The Physics of MusicCharles H. Bennett

Bennett Conservatory of MusicApril 12, 2015

What is sound?Aristotle understood the gist of it over 2000 years ago.

“Sound takes place when bodies strike the air, . . . by its being moved in a corresponding manner; the air being contracted and expanded and overtaken, and again struck by the impulses of the breath and the strings, for when air falls upon and strikes the air which is next to it, the air is carried forward with an impetus, and that which is contiguous to the first is carried onward; so that the same voice spreads every way as far as the motion of the air takes place.” —Aristotle (384–322 BCE), Treatise on Sound and Hearing

But light was poorly understood until much more recently.(see me after class)

Click to show Aristotle’s correct idea of sound propagation( You may have to click it a few times)

Physical versus Perceptual properties of Sound

Loudness ---------- Amplitude Pitch ---------- FrequencyTimbre ---------- Wave shape

If the vibrations repeat, exactly or approximately, between about 40 to 15,000 times per second, the sound is perceived as a musical tone with a definite pitch. Otherwise it is perceived as a non-musical sound.

We can see this on an Oscilloscope display, showing air pressure versus time in milliseconds, or a Sound Spectrograph display, showing frequency and intensity versus time in seconds.

Most musical instruments, except tuning forks, produce a mix of frequencies. For string and wind instruments these are whole number multiples of some lowest, or “fundamental” frequency.

Mathematically, the repeat frequency, which corresponds to the pitch, is the Greatest Common Divisor of all the frequencies the mixture.

For example the B above middle C corresponds to 500 vibrations per second. A tuning for tuned to that note would produce only that frequency. But a piano playing that note also produces 1000, 1500, 2000 frequencies. If the 500 frequency were removed, the note would still sound like the B above middle C, but with a different timbre or tone quality, because 500 is the greatest common divisor of 1000, 1500, and 2000. If the 1500 frequency were removed, what pitch would the remaining mix of 1000 and 2000 vibrations have?

First 6 Normal Modes of vibration of a string. n’th mode frequency is n times the fundamental.

Red dots denote nodes, i.e. places on the stringthat don’t move.

n=1

n=3

n=5

n=2

n=4

n=6

Harmonics on a string instrument:

Placing a finger gently against the string damps out all modes except those with a node at that position.

So, placing a finger half way along the string leaves only the even modes sounding. The pitch is an octave above the fundamental.

Placing a finger 1/3 of the way along the string leaves only the modes divisible by 3 sounding (e.g. n=3 and 6). The pitch is an octave and a fifth above the fundamental.

n=1

n=3

n=5

n=2

n=4

n=6

Harmonious musical intervals correspond to small whole number frequency ratios.

Octave 2:1Fifth 3:2Fourth 4:3Major third 5:4Minor third 6:5

Using the “circle of fifths” one can tune a piano by alternately going up a fifth (3/2) and down a fourth (3/4) until you have gone up a whole octave.

But the resulting octave will be a little sharp, because (9/8)6 = 2.0273 is a little more than 2.Click speaker to hear the 1% difference.

Perfect Tempered 2:1 = 2.000 2.00003:2 = 1.500 1.49834:3 = 1.333 1.33485:4 = 1.250 1.25996:5 = 1.200 1.1892

To mitigate this problem a slightly flattened fifth is used instead, causing the octave to be in perfectly in tune but the throwing the other intervals slightly out of tune. This is called “Equal Temperament.”

Unlike a string or wind instrument, a drumhead has complicated vibration modes, whose frequencies are not integer multiples of the fundamental. Therefore most drums are not perceived as having a definite pitch.Click picture for a movie clip of drum vibration modes (ignore movie audio)

Indian Tabla A pair of drums. Each drumhead is thickened witha dark spot of dried paste to shift the normal mode frequencies. In the smaller drum, the centered spot causes most of the frequencies to lie at near-whole number ratios, giving the smaller tabla a bell-like sound with a definite pitch.

UNSW animations and movies of wave refl and superposhttp://www.animations.physics.unsw.edu.au/jw/waves_superposition_reflection.htm#pulseswave or tone files in MyDocuments. Dancing hair cell, http://auditoryneuroscience.com/acoustics/sound_propagationHearing missing fundamentalMosquito ringtones http://www.noiseaddicts.com/2011/06/mosquito-ringtones/ Tabla 145 02 M idea on http://www.soundsnap.com/tags/tablaTympani musical element tympani roll and hit httptp://www.soundsnap.com/search/audio/tympani/score

Leftovers and referencesEar largely insensitive to phase, <detuned waves on scope>Sound spectrogram looking again at whistle, sing, pot lid, and various instruments: Violin and harmonics French horn, clarinetSound propagation in 1d, 2d, and 3d. Trouble for whale communication Mosquito ring tones