Yahoo Tuning Groups Ultimate Backup tuning-math Surprising conjecture about rank-3 distributionally even scales

I wrote a significantly more efficient algorithm to generate rank-3 distributionally even scale patterns (using "dynamic programming").

Following Paul, I'm calling them "distributionally even" now, rather than "generalized Myhill", because the definition of the original Myhill property is "EXACTLY two specific intervals per generic interval", but I want "AT MOST n specific intervals per generic interval". "Rank-N distributionally even" seems the best description.

Anyway, the new algorithm goes significantly faster, so I already have the data for scales up to and including 24 steps. The data are given at the end of this post. I think the data up to 31 steps will be available in a few hours or so.

But what I really want to talk about is an obvious pattern showing up in the data, which leads me to a conjecture:

CONJECTURE 1. For any triple of positive integers (a,b,c), a distributionally even scale with a small steps, b medium steps, and c large steps exists if and only if EITHER:
* Two of the numbers a, b, and c are equal, OR
* (a,b,c) is a multiple of some permutation of (1,2,4)

I have no idea how to prove this yet, but if you look at the data it's just staring you in the face. Does anyone have a clue why it should be true?

If true, this would mean that the scale pattern "aabacab" (along with its repetitions, "aabacabaabacab" etc.), as exemplified by the marvel heptatonic scale, is *unique* in that it's only possible rank-3 distributionally even scale with different numbers of all three steps.

Keenan

Data for scales up to 24 steps follows. First come the numbers of a's, b's and c's, then the possible patterns. The sorting order is first by total number of steps, then lexicographically.

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> Anyway, the new algorithm goes significantly faster, so I already have the data for scales up to and including 24 steps. The data are given at the end of this post. I think the data up to 31 steps will be available in a few hours or so.

Here's the data up to 31 steps. No counterexamples to my conjecture so far.

