The harmonic progression, or proportion, looks like this:

a—b:b—c::a:c

the formula for which is:

b = 2ac/(a+c)

which results in the harmonic, or musical, proportion:

1, 4/3, 3/2, 2

where 1 is the fundamental, 4/3 is a fourth, 3/2 is a fifth, and 2 the octave above 1.

If you’re interested in a full discussion of the math behind this, see Richard Lawlor’s Sacred Geometry ¹. For a fascinating and in-depth history of how geometry becomes music and the development of our octave, see Richard Merrick’s Interference ². For my purposes, I’m interested in how this might get mapped to the color wheel, music becomes color becomes music. And when I began playing with this idea, it made more sense to use four primaries for painting (red, yellow, green, blue) instead of the traditional three color primaries (red, yellow, blue).

A short digression before I go on:

Color is vibration, as music is vibration. In the visible light spectrum color moves from violet (the shortest wavelength) through blue, then green, yellow, orange, and red (the longest wavelength). The difference between the colors of visible light (additive color) and the colors mixed for painting (subtractive color) can be charted as:

The primaries of additive color are red, green and blue; of subtractive color are magenta, cyan and yellow (even though this isn’t really reflected – no pun intended – in traditional color wheels or in most discussions of color theory). When two of the additive color primaries are mixed, these are called secondaries. The secondaries of additive color are the primaries of subtractive color, and vice versa. I find this fascinating.

But, as I said, I’m basing my colors for Fourths and Fifths on primaries of red, yellow, green and blue. I have planned three triads (triptychs) that will look something like this:

R → Y → G, where yellow is the fundamental and red is the octave below, green the octave above. The fourth of the R-Y octave will be a red-orange. The fifth of the Y-G octave will be a yellow-green.

And so on. Now it’s time to get into the studio.

References / Additional Reading

1. Johannes Itten, The Elements of Color (John Wiley & Sons, 2003)
2. Mark David Gottsegen, The Painter’s Handbook (Watson-Guptill Publications, 2006)
3. Paul Zelanski and Mary Pat Fisher, Color (Prentice Hall, 2003)
4. Lawrence D. Woolf, It’s a Colorful Life (General Atomics Sciences Education Foundation, 2000)
5. Ian Stewart and Martin Golubitsky, Fearful Symmetry: Is God a Geometer? (Dover Publications, 2011)

I left off with the arithmetic progression. Now for the second of the three: Geometric. Again, this is a three-term proportion where a > b > c. In the language of ratios, a and c are the extremes and b is the mean. As with an arithmetic progression, begin with two of the differences in the numbers: a—b and b—c. In a geometric or harmonic progression, these differences are to each other in the same way as one of these numbers is to one of the other numbers, not as one of the numbers is to itself as in an arithmetic proportion.

The geometric proportion is expressed as:

a—b and b—c::a:b

The solution for a geometric proportion where the mean term is b is b² = ac or b = √ac. Using this formula for the extremes of 4 and 16, the mean is 8. The geometric progression is 4, 8, 16. This proportion is also expressed as the golden mean:

a:b::b:c

If this were mapped to a color wheel, this could be two primaries (the two extremes) and their secondary (the mean): red is to orange as orange is to yellow.

With the Fourths and Fifths series, however, I am playing with the idea of mapping the musical proportion to color. The progression for the musical proportion, based on an octave, is 1 (the fundamental), 4/3 (a fourth), 3/2 (a fifth), 2 (octave above 1). That idea will be the third post in this series.

As I worked through the color relationships for the new body of work, I felt there might be some such relationship. In a previous post, The closed circle and the infinite loop, my first instinct was that the colors for the series somehow related to thirds and fifths in music. I now amend this to fourths and fifths after reading Lawlor.

This is based on the concept of “mediating proportions” – binding two extremes through a single mean term.

There are three such mediating proportions: arithmetic, geometric and harmonic. It’s the latter I’m trying to puzzle through (which means I’ll need to relearn everything I’ve forgotten about music theory). So between music theory and color theory is there a correspondence we can define and talk about? It’s all vibration, right?

And now the math behind these progressions. First, these proportions are all three-term proportions, a group of three unequal numbers where a > b > c. In the language of ratios a and c are the extremes and b is the mean. The relationship between these numbers is that “two of their differences are to each other in the same relationship as one of these numbers is to itself…” (arithmetic) or is in the same relationship as one of these numbers is to one of the other numbers (geometric and harmonic). The two differences here are a—b and b—c.

Arithmetic Progression
In an arithmetic progression a—b is to b—c as a is to a, b is to b, c is to c:

a—b:b—c::a:a, b:b, c:c

To take a simple example, let’s say the extremes of this proportion are 3 and 7. To find the mean, add the two extremes and divide by two:

b = (a+c)/2

The mean term is 5 and the arithmetic progression is 3, 5, 7.

Next: the geometric and harmonic progressions.

