Spiral of 5ths and chords

Spiral of 5ths Outside 12-equal Temperament

The purpose of the spiral is to illustrate the difference between notes created with sharps and flats, notes that would coincide in 12-equal temperament. Here I am specifically referring to notes generated by taking the 3rd harmonic as you rotate clockwise (Pythagorean scale) or an approximation of the 3rd harmonic (the various meantone scales, like 1/2-comma, 2/7-comma, etc).

This diagram is discussed extensively on my Temperaments page.

It is unnecessary to redraw all of the diagrams from the Circle of 5ths page; here I will just draw those which are significantly different.
  1. C augmented. Chain of Major 3rds (5th harmonics).
  2. C diminished Counter-clockwise chain of minor 3rds (or clockwise chain of major thirds ).
  3. C diminished 6. Again a chain of scale 3rds. The red shape is what I take the standard chord to be, but the green shape might be more natural since it continues the chain down from c. Why not go to bbb instead of a? The dotted purple line is just making the shapes closed, like they were in the Circle of 5ths diagrams. This line is dotted because it cuts across branches.
  4. Observe that f-c-g is a chain of 3rd harmonics!
  5. Reflection of C major to C minor. (If this diagram is unclear, please refer to the Circle of 5ths page for insight).
    • This illustrates that reflections must be done by counting points around the spiral in this representation, since this picture distorts the circularity.
    • Note also the 2 tri-tone cut lines. The notes c# and f# are excluded because they are outside both key regions.

  6. The resolution of the tri-tone is quite different here than in 12-equal. If b-f resolves outward, it goes to f#-bb and since bb does not equal a# here, this interval is not a Major 3rd.