This project is to demonstrate the astounding geometry of three intersecting, orthogonal, golden rectangles and an inscribed regular icosahedron, as well as the three interrelated musical tonalities of G, Eb, and B

Golden Rectangles

Golden rectangles are rectangles such that the ratio of length to width is 1 to phi, where phi is approximately 1.618033988749895, otherwise known as the Golden Ratio, the Golden Mean, or the Golden Section.

Regular Icosahedron

A regular icosahedron is one of the five Platonic solids or polyhedra, consisting of 20 regular equilateral triangles.

Giant Steps, Mark Turner, and Me

"Giant Steps" was John Coltrane's seminal composition from the 1960 album of the same name. It represents a breakthru of Trane-ian proportions, creating a completely new harmonic pathway for modern jazz, distinct from the type of harmonic motion that was typical of the previous decades in jazz and classical music. Instead of songs being based on a single tonal center (the "root" or "tonic") and chords moving only to keys nearby on the circle of fifths, "Giant Steps" features no set tonal center, rather a succession of quickly shifting chords separated by major thirds (a distance of 4 semitones away). This type of harmony divides the 12-tone scale into three equal portions, much like you can divide the color wheel into three equal portions by choosing three triadic colors.

Saxophonist Mark Turner's opening statement on his 1998 In This World album, the stunning original composition "Mesa", begins with a drone in the key of B, accompanied by the same 6-note melodic cell being repeated in the successive keys of B, Eb, and G -- the same major-third harmonic motion as in "Giant Steps". The 6-note melodic cell represents what could be called a Mixolydian Pentatonic tonality in the Western system. Similar scales or tonalities exist around the world, such as the Raga Savethri in the South Indian Carnatic system. I've always loved the haunting symmetry and hidden beauty of "Mesa" as well as "Giant Steps" and this is my attempt to merge triadic color theory, "Giant Steps" harmony, and the properties of three intersecting golden rectangles.