What a Tulip, an Orange, the Color Wheel and the Circle of Fifths Have in Common

In the closed systems of a tulip, an orange, the color wheel or the circle of fifths we can continually find patterns which correlate their structures. The geometric shapes which emerge in the numbers that govern their architecture begin to tell the story of their kinships. These correlations take us to the border of the known and the unknown where physics, speculation and spirituality meet.

Art & Music

I had started painting again after many years and was making connections between what I was learning about the color wheel and what I knew about the tertiary structures of three, six and twelve that are built into the circle of fifths. The triangle, I thought, is the genitor which becomes the six and the twelve of both the wheel and the circle. In the midst of my thought, I saw this tulip in my kitchen.

Nature Imitates Nature

Here was the form again…complete! In the center is a triangular stigma (genetrix) surrounded by six stamens which are the male pollinators. Enclosing these are the six petals (like whole tones), each of which is bisected (making twelve semitones.)

The Rind that Binds

Going on a hunch, I cut this orange in half and sure enough…twelve segments; each half having six, with a twelve-pointed star in the center. Whether the Zodiac, the moon cycles or the twelve nerves that run to your head, this repeated pattern has deeper meanings beyond current exploration.

Our Limited Senses

Commonalities between the color wheel and the circle of fifths are more readily observed. The frequency spectra that can be perceived by the eye and the ear are limited to a tiny fraction of all wavelengths which exist. Since most every thing can be thought of as a vibration with a beginning, a middle and an end (including a musical piece, a human being or a galaxy), it follows that these phenomena we observe are the same thing, modulated on different frequencies as iterations of the one form which permeates all.

Geometry of Color

The colors yellow, blue and red make up the nodes of the equilateral triangle of primary colors on the wheel. These colors combine with one another to make the triangle of secondary colors: green, purple and orange. These two triangles, when superimposed on the wheel, one inverted upon the other, are also known as the symbols of masculinity and femininity and appear in other symbology; most recognizably in the Star of David.

Geometry of Sound

In music, the notes which are an interval of a major third apart (i.e., C, E and Ab), and which sound as an augmented chord, make an equilateral triangle on the circle of fifths. This is the most ambiguous triad, and this division of the circle mirrors the purity of primary colors since there is no perfect fifth to give any one pitch dominion over any other. In contrast, the approach in tonal music is to split the circle in half and use pitches from one side only: the seven pitches of the diatonic scale (omitting the other five). Incidentally, painters often stick to one side of the wheel as a way to give a dominant tonality to a work. Also, each of the primary colors exists in seven of the colors on the wheel and is absent in five, corresponding to diatonicism in music.

Triangular Motion

Triangular harmonic movement is what I use to compose cues for horror scores. Chords built around these nodes create an unnerving and unsettling emotion. This is tied to our understanding that we are not moored to a tonal center, but rather unhinged from a home key. Jazz composers use triangular key movement to the same effect.

Parallel Universes

The inversion of C, E and Ab (geometrically) is the augmented triad Bb, D and F#. This makes the same inverted triangle as we saw with the secondary colors, and produces the aforementioned six-pointed star. These six nodes combine to make the whole-tone scale, which is the most ambiguous of scales with no perfect fifths. It also divides the chromatic scale into two interlocking six-note scales. Conversely, these can be seen as four triangles; the other two being C#, F, A and Eb, G, B.

Chromatic Spectrum of Color

If we make the six tertiary colors by combining each primary color (yellow, blue or red) with each adjacent secondary color (green, purple or orange), we complete the twelve-color wheel (i.e., yellow, yellow-green, green, blue-green, blue, etc.,). These twelve correspond to the twelve tones of the chromatic scale.

A Study in Opposites

A full, explorative comparison of the color wheel with the circle of fifths would be exhaustive. Here I will focus on a specific aspect, which is the correlation of dissonance in the mixture of nodes from opposite sides of the wheel/circle. 

The Dissonance of Pitch Poles

In music, the note on the opposite side of the circle is called a tritone (three-tone), which is the distance of six half steps. This “devil’s interval,”built on the strange number six, is the most dissonant. The tritone has been thought to split dark from light, warm from cold, yin from yang, order from chaos, or good from evil. Incidentally, it is the central harmonic tension of music which seeks the consonance of the perfect fifth, which is seven half steps.

There are six tritone intervals (and their six inversions):

• C-F#     F#-C
• G-C#    C#-G
• D-Ab    Ab-D
• A-Eb    Eb-A
• E-Bb    Bb-E
• B-F       F-B

The Dissonance of Color Poles

In painting, the node opposite any node on the wheel is called “complementary” and is said to “neutralize” the other node. Therefore, a color which is “too purple” can be neutralized by yellow which exists opposite purple. This will gray the purple, masking the red-blue purity with the dissonance of yellow.

The Science of Color and Sound

Though I don’t claim expertise in the science of paint, the mixture of pigments, media, and the resultant chemical reactions make for difficulty in quantifying the math of frequency modulation between nodes. In the physics of music however, two pure sine tones that are six half steps apart can be analyzed through spectrometers and equations based on the harmonic series. It’s my contention that the composition of the six grays, if controlled as in the sine tone example, parallels that of the six tritones.

The Waveforms of Art and Music

Technology will emerge to deepen our understanding of these complex waveform interactions. While the eye sees hue, tint and shade; the ear hears timbre, treble and bass. The fact of one shows itself in the other. In the drawing above, the yellow-gray produced by mixing yellow and purple (2Y + (1R+1B)) has twice the yellow. The C-F# interval, in turn, with twice the emphasis on the C, contains an analogous relative dissonance and frequency balance. In other words, a yellow-gray and a C-heavy tritone are essentially the same thing viewed through a different lens.

Practical Applications

Technology opens new domains of cross pollination and creativity where disciplines combine and new tools are invented. Given that the node combinations of music can be correlated to the node combinations in art, I could imagine a painting being made from musical intervals, or conversely, music being made from a string of colors. The deeper we search the relationships, the more similarities we find, and the closer we come to an understanding of ourselves, our experience and our world.