a common space for harmonic overtones 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 harm