Tonal Donut Download

Tonal Donut is a program resulted from considerations based on mathematical Group Theory and helps the viusualisation of group theory in Music.

The musical scale of the western hemisphere comprises of 12 semi tones.

These 12 tones can be viewed as elements of the cyclic group Z12. Its shown that Z12 is the direct products of the groups Z3 and Z4. This is not trivial, for example, Z2 and Z6 do not yield Z12.

Which musical implications this fact may have?
The mathematical proof is rather simple, its interpretation in musical terms is less simple. The result of the observations appeared intriguing at first sight, but eventually logical. Z3 represents the group of the three diminished chords (stacked minor thirds) and Z4 represents the four augmented chords (stacked major thirds). For the sake of completeness we would like to mention that Z2 represents the group of the whole tone scales.