Music 1 - Notes, Harmonics

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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Let us examine "harmonics", visible on a string (as demonstrated in class).  Harmonics are wave shapes produced that have a maximum amplitude under given conditions (tension in string, length of string, composition of string, etc.).  Every stretched string has a particular lowest frequency at which it will naturally resonate or vibrate.  However, there are also higher frequencies that will also give "harmonics" - basically, pretty wave shapes (also known as "standing waves").  These higher frequencies are integer multiples of the lowest frequency.

So, if the frequency of the lowest frequency is 10 Hz (for an N = 1 harmonic), the next harmonic (N = 2) occurs at 20 Hz.  N = 3 is at 30 Hz, and so on.

For those of you who play guitar, you know that you get harmonics on certain frets.  In the exact center of the neck (12th fret) you get a harmonic (the 2nd one) and the frequency is twice that of the open string - one octave above, as we will discuss.