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The Bridges Conference: Mathematical Connections
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Held at the Logan Hall at the Institute of Education, London WC1H 0AL.
This event featured a combination of musical performance and short lectures, all aiming to illustrate - through music - how mathematics is intertwined with human activity and creativity.
The contributors were:
This event was sponsored by Sibelius Software - The world market leader in software for writing, teaching and publishing music.
Franz Schubert's setting of Goethe's ballad Erlkönig, written in 1815, is one of the first works to show the power and originality of his musical imagination. A characteristic feature of Schubert's musical idiom is the way in which he combines major and minor music elements associated with the same tonic note. A mathematical model based solely on the conventional cycle of fifths cannot do justice to this musical device. This presentation will demonstrate one way in which the classical model of keys might be adapted to convey Schubert's innovative harmonic style. It features animation of a model constructed using principles and tools developed by the author and his collaborators. These principles serve to connect model making, music making, and making mathematics with a primitive but primary notion of making meaning.
In 1967 I wrote a program to compose electronic music to be realised at Peter Zinovieff's computer controlled electronic music studio in Putney under control of a PDP-8. Called ZASP, the piece won second prize at the International Computer Music Contest at IFIP 68. This led to the formation of the Computer Arts Society (CAS), which was revived three years ago. I currently edit PAGE, bulletin of the CAS and continue to collaborate with Peter Zinovieff.
In 2004 I wrote a program to enumerate all the chords there are using the 12 notes of the diatonic scale, ignoring octave transposition. There are 352 different chords, from the null chord having no notes, to the chord of all 12 notes which Beethoven used in the finale of the Choral Symphony. There is a one-to-one correspondence between the chords and the necklaces of 12 beads, each bead either black or white. Necklaces that are different when turned over are counted separately. This year I composed the piece to be played, All352Chords, which uses each chord once. It is for four hands at one keyboard and will be played from a midi file. Different schemes are used to determine the rhythms and to distribute the notes among the four hands.
Steve Reich's Clapping Music consists of a rhythmic pattern played by two performers each clapping the rhythm with their hands. One performer repeats the pattern unchangingly throughout the piece, while the other shifts the pattern by one unit of time after a certain fixed number of repetitions. This shifting continues until the the performers are once again playing in unison, which signals the end of the piece.
Mathematically, this process of shifting one pattern over all positions of another pattern, appears in a variety of applications, depending on the type of function that is computed at each position of the shift, and what action is taken using the resulting computations. For example, in computer vision, the function computed at each position is the amount of agreement between the two patterns, and the action taken is the selection of the position which maximizes this agreement. This process is called template matching. In the performance of the piece the points in time at which the two patterns agree are accented by the fact that the corresponding claps of the two performers occur simultaneously. In this presentation we we first illustrate several such mathematical applications of the Clapping Music process, and then we perform the piece.
H. S. M. (Donald) Coxeter is well known for his work in geometry, particularly the theory of polytopes, and was in his youth a budding musician. Both of his parents were artists, his mother a painter and his father a sculptor and baritone singer. He learned to play the piano before he was 10 and began composing about the age of 12. He wrote a variety of types of music including songs, incidental music, a string quartet and numerous short pieces for piano. He was discouraged as a youth from pursuing music seriously, probably unfairly stifling the career of a young untrained composer. All of the compositions that I have been able to analyse were written in the styles of the Baroque, Classical and late Romantic periods, and do not show any exploration into the styles of the early 20th century. In his titles, Coxter often included a Roman numeral followed by an opus number. This writer has been unable to determine the significance of these numbers and if they have any relation to each other. It does appear that they are correct in terms of chronology, but if they have any significance as to Coxeter's total musical output it is unknown at this time. The search for more of his music is ongoing.
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