THE GEOMETRY OF SOUNDMUSIC & MATHEMATICS

MUSIC & MATHEMATICS

SO FAR WE’VE LOOKED AT…▸ The origins of tuning theory - i.e. how to organize harmonic

sound

▸ Pythagorean tuning

▸ Seems to be both psychological and physical precedent for the way(s) we structure music

▸ Preferences for symmetry, “even” ratios, overtone series, etc.

▸ The algebra of music theory

▸ Ways of analyzing the structure of (most) Western music

▸ Symmetries on Z12

MUSIC & MATHEMATICS

TODAY▸ Complement our algebraic

understanding of how to structure music with the geometry of sound.

▸ Goal: create music from first principles

▸ Waves and harmonics

▸ Fourier Theory

▸ Analysis of acoustic instruments

▸ Synthesis of virtual instruments

MUSIC & MATHEMATICS

WHAT IS SOUND?▸ Vibrations through the air

▸ The mean velocity of air molecules at room temperature is 450-500 m/s (~1000 mph).

▸ The mean free path of an air molecule is about 6x10-8 m

▸ This is why air molecules don't fall down - so the effect of gravity on air takes the form of a gradation of pressure.

▸ When an object vibrates, it causes waves of increased and decreased pressure. These waves are perceived by our ears as sound.

MUSIC & MATHEMATICS

WAVES▸ Tempting to use ocean waves as analogy

▸ Transverse vs longitudinal waves

MUSIC & MATHEMATICS

DUAL CONCEPTIONS

MUSIC & MATHEMATICS

MAIN ATTRIBUTES OF SOUND (WAVES)Perceptual PhysicalLoudness Amplitude

Pitch FrequencyTimbre SpectrumLength Duration

▸ Note: most vibrations do not consist of a single frequency

▸ E.g. the phenomenon of the missing fundamental.

▸ Recall: an instrument that produces a discernible pitch resonates at every integer multiple of the “fundamental” frequency.

▸ http://teropa.info/harmonics-explorer/

MUSIC & MATHEMATICS

THE HUMAN EAR

Ear Drum (tympanic membrane)

Cochlea

MUSIC & MATHEMATICS

COCHLEA▸ Separates sounds into frequency components before

passing to nerve pathways

▸ Twists 2-3/4 times around central axis. Unrolled…

▸ The cochlea is like our inner EQ

About 33mm ~ 1in long

Hairs connected to nerves

MUSIC & MATHEMATICS

WHAT HAPPENS WHEN SOUND ENTERS THE EAR?▸ Sound wave is focused into the meatus, where it vibrates

the ear drum

▸ Hammer, anvil and stapes move as a system of levers

▸ the stapes alternately pushes and pulls the membrana tympani secundaria in rapid succession.

▸ This causes fluid waves to flow back and forth round the length of the cochlea, in opposite directions in the scala vestibuli and the scala tympani,

▸ Basilar membrane to moves up and down.

MUSIC & MATHEMATICS

OHM AND HEMHOLTZ▸ Ohm’s acoustic law: the ear picks out frequency components of an incoming

sound

▸ Hemholtz: the place theory of pitch perception

▸ Consider a pure sine wave transmitted by the stapes:

▸ Speed of the wave of fluid in the cochlea at any particular point depends on the frequency of the vibration and on the area of cross-section of the cochlea at that point, as well as the stiffness and density of the basilar membrane.

▸ Speed of travel decreases towards the apical end, and falls to almost zero at the point where the narrowness causes a wave of that frequency to be too hard to maintain.

▸ Just to the wide side of that point, the basilar membrane will have to have a peak of amplitude of vibration in order to absorb the motion.

MUSIC & MATHEMATICS

WHY SINE WAVES?▸ This differential equation represents what happens when

an object is subject to a force towards an equilibrium position, the magnitude of the force being proportional to the distance from equilibrium.

