What humans seem to like is a balance between simplicity and complexity. This is similar to a fractal: it repeats but in an ever changing way. It’s probably not good if the pattern is only identifiable after looking at hundreds or thousands of members in the sequence. So, not every sequence is going to be useable for music that would be appreciable by humans.
Here is an interesting example of a pattern that repeats in an ever changing way:
10
01
10/01
01/10
10/01/01/10
01/10/10/01
If you keep repeating these patterns in this manner, ironically, you end up with a pattern that never repeats.
Also, we can take note lengths such as short and long and combine them into a structured but never repeating pattern. This is only one of several ways to do this. And both this and the above pattern have already been used by many composers without recourse to a computer. Just an interest in mathematics.
SS,L
SSS,SL,LS
SSSS,SSL,SLS,LSS,LL
SSSSS,SSSL,SSLS,SLSS,SLL,LSSS,LSL,LSS
So, in the first links in this thread there are mathematical sequences found in "nature". One group which seemed promising for a pitch sequence was the Sprague Grundy Values for Dawson’s Chess if we assign a number to mean pitches and perhaps the zeros mean rests. I thought it would be good for that because the first line of the string would primarily create a melody that keeps in the range of steps and thirds.
It’s like this;
01120311033224052233011302110 etc...
It has a cousin called the Sprague-Grundy Values for games of Kayles. I thought it would be good for creating both a pitch series and rhythms where if each rhythm from a chart is assigned a number. The most important rhythms being a 1 etc... You can click the chart to see it larger.
Sprague-Grundy for games of Kayles goes; 01231432142641271432146741265 etc...
The Class Number of Q Series would produce rhythm patterns of ever increasing complexity.
111221212424142366434422648445264423688818474etc...
The Infinite Juggling Sequence seems like it would also be good for rhythms because there would be a balance of repetition and change.
3342333342423411etc...
In "The Number of Factorizations of n into Prime Powers Greater than 1", the repetitions of 1 while the series progresses would help create a sense of unity;
1112111321121115121211321321etc...
This Infinite Fibonacci Word (there are several) would be good for alternating two pitches or two rhythms in a complex manner;
1011010110110101101011011010110110101101011011010110 etc..
For "The Number of Segments needed to represent ’n’ on a Calculator Display" could be an ever rising pitch pattern or an ever increasingly complex rhythm scheme;
6 2 5 5 4 5 6 3 7 6 8 4 7 7 6 7 8 5 9 8 11 7 10 10 9 10 11 8 etc...
As would "The Number of Letters in English names of length ’n’";
4 3 3 5 4 4 3 5 5 4 3 6 6 8 8 7 7 9 8 8 6 9 9 11 10 10 9 11
I forget the name but the code for this one in the second link is AO72203 would produce only stepwise pitch motion if each number is assigned a pitch;
1 2 1 2 1 2 3 2 1 2 3 4 3 2 1 2 3 4 5 4 3 4 3 2 1 2 3 4 5 6 7 6 5 4 3 4 3 4 3 2 1 etc...
as would "The Number of Runs in a Binary Expansion of n";
1 2 1 2 3 2 1 2 3 4 3 2 3 2 1 2 3 4 3 4 5 4 3 2 3 4 3 2 3 2 1 etc...
and "Roman Numbers for n" with occasional motion by third and even more rare by fourth;
1 2 3 2 1 2 3 4 2 1 2 3 4 3 2 3 4 5 3 2 3 4 5 4 3 4 5 6 4 3 4 5 6 7 5 2
Some Cellular automata would produce various patterns, some that involve simultanaeities where some voices would linger, some drop out, and some appear. It could be interesting for experimental voice motion/chord construction.
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