bmswhatis

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What is…the big deal about math and music?

Aaron GreiciusHumboldt-Universität zu Berlin

Berlin Mathematical School, May 2010

+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

+Pitch and timbre

Connection traces back to Pythagoras

Music as mathematical object

+Pitch and timbre

Consequences 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’.

Connection traces back to Pythagoras

Observation Strike two instruments at the same time The interval sounds more consonant when the ratios of

their two marked measurements are simple E.g. 16/8=2/1 and 6/4=3/2 yield consonant intervals, while

16/9 yields a dissonant one


+Pitch and timbre

Harmonic series: Begin with a fundamental frequency or pitch f and take successive multiples (the partials) to yield the sequence f, 2f, 3f, 4f,…

These simple ratios correspond to

simple ratios of the corresponding frequencies (which we will equate

with pitch) of the sounded notes.


+Harmonic series

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.


+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.


+Rhythm and melody

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.


+Rhythm and melody

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.


+Goldberg Variations, Var. 21J.S. Bach


+Vingt Regards,V. Regard du Filssur le FilsOlivier Messiaen


+Das Wohltemperierte Klavier,Zweiter Teil, Fuga IIJ.S. Bach


+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

AtmosphèresG. Ligeti

+Die Nebensonnen,Die WinterreiseF. Schubert


+L’Isle JoyeuseC. Debussy


+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


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”.


Music


Begin with a formal language L. E.g. the propositional calculus, which has expressions of the

form P, Q, P∧Q, PQ, 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 PQ is in our

theory, then we can also add Q to our theory.


Formalist view of mathematics


A proof in 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


Formalist view of mathematics


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.


Formalist view of music


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.


Math and music

+Economy, elegance and surprise

“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


Toward a shared aesthetics