Equal Temperament through Geometry and Metrology

The form of musical scale we use today is the (apparently modern) equal tempered scale. Its capabilities express well the new mind’s freedom of movement in that it allows us to change key to play compositions that move between alternative frameworks. This possibility was known to ancient tuning theory, could be approximated within Just intonation’s chromatic notes and was discussed by Plato as forming the constitution of one of his harmonic city states called Magnesia.

Relationship of the Equal Temperament Keyboard to the (logarithmic base-2) tone circle of an octave. We choose D (the Dorian scale) because it is symmetrical on both keyboard and tone circle. Equal Temperament supplies tones which enable any scale to be played starting from any note, though it is the Ionian (C-major) which defines modern key signatures.
Continue reading “Equal Temperament through Geometry and Metrology”

Earth and Moon within Westminster’s Coronation Pavement

Our modern globes are based upon political boundaries and geographical topography yet they have geometrical predecessors, which described the world as an image, diagram or schemata. The original idea for the form of the world was summarised within a simple two dimensional geometry, like an eastern mandala or yantra.

Such a diagram was built into the Cosmati pavement at Westminster Abbey, built by Henry III and dedicated to the Saxon King and Saint Edward the Confessor. This exotic pavement became the focus for the Coronations of subsequent English then British monarchs.

Figure 1 Photo of the Cosmati Pavement at Westminster Abbey
[Copyright: Dean and Chapter of Westminster]
Continue reading “Earth and Moon within Westminster’s Coronation Pavement”

Musical Tones of the Outer Planets

My crucial entré to planetary harmony came when I noticed musical ratios in the synodic time periods of Jupiter and Saturn relative to the lunar year. This approach differs from the norms for “harmonies of the spheres” (a.k.a. Musica Universalis) which are geometrical and spatial, rather than temporally harmonic.

The planetary harmony I found within synodic periods became the subject of my new book The Harmonic Origins of the World (pub. 2018). These synodic ratios have been parts of my work from c. 2000, then expressed as “matrix diagrams” (Matrix of Creation, figure 2 below). In my new book, I show how ancient tuning theory seems to have presented the same information, in a different type of matrix (see figure 4).

Below I connect the outer planets using two additional (and useful) kinds of diagram, the right-angled triangle (figure 1) and the Pentad (figure 5), the latter developed in the 20th century within a discipline called Systematics. 

Figure 1 The harmonic ratios between the nearest two outer planets and the lunar year. The four square rectangle with side length of four, when equal to the lunar year gives, geometrically, the solar year as its diagonal length. The outer planetary synods are longer than the solar year as the planets have moved ahead of their last opposition to the sun. Such oppositions are marked by an outer planet appearing to travel in a loop, amongst the stars
Continue reading “Musical Tones of the Outer Planets”