Mathematically Modelling Progression and Resolution

Partch makes an offhand statement that a move from one of his 43-tone notes to a certain other is a move from dissonance towards resolution. This was the first moment in which I saw the link between consonance and resolution.

Resolution is when a melody or harmony comes cleanly to an end. Typically it means returning to the root note or chord at the end of a phrase. The second most resolved place to finish a melody is on the fifth. Progression is when you move away from the root note, developing a melody or harmony into something interesting. Guitarists often say that progression is when you move forward around the circle of fifths, and resolution is when you move back. In both cases, what you are doing is controlling the amount of dissonance relative to the tonic. More dissonance is progression, less dissonance is resolution. In the case of the circle of fifths, you are both playing with powers of 3/2 and increasing the dissonant error in your tuning when you move forwards.

We can do maths with this. Let's assume a simple single note melody (if you like you can imagine a single note drone underneath).

At any time the dissonance is equal to the product of the ratio of the interval between the current note and the tonic.

Let D = dissonance, o = odentity (the numerator of the ratio), u = udentity (the denominator of the ratio)

D = ou

The progression or resolution of a melody is the rate of change of D by time.

Let m = progression/resolution, t = time

m = ΔD/Δt

As an example, let's assume a melody playing four equal-length notes on C, G, D and C. As the melody plays the second note, G, the dissonance is equal to 2x3 = 6 (D = ou). When it was playing the first note, D = ou = 1x1 = 1. So the change in D is 5, and the change in t is 1 unit. m = 5/1 = 5. Coming to the third note, D = ou = 9*8 = 72. m = (72-5)/1 = 67. Finally, at the last note, m = (1-72)/1 = -71.

time, t | dissonance, D | progression/resolution, m
0       | 0             | 0
1       | 1             | 1
2       | 6             | 5
3       | 72            | 67
4       | 1             | -71

The negative value of m at t = 4 indicates that it is resolution, not progression, that is occuring. It will be noted the m is not zero-sum, just as in musical practice.

I believe that it's not possible to do proper calculus with this because the series involved are discrete and not continuous.

Obviously this is an abstraction, in real music you do not get isolated one note sine-wave melodies. We need to take into account:
Polyphony
Moving root notes ("key changes")
Timbre
But none of these problems are insurmountable.

It should be obvious that by reversing this method, it would be possible to design a melody or harmonic sequence in which exactly the desired degree of progression and resolution are manufactured. Perhaps you are trying to illustrate a story or lyrics and you want your song to have a certain shape. No longer do you have to have a memory for trial-and-error and tradition generated chord sequences. You could trivially design a program to create them for you, perfectly. I imagine such a program would require a defined set of notes (a key) and some rhythmic ideas to get going, but you could really come up with some cunning harmonies far more complex than a human can efficiently design.