I mentioned this to a new Facebook friend who asked about my practice and who, synchronistically, works with color and sound. He recommended a book to me, Interference: A Grand Scientific Musical Theory. I have worked some with interference media and am fascinated by interference in waveforms and cymatics, so I was intrigued. Also synchronistically another friend is working with fluid dynamics which as far as I know, and I don’t know much, is related to chaos theory. Everything came together in my mind in an ‘ah-ha’ moment where I saw the connections in all these, and then, of course, immediately lost it. Color harmony, musical harmony, symmetry in chaos…

I began reading Interference this morning. Besides rekindling memories of studying piano and the point at which the patterns my hands made on the keyboard were beginning to make sense, this is what struck me:

The Pythagoreans were searching for a unified theory of everything, just like some physicists today. What is the underlying harmony behind the veil of our senses which unites all phenomena? So they, the Pythagoreans that is, came up with the idea of stacking musical fifths, believing that “a stack of five perfects 5ths’ should close to form a pentagram at the third octave.” This is based on their association of the geometry of sound with certain regular shapes, in this case the pentagram, an important form in sacred geometry. Well, it didn’t work. Turns out there’s a gap. The circle is not closed, but is an infinite spiral, like the famous Nautilus. And so it’s my guess, at this point in my reading, that the gap is equal to the phi ratio.

Somehow in my mind this is all related to mixing colors. Stay tuned.

One of the essential questions, and I think especially so with monochromes, is how the paint is applied to the surface. I am now actively studying other ways and other vehicles for applying paint which will allow me to work looser and acknowledge the essential messy-ness of paint in preparation for my next series of work. It’s good to be in the studio playing with ideas. A welcome change from the pressure of creating a final work…and constructing a painting is hard work! I have found an artist who makes beautiful Baltic birch panels for me and I am developing a love affair with calcium carbonate.

The end result of this play will be a new series of paintings whose working title is “Seconds and Thirds” based on color theory. More later as the series unfolds.

“…when we refuse what has been offered to the empty heart, when possible futures are given and not acted upon, then the imagination recedes. And without the imagination we can do no more than spin the future out of the logic of the present; we will never be led into new life because we can work only from the known.” (p. 252, second Vintage Books edition, 2007)

“Every artist secretly hopes his art will make him attractive. Sometimes he or she imagines it is a lover, a child, a mentor, who will be drawn to the work. But alone in the workshop it is the soul itself the artist labors to delight. The labor of gratitude is the initial food we offer the soul in return for its gifts, and if it accepts our sacrifice we may be…drawn into a gifted state – out of time, coherent, ‘in place.’” (p. 249, second Vintage Books edition, 2007)

Finding one’s place and putting down roots has much to recommend it. I have tried – in San Francisco, in Santa Fe, in Marfa – yet I am restless and sooner or later my restlessness uproots me like a tumbleweed. Then I look to the next horizon. I have set my sights on something other than this small, dusty burb, yearning for engagement with a larger vision. (Slowly I begin to understand that I carry home with me.)

Like Santa Fe and, before it, San Francisco, there is Marfa and there is the myth of Marfa created in part by the seemingly endless articles in the New York Times. Yes, Chinati is incredible. But incredible also are the greetings and genuine smiles of the three women tellers at Marfa National Bank, impromptu gatherings at the post office, listening to the stories of those for whom Marfa has been home for generations. Marfa was for me a retreat, ironically from the art world as it turned out. A place where I remembered that creating is process, not result and with that remembrance happily tucked into my pocket was able to enter the studio again after an almost three-year hiatus. I am glad Marfa was that place of retreat and reflection. I am now ready to move on.

“He was working for himself now, no longer burdened by the threat of other people’s opinions, and that alone was enough to produce a fundamental change in how he approached his art. For the first time in his life, he stopped worrying about results, and as a consequence the terms “success” and “failure” had suddenly lost their meaning for him. The true purpose of art was not to create beautiful objects, he discovered. It was a method of understanding, a way of penetrating the world and finding one’s place in it…”

She asked me where I found the patience to do the work, a question I don’t think I’ve been asked. I did not have a ready answer. Since then I’ve been thinking about her question. Where do I find the patience, being a restless, anxious type? I’m thinking patience is akin to focus and that focus requires a quiet mind, so the question has morphed into “How do I quiet my mind.” The best answer I have for that is to give the mind something to do while I do something else.

All artists have various tricks and techniques to get into the studio and work. Mine has been to clean and organize the studio or clean the house before I start a session in the studio. Here are some other ways I’ve found to quiet the mind and, therefore, give me patience:

Counting
Washing dishes
Weeding the garden
Repeating the same task over and over (pretty much how I do a painting)
Taking a really deep breath before I touch brush to canvas
Playing with my dog, Zack
Sweeping the floor

The subjects of my paintings are color and perception. The six panels of Work Group1 (2008) continue my investigation into subtle differences of light and color (color being a quality of light) through the use of multiples.

The work is divided into three diptychs. Each diptych’s final color is the same, only the underpainting differs. The final colors were applied thinly and loosely using transparent or semi-transparent colors. This causes the top color to mix with the color underneath in the viewer’s perception, subtly altering the perceived hue. The color perceived does not actually exist on the surface of the painting.

Another result of working loosely is that the sufrace color appears flat at a distance, yet closer is actually many colors which distance blurs.

Installation View

Installation view