From 7 to 12

There are literally hundreds of ways to tune an instrument by dividing the octave circle in twelve halftones. But whether your tuning system is 'equal tempered', 'just', 'mean tone' or one of the many 'well tempered' tunings, the pattern behind the division of the octave in twelve halftone steps, the universal structure, derives from combining the two major forces in music:
- The octave as the frame of all musical scales, the 'identity interval', ratio 1:2,
- The perfect fifth as the 'generator interval', ratio 2:3.
In the animation the octave covers one winding of a spiral, so in this perfect analogy between aural and visual presentation you return to the same quality on a new level.
The perfect fifth covers 210.6 degrees.

The issue here is NOT to advocate Pythagorean tuning but to illustrate how this universal pattern behind all tonal systems is created.
Furthermore this will shed light upon the contractive and expansive forces of musical intervals which govern all harmonic and melodic dynamics in music.
An interval tends to resolve its tensions either outwardly (generally major and augmented intervals) or inwardly (generally minor and diminished intervals).
As it may be clarified from the animation, the 12 halftones generated by 12 perfect fifths are not evenly distributed in the octave circle -- the generation principle gives birth to 'minor' halftones (dia-) and 'major' halftones (dia+), so not only the 7-tonality is diatonic, but the 12-tonality from its root may be considered diatonic as well.
This diatonic trait may reveal that the breakdown of tonality in early 20th century could just as well have been considered an invitation to new opportunities with a wider frame of tonality -- so far this invitation has been left virtually unanswered.

The animations of both the 7-tonality and the 12-tonality have three stages:
- Generation by sequence of perfect fifths:
7-tonality: 00:00, 12-tonality: 02:00
- Octavation:
7-tonality: 00:58, 12-tonality: 03:20
- Scale:
7-tonality: 01:44, 12-tonality: 04:40

The piano sounds of the animation are tuned to Pythagorean tuning.

D has been chosen as central note for the sake of symmetry.

More to be found (in danish) at Knud Brant Nielsens website http://www.tolvtonalitet.dk