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 norm 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. *