(1, 1, 1) ['abc']
(2, 1, 1) ['abac']
(2, 2, 1) ['aabcb', 'ababc']
(3, 1, 1) ['aabac']
(2, 2, 2) ['abacbc', 'abcabc']
(3, 2, 1) []
(4, 1, 1) ['aabaac']
(3, 2, 2) ['abacabc']
(3, 3, 1) ['abababc']
(4, 2, 1) ['aabacab']
(5, 1, 1) ['aaabaac']
(3, 3, 2) ['abacbabc']
(4, 2, 2) ['abacabac']
(4, 3, 1) []
(5, 2, 1) []
(6, 1, 1) ['aaabaaac']
(3, 3, 3) ['abacbacbc', 'abcabcabc']
(4, 3, 2) []
(4, 4, 1) ['ababababc']
(5, 2, 2) ['aabacabac']
(5, 3, 1) []
(6, 2, 1) []
(7, 1, 1) ['aaaabaaac']
(4, 3, 3) ['abacbacabc']
(4, 4, 2) ['aabcbaabcb', 'ababacbabc', 'ababcababc']
(5, 3, 2) []
(5, 4, 1) []
(6, 2, 2) ['aabaacabac', 'aabacaabac']
(6, 3, 1) []
(7, 2, 1) []
(8, 1, 1) ['aaaabaaaac']
(4, 4, 3) ['abacbabcabc']
(5, 3, 3) ['abacabacabc']
(5, 4, 2) []
(5, 5, 1) ['abababababc']
(6, 3, 2) []
(6, 4, 1) []
(7, 2, 2) ['aabaacaabac']
(7, 3, 1) []
(8, 2, 1) []
(9, 1, 1) ['aaaaabaaaac']
(4, 4, 4) ['abacbacbacbc', 'abacbacbcabc', 'abacbcabacbc', 'abcabcabcabc']
(5, 4, 3) []
(5, 5, 2) ['ababacbababc']
(6, 3, 3) ['abacabacabac']
(6, 4, 2) []
(6, 5, 1) []
(7, 3, 2) []
(7, 4, 1) []
(8, 2, 2) ['aabaacaabaac']
(8, 3, 1) []
(9, 2, 1) []
(10, 1, 1) ['aaaaabaaaaac']
(5, 4, 4) ['abacbacabcabc']
(5, 5, 3) ['ababcabacbabc']
(6, 4, 3) []
(6, 5, 2) []
(6, 6, 1) ['ababababababc']
(7, 3, 3) ['aabacabacabac']
(7, 4, 2) []
(7, 5, 1) []
(8, 3, 2) []
(8, 4, 1) []
(9, 2, 2) ['aaabaacaabaac']
(9, 3, 1) []
(10, 2, 1) []
(11, 1, 1) ['aaaaaabaaaaac']
(5, 5, 4) ['abacbacbabcabc']
(6, 4, 4) ['abacabacbacabc', 'abacabcabacabc']
(6, 5, 3) []
(6, 6, 2) ['abababacbababc', 'abababcabababc']
(7, 4, 3) []
(7, 5, 2) []
(7, 6, 1) []
(8, 3, 3) ['aabacabaacabac']
(8, 4, 2) ['aabacabaabacab']
(8, 5, 1) []
(9, 3, 2) []
(9, 4, 1) []
(10, 2, 2) ['aaabaaacaabaac', 'aaabaacaaabaac']
(10, 3, 1) []
(11, 2, 1) []
(12, 1, 1) ['aaaaaabaaaaaac']
(5, 5, 5) ['abacbacbacbacbc', 'abacbacbacbcabc', 'abacbacbcabacbc', 'abcabcabcabcabc']
(6, 5, 4) []
(6, 6, 3) ['aabcbaabcbaabcb', 'ababacbabacbabc', 'ababacbabcababc', 'ababcababcababc']
(7, 4, 4) ['abacabacabacabc']
(7, 5, 3) []
(7, 6, 2) []
(7, 7, 1) ['abababababababc']
(8, 4, 3) []
(8, 5, 2) []
(8, 6, 1) []
(9, 3, 3) ['aabaacabaacabac', 'aabaacabacaabac', 'aabacaabacaabac']
(9, 4, 2) []
(9, 5, 1) []
(10, 3, 2) []
(10, 4, 1) []
(11, 2, 2) ['aaabaaacaaabaac']
(11, 3, 1) []
(12, 2, 1) []
(13, 1, 1) ['aaaaaaabaaaaaac']
(6, 5, 5) ['abacbacbacabcabc']
(6, 6, 4) ['abacbabcabacbabc']
(7, 5, 4) []
(7, 6, 3) []
(7, 7, 2) ['abababacbabababc']
(8, 4, 4) ['abacabacabacabac']
(8, 5, 3) []
(8, 6, 2) []
(8, 7, 1) []
(9, 4, 3) []
(9, 5, 2) []
(9, 6, 1) []
(10, 3, 3) ['aabaacabaacaabac']
(10, 4, 2) []
(10, 5, 1) []
(11, 3, 2) []
(11, 4, 1) []
(12, 2, 2) ['aaabaaacaaabaaac']
(12, 3, 1) []
(13, 2, 1) []
(14, 1, 1) ['aaaaaaabaaaaaaac']
(6, 6, 5) ['abacbacbabcabcabc']
(7, 5, 5) ['abacabcabacbacabc']
(7, 6, 4) []
(7, 7, 3) ['ababacbababcababc']
(8, 5, 4) []
(8, 6, 3) []
(8, 7, 2) []
(8, 8, 1) ['ababababababababc']
(9, 4, 4) ['aabacabacabacabac']
(9, 5, 3) []
(9, 6, 2) []
(9, 7, 1) []
(10, 4, 3) []
(10, 5, 2) []
(10, 6, 1) []
(11, 3, 3) ['aabaacaabaacaabac']
(11, 4, 2) []
(11, 5, 1) []
(12, 3, 2) []
(12, 4, 1) []
(13, 2, 2) ['aaaabaaacaaabaaac']
(13, 3, 1) []
(14, 2, 1) []
(15, 1, 1) ['aaaaaaaabaaaaaaac']
(6, 6, 6) ['abacbacbacbacbacbc', 'abacbacbacbacbcabc', 'abacbacbacbcabacbc', 'abacbacbacbcabcabc', 'abacbacbcabacbacbc', 'abacbacbcabacbcabc', 'abacbcabacbcabacbc', 'abcabcabcabcabcabc']
(7, 6, 5) []
(7, 7, 4) ['ababcabacbabacbabc']
(8, 5, 5) ['abacabacbacabacabc']
(8, 6, 4) []
(8, 7, 3) []
(8, 8, 2) ['ababababacbabababc', 'ababababcababababc']
(9, 5, 4) []
(9, 6, 3) []
(9, 7, 2) []
(9, 8, 1) []
(10, 4, 4) ['aabacabaacabacabac', 'aabacabacaabacabac']
(10, 5, 3) []
(10, 6, 2) []
(10, 7, 1) []
(11, 4, 3) []
(11, 5, 2) []
(11, 6, 1) []
(12, 3, 3) ['aabaacaabaacaabaac']
(12, 4, 2) []
(12, 5, 1) []
(13, 3, 2) []
(13, 4, 1) []
(14, 2, 2) ['aaaabaaaacaaabaaac', 'aaaabaaacaaaabaaac']
(14, 3, 1) []
(15, 2, 1) []
(16, 1, 1) ['aaaaaaaabaaaaaaaac']
(7, 6, 6) ['abacbacbacabcabcabc']
(7, 7, 5) ['abacbabcabacbabcabc']
(8, 6, 5) []
(8, 7, 4) []
(8, 8, 3) ['abababcababacbababc']
(9, 5, 5) ['abacabacabacabacabc']
(9, 6, 4) []
(9, 7, 3) []
(9, 8, 2) []
(9, 9, 1) ['abababababababababc']
(10, 5, 4) []
(10, 6, 3) []
(10, 7, 2) []
(10, 8, 1) []
(11, 4, 4) ['aabacaabacabaacabac']
(11, 5, 3) []
(11, 6, 2) []
(11, 7, 1) []
(12, 4, 3) []
(12, 5, 2) []
(12, 6, 1) []
(13, 3, 3) ['aaabaacaabaacaabaac']
(13, 4, 2) []
(13, 5, 1) []
(14, 3, 2) []
(14, 4, 1) []
(15, 2, 2) ['aaaabaaaacaaaabaaac']
(15, 3, 1) []
(16, 2, 1) []
(17, 1, 1) ['aaaaaaaaabaaaaaaaac']
(7, 7, 6) ['abacbacbacbabcabcabc']
(8, 6, 6) ['abacbacabcabacbacabc']
(8, 7, 5) []
(8, 8, 4) ['aabcbaabcbaabcbaabcb', 'ababacbabacbabacbabc', 'ababacbabacbabcababc', 'ababacbabcababacbabc', 'ababacbabcababcababc', 'ababcababcababcababc']
(9, 6, 5) []
(9, 7, 4) []
(9, 8, 3) []
(9, 9, 2) ['ababababacbababababc']
(10, 5, 5) ['abacabacabacabacabac']
(10, 6, 4) []
(10, 7, 3) []
(10, 8, 2) []
(10, 9, 1) []
(11, 5, 4) []
(11, 6, 3) []
(11, 7, 2) []
(11, 8, 1) []
(12, 4, 4) ['aabaacabaacabaacabac', 'aabaacabaacabacaabac', 'aabaacabacaabaacabac', 'aabaacabacaabacaabac', 'aabacaabacaabacaabac']
(12, 5, 3) []
(12, 6, 2) []
(12, 7, 1) []
(13, 4, 3) []
(13, 5, 2) []
(13, 6, 1) []
(14, 3, 3) ['aaabaacaabaaacaabaac']
(14, 4, 2) []
(14, 5, 1) []
(15, 3, 2) []
(15, 4, 1) []
(16, 2, 2) ['aaaabaaaacaaaabaaaac']
(16, 3, 1) []
(17, 2, 1) []
(18, 1, 1) ['aaaaaaaaabaaaaaaaaac']
(7, 7, 7) ['abacbacbacbacbacbacbc', 'abacbacbacbacbacbcabc', 'abacbacbacbacbcabacbc', 'abacbacbacbacbcabcabc', 'abacbacbacbcabacbacbc', 'abacbacbacbcabacbcabc', 'abacbacbacbcabcabacbc', 'abacbacbcabacbacbcabc', 'abacbacbcabacbcabacbc', 'abcabcabcabcabcabcabc']
(8, 7, 6) []
(8, 8, 5) ['ababcabacbabcabacbabc']
(9, 6, 6) ['abacabacbacabacbacabc', 'abacabacbacabcabacabc', 'abacabcabacabcabacabc']
(9, 7, 5) []
(9, 8, 4) []
(9, 9, 3) ['abababacbababacbababc', 'abababacbababcabababc', 'abababcabababcabababc']
(10, 6, 5) []
(10, 7, 4) []
(10, 8, 3) []
(10, 9, 2) []
(10, 10, 1) ['ababababababababababc']
(11, 5, 5) ['aabacabacabacabacabac']
(11, 6, 4) []
(11, 7, 3) []
(11, 8, 2) []
(11, 9, 1) []
(12, 5, 4) []
(12, 6, 3) ['aabacabaabacabaabacab']
(12, 7, 2) []
(12, 8, 1) []
(13, 4, 4) ['aabaacabaacaabacaabac']
(13, 5, 3) []
(13, 6, 2) []
(13, 7, 1) []
(14, 4, 3) []
(14, 5, 2) []
(14, 6, 1) []
(15, 3, 3) ['aaabaaacaabaaacaabaac', 'aaabaaacaabaacaaabaac', 'aaabaacaaabaacaaabaac']
(15, 4, 2) []
(15, 5, 1) []
(16, 3, 2) []
(16, 4, 1) []
(17, 2, 2) ['aaaaabaaaacaaaabaaaac']
(17, 3, 1) []
(18, 2, 1) []
(19, 1, 1) ['aaaaaaaaaabaaaaaaaaac']
(8, 7, 7) ['abacbacbacbacabcabcabc']
(8, 8, 6) ['abacbabcabacbacbabcabc', 'abacbabcabcabacbabcabc']
(9, 7, 6) []
(9, 8, 5) []
(9, 9, 4) ['ababacbabacbababcababc']
(10, 6, 6) ['abacabacabacbacabacabc', 'abacabacabcabacabacabc']
(10, 7, 5) []
(10, 8, 4) []
(10, 9, 3) []
(10, 10, 2) ['abababababacbababababc', 'abababababcabababababc']
(11, 6, 5) []
(11, 7, 4) []
(11, 8, 3) []
(11, 9, 2) []
(11, 10, 1) []
(12, 5, 5) ['aabacabacabaacabacabac']
(12, 6, 4) []
(12, 7, 3) []
(12, 8, 2) []
(12, 9, 1) []
(13, 5, 4) []
(13, 6, 3) []
(13, 7, 2) []
(13, 8, 1) []
(14, 4, 4) ['aabaacaabaacabaacaabac', 'aabaacaabacaabaacaabac']
(14, 5, 3) []
(14, 6, 2) []
(14, 7, 1) []
(15, 4, 3) []
(15, 5, 2) []
(15, 6, 1) []
(16, 3, 3) ['aaabaaacaabaaacaaabaac']
(16, 4, 2) []
(16, 5, 1) []
(17, 3, 2) []
(17, 4, 1) []
(18, 2, 2) ['aaaaabaaaaacaaaabaaaac', 'aaaaabaaaacaaaaabaaaac']
(18, 3, 1) []
(19, 2, 1) []
(20, 1, 1) ['aaaaaaaaaabaaaaaaaaaac']
(8, 8, 7) ['abacbacbacbabcabcabcabc']
(9, 7, 7) ['abacbacabcabacbacabcabc']
(9, 8, 6) []
(9, 9, 5) ['ababcababcabacbabacbabc']
(10, 7, 6) []
(10, 8, 5) []
(10, 9, 4) []
(10, 10, 3) ['abababacbabababcabababc']
(11, 6, 6) ['abacabacabacabacabacabc']
(11, 7, 5) []
(11, 8, 4) []
(11, 9, 3) []
(11, 10, 2) []
(11, 11, 1) ['abababababababababababc']
(12, 6, 5) []
(12, 7, 4) []
(12, 8, 3) []
(12, 9, 2) []
(12, 10, 1) []
(13, 5, 5) ['aabacabaacabacaabacabac']
(13, 6, 4) []
(13, 7, 3) []
(13, 8, 2) []
(13, 9, 1) []
(14, 5, 4) []
(14, 6, 3) []
(14, 7, 2) []
(14, 8, 1) []
(15, 4, 4) ['aabaacaabaacaabaacaabac']
(15, 5, 3) []
(15, 6, 2) []
(15, 7, 1) []
(16, 4, 3) []
(16, 5, 2) []
(16, 6, 1) []
(17, 3, 3) ['aaabaaacaaabaaacaaabaac']
(17, 4, 2) []
(17, 5, 1) []
(18, 3, 2) []
(18, 4, 1) []
(19, 2, 2) ['aaaaabaaaaacaaaaabaaaac']
(19, 3, 1) []
(20, 2, 1) []
(21, 1, 1) ['aaaaaaaaaaabaaaaaaaaaac']
(8, 8, 8) ['abacbacbacbacbacbacbacbc', 'abacbacbacbacbacbacbcabc', 'abacbacbacbacbacbcabacbc', 'abacbacbacbacbacbcabcabc', 'abacbacbacbacbcabacbacbc', 'abacbacbacbacbcabacbcabc', 'abacbacbacbacbcabcabacbc', 'abacbacbacbacbcabcabcabc', 'abacbacbacbcabacbacbacbc', 'abacbacbacbcabacbacbcabc', 'abacbacbacbcabacbcabacbc', 'abacbacbacbcabacbcabcabc', 'abacbacbacbcabcabacbacbc', 'abacbacbacbcabcabacbcabc', 'abacbacbcabacbacbcabacbc', 'abacbacbcabacbcabacbcabc', 'abacbacbcabacbcabcabacbc', 'abacbacbcabcabacbacbcabc', 'abacbcabacbcabacbcabacbc', 'abcabcabcabcabcabcabcabc']
(9, 8, 7) []
(9, 9, 6) ['abacbabcabacbabcabacbabc']
(10, 7, 7) ['abacabcabacbacabacbacabc']
(10, 8, 6) []
(10, 9, 5) []
(10, 10, 4) ['ababacbababcababacbababc']
(11, 7, 6) []
(11, 8, 5) []
(11, 9, 4) []
(11, 10, 3) []
(11, 11, 2) ['abababababacbabababababc']
(12, 6, 6) ['abacabacabacabacabacabac']
(12, 7, 5) []
(12, 8, 4) []
(12, 9, 3) []
(12, 10, 2) []
(12, 11, 1) []
(13, 6, 5) []
(13, 7, 4) []
(13, 8, 3) []
(13, 9, 2) []
(13, 10, 1) []
(14, 5, 5) ['aabacaabacabaacabaacabac']
(14, 6, 4) []
(14, 7, 3) []
(14, 8, 2) []
(14, 9, 1) []
(15, 5, 4) []
(15, 6, 3) []
(15, 7, 2) []
(15, 8, 1) []
(16, 4, 4) ['aabaacaabaacaabaacaabaac']
(16, 5, 3) []
(16, 6, 2) []
(16, 7, 1) []
(17, 4, 3) []
(17, 5, 2) []
(17, 6, 1) []
(18, 3, 3) ['aaabaaacaaabaaacaaabaaac']
(18, 4, 2) []
(18, 5, 1) []
(19, 3, 2) []
(19, 4, 1) []
(20, 2, 2) ['aaaaabaaaaacaaaaabaaaaac']
(20, 3, 1) []
(21, 2, 1) []
(22, 1, 1) ['aaaaaaaaaaabaaaaaaaaaaac']
(9, 8, 8) ['abacbacbacbacabcabcabcabc']
(9, 9, 7) ['abacbabcabcabacbacbabcabc']
(10, 8, 7) []
(10, 9, 6) []
(10, 10, 5) ['aabcbaabcbaabcbaabcbaabcb', 'ababacbabacbabacbabacbabc', 'ababacbabacbabacbabcababc', 'ababacbabacbabcababacbabc', 'ababacbabacbabcababcababc', 'ababacbabcababacbabcababc', 'ababacbabcababcababcababc', 'ababcababcababcababcababc']
(11, 7, 7) ['abacabacbacabacabcabacabc']
(11, 8, 6) []
(11, 9, 5) []
(11, 10, 4) []
(11, 11, 3) ['ababababcabababacbabababc']
(12, 7, 6) []
(12, 8, 5) []
(12, 9, 4) []
(12, 10, 3) []
(12, 11, 2) []
(12, 12, 1) ['ababababababababababababc']
(13, 6, 6) ['aabacabacabacabacabacabac']
(13, 7, 5) []
(13, 8, 4) []
(13, 9, 3) []
(13, 10, 2) []
(13, 11, 1) []
(14, 6, 5) []
(14, 7, 4) []
(14, 8, 3) []
(14, 9, 2) []
(14, 10, 1) []
(15, 5, 5) ['aabaacabaacabaacabaacabac', 'aabaacabaacabaacabacaabac', 'aabaacabaacabacaabaacabac', 'aabaacabaacabacaabacaabac', 'aabaacabacaabaacabacaabac', 'aabaacabacaabacaabacaabac', 'aabacaabacaabacaabacaabac']
(15, 6, 4) []
(15, 7, 3) []
(15, 8, 2) []
(15, 9, 1) []
(16, 5, 4) []
(16, 6, 3) []
(16, 7, 2) []
(16, 8, 1) []
(17, 4, 4) ['aaabaacaabaacaabaacaabaac']
(17, 5, 3) []
(17, 6, 2) []
(17, 7, 1) []
(18, 4, 3) []
(18, 5, 2) []
(18, 6, 1) []
(19, 3, 3) ['aaaabaaacaaabaaacaaabaaac']
(19, 4, 2) []
(19, 5, 1) []
(20, 3, 2) []
(20, 4, 1) []
(21, 2, 2) ['aaaaaabaaaaacaaaaabaaaaac']
(21, 3, 1) []
(22, 2, 1) []
(23, 1, 1) ['aaaaaaaaaaaabaaaaaaaaaaac']
(9, 9, 8) ['abacbacbacbacbabcabcabcabc']
(10, 8, 8) ['abacbacabcabacbacbacabcabc', 'abacbacabcabcabacbacabcabc']
(10, 9, 7) []
(10, 10, 6) ['ababcabacbabacbabcabacbabc', 'ababcabacbabcababcabacbabc']
(11, 8, 7) []
(11, 9, 6) []
(11, 10, 5) []
(11, 11, 4) ['abababcababacbababacbababc']
(12, 7, 7) ['abacabacabacbacabacabacabc']
(12, 8, 6) []
(12, 9, 5) []
(12, 10, 4) []
(12, 11, 3) []
(12, 12, 2) ['ababababababacbabababababc', 'ababababababcababababababc']
(13, 7, 6) []
(13, 8, 5) []
(13, 9, 4) []
(13, 10, 3) []
(13, 11, 2) []
(13, 12, 1) []
(14, 6, 6) ['aabacabacabaacabacabacabac', 'aabacabacabacaabacabacabac']
(14, 7, 5) []
(14, 8, 4) []
(14, 9, 3) []
(14, 10, 2) []
(14, 11, 1) []
(15, 6, 5) []
(15, 7, 4) []
(15, 8, 3) []
(15, 9, 2) []
(15, 10, 1) []
