Math of Music

Have you ever wondered why our musical scales have twelve notes and not ten or eight? Why some notes sounds good or melodic together? Why most of us are able to identify when a note is out of place in a song even though most of us musically illiterate? Well folks, the answer lies in - Frequencies and ratios. 

In Western music, the primary reference tone is the pitch A - a frequency of 440 Hz. The next higher A in the musical scale is at frequency 880 Hz or 440 X 2 Hz. the difference of 440 Hz makes the two A an octave apart. Every octave is divided into twelve notes, the frequencies of which are mentioned in the table below.

A	440
B flat	466
B	494
C	523
C sharp	554
D	587
D sharp	622
E	659
F	698
F sharp	740
G	784
A flat	831
A	880

Lets consider the major triad - A, C# and E or the A chord.

Notice that A is 440 Hz. Now 440 X 3 = 1320 and 1320 / 2 = 660 ~ frequency for E. I.E. A and E are harmonically related  - 440 X 3 is E one octave above the note A - 440. 

For somewhat more arcane reasons, the interval between A and E, which is a frequency rise of 3/2, is called a fifth. The note 3/2 above E has frequency 988, which is an octave above B-494. Another 3/2 above that is approximately F sharp (740 Hz). Continuing in this fashion, multiplying frequencies by 3/2, and then possibly dividing by two, you can approximately trace the twelve notes of the scale. This progression is called the circle of fifths. The notion of key in music and a scale are based on this circle of fifths.

How is C# then part of the major triad?

440 × 5 ≈ 554 × 4 or C# is 3 octaves above A - 440. 

So A C# and E are harmonically related. Hence when you play the notes A C# and E together you find the resultant sound "musical".

In short,the human ear hears frequencies   and frequencies that are harmonically related tend to sound good together - they are consonant rather than dissonant. People prefer musical scales that have many consonant intervals. And such scales contains frequencies which equal the ratio of small integers ( 3/2  and  5/4 as in the case of major triads ). 

There are seven basic consonant intervals in the musical world - 

Basic Consonant Intervals

2/1	octave	harmonic inverse of 1/1
3/2	perfect fifth	harmonic inverse of 4/3
4/3	perfect fourth	harmonic inverse of 3/2
5/3	major sixth	harmonic inverse of 6/5
5/4	major third	harmonic inverse of 8/5
6/5	minor third	harmonic inverse of 5/3
8/5	minor sixth	harmonic inverse of 5/4

But why the number twelve in octave - not 20 or 10?

The answer is  - The twelve-tone equal-tempered scale is the smallest equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation — within one percent.

This can bee seen below for the note C  

Number of Semitones	Interval Name	Notes	Consonant?	Simple ratio	Twelve notes Scale	Difference
0	unison	C-C	Yes	1/1=1.000	20/12=1.000	0.0%
1	semitone	C-C#	No	16/15=1.067	21/12=1.059	0.7%
2	whole tone	C-D	No	9/8=1.125	22/12=1.122	0.2%
3	minor third	C-Eb	Yes	6/5=1.200	23/12=1.189	0.9%
4	major third	C-E	Yes	5/4=1.250	24/12=1.260	0.8%
5	perfect fourth	C-F	Yes	4/3=1.333	25/12=1.335	0.1%
6	tritone	C-F#	No	7/5=1.400	26/12=1.414	1.0%
7	perfect fifth	C-G	Yes	3/2=1.500	27/12=1.498	0.1%
8	minor sixth	C-Ab	Yes	8/5=1.600	28/12=1.587	0.8%
9	major sixth	C-A	Yes	5/3=1.667	29/12=1.682	0.9%
10	minor seventh	C-Bb	No	9/5=1.800	210/12=1.782	1.0%
11	major seventh	C-B	No	15/8=1.875	211/12=1.888	0.7%
12	octave	C-C'	Yes	2/1=2.000	212/12=2.000	0.0%

My favorite melodic scale is B-minor. Here is a link for a B-minor arpeggio played over the Piano.