Lesser diesis, syntonic comma and Pythagorean comma

Cindarellas half sisters: "Cut a heel and a toe!"

The partials #1 and #2 of the harmonic series form the mother octave, the frame of the tonal patterns. The spiral is suitable as analogy of the octave principle. One winding of the spiral represents one tonal octave: An already established tonal quality is repeated on a lower or higher level. Visually and aurally obvious.

When the further elements of the series are added and used for generating tonal systems, problems may arise:

- 7 octaves almost equal 12 perfect fifths (3rd partial of the harmonic series) but only almost!
The deviation is called the Pythagorean comma.

- Three just major thirds (5th partial of the harmonic series) almost equal one octave but only almost!
This deviation is called lesser diesis.

- And the major third generated by perfect fifths deviates quite a bit from the major third of the harmonic series.
This deviation is called the syntonic comma.

There are more commas in the field of tonality but these are fundamental and involves the three primary harmonic factors:

2 = octave
3 = perfect fifth
5 = just major third

What is really remarkable is that with the equal tempered octave we assume that the yellow dots fall on the blue radii and the violet arms are straight and convergent with 12 o'clock, 4 o'clock and 8 o'clock!!

Feel free to grasp, feel free to share!