Clockface: circle becomes spiral becomes cylinder. This clockface pattern can be mirror imaged and/or developed into a spiral as shown here.Try imagining it as a three dimensional cylinder with the spiral pattern wrapped around the surface, as described in the recipe.The views from opposite ends will be mirror images.The next stage of visualisation requires you to see that a longitudinal cross-section projection is a sinewave.So is a sinewave the shadow cast by a more this more complex shape?

I am skeptical about this method of mapping harmonics, as it seems to me that "harmonics" should be at the pitches which you can generate on
stringed and other instruments. e.g. by touching the strings lightly, or blowing and over blowing.

The usual theory is demonstrated using the image of a sine wave, and how the peaks of waves coincide at integer ratios.e.g. 440 A and 660 E.

My own belief is that the traditional model is static and mathematically simplistic.There seems to be a more subtle (yet also simple) underlying pattern, judging from my research and experiments into John 'Longitude' Harrison's ideas.

Philosophically, you could consider the "sine wave" as a two dimensional representation of a three dimensional coiled spring pattern. (Shades of Plato ????;-)

Imagine shining a light through a stretched spring. The resultant shadow cast by the spring could be seem as a sine wave, when viewed from the
appropriate angle on a background screen. We know that the spring is 3 dimensional, "yet we continue to measure the shadow" ????

I feel that there is much more to this question that most people realise. Harmonics in the real physical world tend to also contain vibrato. I suggest that they can be mapped in a better way which accounts for vibrato, and the resultant beating. Read between the lines on the site, for I suspect that the "harmonics" are in reality at the octaves of the notes generate using intervals derives from pi. It sounds musically correct, yet I can't yet prove it scientifically. Hence I am following a "different" paradigm of what constitutes a "harmonic" in the real world.

The diagram below may assist in visualising how the spiral pattern used for mapping LucyTuning connects with the chain of fourths and fifths used for scalemaking and scalecoding.

The colored strip on the left represents the chain arranged in steps of fifths down the page.
The list of scale positions to the extreme left shows how the notes are assigned for the table of tunings for MIDI keyboards as twelve contiguous steps for tuning table 2 (three flats and two sharps).

The chords illustrated are the four main patterns used in Western harmony (Major, Augmented, minor and diminished). The chords are expressed with C as the tonic. Transposition may be demonstrated by "sliding" the scale positions up or down in relation to the colored column for various tonics. By saving this page and opening the .gif in any paint program the colored column or the positions may be cut and re-pasted to show other keys and tonics.