Distorted guitars and independence
I should be going to bed, but for some reason I'm still awake, so I should note something I found a few days ago.
But first, a little background. Why do amplified distorted guitars sound so interesting, and so different from most existing instruments? The strings themselves vibrate in ways that are very similar to the strings of an acoustic guitar (minor differences, but nothing important for this discussion). In both cases, the vibrations occur at several frequencies simultaneously. The beauty of distortion is what it does to the set of frequencies.
When an electric guitar's vibrations are picked up by their, um, pickups, they may be subjected to nonlinearities (so that if a string vibrates with twice as much amplitude as normal, it is not necessarily represented with an electric signal with twice as much amplitude). The really fun thing about a signal with multiple frequencies subjected to nonlinearities is that the set of frequencies you get out of the process is vastly enriched.
For every pair of frequencies present in the original signal, the output will contain not just the original pair, but also the sum of the original frequencies and the difference of the original frequencies. Those pairs of frequencies could be represented as distinct entries in a square table with as many rows and columns as there were frequencies in the original input, so by distorting your input you're basically squaring the number of frequencies present.
Now, distorting a plucked electric guitar string doesn't really end up squaring the number of frequencies present, because a lot of the entries in your square table of frequencies end up being duplicates. That's because the original frequencies present in the input are in a sense not very independent from one another: they are all pretty close to multiples of a single frequency (the "fundamental frequency") of the string that's vibrating. Taking sums and differences from a such a series of numbers produces a lot of duplicates (e.g. the frequencies 100, 200, 300 have sums and differences 0, 100, 200, 300, 400, 500, and 600) But you still end up generating plenty of new frequencies when you strum a chord, because when more than one string is vibrating, the entire set of input frequencies is no longer a boring set of multiples of a single fundamental.
One last note before moving on. Real distortion doesn't just introduce sum and difference frequencies; it also generates multiples of those sum and difference frequencies. A ton of frequencies can get generated with very simple input.
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Sorry that took a while, but that was all one chunk in my brain and so it's been used as a building block for thinking about other things. Which brings us to what I found recently.
Imagine trying to build a sequence of whole numbers that are as independent from one another as possible. Certainly, no number in the sequence should be a multiple of any other number in the sequence. Here's the twist. No number in the sequence should be a multiple of the sum or difference of any pair of multiples of earlier numbers in the sequence. We'll call such a sequence "maximally independent" (a term I just made up; the definition is essentially a kind of integer-flavored linear independence)
Let's say we started out with 2 and 3; they're not multiples of one another. Unfortunately, that's as long as that sequence can get. We can't add any more numbers to it, because 3-2=1 and any other number we could think of adding to the sequence will be a multiple of 1.
Because this post is already plenty long, I will conclude with two conjectures about these sequences, without rigorous proof (but I am nearly certain of both).
1. There is no maximally independent sequence that increases on to infinity. (Argument: consider the first/smallest element E of a candidate infinite sequence. What do the rest of the elements of the sequence look like modulo E? Suppose two larger elements of the sequence were congruent mod E, then maximal independence is violated; so every element of the sequence must be distinct mod E, but there are only E equivalence classes mod E, so the sequence is at most E in length!)
2. Maximally independent sequences of arbitrary length may be constructed. (I have a method, but I'm curious to see if anyone else comes up with one).
But first, a little background. Why do amplified distorted guitars sound so interesting, and so different from most existing instruments? The strings themselves vibrate in ways that are very similar to the strings of an acoustic guitar (minor differences, but nothing important for this discussion). In both cases, the vibrations occur at several frequencies simultaneously. The beauty of distortion is what it does to the set of frequencies.
When an electric guitar's vibrations are picked up by their, um, pickups, they may be subjected to nonlinearities (so that if a string vibrates with twice as much amplitude as normal, it is not necessarily represented with an electric signal with twice as much amplitude). The really fun thing about a signal with multiple frequencies subjected to nonlinearities is that the set of frequencies you get out of the process is vastly enriched.
For every pair of frequencies present in the original signal, the output will contain not just the original pair, but also the sum of the original frequencies and the difference of the original frequencies. Those pairs of frequencies could be represented as distinct entries in a square table with as many rows and columns as there were frequencies in the original input, so by distorting your input you're basically squaring the number of frequencies present.
Now, distorting a plucked electric guitar string doesn't really end up squaring the number of frequencies present, because a lot of the entries in your square table of frequencies end up being duplicates. That's because the original frequencies present in the input are in a sense not very independent from one another: they are all pretty close to multiples of a single frequency (the "fundamental frequency") of the string that's vibrating. Taking sums and differences from a such a series of numbers produces a lot of duplicates (e.g. the frequencies 100, 200, 300 have sums and differences 0, 100, 200, 300, 400, 500, and 600) But you still end up generating plenty of new frequencies when you strum a chord, because when more than one string is vibrating, the entire set of input frequencies is no longer a boring set of multiples of a single fundamental.
One last note before moving on. Real distortion doesn't just introduce sum and difference frequencies; it also generates multiples of those sum and difference frequencies. A ton of frequencies can get generated with very simple input.
--------------------
Sorry that took a while, but that was all one chunk in my brain and so it's been used as a building block for thinking about other things. Which brings us to what I found recently.
Imagine trying to build a sequence of whole numbers that are as independent from one another as possible. Certainly, no number in the sequence should be a multiple of any other number in the sequence. Here's the twist. No number in the sequence should be a multiple of the sum or difference of any pair of multiples of earlier numbers in the sequence. We'll call such a sequence "maximally independent" (a term I just made up; the definition is essentially a kind of integer-flavored linear independence)
Let's say we started out with 2 and 3; they're not multiples of one another. Unfortunately, that's as long as that sequence can get. We can't add any more numbers to it, because 3-2=1 and any other number we could think of adding to the sequence will be a multiple of 1.
Because this post is already plenty long, I will conclude with two conjectures about these sequences, without rigorous proof (but I am nearly certain of both).
1. There is no maximally independent sequence that increases on to infinity. (Argument: consider the first/smallest element E of a candidate infinite sequence. What do the rest of the elements of the sequence look like modulo E? Suppose two larger elements of the sequence were congruent mod E, then maximal independence is violated; so every element of the sequence must be distinct mod E, but there are only E equivalence classes mod E, so the sequence is at most E in length!)
2. Maximally independent sequences of arbitrary length may be constructed. (I have a method, but I'm curious to see if anyone else comes up with one).