(16, 5, 5) ['aabaacabaacabaacaabacaabac']
(16, 6, 4) []
(16, 7, 3) []
(16, 8, 2) []
(16, 9, 1) []
(17, 5, 4) []
(17, 6, 3) []
(17, 7, 2) []
(17, 8, 1) []
(18, 4, 4) ['aaabaacaabaaacaabaacaabaac', 'aaabaacaabaacaaabaacaabaac']
(18, 5, 3) []
(18, 6, 2) []
(18, 7, 1) []
(19, 4, 3) []
(19, 5, 2) []
(19, 6, 1) []
(20, 3, 3) ['aaaabaaacaaabaaaacaaabaaac']
(20, 4, 2) []
(20, 5, 1) []
(21, 3, 2) []
(21, 4, 1) []
(22, 2, 2) ['aaaaaabaaaaaacaaaaabaaaaac', 'aaaaaabaaaaacaaaaaabaaaaac']
(22, 3, 1) []
(23, 2, 1) []
(24, 1, 1) ['aaaaaaaaaaaabaaaaaaaaaaaac']
(9, 9, 9) ['abacbacbacbacbacbacbacbacbc', 'abacbacbacbacbacbacbacbcabc', 'abacbacbacbacbacbacbcabacbc', 'abacbacbacbacbacbacbcabcabc', 'abacbacbacbacbacbcabacbacbc', 'abacbacbacbacbacbcabacbcabc', 'abacbacbacbacbacbcabcabacbc', 'abacbacbacbacbacbcabcabcabc', 'abacbacbacbacbcabacbacbacbc', 'abacbacbacbacbcabacbacbcabc', 'abacbacbacbacbcabacbcabacbc', 'abacbacbacbacbcabacbcabcabc', 'abacbacbacbacbcabcabacbacbc', 'abacbacbacbacbcabcabacbcabc', 'abacbacbacbacbcabcabcabacbc', 'abacbacbacbcabacbacbacbcabc', 'abacbacbacbcabacbacbcabacbc', 'abacbacbacbcabacbacbcabcabc', 'abacbacbacbcabacbcabacbacbc', 'abacbacbacbcabacbcabacbcabc', 'abacbacbacbcabacbcabcabacbc', 'abacbacbacbcabcabacbacbcabc', 'abacbacbacbcabcabacbcabacbc', 'abacbacbacbcabcabacbcabcabc', 'abacbacbcabacbacbcabacbacbc', 'abacbacbcabacbacbcabacbcabc', 'abacbacbcabacbacbcabcabacbc', 'abacbacbcabacbcabacbacbcabc', 'abacbacbcabacbcabacbcabacbc', 'abcabcabcabcabcabcabcabcabc']
(10, 9, 8) []
(10, 10, 7) ['abacbabcabacbabcabacbabcabc']
(11, 8, 8) ['abacabcabacbacabcabacbacabc']
(11, 9, 7) []
(11, 10, 6) []
(11, 11, 5) ['ababacbabacbababcababcababc']
(12, 8, 7) []
(12, 9, 6) []
(12, 10, 5) []
(12, 11, 4) []
(12, 12, 3) ['ababababacbabababacbabababc', 'ababababacbabababcababababc', 'ababababcababababcababababc']
(13, 7, 7) ['abacabacabacabacabacabacabc']
(13, 8, 6) []
(13, 9, 5) []
(13, 10, 4) []
(13, 11, 3) []
(13, 12, 2) []
(13, 13, 1) ['abababababababababababababc']
(14, 7, 6) []
(14, 8, 5) []
(14, 9, 4) []
(14, 10, 3) []
(14, 11, 2) []
(14, 12, 1) []
(15, 6, 6) ['aabacabaacabacabaacabacabac', 'aabacabaacabacabacaabacabac', 'aabacabacaabacabacaabacabac']
(15, 7, 5) []
(15, 8, 4) []
(15, 9, 3) []
(15, 10, 2) []
(15, 11, 1) []
(16, 6, 5) []
(16, 7, 4) []
(16, 8, 3) []
(16, 9, 2) []
(16, 10, 1) []
(17, 5, 5) ['aabaacaabacaabaacabaacaabac']
(17, 6, 4) []
(17, 7, 3) []
(17, 8, 2) []
(17, 9, 1) []
(18, 5, 4) []
(18, 6, 3) []
(18, 7, 2) []
(18, 8, 1) []
(19, 4, 4) ['aaabaacaaabaacaabaaacaabaac']
(19, 5, 3) []
(19, 6, 2) []
(19, 7, 1) []
(20, 4, 3) []
(20, 5, 2) []
(20, 6, 1) []
(21, 3, 3) ['aaaabaaaacaaabaaaacaaabaaac', 'aaaabaaaacaaabaaacaaaabaaac', 'aaaabaaacaaaabaaacaaaabaaac']
(21, 4, 2) []
(21, 5, 1) []
(22, 3, 2) []
(22, 4, 1) []
(23, 2, 2) ['aaaaaabaaaaaacaaaaaabaaaaac']
(23, 3, 1) []
(24, 2, 1) []
(25, 1, 1) ['aaaaaaaaaaaaabaaaaaaaaaaaac']
(10, 9, 9) ['abacbacbacbacbacabcabcabcabc']
(10, 10, 8) ['abacbacbabcabcabacbacbabcabc']
(11, 9, 8) []
(11, 10, 7) []
(11, 11, 6) ['ababcababcabacbabacbabacbabc']
(12, 8, 8) ['abacabacbacabacbacabacbacabc', 'abacabacbacabacbacabcabacabc', 'abacabacbacabcabacabacbacabc', 'abacabacbacabcabacabcabacabc', 'abacabcabacabcabacabcabacabc']
(12, 9, 7) []
(12, 10, 6) []
(12, 11, 5) []
(12, 12, 4) ['abababacbababacbababacbababc', 'abababacbababacbababcabababc', 'abababacbababcabababacbababc', 'abababacbababcabababcabababc', 'abababcabababcabababcabababc']
(13, 8, 7) []
(13, 9, 6) []
(13, 10, 5) []
(13, 11, 4) []
(13, 12, 3) []
(13, 13, 2) ['ababababababacbababababababc']
(14, 7, 7) ['abacabacabacabacabacabacabac']
(14, 8, 6) []
(14, 9, 5) []
(14, 10, 4) []
(14, 11, 3) []
(14, 12, 2) []
(14, 13, 1) []
(15, 7, 6) []
(15, 8, 5) []
(15, 9, 4) []
(15, 10, 3) []
(15, 11, 2) []
(15, 12, 1) []
(16, 6, 6) ['aabacabaacabacaabacabaacabac']
(16, 7, 5) []
(16, 8, 4) ['aabacabaabacabaabacabaabacab']
(16, 9, 3) []
(16, 10, 2) []
(16, 11, 1) []
(17, 6, 5) []
(17, 7, 4) []
(17, 8, 3) []
(17, 9, 2) []
(17, 10, 1) []
(18, 5, 5) ['aabaacaabaacabaacaabaacaabac']
(18, 6, 4) []
(18, 7, 3) []
(18, 8, 2) []
(18, 9, 1) []
(19, 5, 4) []
(19, 6, 3) []
(19, 7, 2) []
(19, 8, 1) []
(20, 4, 4) ['aaabaaacaabaaacaabaaacaabaac', 'aaabaaacaabaaacaabaacaaabaac', 'aaabaaacaabaacaaabaaacaabaac', 'aaabaaacaabaacaaabaacaaabaac', 'aaabaacaaabaacaaabaacaaabaac']
(20, 5, 3) []
(20, 6, 2) []
(20, 7, 1) []
(21, 4, 3) []
(21, 5, 2) []
(21, 6, 1) []
(22, 3, 3) ['aaaabaaaacaaabaaaacaaaabaaac']
(22, 4, 2) []
(22, 5, 1) []
(23, 3, 2) []
(23, 4, 1) []
(24, 2, 2) ['aaaaaabaaaaaacaaaaaabaaaaaac']
(24, 3, 1) []
(25, 2, 1) []
(26, 1, 1) ['aaaaaaaaaaaaabaaaaaaaaaaaaac']
(10, 10, 9) ['abacbacbacbacbabcabcabcabcabc']
(11, 9, 9) ['abacbacabcabcabacbacbacabcabc']
(11, 10, 8) []
(11, 11, 7) ['ababcabacbabcabacbabcabacbabc']
(12, 9, 8) []
(12, 10, 7) []
(12, 11, 6) []
(12, 12, 5) ['ababacbababcababacbababcababc']
(13, 8, 8) ['abacabacabcabacabacbacabacabc']
(13, 9, 7) []
(13, 10, 6) []
(13, 11, 5) []
(13, 12, 4) []
(13, 13, 3) ['ababababacbababababcababababc']
(14, 8, 7) []
(14, 9, 6) []
(14, 10, 5) []
(14, 11, 4) []
(14, 12, 3) []
(14, 13, 2) []
(14, 14, 1) ['ababababababababababababababc']
(15, 7, 7) ['aabacabacabacabacabacabacabac']
(15, 8, 6) []
(15, 9, 5) []
(15, 10, 4) []
(15, 11, 3) []
(15, 12, 2) []
(15, 13, 1) []
(16, 7, 6) []
(16, 8, 5) []
(16, 9, 4) []
(16, 10, 3) []
(16, 11, 2) []
(16, 12, 1) []
(17, 6, 6) ['aabacaabacaabacabaacabaacabac']
(17, 7, 5) []
(17, 8, 4) []
(17, 9, 3) []
(17, 10, 2) []
(17, 11, 1) []
(18, 6, 5) []
(18, 7, 4) []
(18, 8, 3) []
(18, 9, 2) []
(18, 10, 1) []
(19, 5, 5) ['aabaacaabaacaabaacaabaacaabac']
(19, 6, 4) []
(19, 7, 3) []
(19, 8, 2) []
(19, 9, 1) []
(20, 5, 4) []
(20, 6, 3) []
(20, 7, 2) []
(20, 8, 1) []
(21, 4, 4) ['aaabaaacaabaaacaaabaacaaabaac']
(21, 5, 3) []
(21, 6, 2) []
(21, 7, 1) []
(22, 4, 3) []
(22, 5, 2) []
(22, 6, 1) []
(23, 3, 3) ['aaaabaaaacaaaabaaaacaaaabaaac']
(23, 4, 2) []
(23, 5, 1) []
(24, 3, 2) []
(24, 4, 1) []
(25, 2, 2) ['aaaaaaabaaaaaacaaaaaabaaaaaac']
(25, 3, 1) []
(26, 2, 1) []
(27, 1, 1) ['aaaaaaaaaaaaaabaaaaaaaaaaaaac']
(10, 10, 10) ['abacbacbacbacbacbacbacbacbacbc', 'abacbacbacbacbacbacbacbacbcabc', 'abacbacbacbacbacbacbacbcabacbc', 'abacbacbacbacbacbacbacbcabcabc', 'abacbacbacbacbacbacbcabacbacbc', 'abacbacbacbacbacbacbcabacbcabc', 'abacbacbacbacbacbacbcabcabacbc', 'abacbacbacbacbacbacbcabcabcabc', 'abacbacbacbacbacbcabacbacbacbc', 'abacbacbacbacbacbcabacbacbcabc', 'abacbacbacbacbacbcabacbcabacbc', 'abacbacbacbacbacbcabacbcabcabc', 'abacbacbacbacbacbcabcabacbacbc', 'abacbacbacbacbacbcabcabacbcabc', 'abacbacbacbacbacbcabcabcabacbc', 'abacbacbacbacbacbcabcabcabcabc', 'abacbacbacbacbcabacbacbacbacbc', 'abacbacbacbacbcabacbacbacbcabc', 'abacbacbacbacbcabacbacbcabacbc', 'abacbacbacbacbcabacbacbcabcabc', 'abacbacbacbacbcabacbcabacbacbc', 'abacbacbacbacbcabacbcabacbcabc', 'abacbacbacbacbcabacbcabcabacbc', 'abacbacbacbacbcabacbcabcabcabc', 'abacbacbacbacbcabcabacbacbacbc', 'abacbacbacbacbcabcabacbacbcabc', 'abacbacbacbacbcabcabacbcabacbc', 'abacbacbacbacbcabcabacbcabcabc', 'abacbacbacbacbcabcabcabacbacbc', 'abacbacbacbacbcabcabcabacbcabc', 'abacbacbacbcabacbacbacbcabacbc', 'abacbacbacbcabacbacbacbcabcabc', 'abacbacbacbcabacbacbcabacbacbc', 'abacbacbacbcabacbacbcabacbcabc', 'abacbacbacbcabacbacbcabcabacbc', 'abacbacbacbcabacbcabacbacbcabc', 'abacbacbacbcabacbcabacbcabacbc', 'abacbacbacbcabacbcabacbcabcabc', 'abacbacbacbcabacbcabcabacbacbc', 'abacbacbacbcabacbcabcabacbcabc', 'abacbacbacbcabacbcabcabcabacbc', 'abacbacbacbcabcabacbacbacbcabc', 'abacbacbacbcabcabacbacbcabacbc', 'abacbacbacbcabcabacbacbcabcabc', 'abacbacbacbcabcabacbcabacbacbc', 'abacbacbacbcabcabacbcabacbcabc', 'abacbacbacbcabcabacbcabcabacbc', 'abacbacbcabacbacbcabacbacbcabc', 'abacbacbcabacbacbcabacbcabacbc', 'abacbacbcabacbacbcabcabacbcabc', 'abacbacbcabacbcabacbacbcabacbc', 'abacbacbcabacbcabacbcabacbcabc', 'abacbacbcabacbcabacbcabcabacbc', 'abacbacbcabacbcabcabacbacbcabc', 'abacbcabacbcabacbcabacbcabacbc', 'abcabcabcabcabcabcabcabcabcabc']
(11, 10, 9) []
(11, 11, 8) ['abacbabcabacbacbabcabacbabcabc']
(12, 9, 9) ['abacbacabcabacbacabcabacbacabc']
(12, 10, 8) []
(12, 11, 7) []
(12, 12, 6) ['aabcbaabcbaabcbaabcbaabcbaabcb', 'ababacbabacbabacbabacbabacbabc', 'ababacbabacbabacbabacbabcababc', 'ababacbabacbabacbabcababacbabc', 'ababacbabacbabacbabcababcababc', 'ababacbabacbabcababacbabacbabc', 'ababacbabacbabcababacbabcababc', 'ababacbabacbabcababcababacbabc', 'ababacbabacbabcababcababcababc', 'ababacbabcababacbabcababacbabc', 'ababacbabcababacbabcababcababc', 'ababacbabcababcababacbabcababc', 'ababacbabcababcababcababcababc', 'ababcababcababcababcababcababc']
(13, 9, 8) []
(13, 10, 7) []
(13, 11, 6) []
(13, 12, 5) []
(13, 13, 4) ['abababacbababacbabababcabababc']
(14, 8, 8) ['abacabacabacabacbacabacabacabc', 'abacabacabacabcabacabacabacabc']
(14, 9, 7) []
(14, 10, 6) []
(14, 11, 5) []
(14, 12, 4) []
(14, 13, 3) []
(14, 14, 2) ['abababababababacbababababababc', 'abababababababcabababababababc']
(15, 8, 7) []
(15, 9, 6) []
(15, 10, 5) []
(15, 11, 4) []
(15, 12, 3) []
(15, 13, 2) []
(15, 14, 1) []
(16, 7, 7) ['aabacabacabacabaacabacabacabac']
(16, 8, 6) []
(16, 9, 5) []
(16, 10, 4) []
(16, 11, 3) []
(16, 12, 2) []
(16, 13, 1) []
(17, 7, 6) []
(17, 8, 5) []
(17, 9, 4) []
(17, 10, 3) []
(17, 11, 2) []
(17, 12, 1) []
(18, 6, 6) ['aabaacabaacabaacabaacabaacabac', 'aabaacabaacabaacabaacabacaabac', 'aabaacabaacabaacabacaabaacabac', 'aabaacabaacabaacabacaabacaabac', 'aabaacabaacabacaabaacabaacabac', 'aabaacabaacabacaabaacabacaabac', 'aabaacabaacabacaabacaabaacabac', 'aabaacabaacabacaabacaabacaabac', 'aabaacabacaabaacabacaabaacabac', 'aabaacabacaabaacabacaabacaabac', 'aabaacabacaabacaabaacabacaabac', 'aabaacabacaabacaabacaabacaabac', 'aabacaabacaabacaabacaabacaabac']
(18, 7, 5) []
(18, 8, 4) []
(18, 9, 3) []
(18, 10, 2) []
(18, 11, 1) []
(19, 6, 5) []
(19, 7, 4) []
(19, 8, 3) []
(19, 9, 2) []
(19, 10, 1) []
(20, 5, 5) ['aabaacaabaacaabaacaabaacaabaac']
(20, 6, 4) []
(20, 7, 3) []
(20, 8, 2) []
(20, 9, 1) []
(21, 5, 4) []
(21, 6, 3) []
(21, 7, 2) []
(21, 8, 1) []
(22, 4, 4) ['aaabaaacaaabaaacaabaaacaaabaac', 'aaabaaacaaabaacaaabaaacaaabaac']
(22, 5, 3) []
(22, 6, 2) []
(22, 7, 1) []
(23, 4, 3) []
(23, 5, 2) []
(23, 6, 1) []
(24, 3, 3) ['aaaabaaaacaaaabaaaacaaaabaaaac']
(24, 4, 2) []
(24, 5, 1) []
(25, 3, 2) []
(25, 4, 1) []
(26, 2, 2) ['aaaaaaabaaaaaaacaaaaaabaaaaaac', 'aaaaaaabaaaaaacaaaaaaabaaaaaac']
(26, 3, 1) []
(27, 2, 1) []
(28, 1, 1) ['aaaaaaaaaaaaaabaaaaaaaaaaaaaac']
(11, 10, 10) ['abacbacbacbacbacabcabcabcabcabc']
(11, 11, 9) ['abacbacbabcabcabacbacbabcabcabc']
(12, 10, 9) []
(12, 11, 8) []
(12, 12, 7) ['ababcabacbabacbabcababcabacbabc']
(13, 9, 9) ['abacabcabacabcabacbacabacbacabc']
(13, 10, 8) []
(13, 11, 7) []
(13, 12, 6) []
(13, 13, 5) ['abababcababacbababcababacbababc']
(14, 9, 8) []
(14, 10, 7) []
(14, 11, 6) []
(14, 12, 5) []
(14, 13, 4) []
(14, 14, 3) ['abababababcababababacbababababc']
(15, 8, 8) ['abacabacabacabacabacabacabacabc']
(15, 9, 7) []
(15, 10, 6) []
(15, 11, 5) []
(15, 12, 4) []
(15, 13, 3) []
(15, 14, 2) []
(15, 15, 1) ['abababababababababababababababc']
(16, 8, 7) []
(16, 9, 6) []
(16, 10, 5) []
(16, 11, 4) []
(16, 12, 3) []
(16, 13, 2) []
(16, 14, 1) []
(17, 7, 7) ['aabacabacaabacabacabaacabacabac']
(17, 8, 6) []
(17, 9, 5) []
(17, 10, 4) []
(17, 11, 3) []
(17, 12, 2) []
(17, 13, 1) []
(18, 7, 6) []
(18, 8, 5) []
(18, 9, 4) []
(18, 10, 3) []
(18, 11, 2) []
(18, 12, 1) []
(19, 6, 6) ['aabaacabaacabaacaabacaabacaabac']
(19, 7, 5) []
(19, 8, 4) []
(19, 9, 3) []
(19, 10, 2) []
(19, 11, 1) []
(20, 6, 5) []
(20, 7, 4) []
(20, 8, 3) []
(20, 9, 2) []
(20, 10, 1) []
(21, 5, 5) ['aaabaacaabaacaabaacaabaacaabaac']
(21, 6, 4) []
(21, 7, 3) []
(21, 8, 2) []
(21, 9, 1) []
(22, 5, 4) []
(22, 6, 3) []
(22, 7, 2) []
(22, 8, 1) []
(23, 4, 4) ['aaabaaacaaabaaacaaabaaacaaabaac']
(23, 5, 3) []
(23, 6, 2) []
(23, 7, 1) []
(24, 4, 3) []
(24, 5, 2) []
(24, 6, 1) []
(25, 3, 3) ['aaaaabaaaacaaaabaaaacaaaabaaaac']
(25, 4, 2) []
(25, 5, 1) []
(26, 3, 2) []
(26, 4, 1) []
(27, 2, 2) ['aaaaaaabaaaaaaacaaaaaaabaaaaaac']
(27, 3, 1) []
(28, 2, 1) []
(29, 1, 1) ['aaaaaaaaaaaaaaabaaaaaaaaaaaaaac']

