musings: major scales and harmonic cubes PDF Print E-mail
Written by D. Bond   
Wednesday, 02 June 2010 09:20

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 cross-multiplied 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. Cross-multiplied 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.

Last Updated on Thursday, 10 June 2010 14:57