- Peter Ivan Edwards
Uniform Waves: Part 3, in which notes are compared to fruits and the Law of Large Numbers is explain
Updated: Sep 20, 2021
In the previous post, I created some movie music with a collection of 6 randomly generated notes. How is it possible to use randomly generated notes to do that? Effectively, a small collection that results from random sampling of a larger collection can expose neither all potential outcomes nor the likelihood of the potential outcomes. Another way to put it, the random nature of the selection process cannot be experienced. To understand this, imagine that there are only 12 fruits in the world and, while each is a favorite of someone, they are all equally preferred. That is, fruit 1 and fruit 2 are favored by different people but in equal degrees of roughly 8.33% of the population. Of course, you usually run a survey to deduce these weightages, but we need to put the cart before the horse for this mental exercise because we need to be able to see how a small dataset can lead to a big inaccuracy. Note that this scenario is what we have when we use a uniform distribution in randomly choosing from 12 notes. We have 12 of them and they are all equally likely to be selected.
You poll 6 people and your results look like the following (which are the same as the Random Potter melody):
Only 5 of the fruits - Fruits 1, 4, 5, 8, & 9 - are eaten. All other fruits aren't liked by people so there is no need to keep harvesting them. Turn those fields into forests for our carbon capture initiative! "But wait," shout the lobbyists for farmers of Fruits 0, 2, 3, 6, 7, 10, and 11, "We sell tons of our product each year. There must be people who like them. You need to collect more data." And so you do. You collect 48 total responses. Those look like the following (which are the same as the 48 notes randomly selected using a uniform distribution):
Clearly, there are winners and losers but you have discovered that all fruits are consumed. Strikingly, you discovered that Fruit 6, which you thought no one consumed, is actually really loved! But now producers of Fruits 4 and 9 are a bit perturbed. The first results showed that they were clearly favorites. These more recent results show that they aren't liked as much as several others. Needly to say, more lobbyists descend upon our weary lawmakers, who secretly love all of the attention and campaign contributions. Further funding is allocated for your survey and you go out and collect 1,000,000 surveys and your results look like the following: (These are actual results of a program I ran to simulate the scenario)
Note that 1,000,000 divided by 12 roughly equals 83,333. Hence, upon further inspection it turns out that all 12 fruits are essentially equally liked, but we only know this because the sampling size was large enough to reveal it. This is what statisticians call the Law of Large Numbers. Just as a sample size of 6 people distorted the reality of our fruit preferences, a sampling of 6 notes from 12 where all are equally likely to be selected distorts the quality of randomness applied in the generation process. We read into 6 randomly chosen notes something other than the statistical experience of 12 equally likely notes being randomly chosen. We look at the series' contour and pattern, and place it in a familiar framework, that is, we look at it as a melody and deduce a way in which it is projecting a harmonic progression.
But the Random Potter passage further destroys a sense of randomness through the most common of musical devices: repetition. (In the passage, transposition is also incorporated.) You may recall that this is also employed in the 3rd section of Shuffled, in which a 2-voice contrapuntal passage is continually interrupted by a fixed 12-note series. In that work, the 12-note series takes on a melodic quality because it is so familiar by the end of the passage. We'll explore this further in the next post when beginning to create the work Uniform Waves.