Keenan wrote:

>Following Paul, I'm calling them "distributionally even" now, rather
>than "generalized Myhill", because the definition of the original
>Myhill property is "EXACTLY two specific intervals per generic
>interval", but I want "AT MOST n specific intervals per generic
>interval". "Rank-N distributionally even" seems the best description.

Why at most? You'll get scales of ranks < n in your results.

I think you mean you want exactly n per period.

>But what I really want to talk about is an obvious pattern showing up
>in the data, which leads me to a conjecture:
>CONJECTURE 1. For any triple of positive integers (a,b,c), a
>distributionally even scale with a small steps, b medium steps, and c
>large steps exists if and only if EITHER:
>* Two of the numbers a, b, and c are equal, OR
>* (a,b,c) is a multiple of some permutation of (1,2,4)
>I have no idea how to prove this yet, but if you look at the data it's
>just staring you in the face. Does anyone have a clue why it should be true?

No, but that's interesting.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> >Following Paul, I'm calling them "distributionally even" now, rather
> >than "generalized Myhill", because the definition of the original
> >Myhill property is "EXACTLY two specific intervals per generic
> >interval", but I want "AT MOST n specific intervals per generic
> >interval". "Rank-N distributionally even" seems the best description.
>
> Why at most? You'll get scales of ranks < n in your results.
>
> I think you mean you want exactly n per period.

No, it needs to be "at most", for the following reasons:

The first reason applies to the rank 2 case as well. If your period is a fraction of an octave, then strictly speaking your scale never has "Myhill's property", because there is only one specific interval representing the generic interval of a period. For example, the symmetric decatonic scale does not have Myhill's property because the interval of a decatonic "sixth" is always a neutral sixth (7/5~10/7), never a diminished or augmented sixth. Myhill requires that there be exactly two kinds, so since in this case there's only one kind, the scale doesn't have Myhill's property.

You might consider fixing this problem by redefining GMP as "exactly n specific intervals per generic interval, excepting multiples of the period", but that still isn't good enough for general rank, for a different reason that only pops up for rank >= 3.

Consider the scale pattern "abacbabc". If a, b, and c are incommensurate step sizes, this is definitely a rank 3 scale, and it is distributionally even, but consider its interval assortment:

1 step: a, b, c
2 steps: a+b, a+c, b+c
3 steps: 2a+b, a+2b, a+b+c
4 steps: 2a+b+c a+2b+c
...

There are only two different kinds of 4-step intervals, even though 4 steps is not the period. Your definition would exclude this scale pattern, but I want to include it because it's rank 3 and the numbers of specific intervals are limited to *at most* 3 (even though sometimes it's just 2).

Keenan

Kenean wrote:

>Consider the scale pattern "abacbabc". If a, b, and c are
>incommensurate step sizes, this is definitely a rank 3 scale,
>There are only two different kinds of 4-step intervals, even though 4
>steps is not the period.

I guess if you have < 2, that interval becomes the period.
But < 3 can be 2, and that doesn't make complete periodicity.
I suppose it makes semi-periodicity, but does that count
for anything?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Kenean wrote:
>
> >Consider the scale pattern "abacbabc". If a, b, and c are
> >incommensurate step sizes, this is definitely a rank 3 scale,
> >There are only two different kinds of 4-step intervals, even though 4
> >steps is not the period.
>
> I guess if you have < 2, that interval becomes the period.
> But < 3 can be 2, and that doesn't make complete periodicity.
> I suppose it makes semi-periodicity, but does that count
> for anything?

My only point is that such scales are no less regular or even, and no worse MOS analogues, than ones with exactly 3 kinds across the board, so we should talk about some property that doesn't exclude them. Hence DE instead of Myhill.

Keenan

>> I guess if you have < 2, that interval becomes the period.
>> But < 3 can be 2, and that doesn't make complete periodicity.
>> I suppose it makes semi-periodicity, but does that count
>> for anything?
>
>My only point is that such scales are no less regular or even, and no
>worse MOS analogues, than ones with exactly 3 kinds across the board,
>so we should talk about some property that doesn't exclude them. Hence
>DE instead of Myhill.

MOS are connected to rank 2 temperaments through the Hypothesis.
Their regularity is secondary. What makes just as good a MOS
analog isn't clear to me, but your method of generating scales is
certainly a good one.

-Carl

I wrote:

>MOS are connected to rank 2 temperaments through the Hypothesis.
>Their regularity is secondary. What makes just as good a MOS
>analog isn't clear to me, but your method of generating scales is
>certainly a good one.

As I think about it, I'm wondering if rank 3 scales should have
3GMP (or Tryhill or whatever you want to call it) at all...

With rank 2 epimorphic scales, any monzo, interpreted as a vector,
corresponds to one generic scale interval regardless of how it's
translated through the lattice. Its specific size depends on
whether it crosses the block boundary defined by the untempered
comma (with respect to the origin).

With rank 3 epimorphic scales there are two different untempered
commas and a vector may cross the boundary of either, or both,
or neither, seemingly producing up to FOUR sizes for each generic
interval.

Consider consecutive powers of a monzo m, that is shorter than
either of the two untempred commas of the system, a & b. Its
1st power (from the origin) crosses no boundaries and produces
the smallest specific size of the corresponding generic
interval, g. Its 2nd power crosses the boundary for a, giving
g + a. Its 3rd power crosses the b boundary, giving g + b, and
its 4th power spans both boundaries, giving g + a + b.

No? -Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> As I think about it, I'm wondering if rank 3 scales should have
> 3GMP (or Tryhill or whatever you want to call it) at all...
>
> With rank 2 epimorphic scales, any monzo, interpreted as a vector,
> corresponds to one generic scale interval regardless of how it's
> translated through the lattice. Its specific size depends on
> whether it crosses the block boundary defined by the untempered
> comma (with respect to the origin).
>
> With rank 3 epimorphic scales there are two different untempered
> commas and a vector may cross the boundary of either, or both,
> or neither, seemingly producing up to FOUR sizes for each generic
> interval.

Aha! You're making the same mistake I've already made a bunch of times in thinking about this: you're unconsciously assuming that, for a given lattice, there's some preferred lattice basis that gives you a unique grid, or boundaries, that go along with that lattice. But this simply isn't true.

If a and b are commas (which form "boundaries" of some fundamental domain), then a+b is also a comma, and its boundaries do not coincide with those of either a or b. Same with a-b, 2a+b, a+2b, and so on (any linear combination of a and b with coprime integer coefficients). The basis {a,a+b} is no less intrinsic or fundamental than the basis {a,b}, but the boundaries of its cells are in different places.

If your octave-reduced pitch lattice is 1D there *is* a preferred lattice basis, because for a 1D lattice there are only two possible bases, {a} and {-a}, where a is the unique generator of the lattice. But for 2D and all higher dimensions there are infinitely many lattice bases that are equally valid.

