Alaudin Mathieu's Lattice of Tones
Inventing "Matchstick Harmony"

Having experimented with connecting diatonic harmonies through keeping a common tone between voices, and then by expanding this technique through the use of secondary dominants, we can now experiment with a technique that will take you outside the realms of conventional harmonic thinking. This process is akin to free association in creative writing, where you allow an image to somewhat randomly take you to another image, and then to another, without too much concern for logic or control. Something like what might occur in dream...

...I go for a walk to the frozen lake near my home. The geese are flying above the lake and I think of airplanes, and also ice cream cones. Now I see ice cream cones flying in formation, towards the south. Now there are people in Hawaiian shirts sitting on top of each of the ice cream cones. I imagine them reaching Florida, where the fall off and then the ice cream cones melt and turn into lawn chairs. The lawn chairs, remembering their experience of flying, grow restless but can only turn into crodiles, which then eat the tourists who fell asleep on them. The crodiles race off towards the lagoon, but before they reach the lagoon they have become boxcars with hoboes on them, and then...Well, you take it from there.

This is what composing "matchstick harmony" along the lattice of tones is like. One starts with a C major triad and begins to move by connecting tones. Before too long, one has strayed far from a sense of "C major" in a diatonic sense, and yet each move has been a smooth and "associated" journey from the previous harmony.

What is matchstick harmony? It is an idea that comes from my favorite book on harmony, Alaudin Mathieu's Harmonic Experience. This work introduced me to the concept of Harmonic Polarity, and its way of organizing simple triadic structures is fascinating, and quite useful to the modern composer.

We have seen some of this information before, but it is worth reviewing and then taking to the next level. The following comes directly from Mathieu's work, although it is a simplification of what is so beautifully presented there.

In Book 1, during the first semester, we identified in chapter 4D a central spine, form by perfect 3:2 fifths above and below a central tone, which we choose to be C:

Above and below this central spine, we created perfectly tuned 5:4 major thirds, creating overtonal and recprocal thirds. This gives us the 12 tones of the chromatic scale perfectly tuned within a 5-limit, justly tuned system.

Overtonal thirds:

Reciprocal thirds:

This creates a lattice of tones, which he draws out connected by "matchstick" lines which form interlocking trianges. Within these triangles are major and minor triads that form the basis of "matchstick harmony."

I will not reproduce his lattice here, so as to protect the integrity of his work. You have been given a replica of his lattice in order to complete your assignment.

The process here is to, as the cliche goes, think outside the box of diatonic harmony while respecting the beauty of justly tuned triads. All one needs to do is begin on C major and move along by the connecting matchsticks, discovering the close connection between all harmonies with the key of C major. Mathieu's point here is that if one imagines that these are tuned in just intonation, even if they are in fact tuned to equal temperament, one will achieve a completely clear and unambiguous harmony.

I will give you a paper handout to complete the rest of this information, taken from Mathieu's opus.

NEXT: CHAPTER 4 and the Chorales of Bach.


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