A very short but sweet observation follows after considering harmonic cubes: the major scale is a basic harmonic cube that can be easily constructed using the relationship of the H[3,3] matrix crossmultiplied by the major [4,5,6]. To simplify we need only consider the basic 2/3, 3/3, 3/2 series as the base set*, giving the classical IV, I, V relationship. Crossmultiplied by the 4/4, 5/4, 6/4 major triad and reduced to simplest terms (D=1/1) we arrive at: 2/3, 5/6, 1/1 = G B D 1/1, 5/4, 3/2 = D F# A 3/2, 15/8, 9/4 = A C# E Rearranging this and transposing for octave gives a simple scale: G, A, B, C#, D, E, F#, G, A, 2/3, 3/4, 5/6, 15/16, 1/1, 9/8, 5/4, 4/3, 3/2 Granted, nothing new here. Another way to generate this is to consider the minor triad as 6/6, 5/6, 4/6 2/3 5/8 4/9 = G Eb C 1/1 5/6 2/3 = D Bb G 3/2 5/4 1/1 = A F D G, A, Bb, C, D, Eb, F, G, A 2/3, 3/4, 5/6, 8/9, 1/1, 5/3, 5/4, 4/3, 3/2 Knowing that [1, 2, 3, 4, 5, 6] can be reduced to [4, 5, 6] we find that this collection is the Harmonic Matrix[3,3] multiplied by ustring[1/6, 2/6, 3/6, 4/6, 5/6, 6/6, 6/5, 6/4, 6/3, 6/2, 6/1] or more simply [4/6, 5/6, 6/6, 6/5, 6/4]. Within this surface, both scales above are represented. * the relationship may also be represented as a range of a perfect fourth above and below 1/1. Thus 3/4, 1/1, 4/3.