> Consider consecutive powers of a monzo m, that is shorter than
> either of the two untempred commas of the system, a & b. Its
> 1st power (from the origin) crosses no boundaries and produces
> the smallest specific size of the corresponding generic
> interval, g. Its 2nd power crosses the boundary for a, giving
> g + a. Its 3rd power crosses the b boundary, giving g + b, and
> its 4th power spans both boundaries, giving g + a + b.
>
> No? -Carl

According to the above this doesn't really make sense because {a,b} is just one possible choice of basis commas. If you make some other choice, for example {a,a+b}, then the boundaries are different and a vector that crossed some in the first case may not cross any in the second, or vice versa.

Keenan

>> Consider consecutive powers of a monzo m, that is shorter than
>> either of the two untempred commas of the system, a & b. Its
>> 1st power (from the origin) crosses no boundaries and produces
>> the smallest specific size of the corresponding generic
>> interval, g. Its 2nd power crosses the boundary for a, giving
>> g + a. Its 3rd power crosses the b boundary, giving g + b, and
>> its 4th power spans both boundaries, giving g + a + b.
>>
>> No? -Carl
>
>According to the above this doesn't really make sense because {a,b} is
>just one possible choice of basis commas. If you make some other
>choice, for example {a,a+b}, then the boundaries are different and a
>vector that crossed some in the first case may not cross any in the
>second, or vice versa.

Sure, these are valid comma bases for a temperament.
What's wrong with there being different scales too?

-Carl

Keenan wrote:

>If a and b are commas (which form "boundaries" of some fundamental
>domain), then a+b is also a comma, and its boundaries do not coincide
>with those of either a or b. Same with a-b, 2a+b, a+2b, and so on (any
>linear combination of a and b with coprime integer coefficients). The
>basis {a,a+b} is no less intrinsic or fundamental than the basis
>{a,b}, but the boundaries of its cells are in different places.

Only a few linear combinations are possible, because it must
be possible to fit v (the vector representing the generic interval
in question) inside the block without without crossing any
boundaries. IOW, the a-component of v must be < a itself, and
likewise for b. I think this just means only coefficients of 1
are possible.

That means every generic interval list starts (v, ...) except for
(v+a,) and (v+b,) which are unisons/octaves.

(v, v+a) and (v, v+b) are parallel to a and b respectively and
therefore subdivide the octave.

Everything else must eventually return to the starting position
in the block, at which time it will earn a (..., v+a+b).
There are (v, v+a, v+a+b) and (v, v+b, v+a+b) and
(v, v+a, v+b, v+a+b).

I don't think we encounter negatives because they're only
meaningful if there are negative and positive terms for the same
comma in the same generic interval, and that would mean the
vector isn't straight. Or something. This is off-the-cuff
at 1am, but it produces the same conclusion as last time: rank 3
scales can have up to FOUR sizes in each interval class.
That's consistent with what Gene and Graham came up with in
the Starling scale contest.

Like you said last time, many different commas lists can
generate the kernel of a temperament T. Each corresponds to
a different periodicity block, and therefore scale. My claim
is that when they are tempered by T (with pure octaves) and T
is rank-3, they will have at most four sizes per interval class.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> Only a few linear combinations are possible, because it must
> be possible to fit v (the vector representing the generic interval
> in question) inside the block without without crossing any
> boundaries. IOW, the a-component of v must be < a itself, and
> likewise for b. I think this just means only coefficients of 1
> are possible.
>
> That means every generic interval list starts (v, ...) except for
> (v+a,) and (v+b,) which are unisons/octaves.
>
> (v, v+a) and (v, v+b) are parallel to a and b respectively and
> therefore subdivide the octave.
>
> Everything else must eventually return to the starting position
> in the block, at which time it will earn a (..., v+a+b).
> There are (v, v+a, v+a+b) and (v, v+b, v+a+b) and
> (v, v+a, v+b, v+a+b).
>
> I don't think we encounter negatives because they're only
> meaningful if there are negative and positive terms for the same
> comma in the same generic interval, and that would mean the
> vector isn't straight. Or something. This is off-the-cuff
> at 1am, but it produces the same conclusion as last time: rank 3
> scales can have up to FOUR sizes in each interval class.
> That's consistent with what Gene and Graham came up with in
> the Starling scale contest.
>
> Like you said last time, many different commas lists can
> generate the kernel of a temperament T. Each corresponds to
> a different periodicity block, and therefore scale. My claim
> is that when they are tempered by T (with pure octaves) and T
> is rank-3, they will have at most four sizes per interval class.

I think you're assuming that the scale has to have the specific shape of a parallelogram, when really it could be any fundamental domain of the lattice (a.k.a. tessellating tile), of which parallelograms are only the most simple.

However, I do think your conclusion, that 4 kinds of intervals are required in the generic case, is correct, because it agrees with something I figured out an entirely different way.

Conjecture 2:
Other than the two special cases, "aabcb" and "aabacab", and their repetitions, every N=3 DE scale pattern has some two of the three steps appear in strict alternation. That is, if "a" and "b" are the two steps, then every pair of adjacent "a"s is separated by exactly one "b" (along with any number of "c"s), and every pair of "b"s is separated by exactly one "a".

Implication:
If true, this conjecture would imply that every N=3 DE scale in a rank 3 temperament (other than the two exceptions) lies along two adjacent parallel lines in the lattice. In other words, there's some lattice basis such that the last coordinate of every pitch in the scale is either 0 or 1.

So, if all this is correct, only certain small and quasi-1D sublattices of the 2D lattice can possibly have the N=3 DE property. For an arbitrarily large, truly 2D scale, the best you can hope for is N=4 DE, which agrees with your conclusion.

I conclude by observing that, assuming the above, for certain temperaments it's impossible for an arbitrarily large N=3 DE scale to contain any complete otonal/utonal chords. For example, it's impossible for a large 3DE marvel scale to contain 7-limit tetrads, because the mappings of 1, 3, 5 and 7 don't lie on two parallel lines in the lattice. So marvel heptatonic may be the largest marvel scale that is both 3DE and contains 7-limit tetrads.

For starling it is possible though, at least in the 7 odd limit. (The generator of the parallel lines is 6/5, so it ends up looking like two superimposed keemun scales.)

If you want 9-limit pentads, the condition is even more strict: the lines have to be parallel to the one containing 1/1-3/2-9/8. In this case it's really easy to tell whether a temperament will work just by looking at the reduced mapping: the last row has to contain nothing but 0s and 1s. So starling doesn't work in the 9-limit, but other temperaments work: meantone extensions (erato, clio...), archytas, 49/48 planar, jubilee...

Later I'll post some Scala files of example 3DE scales generated this way.

Keenan

Keenan wrote:

>> Only a few linear combinations are possible, because it must
>> be possible to fit v (the vector representing the generic interval
>> in question) inside the block without without crossing any
>> boundaries. IOW, the a-component of v must be < a itself, and
>> likewise for b. I think this just means only coefficients of 1
>> are possible.
>> That means every generic interval list starts (v, ...) except for
>> (v+a,) and (v+b,) which are unisons/octaves.
>> (v, v+a) and (v, v+b) are parallel to a and b respectively and
>> therefore subdivide the octave.
>> Everything else must eventually return to the starting position
>> in the block, at which time it will earn a (..., v+a+b).
>> There are (v, v+a, v+a+b) and (v, v+b, v+a+b) and
>> (v, v+a, v+b, v+a+b).
>> I don't think we encounter negatives because they're only
>> meaningful if there are negative and positive terms for the same
>> comma in the same generic interval, and that would mean the
>> vector isn't straight.
>
>I think you're assuming that the scale has to have the specific shape
>of a parallelogram, when really it could be any fundamental domain of
>the lattice (a.k.a. tessellating tile), of which parallelograms are
>only the most simple.

I think I'm only assuming the block be periodic and epimorphic.
That means the a-ward & b-ward components of v must be < a & b,
and that v must correspond to a generic interval. I don't think
shape matters, at least for rank 3 where, ignoring commas that
vanish, the situation is planar. Specifically, I think any tile
can be transformed into a parallelogram by transposing pitches
by a or b. Doing so may change the pattern of scale steps but
not their number.

>However, I do think your conclusion, that 4 kinds of intervals are
>required in the generic case, is correct, because it agrees with
>something I figured out an entirely different way.
>Conjecture 2:
>Other than the two special cases, "aabcb" and "aabacab", and their
>repetitions, every N=3 DE scale pattern has some two of the three
>steps appear in strict alternation. That is, if "a" and "b" are the
>two steps, then every pair of adjacent "a"s is separated by exactly
>one "b" (along with any number of "c"s), and every pair of "b"s is
>separated by exactly one "a".
>Implication:
>If true, this conjecture would imply that every N=3 DE scale in a rank
>3 temperament (other than the two exceptions) lies along two adjacent
>parallel lines in the lattice. In other words, there's some lattice
>basis such that the last coordinate of every pitch in the scale is
>either 0 or 1.
>So, if all this is correct, only certain small and quasi-1D
>sublattices of the 2D lattice can possibly have the N=3 DE property.
>For an arbitrarily large, truly 2D scale, the best you can hope for is
>N=4 DE, which agrees with your conclusion.
>I conclude by observing that, assuming the above, for certain
>temperaments it's impossible for an arbitrarily large N=3 DE scale to
>contain any complete otonal/utonal chords. For example, it's
>impossible for a large 3DE marvel scale to contain 7-limit tetrads,
>because the mappings of 1, 3, 5 and 7 don't lie on two parallel lines
>in the lattice. So marvel heptatonic may be the largest marvel scale
>that is both 3DE and contains 7-limit tetrads.

Excellent work! If your conjecture 2 is right, the implication
certainly is as well.

>For starling it is possible though, at least in the 7 odd limit. (The
>generator of the parallel lines is 6/5, so it ends up looking like two
>superimposed keemun scales.)

What's the max scale size? I imagine it only works when the
parallel lines are adjacent (separated by a consonance like 6/5)?

>Later I'll post some Scala files of example 3DE scales generated this way.

I'd be more interested to know if your dynamic programming
approach scales to 4DE (how large of a scale can it handle?).

Joint publishing efforts around here have failed, I think,
largely because they tried to disclose the entire regular
mapping paradigm in a single paper. It seems to me that
the Hypothesis (and this new rank 3 extension) is perfectly
publishable and would make a nice three-pager. You, Paul,
Gene and myself could be coauthors, since I first suspected
a relationship between MOS and periodicity blocks, Paul
showed what it was, Gene proved it, you attacked the problem
of 3DE, and we both showed that rank 3 requires 4DE. Gene's
proof by the way is here

/tuning-math/message/966

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> I think I'm only assuming the block be periodic and epimorphic.
> That means the a-ward & b-ward components of v must be < a & b,
> and that v must correspond to a generic interval. I don't think
> shape matters, at least for rank 3 where, ignoring commas that
> vanish, the situation is planar. Specifically, I think any tile
> can be transformed into a parallelogram by transposing pitches
> by a or b. Doing so may change the pattern of scale steps but
> not their number.

Perhaps the easiest way to show you're missing something is to demonstrate a simple scale that is periodic and epimorphic, but has 5 kinds of each interval.

1/1 135/128 5/4 3/2 48/25

I claim this 5-limit scale (or marvel scale, or whatever) is epimorphic under the val <5 8 12|, but has 5 different specific intervals for each generic interval other than the octave.

The scale can indeed be transformed into (the set of lattice points in the interior of) a parallelogram by transposing by commas of the val, but doing that changes the intervals of the scale (because they're only commas of the val, aka unison vectors; not commas of the temperament).

> >For starling it is possible though, at least in the 7 odd limit. (The
> >generator of the parallel lines is 6/5, so it ends up looking like two
> >superimposed keemun scales.)
>
> What's the max scale size? I imagine it only works when the
> parallel lines are adjacent (separated by a consonance like 6/5)?

There is no maximum size, because the scale can extend arbitrarily far along those two parallel lines. It's only forbidden to move away from those lines in a different direction.

For starling the lines aren't separated by 6/5; 6/5 is the generator of one of the lines. The lines extend in the direction of the monzo |1 1 -1>.

> I'd be more interested to know if your dynamic programming
> approach scales to 4DE (how large of a scale can it handle?).

I'll run it for 4DE later. I think it should be able to go up to 30 or 40 notes before becoming unreasonably slow.

> Joint publishing efforts around here have failed, I think,
> largely because they tried to disclose the entire regular
> mapping paradigm in a single paper. It seems to me that
> the Hypothesis (and this new rank 3 extension) is perfectly
> publishable and would make a nice three-pager. You, Paul,
> Gene and myself could be coauthors, since I first suspected
> a relationship between MOS and periodicity blocks, Paul
> showed what it was, Gene proved it, you attacked the problem
> of 3DE, and we both showed that rank 3 requires 4DE. Gene's
> proof by the way is here
>
> /tuning-math/message/966

Cool. Would this be for something like Journal of Mathematics and Music, or what?

Keenan

"Keenan Pepper" <keenanpepper@gmail.com> wrote:

> I think you're assuming that the scale has to have the
> specific shape of a parallelogram, when really it could
> be any fundamental domain of the lattice (a.k.a.
> tessellating tile), of which parallelograms are only the
> most simple.

I don't think it works in general. At least, there have to
be certain constraints under which it works. Aren't the
pentachordal decatonics periodicity blocks but not MOS?
They look like they have Fokker's periodicity property, and
they tessellate the lattice. How about harmonic minor in
meantone?

It looks like Mr Pepper came to the same conclusion while I
was out.

What I've also concluded is that the subset of
rank 2 periodicity blocks that are also MOS is precisely
the same subset that are parallelograms (and therefore
Fokker blocks). This property cascades up to higher ranks.
If and only if the periodicity block is a parallelogram, you
can temper out all but one chromatic unison vectors to get
an MOS scale.

The general rule is that a rank r Fokker block has no more
than 2**(r-1) distinct intervals of each class.

For the proof, let's think about our rank 3 Fokker block
defined by chromatic unison vectors a and b and a diatonic
scale step generator that I don't think has a name called
v. They can be defined so that a, b, and v are all
positive. If they aren't, you can re-define the basis so
that they are. Another constraint is that v > |a+b| > 0 and
there are either 0 or 1 instances of a and b in each scale
step.

So, we have a basis <v a b] and position vectors [x y z>.
We know that x=i for the ith note in the scale. From Paul
Erlich's definition of a Fokker block, y and z both look
like maximally even scales. That means that the scales
defined by [x y> and [y z> are both MOS provided the scales
are properly ascending. This must be the case given either
of the constraints above.

To prove the converse: if [x y> defines an MOS, the y must
look like a distributionally even scale. It needn't be
maximally even. But will be maximally even if each y is
either 0 or 1. For an MOS, there will only be two
different coefficients of y. Let j be the smaller one.
You can define v' = v+j*a. Then [x y> in terms of <v' a']
will be such that the lower value of y is always 0. If
the higher value isn't 1, you multiply a' by that value.
Therefore [x y> is an MOS if an only if it's a Fokker
block. The same applies to [x z> by symmetry.

Now, consider a scale step [x y z>. As it's a scale step,
x=1. As [x y> and [x z> are MOS, there are two possible
values for y and two for z. The number of possible scale
steps is 2*2=4.

For rank r, there are r-1 chromatic unison vectors. If and
only if tempering out all but any 1 of these vectors gives
and MOS, we know it follows the formula for a Fokker
block. Given this, there must be two different possible
numbers of each unison vector for each interval class, and
the number of distinct intervals for each class is no mare
than 2^(r-1).

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> What I've also concluded is that the subset of
> rank 2 periodicity blocks that are also MOS is precisely
> the same subset that are parallelograms (and therefore
> Fokker blocks). This property cascades up to higher ranks.
> If and only if the periodicity block is a parallelogram, you
> can temper out all but one chromatic unison vectors to get
> an MOS scale.

This confused me at first, so let me make sure I understand. Do you mean to say the following?

"If and only if the periodicity block is a parallelogram, for *any* chromatic unison vector in the set, if you temper out all the others you *always* get an MOS scale."

> So, we have a basis <v a b] and position vectors [x y z>.
> We know that x=i for the ith note in the scale. From Paul
> Erlich's definition of a Fokker block, y and z both look
> like maximally even scales. That means that the scales
> defined by [x y> and [y z> are both MOS provided the scales
> are properly ascending. This must be the case given either
> of the constraints above.

This is supposed to say "[x y> and [x z>", right?

Also, I'm assuming you mean "the rank-2 scale defined by [x y> when z is tempered out" and "the rank-2 scale defined by [x z> when y is tempered out". Right?

> For rank r, there are r-1 chromatic unison vectors. If and
> only if tempering out all but any 1 of these vectors gives
> and MOS, we know it follows the formula for a Fokker
> block. Given this, there must be two different possible
> numbers of each unison vector for each interval class, and
> the number of distinct intervals for each class is no mare
> than 2^(r-1).

