Found Mathematical Objects

1
Found Mathematical Objects

• Lecture notes from Tom Johnsons presentation
• Musique, mathématiqus et philosophie seminars
• (January 13, 2001)
• Presentation in ISE 575 class
• January 18, 2007
• by Reid Swanson

2
Overview

• Found objects as Art Music
• Mathematical objects as found objects
• Examples
• Automata
• Loops Cyclic Groups
• Others
• Questions Issues Raised
• Summary

3
The Readymade

• Traditionally Art thought of as solely a creation
  of the Artist
• Duchamp pioneered a change of perception with
  The Fountain

4
The Mathematical Readymade

• Most found objects are physical things
• Several reasons Johnson believes mathematics
  provides good found objects
• Numbers and mathematics are infinite and eternal
• Music linked with numbers should also be infinite
  and eternal
• Mathematics can help enable the Music to do
  what it wants
• Found objects offer purity and objectivity

5
Precedent I

• Iconic Paintings
• considered a symbol not a depiction
• 2 dimensional, without human expression (no
  decoration)
• not signed (the Artist isn't really the creator)

Christ in Hagia Sophia, Istanbul. Deisis mosaic
(detail). 1280
6
Precedent II

• Mosaics
• Similar to iconic paintings...
• ...but also a representation of mathematical
  objects

Alhambra tilings produced all 17 crystallographic
groups proposed by Polya
7
The Examples

• Several types of transformations
• Automata
• Simple example
• Narayana's Cow
• Automata Music
• Pascal's Triangle
• Melodic Loops Cyclic Groups
• Rational Melody 15
• Rythmic Cannons
• Loops fo Orchestra
• Miscellaneous puzzles
• proposed by Thomas Noll

8
The Process

• Constraint based
• Formulate an automata, sequence or problem in
  group theory
• Express solution in musical form
• Simple Example
• the equation
• n -gt n, n1, n
• the sequence generated
• 010 121 010 121 232 121 010 121 010 ...
• define a mapping from numbers to music
• 0C, 1D, 2E, etc
• Show the structure without decorating

9
Other Automata

• Narayana's Cow
• 14th century problem
• basically a modified Fibonacci sequence
• n -gt (n-1) (n-3)
• mothers ¼ notes, daughters 1/8 notes, each new
  generation in a lower scale
• Automata Music for Six Percussion

10
Pascal's Triangle

• Offers another approach for deriving music
• Defines a set of musical objects instead of a
  sequence
• Inspired three pieces

11
The Chord Catalog

• Play all the chords in one octave
• 13 keys in an octave
• 213 total combinations 8192
• includes empty set
• includes single keys
• 8178 valid combinations
• Two logical possibilities for playing order
• start with low and work towards high (or
  vice-a-versa)
• start with two note chords and work toward 13
  note chords

12
Music for 88

• Play all chords that could be performed with only
  two intervals
• the major second and the minor third

13
Triangle De Pascal Modulo Seven

• Map number to notes
• 1-6-1-6 cadences with lines 7, 14, 21
• Has self similar structure

1 11 121 1331 14641 153351 1616161 10000001 1100
00011 1210000121 13310001331 146410014641 15335101
53351 16161611616161 100000020000001 11000002200
00011 12100002420000121 133100026620001331 1464100
215120014641 15335102366320153351 161616125252521
616161 1000000300000030000001
14
Melodic Loops Cyclic Groups

• Another type of mathematically derived music
• Basically a sequence or melody that contains a
  replica of itself at a particular ratio
• 3 examples
• Rational Melody 15
• 2nd example, piece for eight bell instrument
• Loops for Orchestra

15
Rational Melody 15

• 15 note melody whose odd notes (1,3,5 etc.) make
  up the same melody as the full 15.
• sequence
• AGGFGEFDGFEDFDDAGGFGEFDGFEDFDD
• A G G F G E F D G F E D F D D

16
2nd Example

• Start with a set of physical/musical constraints
• new instrument with 8 distinct notes
• pose a problem suited to the instrument
• what eight-note rhythm might fill a 24 note-loop
  as three voices of a cannon
• Find a set of solutions
• pick a suitable one
• X O O O X X X O O X X X O O O X
• has a nice waltz rhythm
• Both pieces divide a cyclic group into similar
  sub-groups

17
Loops for Orchestra

• Another object from group theory
• forms groups and sub groups
• Another instance of defining a set of constraints
  and finding a solution to those constraints
• Find a 21 note-cycle that replicates itself 2-1
• do this by making sure all notes in an orbit are
  coded with the same musical letter

18
More Examples

• Colleague Thomas Noll helps formulate other
  problems and solutions that have led to more
  inspirations.
• how to make rhythmic canons with the voices
  moving at different tempos
• Interlude for Guerino Mazzola
• Use of computers to help solve problems
• Markus Reineke

19
Some Issues Questions

• Back to Duchamp
• Is the object, the urinal, the entire piece?
• There is more to the story
• Submitted as a kind of protest to an un-juried
  art show (and was rejected)
• Signed with a pseudonym
• name most likely a play of words
• Armut meaning poverty
• R. short for Richard, slang for moneybags
• Mott was the name of the urinal vendor
• Removed from it's original context
• Given symbolic meaning

20
Music

• What is Music?
• Suggestion that Music is number
• Johnson operates under this assumption
• is this so?
• what else could it be?
• Debate with Grisey
• Music in the notes
• or Music in the sounds
• some of both?
• neither?

21
Purity

• How to maintain the purity
• Johnson seems to acknowledge there are problems
  in preserving the purity, for example when
  talking about when to start and stop an automata
  piece.
• Doesn't really address many of the issues
  involving the decisions that must be made that
  impact the resulting music.

22
General Decisions

• What decisions have to be made
• what notes to map to?
• how fast?
• how loud?
• what instruments, how many?
• is he the creator or not? (signed pieces)
• what context is the music presented?

23
Specific Decisions

• Some particular decisions made
• how to begin, how to end? (automata)
• Triangle De Pascal has 1-6-1-6 cadences
• what formula, why?
• In 2nd cyclic example the instrument had 8 bells
  so natural to look for an 8 note rhythm, but why
  a 24 note loop?
• which solution?
• In 2nd cyclic example Johnson chose a solution
  that had a waltz rhythm (out of 27 possible)
• human preference vs. purity
• what order
• Pascal's triangle from low to high is logical.
  why?

24
Summary

• Johnson uses mathematical objects to derive new
  works
• Believes in an objectivity and purity associated
  with numbers
• Uses several different mathematical constructs to
  guide the composition process
• Must make many intervening decisions to produce a
  complete piece