Sculpting Time

While designing an orchestral section, I was after a very specific effect: a "structural acceleration" in which the perception of time compresses progressively.

The initial setup was simple. One minute of music (60 seconds) divided into several subsections: the first had to last exactly 12 seconds and the last, 3. The question: how do you calculate the intermediate durations so the evolution is gradual, organic and, above all, adds up to the exact minute?

A decay of that kind —close to a geometric or exponential curve— moved the problem out of composition and into algebra.

Stating the problem

In music we can define arbitrary durations and adjust them by ear. But when the structure has to convey the inevitability of a physical process (a fall, a friction), it pays to model it precisely.

The decay suggested a geometric progression, with three known variables:

• Total duration ($S_n$): 60 seconds.
• First subsection ($a_1$): 12 seconds.
• Last subsection ($a_n$): 3 seconds.

If the total number of sections ($n$) is fixed in advance, the common ratio ($r$) follows from the relation between the extremes:

$$r = \sqrt[n-1]{\frac{a_n}{a_1}}$$

The drawback of this approach is that, when the durations are generated by multiplying successively by $r$, the total sum rarely lands on exactly 60 seconds: it demands a normalisation after the fact to fit the minute. I was looking for a more direct formulation.

Inverting the problem: letting the mathematics decide

One question changed the approach: what if I don't fix the number of sections ($n$) and treat it as one more unknown?

The problem then becomes a system in which both the ratio ($r$) and the number of steps ($n$) have to accommodate the three fixed limits: 12 s, 3 s and 60 s.

The sum of a geometric series can be written in terms of its first and last terms, $S_n = \frac{a_1 - r\,a_n}{1 - r}$. From there the ratio can be isolated without knowing $n$:

$$r = \frac{S_n - a_1}{S_n - a_n}$$

With my values (60, 12 and 3), the ratio comes out to $r = \frac{48}{57} \approx 0.8421$: each subsection lasts 84.21 % of the previous one.

Once the ratio is known, $n$ follows by isolating it with logarithms:

$$n = 1 + \frac{\ln(a_n / a_1)}{\ln(r)}$$

The result: 9.067. That is, the geometric curve joining 12 to 3 seconds and summing to an exact minute requires, in practice, 9 sections. I did not decide the number of events: the temporal limits of the piece determined it.

The resulting table of durations

Since $n$ has to be an integer, I rounded to 9 sections and recalculated the ratio for the fixed extremes: $r = (3/12)^{1/8} \approx 0.8409$. The resulting durations:

1. 12.00 s (start)
2. 10.09 s
3. 8.48 s
4. 7.14 s
5. 6.00 s
6. 5.04 s
7. 4.24 s
8. 3.57 s
9. 3.00 s (end)

The sum comes to 59.57 seconds; the remaining 0.43 from the rounding is absorbed in the fine adjustment of the score. Note, in passing, that the central section lasts exactly 6 seconds —the geometric mean of 12 and 3— a symmetry characteristic of this kind of progression.

Conclusion of this first stage

What began as a doubt about durations ended as a compositional procedure: by letting the temporal limits determine the number of events, the form acquired a consistent internal proportion, with no arbitrary decisions in between.

This is only the temporal skeleton. The next step was to carry the same logic of geometric expansion into the orchestration: how to move from a single instrument to a mass of 20 players following the same curve, and how to expand register and noise along with it. That is for the next entry.