This mostly makes sense to me and I'm practically convinced that it really is 2^(r-1) for parallelotope periodicity blocks in rank r.

But why is this bound not always tight?

I just posted a bunch of scales that have only 3 intervals per class, even though they're rank 3 and according to you there's no reason to expect less than 4 intervals per class to be possible. Why are these cases special?

In particular, what is so special about the 5-pitch-per-period scale "aabcb" (corresponding to a plus sign) and the 7-pitch-per-period scale "aabacab" (corresponding to a 6-around-1 hexagonal pattern)? I have a strong suspicion that the resemblances to the root systems "A1 x A1" and "A2" is more than coincidence.

Keenan

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> Joint publishing efforts around here have failed, I think,
> largely because they tried to disclose the entire regular
> mapping paradigm in a single paper.

I've been finding the Xenwiki a pretty good outlet.

Keenan wrote:

>> I think I'm only assuming the block be periodic and epimorphic.
[snip]
>Perhaps the easiest way to show you're missing something is to
>demonstrate a simple scale that is periodic and epimorphic, but has 5
>kinds of each interval.
>1/1 135/128 5/4 3/2 48/25
>I claim this 5-limit scale (or marvel scale, or whatever) is
>epimorphic under the val <5 8 12|, but has 5 different specific
>intervals for each generic interval other than the octave.

That's non-convex and therefore not a "block"...

>The scale can indeed be transformed into (the set of lattice points in
>the interior of) a parallelogram by transposing by commas of the val,
>but doing that changes the intervals of the scale (because they're
>only commas of the val, aka unison vectors; not commas of the temperament).

...but you and Graham are right -- MODMOS like the harmonic
minor scale prove as much even in the rank 2 case.

Graham's 2^n rule follows just as easily, since each comma
can either be in or out of the vector -- I got that right.

>There is no maximum size, because the scale can extend arbitrarily far
>along those two parallel lines. It's only forbidden to move away from
>those lines in a different direction.

Yes, of course.

>I'll run it for 4DE later. I think it should be able to go up to 30 or
>40 notes before becoming unreasonably slow.

That'd be good enough.

>> /tuning-math/message/966
>
>Cool. Would this be for something like Journal of Mathematics and
>Music, or what?

Whoever will publish it, of course. It seems there ought to
be interest, since the original periodicity block work
http://www.huygens-fokker.org/docs/fokkerpb.html
was published (in a conference proceedings) and plenty has been
published about DE/MOS
/tuning-math/files/CarlLumma/CloughEtAl-Taxonomy.pdf
Citations there go to Journal of Music Theory, Music Theory Spectrum,
CMJ, Perspectives of New Music, American Mathematical Monthly...

Recently, the Thumbtronics gang got traction in CMJ and Journal of
Mathematics and Music.

There was also this
http://staff.science.uva.nl/~rens/convex_scales.pdf
which made it into the Journal of New Music Research.

-Carl

>> Joint publishing efforts around here have failed, I think,
>> largely because they tried to disclose the entire regular
>> mapping paradigm in a single paper.
>
>I've been finding the Xenwiki a pretty good outlet.

You have indeed and it will stand a long time.

-Carl

>That's non-convex and therefore not a "block"...
>
>>The scale can indeed be transformed into (the set of lattice points in
>>the interior of) a parallelogram by transposing by commas of the val,
>>but doing that changes the intervals of the scale (because they're
>>only commas of the val, aka unison vectors; not commas of the temperament).
>
>...but you and Graham are right -- MODMOS like the harmonic
>minor scale prove as much even in the rank 2 case.

It's worth pointing out that any rank 3 epimorphic collection
can be made into a parallelogram through transposition by
unisons. Indeed this changes the intervals, and in a good way.
It never increases and usually decreases their number. It also
reduces the number of pure chords. This points to the next
question, about the analog of the MOS series. Clearly this
procedure is related. There should also be a size limit, when
adding more pitches will inevitably produce an interval smaller
than one of the uvs, breaking epimorphism.

-Carl

"Keenan Pepper" <keenanpepper@gmail.com> wrote:

> This confused me at first, so let me make sure I
> understand. Do you mean to say the following?
>
> "If and only if the periodicity block is a parallelogram,
> for *any* chromatic unison vector in the set, if you
> temper out all the others you *always* get an MOS scale."

Yes, that's it. And let's be clear that it doesn't follow
that all rank 3 periodicity blocks with no more than 4
distinct intervals per class will be of this type. That's
an open question.

> > So, we have a basis <v a b] and position vectors [x y
> > z>. We know that x=i for the ith note in the scale.
> > z>From Paul
> > Erlich's definition of a Fokker block, y and z both look
> > like maximally even scales. That means that the scales
> > defined by [x y> and [y z> are both MOS provided the
> > scales are properly ascending. This must be the case
> > given either of the constraints above.
>
> This is supposed to say "[x y> and [x z>", right?

Yes.

> Also, I'm assuming you mean "the rank-2 scale defined by
> [x y> when z is tempered out" and "the rank-2 scale
> defined by [x z> when y is tempered out". Right?

Yes, or scales with a strange period.

> This mostly makes sense to me and I'm practically
> convinced that it really is 2^(r-1) for parallelotope
> periodicity blocks in rank r.
>
> But why is this bound not always tight?

Sometimes the different MOS scales will line up so that one
of the possible combinations of steps never occurs. You're
finding out how rare such cases are.

Well, I can see that if you have one MOS with 3 large steps
and another with 6 small steps, it could be possible to
match each of the 3 large steps in one scale with a small
step in the other. Then you'll never get two large steps
coming together. Maybe you can find a rule by generalizing
that.

Graham

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> That's non-convex and therefore not a "block"...

Okay. If you add "convex" to the requirements (making it "periodic, epimorphic, and convex"), then I think I believe it.

Keenan

I wrote:
>It's worth pointing out that any rank 3 epimorphic collection
>can be made into a parallelogram through transposition by
>unisons. Indeed this changes the intervals, and in a good way.
>It never increases and usually decreases their number. It also
>reduces the number of pure chords.

I meant *increases* the number of pure chords, of course.

>This points to the next question, about the analog of
>the MOS series.

There are a few things to consider here:

1. MODMOS

This is pretty well in the bag. You can transpose notes by
unison vectors. It's liable to increase the Rothenberg mean
variety of the scale, but it might net you tetrachordality
as a prize. Mike is the resident expert, though I don't
know if he's considered rank > 2 MODMOS. In any case I take
back what I said about it having to do with the (generalized)
MOS series.

2. NMOS

For a rank 2 temperament T we can write T[n], where n is the
number of generators used. When n isn't a MOS point some
weird stuff happens. For a rank 3 temperament we can write
T[n,m] and nobody knows where the 'MOS points' are. But
trying some values and checking for the weird stuff might
give us some hints. First we need to characterize "weird
stuff", which I will attempt in a subsequent message unless
someone beats me to it.

3. change of basis

This was Keenan's complaint about the block method - what
happens when the "chromatic" (free) unison vectors change?
Well, it produces a list of vals, each of which send the
"commatic" (given) uvs to zero. Previously I suggested that
the lowest-badness among these (in the relevant prime limit)
might give something like MOS points...

>There should also be a size limit, when adding more pitches
>will inevitably produce an interval smaller than one of the
>uvs, breaking epimorphism.

Just putting this here so I don't forget about it. This
might hold only for JI blocks; since the given comma is
tempered out nothing can be smaller, and the free commas
should shrink as the block size increases...

-Carl

On Thu, Jul 28, 2011 at 12:05 AM, Carl Lumma <carl@lumma.org> wrote:
>
> >This points to the next question, about the analog of
> >the MOS series.
>
> There are a few things to consider here:
>
> 1. MODMOS
>
> This is pretty well in the bag. You can transpose notes by
> unison vectors. It's liable to increase the Rothenberg mean
> variety of the scale, but it might net you tetrachordality
> as a prize. Mike is the resident expert, though I don't
> know if he's considered rank > 2 MODMOS. In any case I take
> back what I said about it having to do with the (generalized)
> MOS series.

No, not yet. I'm just getting back into Philly and am going through
this discussion now, so I'm still behind on what's going on.

The chroma in MODMOS is c=L-s, which enables you to think about unison
vectors without actually have to deal with mappings. Whether we're
talking about mohajira or dicot (or magic, for that matter), the
unison vector for the 3L4s MOS will always correspond to L-s, but the
JI mapping for the chroma will be contingent on the mapping matrix for
the temperament.

So if we're looking at rank-3 MODMOS, with intervals L, m, and s,
there will be three chromata to consider - L-m, m-s, and L-s. Of
those, L-s is likely to have some strange properties as it pertains to
propriety. My hunch is that if L-s > m, the scale will be improper if
there is more than one s interval per period, although I haven't
formally worked it out yet. But since propriety seems to be secondary
to what everyone's trying to work out, this might be a bit too far off
to matter now.

Since you mentioned tetrachordality, I've since discovered that there
are omnitetrachordal scales are not always alterations of MOS's, and
hence not MODMOS's at all. For example, the omnitetrachordal scale for
semaphore is actually 10 notes, and can be thought of as two chains of
meantone[5] offset by 64/63 - however, semaphore's MOS's are generated
at 9 and 14 notes, so this doesn't have anything to do with that.

There seem to be "moments of omnitetrachordality," as there are
"moments of symmetry," which are predicated on the concept of there
being a generator, a subperiod (usually a 4/3), and a period. The
subperiod is an MOS with respect to the generator, and the generator
is an MOS with respect to each "cell" in the subperiod's MOS pattern.
Hence, I have a vague conjecture that omnitetrachordal scales are
tempered versions of rank-3 MOS's (is that what we're calling them
now, not 3GMP anymore?), which I'll post more on when I work it out.

-Mike

Hi Mike,

>The chroma in MODMOS is c=L-s, which enables you to think about unison
>vectors without actually have to deal with mappings. Whether we're
>talking about mohajira or dicot (or magic, for that matter), the
>unison vector for the 3L4s MOS will always correspond to L-s, but the
>JI mapping for the chroma will be contingent on the mapping matrix for
>the temperament.

Yeah, MOS correspond to periodicity blocks whose chroma (untempered
commas) separate the intervals of each interval class. Rank 2 MOS
have only one such comma (which you call c) so they only have two
sizes of each generic interval. Rank 3 MOS have two chroma and
therefore up to four sizes of each generic interval. The rule, first
stated by Graham, is 2^(rank-1) sizes per generic interval.

>There seem to be "moments of omnitetrachordality," as there are
>"moments of symmetry," which are predicated on the concept of there
>being a generator, a subperiod (usually a 4/3), and a period. The
>subperiod is an MOS with respect to the generator, and the generator
>is an MOS with respect to each "cell" in the subperiod's MOS pattern.
>Hence, I have a vague conjecture that omnitetrachordal scales are
>tempered versions of rank-3 MOS's (is that what we're calling them
>now, not 3GMP anymore?), which I'll post more on when I work it out.

Interesting. Keenan's 3DE scales are rank 3, but only a small
subset of possible rank 3 scales, where one comma subdivides
another or something... producing an effect akin to rank 2
temperaments where the period subdivides the octave...

I think it's best to avoid the term "3GMP" for now.

-Carl

On Thu, Jul 28, 2011 at 12:58 AM, Carl Lumma <carl@lumma.org> wrote:
>
> Yeah, MOS correspond to periodicity blocks whose chroma (untempered
> commas) separate the intervals of each interval class. Rank 2 MOS
> have only one such comma (which you call c) so they only have two
> sizes of each generic interval. Rank 3 MOS have two chroma and
> therefore up to four sizes of each generic interval. The rule, first
> stated by Graham, is 2^(rank-1) sizes per generic interval.

Yes, good point, so a brief correction - there's L-m, m-s, and L-s,
but L-s is by definition equal to L-m + m-s. I'll call L-m = c_u for
the "upper chroma," m-s = c_l for the "lower chroma," and L-s = c_o
for the "outer chroma." So we end up with I guess the fundamental
identity for rank-3 MODMOS, which is that c_o = c_u + c_l. So LmsLmLs,
mapped as the 5-limit JI scale, c_l would be 25/24, c_u would be
81/80, and c_o would be 135/128.

> >There seem to be "moments of omnitetrachordality," as there are
> >"moments of symmetry," which are predicated on the concept of there
> >being a generator, a subperiod (usually a 4/3), and a period. The
> >subperiod is an MOS with respect to the generator, and the generator
> >is an MOS with respect to each "cell" in the subperiod's MOS pattern.
> >Hence, I have a vague conjecture that omnitetrachordal scales are
> >tempered versions of rank-3 MOS's (is that what we're calling them
> >now, not 3GMP anymore?), which I'll post more on when I work it out.
>
> Interesting. Keenan's 3DE scales are rank 3, but only a small
> subset of possible rank 3 scales, where one comma subdivides
> another or something... producing an effect akin to rank 2
> temperaments where the period subdivides the octave...

If one comma subdivides another, wouldn't we be dealing with torsion
or something? I guess I'll figure it out when I read through the
thread.

> I think it's best to avoid the term "3GMP" for now.

OK, so 3DE is the term now? Looks like Paul's finally won this battle...

-Mike

Mike wrote:
>Yes, good point, so a brief correction - there's L-m, m-s, and L-s,
>but L-s is by definition equal to L-m + m-s. I'll call L-m = c_u for
>the "upper chroma," m-s = c_l for the "lower chroma," and L-s = c_o
>for the "outer chroma." So we end up with I guess the fundamental
>identity for rank-3 MODMOS, which is that c_o = c_u + c_l. So LmsLmLs,
>mapped as the 5-limit JI scale, c_l would be 25/24, c_u would be
>81/80, and c_o would be 135/128.

Did you catch the part where there are four sizes of interval?

>OK, so 3DE is the term now? Looks like Paul's finally won this battle...

Paul's won nothing of the kind. Distributional evenness is a
property Keenan's scales have and that's what he's been calling
them. It's not completely clear what rank-3 MOS are yet or
what they should be called when they're understood.

-Carl

On Thu, Jul 28, 2011 at 3:55 AM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
> >Yes, good point, so a brief correction - there's L-m, m-s, and L-s,
> >but L-s is by definition equal to L-m + m-s. I'll call L-m = c_u for
> >the "upper chroma," m-s = c_l for the "lower chroma," and L-s = c_o
> >for the "outer chroma." So we end up with I guess the fundamental
> >identity for rank-3 MODMOS, which is that c_o = c_u + c_l. So LmsLmLs,
> >mapped as the 5-limit JI scale, c_l would be 25/24, c_u would be
> >81/80, and c_o would be 135/128.
>
> Did you catch the part where there are four sizes of interval?

Yes. Intervals in the same interval class can differ by L-m, m-s, and
L-s. That gives you three possible chromatic alterations, bringing you
to four interval sizes total. However, there are only two orthogonal
chroma, because c_o = c_u + c_l. I suppose we could call c_u and c_l
orthonormal, in a sense. This is what you said earlier translated away
from the lattice paradigm and into the MODMOS paradigm:

Carl wrote:
> With rank 3 epimorphic scales there are two different untempered
> commas and a vector may cross the boundary of either, or both,
> or neither, seemingly producing up to FOUR sizes for each generic
> interval.

And all of the stuff that I said about c_o = c_u + c_l is me similarly
turning Keenan's statement here into the same paradigm:

Keenan wrote:
> If a and b are commas (which form "boundaries" of some fundamental domain), then a+b is also a
> comma, and its boundaries do not coincide with those of either a or b. Same with a-b, 2a+b, a+2b,
> and so on (any linear combination of a and b with coprime integer coefficients). The basis {a,a+b} is
> no less intrinsic or fundamental than the basis {a,b}, but the boundaries of its cells are in different
> places.

I'm still trying to figure out exactly why some of these end up
producing only three sizes, though. The 5-limit JI major scale is one
of those. You've made reference to the commas subdividing each other,
although I'm still not sure I get it. I'll reply with specific
questions to posts about things I'm confused about.

> >OK, so 3DE is the term now? Looks like Paul's finally won this battle...
>
> Paul's won nothing of the kind. Distributional evenness is a
> property Keenan's scales have and that's what he's been calling
> them. It's not completely clear what rank-3 MOS are yet or
> what they should be called when they're understood.

Are you saying that an imprint like xL+ym+zs can lead to more than
one DE scale, but that there might exist one that's more symmetric
than the others in an as of yet undefined way, and scales with that
property will get the MOS title?

-Mike

Mike wrote:
>Yes. Intervals in the same interval class can differ by L-m, m-s, and
>L-s. That gives you three possible chromatic alterations, bringing you
>to four interval sizes total. However, there are only two orthogonal
>chroma, because c_o = c_u + c_l. I suppose we could call c_u and c_l
>orthonormal, in a sense. This is what you said earlier translated away
>from the lattice paradigm and into the MODMOS paradigm:

L, m, s = 3

>I'm still trying to figure out exactly why some of these end up
>producing only three sizes, though. The 5-limit JI major scale is one
>of those. You've made reference to the commas subdividing each other,
>although I'm still not sure I get it. I'll reply with specific
>questions to posts about things I'm confused about.

