Where Math meets Music
Now let’s
look at two notes that sound terrible together, C and F#:
Do you notice
the difference between these two? Why is the first ‘consonant’ and the second
‘dissonant’? Notice how in the first
graphic there is a repeating pattern: every 3^{rd} wave of the G
matches up with every 2^{nd} wave of the C (and in the second graphic
how there is no pattern). This is the secret for creating pleasing sounding note combinations:
Frequencies that match up at regular intervals (*  Please see footnote
about complications to this rule).
Now let’s look at
a chord, to find out why it’s notes sound good together.
Here are the frequencies of the notes in the C Major chord (starting
at middle C):
C – 261.6
Hz
E – 329.6
Hz
G – 392.0
Hz
The ratio of E to C is about 5/4ths. This means that every 5^{th} wave of
the E matches up with every 4^{th} wave of the C. The ratio of G to E is about 5/4ths as well.
The ratio of G to C is about 3/2.
Since every note’s frequency matches up well with every other note’s
frequencies (at regular intervals) they all sound good together!
C – 1
D – 9/8
E – 5/4
F – 4/3
G – 3/2
A – 5/3
B – 17/9
To tell you the truth, these are approximate
ratios. Remember when I said the
ratio of E to C is about 5/4ths?
The actual ratio is not 1.25 (5/4ths) but 1.2599.
Why isn’t this ratio perfect? That’s
a good question. When the 12note
‘westernstyle’ scale was created, they wanted not only the ratios to be
in tune, but they also wanted the notes to go up in equal sized jumps.
Since they couldn’t have both at the same time, they settled on a
compromise. Here are the actual
frequencies for the notes in the C Major Key:
Note 
Perfect Ratio to C 
Actual Ratio to C 
Ratio off by 
Frequency in Hz 
Middle C 



261.6 
D 
9/8 or 1.125 
1.1224 
0.0026 
293.7 
E 
5/4 or 1.25 
1.2599 
0.0099 
329.6 
F 
4/3 or 1.333… 
1.3348 
0.0015 
349.2 
G 
3/2 or 1.5 
1.4983 
0.0017 
392.0 
A 
5/3 or 1.666… 
1.6818 
0.0152 
440.0 
B 
17/9 or 1.888… 
1.8877 
0.0003 
493.9 
Here is the equation
to figure out the Hz of a note:
Hertz (number
of vibrations a second) = 6.875 x 2 ^ ( ( 3 + MIDI_Pitch ) / 12 )
The ^ symbol means ‘to the power of’. The MIDI_Pitch value
is according to the MIDI standard, where middle C equals 60, and the C an
octave below it equals 48. As an example, let’s figure the hertz for middle
C:
Hertz = 6.875
x 2 ^ ( ( 3 + 60 ) / 12 ) = 6.875 x 2 ^ 5.25 = 261.6255
The next note up, C#, is:
Hertz = 6.875 x 2 ^ ( ( 3 + 61 )
/ 12 ) = 277.1826
And the next note, D, is:
Hertz = 6.875
x 2 ^ ( ( 3 + 62 ) / 12 ) = 293.6648
So now you’re thinking that we have a scale that goes up in even steps and has reasonably accurate ratios, we’re all set, right? Actually, there are a lot of dissenting opinions on the subject. Remember those notquiteaccurate ratios? One reason for this was for instruments to be able to be tuned once, and sound reasonably good in all keys. Some of the grumpier musicians still complain, though, saying that equal temperament makes all keys sound equally bad. If you tune to just one particular key, you can get those ratios perfect (since the human ear can detect a difference of 1Hz, being off by several Hz can be a problem!).
Maybe more importantly, though, is that there
are a lot of undiscovered frequency combinations that can’t be played in
the confining 12note system. Many alternative scales used in India have
up to 22 notes per octave. If you’re not satisfied with the standard western
scale, there are lot of alternative tuning methods available, such as 'Just
Intonation' and 'Lucy Tuning'. With modern digital equipment, these alternate
tunings have become much easier to implement. We should hear some new and
incredibly interesting music come out of these tuning methods as they are
gradually accepted into the mainstream.
Here are some links if you’d like to explore this topic further:
Just
vs Equal Temperment – ‘harmonic tuning’ described
A beginner’s guide to
temperament – with a little history
American Festival
of Microtonal Music