The Mathematics of Musical Harmony

Why do we think some musical combinations sound good together?
 


 
Here are four facts about music,

• A musical instrument produces sound waves that cause air to vibrate at a particular frequency;
the pitch of a musical note (is it a "low note" or a "high note") is determined by the rate of vibration.

• When a note with frequency "x" is produced, it has
overtones at multiples of x, at 2x, 3x, 4x, 5x, 6x, 7x, ...
For example, if a musical note is produced by air vibrating 24 times per second,
it will have overtones with sound waves (in air) vibrating at 48, 72, 96, 120, 144, 168, 192,...

• A major chord is formed by playing the 1-and-3-and-5 notes of a major scale at the same time.

• In a Pythagorean Scale, the ratios between the 7 notes of a major scale are
1/1     9/8     5/4     4/3     3/2     5/3     15/8     2/1
If the 1-note is produced by sound waves vibrating at 24 times per second,
the seven scale-notes are:     24     27     30     32     36     40     45     48

 
and a question:
Can you see a mathematical relationship between the overtones of the 1, 3, and 5 notes?
You can play with the numbers and do detective work, then look at the table & explanation below.
 


 
In this table, notice the "matching up" of some overtones: 13, 15, 35, 135.

1st
24     48   72   96   120 144   168   192   216 240   264   288   312     336 360
 etc 
3rd
  30     60   90     120   150   180   210   240     270   300     330   360
etc
5th
    36     72     108   144     180     216   252     288     324     360
etc
match
 
 
 
 
 
15
 
 
 
13
15
 
 
35
 
 
15
13
 
 
 
15
 
 
 
 
 
135
 

When we play any note, the sound-producer (voice, flute, trumpet,...) also produces
all of its overtones.  For example, when we play a note at 24 it's a "package deal" that
also includes its overtones at 48, 72,... and these contribute to the harmony we hear.
When two or more of these notes (at 24, 30, 36, 48,...) are played simultaneously,
our ears perceive the "matching up" of their overtone-waves as a pleasant harmony.

 Due to the physiology of hearing and vision,
our ears hear simultaneous notes as harmony,
but our eyes see simultaneous colors as a new color.

{ In other pages, you can learn more about the science of music and color. }