The reason is that the JI major scale is small - its block is
only one unit tall in the 5-wardly direction. As a result,
vectors that fit in it can only take a few different angles,
none of which produce the (v, v+a, v+b, v+a+b) case (I think).

>> Paul's won nothing of the kind. Distributional evenness is a
>> property Keenan's scales have and that's what he's been calling
>> them. It's not completely clear what rank-3 MOS are yet or
>> what they should be called when they're understood.
>
>Are you saying that an imprint like xL+ym+zs can lead to more than
>one DE scale, but that there might exist one that's more symmetric
>than the others in an as of yet undefined way, and scales with that
>property will get the MOS title?

Not exactly...

-Carl

On Thu, Jul 28, 2011 at 4:41 AM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
> >Yes. Intervals in the same interval class can differ by L-m, m-s, and
> >L-s. That gives you three possible chromatic alterations, bringing you
> >to four interval sizes total. However, there are only two orthogonal
> >chroma, because c_o = c_u + c_l. I suppose we could call c_u and c_l
> >orthonormal, in a sense. This is what you said earlier translated away
> >from the lattice paradigm and into the MODMOS paradigm:
>
> L, m, s = 3

I explored this and it led somewhere interesting. If we start with
four specific interval sizes, called a, b, c, and d, in ascending
order - so that d is largest, and a is smallest - then we get six
chroma:

c_u = d-c : upper
c_m = c-b : middle
c_l = b-a : lower
c_um = d-b : upper mid
c_lm= c-a : lower mid
c_o = d-a : outer

This is reducible to a basis set of three, which we might call c_l,
c_lm, and c_o: the three chroma that transform a into b, c, and d
respectively. Remember again, a is smallest.

For a rank-3 Fokker block, however, we should end up with a basis set
of 2 chromatic unison vectors, not three. This is because in a rank-3
Fokker block, the largest chromatic unison vector should equal the sum
of the two smaller chromatic vectors. So the largest chroma should be
the smaller two put together, meaning that c_o = c_l + c_lm. That
would mean

c_o = c_l + c_lm
c_u + c_m + c_l = c_l + c_l + c_m
c_u = c_l

Which also means that

d-a = b-a + c-a
d-c = b-a
a+d = b+c

etc. So in that case, we can reduce it to a basis set of two. This
assumes for the moment that all of the chromatic vectors are positive.

I'm not sure what this means yet, but I think it means that not all
scales with 4 specific interval sizes per generic interval correspond
to rank-3 Fokker blocks, only those where c_l = c_u. I assume this
means that rank-4 temperaments (or perhaps higher) can also, under
certain conditions, generate scales with 4 specific interval sizes,
analogously to how rank-3 temperaments can generate scales with 3
specific sizes under certain conditions. For blocks like that, there
would be three orthogonal chromatic unison vectors, as opposed to the
two orthogonal chromatic unison vectors in the rank-3 case and the one
that's a combination of both of them. So if this is true, then not all
scales with 4 specific sizes for each generic interval would be
rank-3; only the ones in which d-c = b-a. I don't expect this will
change if you redefine the basis.

I'm still not sure how or if this pertains to rank-3 scales sometimes
ending up with only 3 sizes of step per interval, but it's interesting
nonetheless.

> >I'm still trying to figure out exactly why some of these end up
> >producing only three sizes, though. The 5-limit JI major scale is one
> >of those. You've made reference to the commas subdividing each other,
> >although I'm still not sure I get it. I'll reply with specific
> >questions to posts about things I'm confused about.
>
> The reason is that the JI major scale is small - its block is
> only one unit tall in the 5-wardly direction. As a result,
> vectors that fit in it can only take a few different angles,
> none of which produce the (v, v+a, v+b, v+a+b) case (I think).

That's interesting. Not sure what it means. Are there periodicity
blocks where only one of the vectors produces 4 step sizes and the
rest produce 3?

> >> Paul's won nothing of the kind. Distributional evenness is a
> >> property Keenan's scales have and that's what he's been calling
> >> them. It's not completely clear what rank-3 MOS are yet or
> >> what they should be called when they're understood.
> >
> >Are you saying that an imprint like xL+ym+zs can lead to more than
> >one DE scale, but that there might exist one that's more symmetric
> >than the others in an as of yet undefined way, and scales with that
> >property will get the MOS title?
>
> Not exactly...

So what are you saying then?

-Mike

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > Also, I'm assuming you mean "the rank-2 scale defined by
> > [x y> when z is tempered out" and "the rank-2 scale
> > defined by [x z> when y is tempered out". Right?
>
> Yes, or scales with a strange period.

Ah, I guess I'm mostly thinking of these as scales that might be musically useful, so really complex periods didn't occur to me. This makes sense though.

> > This mostly makes sense to me and I'm practically
> > convinced that it really is 2^(r-1) for parallelotope
> > periodicity blocks in rank r.
> >
> > But why is this bound not always tight?
>
> Sometimes the different MOS scales will line up so that one
> of the possible combinations of steps never occurs. You're
> finding out how rare such cases are.

Actually, just having one of the combinations of steps never occur is not enough for N=3 DE. There will be only three steps, but there still might be four different intervals in some class other than a single step.

> Well, I can see that if you have one MOS with 3 large steps
> and another with 6 small steps, it could be possible to
> match each of the 3 large steps in one scale with a small
> step in the other. Then you'll never get two large steps
> coming together. Maybe you can find a rule by generalizing
> that.

Let's take this as an example. We'll take the two MOSes (3L+4s) and (1L+6s) and combine them to get a rank-3 scale. But there's more than one way to do this, depending on the relative position:

LsLsLss
sLsssss
acababb
(same pattern as aabcbab)

LsLsLss
sssLsss
abacabb
(same pattern as aababcb)

LsLsLss
sssssLs
ababacb
(same pattern as abababc)

LsLsLss
ssssssL
abababc

The first two choices do not yield N=3 DE scales, because although each contains only 3 steps, some larger interval class has 4 different intervals in it. The second two choices do yield an N=3 DE scale.

Keenan

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > Interesting. Keenan's 3DE scales are rank 3, but only a small
> > subset of possible rank 3 scales, where one comma subdivides
> > another or something... producing an effect akin to rank 2
> > temperaments where the period subdivides the octave...
>
> If one comma subdivides another, wouldn't we be dealing with torsion
> or something? I guess I'll figure it out when I read through the
> thread.

I don't think it's related to torsion at all. I think I would describe it like this: Although there are three chromata*, there are never four different intervals in a class because the chromata always appear in a triangle pattern. One chroma is equal to the sum of the other two, and it always appears subdivided by the other two.

* (plural of "chroma")

> > I think it's best to avoid the term "3GMP" for now.
>
> OK, so 3DE is the term now? Looks like Paul's finally won this battle...

Haha, I didn't know there was a battle. He just told me that Myhill was defined as "exactly two" where as DE was "at most two". The thing I want to talk about is "at most N", so it seems natural to call it a generalization of DE rather than a generalization of Myhill.

Keenan

I'm going to be mathematically nitpicky here, because I hate it when people abuse linear algebra terms.

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Yes. Intervals in the same interval class can differ by L-m, m-s, and
> L-s. That gives you three possible chromatic alterations, bringing you
> to four interval sizes total. However, there are only two orthogonal
> chroma, because c_o = c_u + c_l. I suppose we could call c_u and c_l
> orthonormal, in a sense.

Both "orthogonal" and "orthonormal" only make sense in an inner product space, which is a vector space with an inner product and hence also a metric. This would mean we have not only a way to measure the "size" or "length" of a vector, but also a way to measure the "angle" between two of them.

But interval space doesn't come with any natural metric. You can invent different metrics for it, as Gene did and used in the definition of hobbit scales, but there's no one obvious choice of metric. I haven't seen anyone ever define an inner product.

So in conclusion, don't say "orthogonal", say "linearly independent", because that concept makes sense in any vector space.

Keenan

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> The reason is that the JI major scale is small - its block is
> only one unit tall in the 5-wardly direction. As a result,
> vectors that fit in it can only take a few different angles,
> none of which produce the (v, v+a, v+b, v+a+b) case (I think).

This is part of my conjecture about N=3 DE scales. The conjecture says that any N=3 DE scale is either:

* A scale with 5 notes per period of the pattern "aabcb", the shape of whose block is therefore a square around a central point (a.k.a. "plus sign")

* A scale with 7 notes per period of the pattern "aabacab", the shape of whose block is therefore a hexagon around a central point, or

* An arbitrarily large scale that lies entirely on two adjacent lines in the lattice (i.e. "one unit tall" in some direction).

Keenan

Carl asked for some results for N=4 DE scale patterns, so here they are up to 16 notes:

