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Music & Mathematics MUSIC & MATHEMATICS THE GEOMETRY OF SOUND 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 Physical Loudness Amplitude Pitch Frequency Timbre Spectrum Length 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 Cochlea Ear Drum (tympanic membrane) MUSIC & MATHEMATICS COCHLEA ▸ Separates sounds into frequency components before passing to nerve pathways ▸ Twists 2-3/4 times around central axis. Unrolled… Hairs connected to nerves About 33mm ~ 1in long ▸ The cochlea is like our inner EQ 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?.
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