Some guiding thoughts when investigating sequences

You've created the sound for a sequence. Now what?

For each sound you post, please do all of the following and answer all of the questions below as far as you are able.

• Describe the sound: does it have a constant or varying timbre? Is it rhythmic?

• Any horizontal spectral lines? At what frequency? Is 44100/frequency near an integer? For example, if a line appears to be at 7313 hz, we calculated 44100/7313 and find this equals 6.03035..., quite close to an integer (the integer 6, in particular). This suggest that there is a feature of the sound related to 6, such as: all terms are multiples of six, or no terms are multiples of six, or all terms are 1 more than a multiple of 6, etc. We can investigate the sequence with "modulo 6 glasses" on the see if a statement like one of these is true.
  
  If 44100/frequency is close to a half integer (like 5.5, or 7.5) or any simple fraction (like 3.333333 which is approximately the simple fraction 10/3), similar investigations can be done.
• To get more precise frequency information in Audacity, select the sound (all of it or a part of it) and use Analyze-> Plot Spectrum. This tool makes it easier to estimate the frequency more precisely than the spectrogram allows.

• Is the sequence infinite? How do you know? Sometime this is easy, and sometimes this is very difficult. If your sequence is given explicitly by a_n = f(n), where f is an increasing function, then the sequence is infinite (that's an easy one). For other sequences, it may not be so easy, and in fact there are lots of open problems that are equivalent to the question of whether or not a certain sequence is infinite.
• What can we say about the density of the sequence? Let A(x) be the number of terms in the sequence less than x. Give the values of A(10)/10, A(100)/100, A(1000)/1000, ..., A(10^6)/10^6. Can you tell how A(x)/x behaves? Does it tend to a limit? How do you know?
• Residue classes: is the sequence uniformly distributed in all residue classes? That is, does there exist a modulus for which the sequence is not uniformly distributed? Again, these totally experimental calculations are the place to start. For a modulus m calculate the number of terms of the sequence that are congruent to 0 mod m, 1 mod m, ..., m-1 mod m. If the sequence is equally-distributed modulo m, then there should be about 10^6/m terms in each category. If this is not so, then that tells us something useful about the sequence (for example, modulo 6, all primes are congruent to 1 or 5 (except for a couple of exceptions). So the primes are not equally distributed modulo 6). You should look at a few m and assess whether your sequence is equally-distributed modulo each one. Doing m=2,3,4,...,12 would be enough.
• Difference sequences: Given a sequence {a_n}, it can be useful to consider the difference sequence {a_n+1-a_n}. For example, the difference sequence for the primes begins 1,2,2,4,2,4,2,4,6,2,... which, except for the first 1, is all even. We can also notice that there are, at times, very large differences. On the other hand, the beatty sequence for sqrt(5) begins 2,4,6,8,11,... and its difference sequence begins 2,2,2,3,2,2,2,3,2,...: the difference sequence only contains 2s and 3s! Look at the difference sequence for any sequence you create and see if you can observe anything interesting about it (such as a pattern, or a limited set of values (i.e., the values never get larger than some bound).
• In fact, our sound-making code generates the sound but also outputs information on density, residue classes and difference sequences! So definitely check out that output to help you start investigating your sequence.

• Are there natural, disjoint subsequences that we can partition the sequence into? If we can, and we create the sound for each subsequence, the sound of the main sequence is the functional sum of these sounds.