The Sound of Music

Tonight we will chat about the most well-known of mechanical waves:  sound.

Sound is a mechanical wave, meaning that it REQUIRES a medium through which to travel.  Whereas light (or other EM waves) can travel anywhere (more or less), and travel fastest (at the speed of light) through a vacuum, sound is restricted greatly.  It can only travel through a medium, which itself carries the vibrations that are sound waves.

The same characteristics previously discussed still apply:  frequency, wavelength, speed, amplitude, crest, trough.  However, sound itself is a longitudinal wave (jiggling "back and forth") rather than a transverse wave (like EM waves, which vibrate "up and down").

Animation of sound as a mechanical wave:




Music

In western music, we use an "equal tempered (or well tempered) scale."  It has a few noteworthy characteristics;

The octave is defined as a doubling (or halving) of a frequency.

You may have seen a keyboard before.  The notes are, beginning with C (the note immediately before the pair of black keys):

C
C#
D
D#
E
F
F#
G
G#
A
A#
B
C

(Yes, I could also say D-flat instead of C#, but I don't have a flat symbol on the keyboard.  And I don't want to split hairs over sharps and flats - it's not that important at the moment.)

There are 13 notes here, but only 12 "jumps" to go from C to the next C above it (one octave higher).  Here's the problem.  If there are 12 jumps to get to a factor of 2 (in frequency), making an octave, how do you get from one note to the next note on the piano?  (This is called a "half-step" or "semi-tone".)

The well-tempered scale says that each note has a frequency equal to a particular number multiplied by the frequency that comes before it.  In other words, to go from C to C#, multiply the frequency of the C by a particular number.


So, what is this number?  Well, it's the number that, when multiplied by itself 12 times, will give 2.  In other words, it's the 12th root of 2 - or 2 to the 1/12 power.  That is around 1.0594.

So to go from one note to the next note on the piano or fretboard, multiply the first note by 1.0594.  To go TWO semi-tones up, multiply by 1.0594 again - or multiply the first note by 1.0594^2.  Got it?

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