Note all pitches are derived from the one string, using whole number
ratios. These frequencies (or pitches or tones) occur naturally
in all stretched strings. They are therefore in harmony with nature,
hence the term harmonics.
Harmonics 
Note name (pitch) 
Ratio (length of string) 
Fundamental 
Do 
1:1 
1^{st} harmonic 
Dol 
1:2 
2^{nd} harmonic 
Sol 
1:3 
3^{rd} harmonic 
Do 
1:4 
4^{th} harmonic 
Mi 
1:5 
Pitches played together produce the Major chord Do  Mi  Sol
In the above we used ratios based on the fundamental string (note)
Do, which was set to unity (1). But we can derive, just by using the
information we already have, another set of ratios, namely the ratios
between each successive pair of notes. This will determine the interval
between these pairs. To do this, we need to make one of each pair unity.
The first pair we encounter in the above list already has the first
one set to unity. So we can say our first interval has the ratio of
1 : 2.
The next pair of ratios are 1 : 2 and 1 : 3. The interval between this
(which is actually a ratio of ratios) can be calculated as follows:
Make the first one unity and remember the factor needed to make it
unity. ie we need to multiply the 1 : 2 by 2 to get unity. This factor
2 needs to be made common to both. So, to retain the interval we then
need to multiply 1 : 3 by 2 also ie. 2 : 3.
The above argument is made for the other pairs too, this way we can
expand our table above:
Harmonics 
Note 
Ratio (length of string) 
Interval distance 
Interval name 
Note 
Fundamental 
Do 
1:1 


1 * 1/2 = 1/2 
Octave 
Do 
1^{st} harmonic 
Do 
1:2 


2 * 1/3 = 2/3 
Dominant (fifth) 
Sol 
2^{nd} harmonic 
Sol 
1:3 


3 * 1/4 = 3/4 
Subdominant (fourth) 
Fa 
3^{rd} harmonic 
Do 
1:4 


4 * 1/5 = 4/5 
Mediant (third) 
Mi 
4^{th} harmonic 
Mi 
1:5 
