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.

There’s this guy in town who puts his small dog in his white van every morning and evening and drives around town, the dog with his head out the window, ears flapping, and rhythmically yelping, sharp and piercing. Yelp, yelp, yelp, yelp… What the? Is this his idea of taking the dog for a walk?

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.

I’ve made a lot of noise here in Marfa about moving on and even posted here about it. Always on a quest to find that mythical place ‘home’ and never finding it. And then one day on one of my evening walks with Zack, my dog, I got it. I am home, wherever I am, I am home. This may sound quite simplistic and yet for me it had the power of a major breakthrough.

And so, here I am, still in Marfa, but differently. For the time being I’m at peace here. Berlin still beckons me, and I will probably travel there soon, but relocate? Meanwhile…

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.

More about this later as I plan to write about the process of getting from here to there in the hope it might help someone on a similar journey. But this post is called Learning German. I’ve been relying heavily on Google Translate built into Chrome. I’ve tried different plugins for Firefox and Safari, my browsers of choice, but haven’t found anything as good. Anyone out there have a tip, let me know. I can learn words, the tricky part is how to put them together. Apparently the Germans have a much different idea about this than us English-speakers. Besides translating government and real estate websites, I recently tried to find some background information on the musician Uwe Zahn under the pseudonym Arovane and discovered a very curious site, arovane.de, which cannot possibly be his website. The second page is all about the joys of having a ‘Balcony facility.’ Here is an excerpt from the translation results:

“The private balcony has become more and more becoming an important location of the apartment. On it you can enjoy the first rays of spring sunshine to take under the protection of umbrellas in the sun, in Hollywood swings wonderfully relax or spend warm summer nights with friends. Many people have discovered the balcony of their apartment for herself and enjoy the free time of the year like being there – mostly on sun loungers, swing chairs remain on balconies while lack of space, mostly a dream. With a beautiful view of the countryside it is not always necessary to spend their holidays away from home. Because even on the balcony you can drink cocktails and dreaming in the sun. Without much effort, it is possible to make the balcony with a beautiful patio furniture such as teak, plastic or aluminum, for personal oasis. Those who want it, the small accessories or give special emphasis to his balcony.”

Yes…even on the balcony you can drink cocktails and who wouldn’t want to give special emphasis to his (or her) balcony?

“…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.