Spiral Array Model - Equations

Equations

The pitch spiral P, is represented in parametric form by:

P(k) = \begin{bmatrix} x_{k} \\ y_{k} \\ z_{k} \\ \end{bmatrix} = \begin{bmatrix} r sin (k \cdot \pi / 2)\\ r cos (k \cdot \pi / 2) \\ kh \end{bmatrix}

Where k is an integer representing a semitone, r is the radius of the spiral, and h is the "rise" of the spiral

The major chord CM is represented by:

where and

The weights "w" effect how close the center of effect are to the fundamental, major third, and perfect fifth of the chord. By changing the relative values of these weights, the spiral array model effects how "close" the resulting chord is to the three constituent pitches. Generally in western music, the fundamental is given the greatest weight in identifying the chord (w1), followed by the fifth (w2), followed by the third (w3).

The minor chord Cm is represented by:

where and

The weights "u" function similarly to the major chord.

The major key TM is represented by:

where and

Similar to the weights controlling how close constituent pitches are to the center of effect of the chord they produce, the weights "W" control the relative effect of the I, IV, and V chord in determining how close they are to the resultant key.

The minor key Tm is represented by:

where and and and .

Read more about this topic:  Spiral Array Model