▸ Not 100% accurate

▸ Forced damped harmonic motion

MUSIC & MATHEMATICS

MUSIC & MATHEMATICS

VIBRATING STRINGS▸ Single weight in center:

▸ Uniformly distributed weight allows for other vibrational modes

▸ 12th Fret Harmonic (just touch the string at the half way point)

▸ Other harmonics

MUSIC & MATHEMATICS

▸ In general, a plucked string will vibrate with a mixture of all the modes described by multiples of the natural frequency, with various amplitudes.

▸ Those amplitudes will differ depending on the pluck or hammer - e.g. plucking vs. picking vs. finger-picking etc.

▸ Note: k is really κ/m - i.e. the constant of proportionality divided by mass

MUSIC & MATHEMATICS

TRIG IDENTITIES AND BEATS▸ Treble (higher-end) pitches on a piano typically have three

strings. The tenor and bass notes have two and one, respectively.

▸ Suppose a piano tuner plays two of the strings intended for A440Hz and gets two frequencies: 440Hz and 436Hz

▸ Play these two frequencies in our oscillators.

▸ What do you hear?

▸ Change the 436Hz to something else close by and compare.

MUSIC & MATHEMATICS

▸ A piano tuner comparing two of the three strings on the same note of a piano hears five beats a second. If one of the two notes is concert pitch A (440 Hz), what are the possibilities for the frequency of vibration of the other string?

MUSIC & MATHEMATICS

DAMPED HARMONIC MOTION▸ If we take our differential equation for harmonic motion

and add in a term for friction (frictional force is proportional to velocity) we get the differential equation for damped harmonic motion:

MUSIC & MATHEMATICS

EXAMPLE▸ Consider the following equation:

▸ With an appropriate choice of coefficients, we might get answers that are audible. Try some in Wolfram Alpha:

▸ Type “play e^(-at)sin(f 2pi t)” where a is some coefficient and f is the frequency you want to play.

▸ How is the sound different that the plain sine wave?

MUSIC & MATHEMATICS

FOURIER THEORY▸ How can a string vibrate with a

number of different frequencies at the same time?

▸ Decomposition of a periodic wave

▸ Usually Infinite series

▸ Frequencies are the integer multiples of the fundamental frequency of the periodic wave

▸ Each has an amplitude which can be determined as an integral

MUSIC & MATHEMATICS

THE BIG IDEA▸ Fourier introduced the idea that periodic functions can be

analyzed by using trigonometric series.

▸ Periodicity: A function is said to be periodic with period T provided that f(t + T) = f(t)

MUSIC & MATHEMATICS

▸ We can add any combination of sines and cosines to get a function with period 2π:

▸ Determining the coefficients requires some serious cleverness…starting with these integrals:

MUSIC & MATHEMATICS

▸ To use those integrals to simplify things, we multiply f(θ) by cos(mθ)

▸ Which gives us a (gross) way of computing the a’s:

MUSIC & MATHEMATICS

▸ We repeat this to get the b coefficients.

▸ Now suppose the period of our function is T seconds. Then our fundamental frequency is given by v=1/T Hz.

▸ We get the general Fourier Series by substituting θ=2πvT:

MUSIC & MATHEMATICS

AN EXAMPLE!!!▸ The square wave sounds kinda like a clarinet. It’s defined

as follows

▸ Find the Fourier coefficients…

MUSIC & MATHEMATICS

MUSIC & MATHEMATICS

▸ Examine the first few terms of this series.

▸ Plug them into Wolfram and listen

▸ You’ll have to change the frequency to something audible

▸ Go back to our overtone generator and hit “square wave”

▸ Does it match up with what you came up with?

MUSIC & MATHEMATICS

DRUMS▸ Why do (most) percussion instruments admit no

discernible pitch?

▸ Their frequency spectra are irregular. That is to say, the overtones are not integer multiples of the fundamental

▸ Create a damped sound that is percussive in Wolfram

▸ Is it percussive?

MUSIC & MATHEMATICS

NOW WHAT▸ With Fourier analysis at our disposal (and tools that’ll do it

for us), we can decompose sounds into their frequency spectra.

▸ What else do we need in order to synthesize the sounds generated by instruments?

▸ In other words, what are the other, more subtle properties of (musical) sound?