Before proceeding, we consider that touching and bowing the viola string at a point one sixth of the way along the neck produces another tone.
We have reached the sicth harmonic in the harmonic series; the string is broken into sixths.
The figure is shown in blue to signify that the number 6 is a duplication of the first new tone, 3. Thus, the sixth harmonic does nothing to expand the system. Here we reach another stopping point. The first six harmonics are summarized below.
The figure below shows overlapping patterns for the first six harmonics. This figure can serve as a reminder that when the entire length of a string is made to vibrate without touching the string at any point, the string vibrates in all its harmonics at once. Because 5 is the highest number which produces a unique tone, this figure also represents the 5-limit.
By the 16th century, 6 was accepted as defining the upper harmonic limit of Western music, with 5 being the last unique member of the group. The six tones reduce to three tones when octave congruences are omitted. Since the 17th century, these three tones have defined a standard structure in Western music.
Harmonics 4, 5 and 6 together form a standard structure called a Major Triad.
When placed within the boundaries 1 and 2, the tones of the major triad are 1/1, 5/4, and 3/2.
These three tones are called the tonic, the harmonic mediant, and dominant.
The term Major in Major Triad refers to the quality of the third, which is the Small Major Third, 4:5. The major triad also contains a Perfect Fifth, 2:3. A Large Minor Third, 5:6 is incidentally found between the two upper tones of the triad.
Following the idea of the harmonic series in which the tone 4 is a duplication of the source tone or fundamental, the tonic is called the root of the triad. Following the logic of diatonic interval names, the tones above the root are known as the third and fifth.
Although in practice a variety of major thirds are used, the third of a major triad is generally assumed to be the small major third 4:5 rather than the Pythagorean major third 64:81. Next we see how the idea of symmetry results in a triad of another quality.
NEXT: Harmonic Submediant