In my music theory PhD work on scales, I've come across certain classes of musical scales, which I believe might have parallels within mathematics and I hope you can help me learn more about these classes from a mathematical point of view (I am quite ignorant in Mathematics).
In my work, I represent scales as sets on a necklace of cardinality $12$. The nodes of the necklace are therefore marked $0$ through $11$.
Below is the set [0 2 5 7 10]
The structure can manifest in any of $12$ manifestations:
I have found (musically) that certain slices of a structure can reveal the different amounts of information about its rotation.
As you can see above, the combination [0 2] can appear in 3 of the 12 possible rotations of the scale, while the combination [0 4] can only appear in one rotation. Therefore, with respect to the set [0 2 5 7 10], [0 4] is a:
Diagnostic Combination
The scale in question [0 2 5 7 10] has $7$ such diagnostic combinations:
[0 4]
[0 2 4]
[0 4 7]
[0 4 9]
[0 2 4 7]
[0 2 4 9]
[0 4 7 9]
Since all of these combinations contain [0 4], I say that [0 4] is the:
Identity Fragment for that set.
However, there are some sets that while they have Diagnostic Combinations, they don't have an identity fragment.
For instance, the set [0 1 2 3 5] has many diagnostic combinations (for instance [0 1 5] and [0 2 3]), but no fragment is common to them all. So this has no Identity fragment.
And yet, there's another class of sets, sets that have an Identity Fragment (a fragment that appears in all diagnostic combinations), but where the Identity Fragment in itself isn't diagnostic. Such an example is the set [0 1 3 5 6 8 10] where all the diagnostic combinations contain [0 6], but [0 6] by itself is not diagnostic (has $2$ possible manifestations).
I am wondering if there's well-established work on similar concepts within math:
- a so-called Diagnostic Combination
- a so-called Identity Fragment
- And the cases where the Identity Fragment is and isn't also a diagnostic combination.
My gut is telling me that either Combinatorics, and / or Information Theory can help me deepen my exploration of those musical structures. Am I right?
Thanks for the interesting sharing! In my understanding, this is apparently related to the cyclic group (https://en.wikipedia.org/wiki/Cyclic_group) which focuses on a single point, while what you are interested in involves the combination of points.
I would like to formulate your problem as follows: We call the set $\Omega = \{0, 1,2,3,...,11\}$ the full set, and each subset $N \subset \Omega$ represents a pattern. We always sort $N$ so that $N = \{0 \leq N_1 < N_2 < \ldots < N_{|N|} \leq 11\}$.
For the Diagnostic Combination (DC), let $DC(N)$ denote the set of Diagnostic Combination(s) of $N$. Clearly, if $D$ is a DC of $N$, then $(D + x)\mod 12$ is also a DC of $N$, for any $x$.
Therefore, WLOG, we may assume that each element in $DC(N)$ contains $0$. For the simplest case, two-element DCs, we first compute all the pairwise distances between the points in $N$ which is $(|N_j - N_i| \mod 12)_{i \neq j}$. After that, immediately we may have DCs for each unique distance. (We need to be careful here, e.g., for the equivalent distances $x$ and $12 - x$. We just grab the general idea first.) Now we always take closer distances. See the examples you have mentioned first.
(1) $N = \{0, 2,5,7,10\}$. The pairwise distances are $$(2,5,5,2,3,5,4,2,5,3),$$ where $4$ is unique. Therefore, $\{0, 4\}$ is a DC of $N$.
(2) We generalize our preliminary idea and check the example $N = \{0,1,2,3,5\}$. Since you mentioned two three-element DCs, we shall check all tuplet-wise distances.
We see that both $(0,1,5)$ and $(0,2,3)$ (and more) are unique. That's why both of them are DCs.
(3) For your last example $N = \{0, 1, 3, 5, 6, 8, 10\}$, $(0, 6)$ itself is special due to the equivalent distances we have mentioned above in that $12 -6 = 6$.
(4) Regarding the Identity Fragment (IF), we would like to answer first the question: what kinds of $N$'s have an IF? Mathematically, $N$ has an IF if $DC(N)$ has a minimum set that is a subset of each element in $DC(N)$, i.e., the intersection of all DCs is also a DC.
(*) We also have a similar idea Permutation Cycle (https://mathworld.wolfram.com/PermutationCycle.html). In a similar manner, I would like to call the term we are interested in Cyclic Cycle.
(**) In short, a DC is a set of points that can be perfectly matched only once when you rotate the necklace for a whole circle ($12$ steps). Therefore, I do believe that DCs reveal cyclic symmetry.
(***) We may also see the problem backward. We fix a set of points $D$ and study what kind of $N$'s have $D$ as the DC or IF.
(****) Since $2^{12}$ is small, I tried to enumerate all possible cases. First I would like to point out that for $N = \{0, 2, 5, 7, 10\}$ the OP's statement needs to be clarified. Specifically, $\{0, 4\}$ is not an IF strictly since e.g., $\{0,2,5,10\}$ is also a DC of this $N$, but $\{0,2,5,10\}$ is cyclically equivalent to $\{0,2,4,7\}$ by moving $10$ to $0$. In my understanding, the OP's definition of IF is the maximal $\tilde{D}$ such that for each DC, there exists a cyclically equivalent one containing $\tilde{D}$. Following this idea, I wrote the following code.
Below are the full results for all $N$'s with an IF (due to the length limit, please check https://docs.google.com/document/d/1XjAfKi6_hxUcu5QKdZIA2wi06u0s9z7JtkWk5mWlVPI/edit?usp=sharing):