Thursday, June 7, 2018

Seventh Chords

What are seventh chords? Chords that contain the seventh note in a scale starting from the root of the chord! That's not a very practical answer, though, when experimenting with microtonality, where scales are not clearly defined. Here is an alternate definition: a seventh chord is a four note chord constructed by stacking thirds. The root note is the foundation. The next note is a third above the root. Another third above that is some sort of fifth note. Another third above that is some sort of seventh note. Typically the interval of a third is composed of two steps of a scale. So a seventh chord looks like 1-3-5-7 in terms of steps of a scale. But intervals are more fundamental than scales. The notion of a sequence of three intervals, each a third, makes sense even without any scale.

There are two different intervals of a third: a major third and a minor third. This gives eight different types of seventh chords, built from sequences of these different thirds, e.g. major-major-major, major-major-minor, major-minor-major, etc.

To find out how these seventh chords sound in the microtonal system where an octave is divided into 53 equal steps (rather than the conventional 12 steps), I wrote a little program that uses simulated annealing to construct a fractal pattern near a phase transition. The program builds a 12x12x12 toroidal mesh and places a seventh chord at each vertex. The total number of possible chords is 8x53=424. Each vertex can have any one of these 424 chords. Chords at neighboring vertices are probabilistically selected to be similar to each other. Chords are more similar the more notes they have in common.

The result, so far at least, is not very musical! But it's a lot of seventh chords! 1728 sevenths


  1. I recently had a similar go at thirteenth chords and concluded there were really only 28 - for various reasons, including the requirement that the highest note in the thirdy-stack actually be a thirteenth (including being a minor thirteenth).

    For 7ths, although there are - as you say - eight ways to stack thirds, two of 'em might thus be considered illegitimate (4+4+4 athletically skips right over the actual 7th and 3+3+3+3 replaces it with a 6th).

    Bearing this in mind, I'd find enumerating essentially distinct 13ths (including inversions thereof) built from stacked major and minor thirds within 53TET rather terrifying! :)

    1. In 53TET, a minor third is 14 steps and a major third is 17 steps. I don't know what an "actual 7th" ought to be! I don't know music theory at all - this exploration I'm doing is a kind of attempt to understand music theory in a way that works for me. I am slowly discovering that the work needed is much more reshaping my way of thinking rather than reshaping music theory! But I seem to be a tough nut!

      A sixth is probably a minor third below an octave, so that's 53-14=39 in 53TET. 14+14+14 is 42 so that's a decent bit higher than the sixth. 17+17+17=51, a bit short of an octave. So maybe 53TET is rich enough that all eight seventh chords can be heard unambiguously. Of course hearing them is going to depend on context. My toroidal mesh is a mathematical way to create some context. I offer it as an invitation to some brave musical competent to find meaningful musical context for these various microtonal sevenths!

    2. I guess a major seventh in 53TET should be a major third on top of a perfect fifth, i.e. 17+31 = 48. The eight seventh chords I am proposing will have their sevenths at 42, 45, 48, and 51 steps above the root.

    3. In this experiment, I just used whatever necessary inversions to fit all the chords into a single octave span.

    4. You could generate a 12-scale out of 53TET the Pythagorean way by clocking out 12 stacked fifths (keep adding 31 modulo 53) from a tonic at 0, i.e. 0, 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341 to give you 0, 31, 9, 40, 18, 49, 27, 5, 36, 14, 45 and 23. Re-ordered this is 0, 5, 9, 14, 18, 23, 27, 31, 36, 40, 45, 49.
      That would give you minor/major third at 14 and 18 and minor/major 7th at 45 and 49.
      But you appear to be using 5/4 as a frequency ratio (as opposed to Pythagorean 81/64) to get your major third at, as you say, 17. If that's from using direct frequency harmonics, an actual 'seventh harmonic' approach would give you 42 (for a minor 7th) and you'd need to go up to the 15th harmonic for a major 7th at step 48 out of 53.
      But I feel that - however you generate your scale (you need at least 10-tonicity to introduce a minor third and at least 11-tonicity to begin distinguishing major and minor 7ths) - three major thirds will always overshoot both 7ths (ending up too close to the octave to call) and three minor thirds will always undershoot and plonk you into the sixth's 'sphere of influence'.
      On the other hand, he said boldly, that 'sixth' sound has often been used - musically - as a subdominant preparation for a cadential dominant (7th!) approach for resolution. So, functionally, yer 333 could arguably be a kind of a 7th. Everybody wins!

    5. yeah exactly I am a 5-limit fan rather than 3- or 7-limit! These days I am less interesting in the "emic" approach of overtone ratios etc., though of course that is always at the foundation. I am more interested in an "etic" angle, looking at the network generated by some primitive intervals in a pitch set, and how that network can be a basis for structure and pattern.

    6. ooops I think I got emic and etic backwards! That's what I get for trying to look smart!

  2. I created another version, where the toroidal mesh is two dimensional, 40x40, instead. The longer distance allow for a greater range of fluctuation.

  3. and here is an 8x24x24 mesh: