1. What is…the big deal about math and music? Aaron Greicius Humboldt-Universitätzu Berlin Berlin Mathematical School, May 2010

2. My cowardly appeal to authority “Mathematics and music, the most sharply contrasted fields of intellectual activity which can be found, and yet related, supporting each other, as if to show forth the secret connection which ties together all activities of the mind…” − Hermann von Helmholtz “Music is the arithmetic of sounds as optics is the geometry of light.” − Claude Debussy “Music is the pleasure the human soul experiences from counting without being aware that it is counting.” − G.W. von Leibniz “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made of ideas. His patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way.” − G. H. Hardy “I am not saying that composers think in equations or charts of numbers, nor are those things more able to symbolize music. But the way composers think − the way I think − is, it seems to me, not very different from mathematical thinking.'’ − Igor Stravinsky

3. Pitch and timbre Music as mathematical object Connection traces back to Pythagoras

6. Strike two instruments at the same time

7. The interval sounds more consonant when the ratios of their two marked measurements are simple

8. E.g. 16/8=2/1 and 6/4=3/2 yield consonant intervals, while 16/9 yields a dissonant one

9. Connection traces back to PythagorasConsequences Earned music a spot in the classical quadrivium, along with arithmetic, geometry and astronomy. Such notables as Aristotle, Ptolemy and Kepler sought similar simple ratios of integers governing the movements of the planets. Thus the phrase ‘music of the spheres’.

11. Harmonic series Music as mathematical object The pitch with frequency 2f sounds one octave above f. What about the pitch (3/2)f? 3f is an octave plus a perfect 5th above f. The equality (3/2)f=(1/2)3f show us that (3/2)f is 3f reduced by an octave. We get a perfect 5th above f.

12. Pitch class space and harmony Apply Log 12√2(x/C0): R>0--->R. Linearizes the frequency spectrum and divides the octave into 12 equal steps: the equal-tempered scale. Now equate any two frequencies separated by a number of octaves. This is called octave equivalence: “Bring us back to Do”. The resulting space, R/12Z,is called pitch class space. Music as mathematical object

13. Rhythm and melody Music as mathematical object Whereas chords are sets of pitches, melodies are sequences of pitches. We can think of a melody as a sequence of points Pi=(ti, f_i) in the plane. Here the x-axis is time and the y-axis is pitch. “The musical staff was Europe’s first graph.”—A.W. Crosby, from The measure of reality.

14. Rhythm and melody Music as mathematical object As with pitch class circle, we can consider natural operations on the plane. Reflection through a horizontal line yields pitch inversion. Reflection through the vertical y-axis yields the retrograde of the sequence. E.g., the sequence (P1, P2, P3) becomes the sequence (P3, P2, P1). Shrink or expand the time component of the plane, yielding the musical operations of diminution and augmentation.

15. Music as mathematical object Goldberg Variations, Var. 21J.S. Bach

16. Music as mathematical object Goldberg Variations, Var. 21J.S. Bach

17. Music as mathematical object Vingt Regards,V. Regard du Filssur le FilsOlivier Messiaen

18. Music as mathematical object Vingt Regards,V. Regard du Filssur le FilsOlivier Messiaen

19. Music as mathematical object Das WohltemperierteKlavier,ZweiterTeil, Fuga IIJ.S. Bach

20. Music as mathematical object Das WohltemperierteKlavier,ZweiterTeil, Fuga IIJ.S. Bach

21. Music as mathematical object Das WohltemperierteKlavier,ZweiterTeil, Fuga IIJ.S. Bach

22. Musical manifold “From my childhood I can clearly remember the magic emanating from a score which named the instruments, showing exactly what was played by each. Flute, clarinet, oboe--they promised no less than colourful railway tickets or names of places.” —Theodor Adorno, Beethoven: the philosophy of music Represent a musical piece as a surface in 3-space. Let x-axis be time, y-axis be timbre (if you like line the orchestra up along the y-axis), add pitch as the z-axis. The musical piece is then described as a surface z=f(t,y). Music as mathematical object

23. Music as mathematical object AtmosphèresG. Ligeti

24. Music as mathematical object AtmosphèresG. Ligeti

25. Music as mathematical object Die Nebensonnen,Die WinterreiseF. Schubert

26. Music as mathematical object Die Nebensonnen,Die WinterreiseF. Schubert

27. Music as mathematical object L’IsleJoyeuseC. Debussy

28. Music as mathematical object L’IsleJoyeuseC. Debussy

29. Logic, proof and development Mathematical work does not consist solely in the fashioning of clever mathematical objects. The main output of mathematics is sentences, propositions that tell us about mathematical objects, as well as the proofs that show these propositions are true. Mathematical activity consists largely in the fashioning of arguments. Music as mathematical activity Math

30. Logic, proof and development Music as mathematical activity Music As abstract structure, musical piece qualifies as object of mathematical inquiry. As with math, music is not simply a collection of clever inventions to be regarded passively. We attempt to understand music, to “follow it”, to figure out what it is “trying to say”.

31. Logic, proof and development Begin with a formal language L. E.g. the propositional calculus, which has expressions of the form P, Q, P∧Q, PQ, etc. To build a formal theory T in the language L identify a set of sentences of of L as axioms agree on a set of rules of inference use rules of inference to generate the theory from the axioms Example of a rule of inference: Modus ponens. If P is in our theory, and PQ is in our theory, then we can also add Q to our theory. Music as mathematical activity Formalist view of mathematics

32. Logic, proof and development A proofin our theory is a sequence of propositions P1, P2, …, Pn, such that each Pi is either one of our axioms OR obtained from the previous sentences using our rules of inference Music as mathematical activity Formalist view of mathematics

33. Logic, proof and development The sequence P1, P2, …, Pn reminds us of our description of melodies. Begin with basic musical propositions (pitches, motivic cells, harmonic cells, rhythmic cells), our axioms, and agree on rules of inferences for generating more musical material. Development in music is in this sense akin to proof. Music as mathematical activity Formalist view of music

34. Logic, proof and development Bad theories Too little. Theory only contains truisms of the form ‘P or Not(P)’. Too much. Theory generates all possible sentences. Possible musical interpretations Too little. Stasis, repetition, silence. Too much. Entropy, noise, Wagner. Music as mathematical activity Math and music

35. Economy, elegance and surprise Music as mathematical activity Toward a shared aesthetics “As for myself, I experience a sort of terror when, at the moment of setting to work and finding myself before the infinitude of possibilities that present themselves, I have the feeling that everything is permissible to me…” “My freedom thus consists in my moving about within the narrow frame that I have assigned myself for each one of my undertakings.” —Igor Stravinsky “That is, indeed, artistic economy; only such means are to be used as are absolutely necessary for producing a certain effect. Everything else is beside the point, hence crude, can never be beautiful because it is not organic.” “Music is not to be decorative; it is to be true.” —Arnold Schönberg