday, December 6, 2007

dodecahedron


The dodecahedron has 12 faces...one for each note in the chromatic scale. Has the shape ever been used to create music. For example rolling it as a dice? Or even something more structural like using its symmetry somehow to generate lines of music?

My new book is written as 12 chapters, one chapter for each month of the year. I used a spinning dodecahedron at the top of each chapter to keep track of the chapter number.

2 comments:

Dorothy said...

Composers are very well-acquainted with the geometries of 12. Schoenberg was the first to explore these systematically with the idea of ordered rows including all twelve notes that retain their intervallic identity under transposition and inversion. Different composers have exploited particular characteristics of this dodecaphony, and rows that possess particular properties (particularly symmetries) have a certain mystique. I once found a book listing all the possible 'all-interval' rows but often rows are given a particular flavour by the predominance of particular intervals. Hexacords are the more common unit than complete 12-note rows. These can be constructed to relate to each other in particular ways.

Webern was obsessed with rows that produced the greatest possible degree of symmetry and some of his are partitioned into intervallically isomorphic 3-note groups.

'Clock maths, inversion and transposition become second nature when working with 12-note rows. Intervals are understood as equivalent structurally when inverted, with the tritone adopting a special identity since it is its own inversion. You can transform a row by selecting every fifth or seventh element to produce a new row that has different intervals.

I work with 24 notes and I think this is partly because I got bored with the maths of 12 notes!

The 12-note idea finally broke for me when I was writing 'solo for cello'. I had made a matrix of all possible versions (48) of the row. This was my source material for pitch. As I became more interested in the idea of the spiral I started to make spiral movements around the matrix so that you get more 'row' with each turn of the spiral but it's always incomplete.

After that I couldn't deal with the constraints of the matrix and started looking for other ways of generating and organising pitch.

That includes the dice! and simulations of the dice.

Currently I am using a 7-note row (taken from the 24 chromatic, so including quarter tones) and using permutations of this along with inversions.

I am always interested in discovering properties of such sets that make them 'resonate' with something significant mathematically or numerically.

Dorothy said...

What I find fascinating about the dodecahedron is the way the sides are pentagonal. Is there some intrinsic relationship between 5 and 12 that means that five-sided shapes can generate 12 faces?