The Chord of Nature (harmonic series) (4)

The answer:

The coinciding pitches are in red. All the harmonics in the second Chord coincide with alternating harmonics of the first Chord.

The important point is that the reason lower harmonics coincide, is because the fundamentals (lowest harmonics) are placed apart by one of the intervals from within the lower part of the Chord of Nature itself - in this case, the interval between harmonic 1 and harmonic 2 (an octave). This 'agreement' between harmonics turns out to be very important in producing the tangible sonority that occurs with a musical consonance such as the octave.

Coincidences occur with other intervals also, but are fewer. For example, for two notes, one a perfect fifth above the other, the 2nd harmonic of the upper note will coincide with the 3rd harmonic of the lower, but no other coincidences occur until the 4th of the upper with 6th of the lower, and so on.

There is a 'harmonic ratio' associated with each musical interval. Some of these are:

Octave - 2:1

Perfect fifth - 3:2

Perfect fourth - 4:3

Major third - 5:4

These ratios give:

1) The ratios of the lengths of identical strings at the same tension, that will produce the given interval;

2) The ratios of organ pipe lengths tuned apart by these intervals;

(For string and pipe lengths the longest pipe or string produces the lowest note).

3) The numbers of the harmonics in the Chord of Nature that are separated by that particular interval;

4) The frequency ratio of two notes whose pitches are separated by the given interval. (The highest frequency produces the highest note).

5) The numbers of the lowest harmonics that will coincide when two notes tuned apart by that interval are sounding together. The highest number will be the harmonic number of the lowest note, and vice versa.