1 1 1 0 ['abc']
1 1 1 1 ['abcd']
2 1 1 0 ['aabc', 'abac']
2 1 1 1 ['abacd']
2 2 1 0 ['aabcb', 'ababc']
3 1 1 0 ['aaabc', 'aabac']
2 2 1 1 ['abacdc']
2 2 2 0 ['aabccb', 'abacbc', 'abcabc']
3 1 1 1 []
3 2 1 0 ['aaabcb', 'aababc', 'aabacb', 'aabacc', 'aabbac', 'aabcab', 'aabcac', 'ababac']
4 1 1 0 ['aaaabc', 'aaabac', 'aabaac']
2 2 2 1 ['aabcdcb']
3 2 1 1 ['abacabd', 'abacadc']
3 2 2 0 ['aaabccb', 'aabacbc', 'aabcabc', 'aabcacb', 'aabcbac', 'aabcbcb', 'abacabc']
3 3 1 0 ['aababcb', 'aabcbab', 'abababc']
4 1 1 1 []
4 2 1 0 ['aaaabcb', 'aaabacb', 'aaabcab', 'aabaabc', 'aabaacb', 'aabaacc', 'aababac', 'aabacab', 'aabacac']
5 1 1 0 ['aaaaabc', 'aaaabac', 'aaabaac']
2 2 2 2 ['abcdabcd']
3 2 2 1 ['abacbdbc']
3 3 1 1 []
3 3 2 0 ['aabcabcb', 'aabcbacb', 'ababacbc', 'ababcabc', 'abacbabc']
4 2 1 1 ['aabaacdc']
4 2 2 0 ['aaabcacb', 'aabaacbc', 'aabacabc', 'aabcaabc', 'aabcaacb', 'aabcabac', 'abacabac']
4 3 1 0 ['aaabbcbb', 'aabaabcb', 'aabababc', 'aababacb', 'aababcab', 'aababcbb', 'aabacabb', 'aabacacc', 'aabacbab', 'aabbacab', 'aabcabab', 'aabcacac', 'abababac']
5 1 1 1 []
5 2 1 0 ['aaaaabcb', 'aaaabacb', 'aaaabcab', 'aaabaabc', 'aaabaacb', 'aaabaacc', 'aaabacab', 'aaabbaac', 'aaabcaab', 'aaabcaac', 'aabaabac', 'aabaacab', 'aabaacac']
6 1 1 0 ['aaaaaabc', 'aaaaabac', 'aaaabaac', 'aaabaaac']
3 2 2 2 []
3 3 2 1 []
3 3 3 0 ['abacbacbc', 'abcabcabc']
4 2 2 1 []
4 3 1 1 []
4 3 2 0 ['aaabcabcb', 'aaabcbacb', 'aabacbabc', 'aabcaabcb', 'aabcaacbc', 'aabcababc', 'aabcabacb', 'aabcacabc', 'aabcacacb', 'aabcacbac', 'aabcbabcb', 'aabcbcbcb', 'ababacabc', 'ababacbac', 'ababcabac', 'ababcbabc']
4 4 1 0 ['aabababcb', 'aababcbab', 'aabbaabcb', 'aabcbabab', 'ababababc']
5 2 1 1 []
5 2 2 0 ['aaaabcacb', 'aaabacabc', 'aaabaccab', 'aaabcaabc', 'aaabcaacb', 'aaabcabac', 'aabaacabc', 'aabaacbac', 'aababaacc', 'aabacaabc', 'aabacabac', 'aabacacab']
5 3 1 0 ['aaabaabcb', 'aaababacb', 'aaababcab', 'aaabacbab', 'aaabcabab', 'aaabcbaab', 'aabaababc', 'aabaabacb', 'aabaabcab', 'aabaacbab', 'aababaabc', 'aababaacb', 'aabababac', 'aababacab', 'aabacaabb', 'aabacaacc', 'aabacabab', 'aabacacac']
6 1 1 1 []
6 2 1 0 ['aaaaaabcb', 'aaaaabacb', 'aaaaabcab', 'aaaabaacb', 'aaaabacab', 'aaaabcaab', 'aaabaaabc', 'aaabaaacb', 'aaabaaacc', 'aaabaabac', 'aaabaacab', 'aaabaacac', 'aaababaac', 'aaabacaab', 'aaabacaac', 'aabaabaac']
7 1 1 0 ['aaaaaaabc', 'aaaaaabac', 'aaaaabaac', 'aaaabaaac']
3 3 2 2 []
3 3 3 1 []
4 2 2 2 ['abacdabacd', 'abacdabadc']
4 3 2 1 ['abacbadabc']
4 3 3 0 ['aabacbacbc', 'aabacbcabc', 'aabcabacbc', 'aabcabcabc', 'aabcabcacb', 'aabcabcbac', 'aabcacbacb', 'aabcbacabc', 'aabcbacbac', 'abacabacbc', 'abacabcabc', 'abacbacabc']
4 4 1 1 []
4 4 2 0 ['aabcababcb', 'aabcbaabcb', 'aabcbabacb', 'ababacbabc', 'ababcababc']
5 2 2 1 []
5 3 1 1 []
5 3 2 0 ['aaabcabacb', 'aabaacbabc', 'aababacabc', 'aababcaabc', 'aabacababc', 'aabacabacb', 'aabacabacc', 'aabacacabc', 'aabacbaabc', 'aabacbaacb', 'aabbacabac', 'aabcaabcab', 'aabcaabcac', 'aabcaacbac', 'aabcababac', 'aabcabacab', 'aabcabacac', 'aabcacabac', 'ababacabac']
5 4 1 0 ['aabaababcb', 'aabaabcbab', 'aababaabcb', 'aababababc', 'aabababacb', 'aabababcab', 'aabababcbb', 'aababacabb', 'aababacbab', 'aababcabab', 'aababcbabb', 'aabacababb', 'aabacacacc', 'aabacbabab', 'aabaccaacc', 'aabbaabbac', 'aabbacabab', 'aabcababab', 'aabcacacac', 'ababababac']
6 2 1 1 ['aabaacadac']
6 2 2 0 ['aaaabcaacb', 'aaabaacabc', 'aaabacaabc', 'aaabcaaabc', 'aaabcaaacb', 'aaabcaabac', 'aaabcabaac', 'aabaacabac', 'aabacaabac']
6 3 1 0 ['aaabaaabcb', 'aaabaabacb', 'aaabaabcab', 'aaabacbaab', 'aaabcabaab', 'aabaabaabc', 'aabaabaacb', 'aabaababac', 'aabaabacab', 'aabaacabab', 'aabaacacac', 'aababaabac', 'aababaacab']
7 1 1 1 []
7 2 1 0 ['aaaaaaabcb', 'aaaaaabacb', 'aaaaaabcab', 'aaaaabaacb', 'aaaaabacab', 'aaaaabcaab', 'aaaabaaabc', 'aaaabaaacb', 'aaaabaaacc', 'aaaabaacab', 'aaaabacaab', 'aaaabbaaac', 'aaaabcaaab', 'aaaabcaaac', 'aaabaaabac', 'aaabaaacab', 'aaabaaacac', 'aaabaabaac', 'aaabaacaab', 'aaabaacaac']
8 1 1 0 ['aaaaaaaabc', 'aaaaaaabac', 'aaaaaabaac', 'aaaaabaaac', 'aaaabaaaac']
3 3 3 2 []
4 3 2 2 []
4 3 3 1 []
4 4 2 1 []
4 4 3 0 ['aabcabcabcb', 'aabcabcbacb', 'aabcbacabcb', 'aabcbacbacb', 'ababacbacbc', 'ababacbcabc', 'ababcabacbc', 'ababcabcabc', 'ababcacbabc', 'abacbabcabc']
5 2 2 2 []
5 3 2 1 []
5 3 3 0 ['aabacabacbc', 'aabacabcabc', 'aabacbaacbc', 'aabacbacabc', 'aabacbcaabc', 'aabcaabcabc', 'aabcaabcacb', 'aabcaabcbac', 'aabcaacbabc', 'aabcabaacbc', 'aabcabacabc', 'aabcabacbac', 'aabcabcabac', 'aabcbacabac', 'abacabacabc']
5 4 1 1 []
5 4 2 0 ['aaabcbaabcb', 'aababacbabc', 'aababcababc', 'aababcabacb', 'aababcbabcb', 'aabacbaabcb', 'aabacbababc', 'aabacbabacb', 'aabacbabcab', 'aabaccabacc', 'aabbacabbac', 'aabcabaabcb', 'aabcababacb', 'aabcababcab', 'aabcabacbab', 'aabcacabcac', 'aabcacacbac', 'aabcacbacac', 'aabcbababcb', 'aabcbabcbab', 'aabcbcbcbcb', 'ababacababc', 'ababacbabac']
5 5 1 0 ['aababababcb', 'aabababcbab', 'aababcbaabb', 'aababcbabab', 'aabbaabcbab', 'aabcbababab', 'abababababc']
6 2 2 1 []
6 3 1 1 []
6 3 2 0 ['aaaabcabacb', 'aaabcaabacb', 'aaabcabaacb', 'aabaacbaabc', 'aababacaabc', 'aababacabac', 'aabacabaabc', 'aabacabaacb', 'aabacabaacc', 'aabacababac', 'aabacabacab', 'aabacabacac', 'aabacacaabc', 'aabacacabac']
6 4 1 0 ['aaababacbab', 'aaababcabab', 'aabaabaabcb', 'aabaababacb', 'aabaababcab', 'aabaabacbab', 'aabaabcabab', 'aababaababc', 'aababaabacb', 'aababaabcab', 'aababaacbab', 'aabababaabc', 'aabababaacb', 'aababababac', 'aabababacab', 'aababacaabb', 'aababacabab', 'aabacabaabb', 'aabacababab', 'aabacacaacc', 'aabacacacac']
7 2 1 1 []
7 2 2 0 ['aaaaabcaacb', 'aaaabacaabc', 'aaaabcaaabc', 'aaaabcaaacb', 'aaaabcaabac', 'aaabaaacabc', 'aaabaaacbac', 'aaabaacaabc', 'aaabaacabac', 'aaabacaaabc', 'aaabacaabac', 'aaabacaacab', 'aaabacabaac', 'aaabcaabaac', 'aabaacaabac']
7 3 1 0 ['aaaabaaabcb', 'aaaabcbaaab', 'aaabaaabacb', 'aaabaaabcab', 'aaabaabaabc', 'aaabaabaacb', 'aaabaabacab', 'aaabaabcaab', 'aaabaacbaab', 'aaabacabaab', 'aaabcaabaab', 'aaabcaacaac', 'aabaabaabac', 'aabaabaacab', 'aabaababaac', 'aabaacaabab']
8 1 1 1 []
8 2 1 0 ['aaaaaaaabcb', 'aaaaaaabacb', 'aaaaaaabcab', 'aaaaaabaacb', 'aaaaaabacab', 'aaaaaabcaab', 'aaaaabaaacb', 'aaaaabaacab', 'aaaaabacaab', 'aaaaabcaaab', 'aaaabaaaabc', 'aaaabaaaacb', 'aaaabaaaacc', 'aaaabaaabac', 'aaaabaaacab', 'aaaabaaacac', 'aaaabaacaab', 'aaaababaaac', 'aaaabacaaab', 'aaaabacaaac', 'aaabaaabaac', 'aaabaaacaab', 'aaabaaacaac']
9 1 1 0 ['aaaaaaaaabc', 'aaaaaaaabac', 'aaaaaaabaac', 'aaaaaabaaac', 'aaaaabaaaac']
3 3 3 3 ['abcdabcdabcd']
4 3 3 2 []
4 4 2 2 ['abacdcabacdc']
4 4 3 1 []
4 4 4 0 ['aabccbaabccb', 'abacbacbacbc', 'abacbacbcabc', 'abacbcabacbc', 'abcabcabcabc']
5 3 2 2 []
5 3 3 1 []
5 4 2 1 []
5 4 3 0 ['aaabcabcbacb', 'aabacbabcabc', 'aabacbacbabc', 'aabcaabcbacb', 'aabcaacbacbc', 'aabcababcabc', 'aabcababcacb', 'aabcabacabcb', 'aabcabacbabc', 'aabcabacbacb', 'aabcabcaabcb', 'aabcabcaacbc', 'aabcabcababc', 'aabcabcabacb', 'aabcabcacabc', 'aabcabcacbac', 'aabcabcbabcb', 'aabcacabcabc', 'aabcacabcacb', 'aabcacbaabcb', 'aabcacbabacb', 'aabcacbacabc', 'aabcacbacacb', 'aabcacbacbac', 'aabcbaabcbcb', 'aabcbabcabcb', 'aabcbabcbacb', 'aabcbacabacb', 'aabcbacbabcb', 'ababacbacabc', 'ababcabacabc', 'ababcabacbac', 'ababcabcabac', 'ababcabcbabc', 'ababcbabcabc', 'ababcbacbabc', 'abacabacbabc']
5 5 1 1 []
5 5 2 0 ['aababcababcb', 'aababcbaabcb', 'aababcbabacb', 'aabcababcbab', 'aabcbaabcbab', 'aabcbabacbab', 'abababacbabc', 'abababcababc', 'ababacbababc']
6 2 2 2 []
6 3 2 1 []
6 3 3 0 ['aabaacbaacbc', 'aabaacbacabc', 'aabaacbcaabc', 'aabacabaacbc', 'aabacabacabc', 'aabacabcaabc', 'aabcaabcaabc', 'aabcaabcaacb', 'aabcaabcabac', 'aabcabacabac', 'abacabacabac']
6 4 1 1 []
6 4 2 0 ['aaabcabaabcb', 'aaabcababacb', 'aaabcbaaabcb', 'aaabcbaabacb', 'aabaacbababc', 'aabaaccabacc', 'aababaacbabc', 'aababacababc', 'aababcaababc', 'aababcaabacb', 'aababcabaabc', 'aabacabaabcb', 'aabacababacb', 'aabacacabacc', 'aabacbaabacb', 'aabacbaabcab', 'aabacbabaacb', 'aabaccaabacc', 'aabaccabaacc', 'aabbacaabbac', 'aabbacababac', 'aabcabaabcab', 'aabcabaacbab', 'aabcababacab', 'aabcacaabcac', 'aabcacabacac', 'ababacababac']
6 5 1 0 ['aabaababcbab', 'aabaabbaabcb', 'aabaabbacabb', 'aabaabcbaabb', 'aababaababcb', 'aababaabcbab', 'aabababaabcb', 'aabababababc', 'aababababacb', 'aababababcab', 'aababababcbb', 'aabababacabb', 'aabababacbab', 'aabababcabab', 'aabababcbabb', 'aababacababb', 'aababacbabab', 'aababcababab', 'aababcbababb', 'aabacabababb', 'aabacacacacc', 'aabacbababab', 'aabbacababab', 'aabcabababab', 'aabcacacacac', 'abababababac']
7 2 2 1 []
7 3 1 1 []
7 3 2 0 ['aaaabcaabacb', 'aaaabcabaacb', 'aaabaacbaabc', 'aaabacaabacb', 'aaabacabaabc', 'aaabacabaacb', 'aaabacbaaabc', 'aaabacbaaacb', 'aaabcaaabcab', 'aaabcaaacbac', 'aaabcaabaabc', 'aaabcaabaacb', 'aaabcaabacab', 'aaabcaacaabc', 'aaabcaacabac', 'aaabcaacbaac', 'aaabcabaacab', 'aabaabcaabac', 'aabaacabaabc', 'aabaacabaacb', 'aabaacabaacc', 'aabaacababac', 'aabaacabacac', 'aabaacacabac', 'aabaacbaabac', 'aabaaccaabac', 'aababaacabac', 'aababacaabac', 'aabacaabacab', 'aabacaabacac', 'aabacaacabac']
7 4 1 0 ['aaabaabaabcb', 'aaabaabacbab', 'aaabaabcabab', 'aaabaabcbaab', 'aaababaabacb', 'aaababaabcab', 'aaababacbaab', 'aaababcabaab', 'aaabacbaabab', 'aaabbaacaabb', 'aaabcabaabab', 'aaabcbaabaab', 'aabaabaababc', 'aabaabaabacb', 'aabaabaabcab', 'aabaabaacbab', 'aabaababaabc', 'aabaababaacb', 'aabaababacab', 'aabaabacabab', 'aabaabbaabac', 'aabaabcaabab', 'aabaacabaabb', 'aabaacacacac', 'aabaacbaabab', 'aababaababac', 'aababaabacab', 'aababaacabab', 'aabababaabac', 'aabababaacab']
8 2 1 1 []
8 2 2 0 ['aaaaabcaaacb', 'aaaabaacaabc', 'aaaabacaaabc', 'aaaabcaaaabc', 'aaaabcaaaacb', 'aaaabcaaabac', 'aaaabcaabaac', 'aaabaaacabac', 'aaabaacaabac', 'aaabacaaabac', 'aaabacaabaac', 'aabaacaabaac']
8 3 1 0 ['aaaabaaaabcb', 'aaaabaaabacb', 'aaaabaaabcab', 'aaaabaabaacb', 'aaaabaabacab', 'aaaabaabcaab', 'aaaabaacbaab', 'aaaabacabaab', 'aaaabacbaaab', 'aaaabcaabaab', 'aaaabcabaaab', 'aaabaaabaabc', 'aaabaaabaacb', 'aaabaaabacab', 'aaabaaabcaab', 'aaabaaacbaab', 'aaabaabaaabc', 'aaabaabaaacb', 'aaabaabaabac', 'aaabaabaacab', 'aaabaabacaab', 'aaabaacaabab', 'aaabaacaacac', 'aaabaacabaab', 'aaababaabaac', 'aaababaacaab', 'aaabacaabaab', 'aaabacaacaac', 'aabaabaabaac']
9 1 1 1 []
9 2 1 0 ['aaaaaaaaabcb', 'aaaaaaaabacb', 'aaaaaaaabcab', 'aaaaaaabaacb', 'aaaaaaabacab', 'aaaaaaabcaab', 'aaaaaabaaacb', 'aaaaaabaacab', 'aaaaaabacaab', 'aaaaaabcaaab', 'aaaaabaaaabc', 'aaaaabaaaacb', 'aaaaabaaaacc', 'aaaaabaaacab', 'aaaaabaacaab', 'aaaaabacaaab', 'aaaaabbaaaac', 'aaaaabcaaaab', 'aaaaabcaaaac', 'aaaabaaaabac', 'aaaabaaaacab', 'aaaabaaaacac', 'aaaabaaabaac', 'aaaabaaacaab', 'aaaabaaacaac', 'aaaabaabaaac', 'aaaabaacaaab', 'aaaabaacaaac', 'aaabaaabaaac']
10 1 1 0 ['aaaaaaaaaabc', 'aaaaaaaaabac', 'aaaaaaaabaac', 'aaaaaaabaaac', 'aaaaaabaaaac', 'aaaaabaaaaac']
4 3 3 3 []
4 4 3 2 []
4 4 4 1 []
5 3 3 2 []
5 4 2 2 []
5 4 3 1 []
5 4 4 0 ['aabacbacbacbc', 'aabacbacbcabc', 'aabacbcabacbc', 'aabacbcabcabc', 'aabcabacbacbc', 'aabcabacbcabc', 'aabcabcabacbc', 'aabcabcabcabc', 'aabcabcabcacb', 'aabcabcabcbac', 'aabcabcacbacb', 'aabcabcbacabc', 'aabcabcbacbac', 'aabcacbabcacb', 'aabcacbacbacb', 'aabcbacabcabc', 'aabcbacabcbac', 'aabcbacbacabc', 'aabcbacbacbac', 'abacabacbacbc', 'abacabacbcabc', 'abacabcabacbc', 'abacabcabcabc', 'abacabcbacabc', 'abacbacabcabc']
5 5 2 1 []
5 5 3 0 ['aabcababcabcb', 'aabcababcbacb', 'aabcabcbaabcb', 'aabcabcbabacb', 'aabcbaabcbacb', 'aabcbacbabacb', 'ababacbabacbc', 'ababacbabcabc', 'ababacbacbabc', 'ababacbcababc', 'ababcababcabc', 'ababcabacbabc']
6 3 2 2 []
6 3 3 1 []
6 4 2 1 []
6 4 3 0 ['aabacbaacbabc', 'aabcaabcabacb', 'aabcaabcbaacb', 'aabcaacbacabc', 'aabcabaacbabc', 'aabcababacabc', 'aabcabacababc', 'aabcabacabacb', 'aabcabacacabc', 'aabcacabacabc', 'aabcacabacacb', 'aabcbabcbabcb', 'ababacabacabc', 'ababacabacbac', 'ababacabcabac', 'ababacbacabac', 'ababcabacabac', 'ababcbabcbabc']
6 5 1 1 []
6 5 2 0 ['aabaabcbaabcb', 'aababacbababc', 'aababcabaabcb', 'aababcabababc', 'aababcababacb', 'aababcbaabacb', 'aababcbababcb', 'aabacbabaabcb', 'aabacbabababc', 'aabacbababacb', 'aabacbababcab', 'aabcabaabcbab', 'aabcabababacb', 'aabcabababcab', 'aabcababacbab', 'aabcacacabcac', 'aabcacacacbac', 'aabcacacbacac', 'aabcbabababcb', 'aabcbababcbab', 'aabcbcbcbcbcb', 'abababacababc', 'abababacbabac', 'abababcababac', 'abababcbababc']
6 6 1 0 ['aabababababcb', 'aababababcbab', 'aabababcbaabb', 'aabababcbabab', 'aababcbabaabb', 'aababcbababab', 'aabbaabcbabab', 'aabcbabababab', 'ababababababc']
7 2 2 2 []
7 3 2 1 []
7 3 3 0 ['aaabacabacabc', 'aaabacabcaabc', 'aaabcaabacabc', 'aaabcaabcaabc', 'aaabcaabcaacb', 'aaabcaabcabac', 'aaabcaacbaacb', 'aaabcabacaabc', 'aaabcabacabac', 'aabaacabaacbc', 'aabaacabacabc', 'aabaacabcaabc', 'aabaacbaacabc', 'aabaacbaacbac', 'aabaacbacaabc', 'aabaacbacabac', 'aabaacbcaabac', 'aabacaabacabc', 'aabacaabcaabc', 'aabacaabcabac', 'aabacabaacabc', 'aabacabaacbac', 'aabacabacaabc', 'aabacabacabac']
7 4 1 1 []
7 4 2 0 ['aaaabcbaaabcb', 'aaabacbaaabcb', 'aaabacbaabacb', 'aaabacbaabcab', 'aaabcabaaabcb', 'aaabcabaabacb', 'aaabcabaabcab', 'aabaabacababc', 'aabaabacbaabc', 'aabaabcaababc', 'aabaabcaabacb', 'aabaabcabaabc', 'aabaabcabaacb', 'aabaacbaababc', 'aabaacbaabacb', 'aabaacbaabcab', 'aabaacbabaabc', 'aabaacbabaacb', 'aabaacbabacab', 'aabaaccaabacc', 'aabaaccabaacc', 'aababaacbaabc', 'aababacabaabc', 'aababacababac', 'aabacabaabacb', 'aabacabaabcab', 'aabacababaacb', 'aabacabababac', 'aabacababacab', 'aabacacabaacc', 'aabacacabacac', 'aabacacacabac']
7 5 1 0 ['aaabababacbab', 'aaabababcabab', 'aaababacbabab', 'aaababcababab', 'aabaabaababcb', 'aabaabaabcbab', 'aabaababaabcb', 'aabaababacbab', 'aabaababcabab', 'aabaabcbaabab', 'aababaabababc', 'aababaababacb', 'aababaababcab', 'aababaabacbab', 'aababaabcabab', 'aababaacbabab', 'aabababaababc', 'aabababaabacb', 'aabababaabcab', 'aabababaacbab', 'aababababaabc', 'aababababaacb', 'aabababababac', 'aababababacab', 'aabababacaabb', 'aabababacabab', 'aababacabaabb', 'aababacababab', 'aababbaabacab', 'aababbabcbabb', 'aabacababaabb', 'aabacabababab', 'aabacacacaacc', 'aabacacacacac']
8 2 2 1 []
8 3 1 1 []
8 3 2 0 ['aaaaabcaabacb', 'aaaaabcabaacb', 'aaaabcaabaacb', 'aaabaaacbaabc', 'aaabaacabaabc', 'aaabaacbaaabc', 'aaabacaabaabc', 'aaabacaabaacb', 'aaabacaabacab', 'aaabacabaaabc', 'aaabacabaaacb', 'aaabacabaacab', 'aaabcaaacbaac', 'aaabcaabaacab', 'aaabcaacaabac', 'aaabcaacabaac', 'aabaabaacabac', 'aabaabacaabac', 'aabaacabaabac', 'aabaacabaacab', 'aabaacabaacac', 'aabaacacaabac']
8 4 1 0 ['aaabaaabaabcb', 'aaabaaabcbaab', 'aaabaabaaabcb', 'aaabaabaabacb', 'aaabaabaabcab', 'aaabaabacbaab', 'aaabaabcabaab', 'aaababaaabacb', 'aaababaaabcab', 'aaabacbaabaab', 'aaabcabaabaab', 'aabaabaabaabc', 'aabaabaabaacb', 'aabaabaababac', 'aabaabaabacab', 'aabaabaacabab', 'aabaababaabac', 'aabaababaacab', 'aabaabacaabab', 'aabaacabaabab', 'aabaacacaacac']
9 2 1 1 []
9 2 2 0 ['aaaaaabcaaacb', 'aaaaabacaaabc', 'aaaaabcaaaabc', 'aaaaabcaaaacb', 'aaaaabcaaabac', 'aaaabaaacaabc', 'aaaabaaacabac', 'aaaabaacaaabc', 'aaaabaacaabac', 'aaaabacaaaabc', 'aaaabacaaabac', 'aaaabacaaacab', 'aaaabacaabaac', 'aaaabacabaaac', 'aaaabcaaabaac', 'aaaabcaabaaac', 'aaabaaacaabac', 'aaabaaacabaac', 'aaabaacaaabac', 'aaabaacaabaac']
9 3 1 0 ['aaaaabaaaabcb', 'aaaaabcbaaaab', 'aaaabaaaabacb', 'aaaabaaaabcab', 'aaaabaaabaacb', 'aaaabaaabacab', 'aaaabaaabcaab', 'aaaabaacbaaab', 'aaaabacabaaab', 'aaaabcaabaaab', 'aaabaaabaaabc', 'aaabaaabaaacb', 'aaabaaabaabac', 'aaabaaabaacab', 'aaabaaabacaab', 'aaabaaacabaab', 'aaabaabaaabac', 'aaabaabaaacab', 'aaabaabaabaac', 'aaabaabaacaab', 'aaabaacaaabab', 'aaabaacaaacac', 'aaabaacaabaab', 'aaabaacaacaac']
10 1 1 1 []
10 2 1 0 ['aaaaaaaaaabcb', 'aaaaaaaaabacb', 'aaaaaaaaabcab', 'aaaaaaaabaacb', 'aaaaaaaabacab', 'aaaaaaaabcaab', 'aaaaaaabaaacb', 'aaaaaaabaacab', 'aaaaaaabacaab', 'aaaaaaabcaaab', 'aaaaaabaaaacb', 'aaaaaabaaacab', 'aaaaaabaacaab', 'aaaaaabacaaab', 'aaaaaabcaaaab', 'aaaaabaaaaabc', 'aaaaabaaaaacb', 'aaaaabaaaaacc', 'aaaaabaaaabac', 'aaaaabaaaacab', 'aaaaabaaaacac', 'aaaaabaaacaab', 'aaaaabaacaaab', 'aaaaababaaaac', 'aaaaabacaaaab', 'aaaaabacaaaac', 'aaaabaaaabaac', 'aaaabaaaacaab', 'aaaabaaaacaac', 'aaaabaaabaaac', 'aaaabaaacaaab', 'aaaabaaacaaac']
11 1 1 0 ['aaaaaaaaaaabc', 'aaaaaaaaaabac', 'aaaaaaaaabaac', 'aaaaaaaabaaac', 'aaaaaaabaaaac', 'aaaaaabaaaaac']
4 4 3 3 []
4 4 4 2 ['aabcdcbaabcdcb']
5 3 3 3 []
5 4 3 2 []
5 4 4 1 []
5 5 2 2 []
5 5 3 1 []
5 5 4 0 ['aabcabcabcabcb', 'aabcabcabcbacb', 'aabcabcbacabcb', 'aabcabcbacbacb', 'aabcbacabcbacb', 'aabcbacbacbacb', 'ababacbacbacbc', 'ababacbacbcabc', 'ababacbcabacbc', 'ababacbcabcabc', 'ababcabacbacbc', 'ababcabacbcabc', 'ababcabcabacbc', 'ababcabcabcabc', 'ababcabcacbabc', 'ababcacbabcabc', 'ababcacbacbabc', 'abacbabcabacbc', 'abacbabcabcabc', 'abacbacbabcabc']
6 3 3 2 []
6 4 2 2 ['abacabadbacabd', 'abacabdabacabd', 'abacadcabacadc']
6 4 3 1 []
6 4 4 0 ['aaabccbaaabccb', 'aabacbaacbacbc', 'aabacbaacbcabc', 'aabacbacabacbc', 'aabacbacabcabc', 'aabacbcaabacbc', 'aabacbcaabcabc', 'aabacbcabaacbc', 'aabcabaacbacbc', 'aabcabaacbcabc', 'aabcabacabacbc', 'aabcabacabcabc', 'aabcabcaabcabc', 'aabcabcaabcacb', 'aabcabcaabcbac', 'aabcabcaacbabc', 'aabcabcabacabc', 'aabcabcabacbac', 'aabcacbaabcacb', 'aabcbacaabcbac', 'aabcbacabacabc', 'aabcbacabacbac', 'aabcbcbaabcbcb', 'abacabacbacabc', 'abacabcabacabc']
6 5 2 1 []
6 5 3 0 ['aabacbabacbabc', 'aabacbabcababc', 'aabcaabcbaabcb', 'aabcaacbcaacbc', 'aabcababcababc', 'aabcababcabacb', 'aabcabacbaabcb', 'aabcabacbababc', 'aabcabacbabacb', 'aabcacabcacabc', 'aabcacabcacacb', 'aabcacabcacbac', 'aabcacacbacabc', 'aabcacacbacacb', 'aabcacbacacbac', 'aabcbaabcbabcb', 'aabcbababcabcb', 'aabcbabacbabcb', 'aabcbabcababcb', 'aabcbacbababcb', 'ababacabcababc', 'ababacbabacabc', 'ababacbabacbac', 'ababacbabcbabc', 'ababacbacababc', 'ababcababcabac', 'ababcababcbabc', 'ababcabacbabac']
6 6 1 1 []
6 6 2 0 ['aababcabababcb', 'aababcbaababcb', 'aababcbabaabcb', 'aabcbabaabcbab', 'aabcbababacbab', 'abababacbababc', 'abababcabababc']
7 3 2 2 []
7 3 3 1 []
7 4 2 1 []
7 4 3 0 ['aaabcaabcabacb', 'aaabcabacabacb', 'aaabcabacbaacb', 'aababacabacabc', 'aababcaabcaabc', 'aabacabaacbabc', 'aabacababacabc', 'aabacababcaabc', 'aabacabacababc', 'aabacabacabacb', 'aabacabacabacc', 'aabacabacacabc', 'aabacabacbaabc', 'aabacabacbaacb', 'aabacacabacabc', 'aabacbaabcaabc', 'aabacbaacbaabc', 'aabacbaacbaacb', 'aabbacabacabac', 'aabcaabcaabcab', 'aabcaabcaabcac', 'aabcaabcaacbac', 'aabcaabcabaacb', 'aabcaabcabacab', 'aabcaabcacabac', 'aabcaacbacabac', 'aabcababacabac', 'aabcabacababac', 'aabcabacabacab', 'aabcabacabacac', 'aabcabacacabac', 'aabcacabacabac', 'ababacabacabac']
7 5 1 1 []
7 5 2 0 ['aaababcababacb', 'aaabcabababacb', 'aaabcababacbab', 'aaabcbaabaabcb', 'aabaabcabaabcb', 'aabaabcababacb', 'aabaabcbaabacb', 'aababaacbababc', 'aabababacababc', 'aabababcaababc', 'aababacabababc', 'aababacababacb', 'aababacbaababc', 'aababacbaabacb', 'aababcaababcab', 'aababcabaabacb', 'aababcabaabcab', 'aababcbababcbb', 'aabacabababacb', 'aabacababacabb', 'aabacacabacacc', 'aabacacacabacc', 'aabacbaabacbab', 'aabacbabaabcab', 'aabacbabaacbab', 'aabbacabababac', 'aabbacababacab', 'aabcabaabcabab', 'aabcababaacbab', 'aabcabababacab', 'aabcababacabab', 'aabcacaabcacac', 'aabcacabacacac', 'aabcacacabacac', 'abababacababac']
7 6 1 0 ['aababaabababcb', 'aababaababcbab', 'aababaabcbabab', 'aabababaababcb', 'aabababaabcbab', 'aababababaabcb', 'aababababababc', 'aabababababacb', 'aabababababcab', 'aabababababcbb', 'aababababacabb', 'aababababacbab', 'aababababcabab', 'aababababcbabb', 'aabababacababb', 'aabababacbabab', 'aabababcababab', 'aabababcbababb', 'aababacabababb', 'aababacbababab', 'aababcabababab', 'aababcbabababb', 'aabacababababb', 'aabacacacacacc', 'aabacbabababab', 'aabaccaaccaacc', 'aabbaabbaabbac', 'aabbacabababab', 'aabcababababab', 'aabcacacacacac', 'ababababababac']
8 2 2 2 []
8 3 2 1 []
8 3 3 0 ['aaabaacbaacabc', 'aaabaacbacaabc', 'aaabacaabacabc', 'aaabacaabcaabc', 'aaabacabaacabc', 'aaabacabacaabc', 'aaabacabcaaabc', 'aaabcaaabcaabc', 'aaabcaaabcaacb', 'aaabcaaabcabac', 'aaabcaaacbaabc', 'aaabcaabaacabc', 'aaabcaabacaabc', 'aaabcaabacabac', 'aaabcaabacbaac', 'aaabcaabcaabac', 'aaabcabaacaabc', 'aaabcabaacabac', 'aaabcabaacbaac', 'aaabcabacaabac', 'aabaacabaacabc', 'aabaacabaacbac', 'aabaacabacaabc', 'aabaacabacabac', 'aabaacabcaabac', 'aabaacbaacabac', 'aabaacbacaabac', 'aabacaabacaabc', 'aabacaabacabac', 'aabacabaacabac']
8 4 1 1 []
8 4 2 0 ['aaaabcabaaabcb', 'aaaabcbaaaabcb', 'aaaabcbaaabacb', 'aaabacabaabacb', 'aaabacbaaabacb', 'aaabacbaaabcab', 'aaabacbaabaacb', 'aaabcaabaabcab', 'aaabcabaaabcab', 'aaabcabaabacab', 'aabaabaacbaabc', 'aabaabacabaabc', 'aabaabcaabaabc', 'aabaabcaabaacb', 'aabaabcaababac', 'aabaacababaacb', 'aabaacacabaacc', 'aabaacacabacac', 'aabaacbaabaacb', 'aabaacbaabacab', 'aabaaccaabaacc', 'aabaaccaabacac', 'aababaacababac', 'aababacaababac', 'aababacabaabac', 'aabacaabbaacab', 'aabacaacabacac', 'aabacabaabacab', 'aabacacaabacac']
8 5 1 0 ['aaababaabacbab', 'aaababaabcabab', 'aaababacbaabab', 'aaababcabaabab', 'aabaabaabaabcb', 'aabaabaababacb', 'aabaabaababcab', 'aabaabaabacbab', 'aabaabaabcabab', 'aabaababaababc', 'aabaababaabacb', 'aabaababaabcab', 'aabaababaacbab', 'aabaababacabab', 'aabaababcaabab', 'aabaabacbaabab', 'aabaabcabaabab', 'aabaacacacacac', 'aabaacbabaabab', 'aababaababaabc', 'aababaababaacb', 'aababaabababac', 'aababaababacab', 'aababaabacabab', 'aababaacababab', 'aabababaababac', 'aabababaabacab', 'aabababaacabab', 'aababababaabac', 'aababababaacab']
9 2 2 1 []
9 3 1 1 []
9 3 2 0 ['aaaaabcaabaacb', 'aaaabacaabaacb', 'aaaabacabaaacb', 'aaaabcaaabaacb', 'aaaabcaaabacab', 'aaaabcaabaaacb', 'aaaabcaabaacab', 'aaabaaacbaaabc', 'aaabaacabaaabc', 'aaabaacabaaacb', 'aaabaacabaabac', 'aaabacaabaaabc', 'aaabacaabaaacb', 'aaabacaabaabac', 'aaabacaabaacab', 'aaabacaacaabac', 'aaabacaacabaac', 'aabaabaacaabac', 'aabaabaacabaac', 'aabaabacaabaac', 'aabaacaabaacab', 'aabaacaabaacac']
9 4 1 0 ['aaabaaabaaabcb', 'aaabaaabaabacb', 'aaabaaabaabcab', 'aaabaaabacbaab', 'aaabaaabcabaab', 'aaabaabaaabacb', 'aaabaabaaabcab', 'aaabaabaabaabc', 'aaabaabaabaacb', 'aaabaabaabacab', 'aaabaabaabcaab', 'aaabaabaacbaab', 'aaabaabacabaab', 'aaabaabcaabaab', 'aaabaacbaabaab', 'aaababaacaabab', 'aaabacabaabaab', 'aaabcaabaabaab', 'aaabcaacaacaac', 'aabaabaabaabac', 'aabaabaabaacab', 'aabaabaababaac', 'aabaabaacaabab', 'aabaababaabaac']
10 2 1 1 []
10 2 2 0 ['aaaaaabcaaaacb', 'aaaaabaacaaabc', 'aaaaabacaaaabc', 'aaaaabcaaaaabc', 'aaaaabcaaaaacb', 'aaaaabcaaaabac', 'aaaaabcaaabaac', 'aaaabaaaacabac', 'aaaabaaacaabac', 'aaaabaacaaabac', 'aaaabacaaaabac', 'aaaabacaaabaac', 'aaaabacaabaaac', 'aaabaaacaabaac', 'aaabaacaaabaac']
10 3 1 0 ['aaaaabaaaaabcb', 'aaaaabaaaabacb', 'aaaaabaaaabcab', 'aaaaabacbaaaab', 'aaaaabcabaaaab', 'aaaabaaaabaacb', 'aaaabaaaabacab', 'aaaabaaaabcaab', 'aaaabaaabaaabc', 'aaaabaaabaaacb', 'aaaabaaabaacab', 'aaaabaaabacaab', 'aaaabaaabcaaab', 'aaaabaaacbaaab', 'aaaabaacabaaab', 'aaaabacaabaaab', 'aaaabcaaabaaab', 'aaaabcaaacaaac', 'aaabaaabaaabac', 'aaabaaabaaacab', 'aaabaaabaabaac', 'aaabaaabaacaab', 'aaabaaacaabaab', 'aaabaaacaacaac', 'aaabaabaaabaac', 'aaabaabaaacaab']
11 1 1 1 []
11 2 1 0 ['aaaaaaaaaaabcb', 'aaaaaaaaaabacb', 'aaaaaaaaaabcab', 'aaaaaaaaabaacb', 'aaaaaaaaabacab', 'aaaaaaaaabcaab', 'aaaaaaaabaaacb', 'aaaaaaaabaacab', 'aaaaaaaabacaab', 'aaaaaaaabcaaab', 'aaaaaaabaaaacb', 'aaaaaaabaaacab', 'aaaaaaabaacaab', 'aaaaaaabacaaab', 'aaaaaaabcaaaab', 'aaaaaabaaaaabc', 'aaaaaabaaaaacb', 'aaaaaabaaaaacc', 'aaaaaabaaaacab', 'aaaaaabaaacaab', 'aaaaaabaacaaab', 'aaaaaabacaaaab', 'aaaaaabbaaaaac', 'aaaaaabcaaaaab', 'aaaaaabcaaaaac', 'aaaaabaaaaabac', 'aaaaabaaaaacab', 'aaaaabaaaaacac', 'aaaaabaaaabaac', 'aaaaabaaaacaab', 'aaaaabaaaacaac', 'aaaaabaaacaaab', 'aaaaabaabaaaac', 'aaaaabaacaaaab', 'aaaaabaacaaaac', 'aaaabaaaabaaac', 'aaaabaaaacaaab', 'aaaabaaaacaaac']
12 1 1 0 ['aaaaaaaaaaaabc', 'aaaaaaaaaaabac', 'aaaaaaaaaabaac', 'aaaaaaaaabaaac', 'aaaaaaaabaaaac', 'aaaaaaabaaaaac', 'aaaaaabaaaaaac']
4 4 4 3 []
5 4 3 3 []
5 4 4 2 []
5 5 3 2 []
5 5 4 1 []
5 5 5 0 ['abacbacbacbacbc', 'abacbacbacbcabc', 'abacbacbcabacbc', 'abcabcabcabcabc']
6 3 3 3 ['abacdabacdabacd', 'abacdabacdabadc']
6 4 3 2 []
6 4 4 1 []
6 5 2 2 []
6 5 3 1 []
6 5 4 0 ['aabacbacbabcabc', 'aabcabacbabcabc', 'aabcabacbacbabc', 'aabcabcaabcbacb', 'aabcabcaacbacbc', 'aabcabcababcabc', 'aabcabcababcacb', 'aabcabcabacabcb', 'aabcabcabacbabc', 'aabcabcabacbacb', 'aabcabcacabcabc', 'aabcabcacbacabc', 'aabcabcacbacbac', 'aabcabcbabcabcb', 'aabcabcbabcbacb', 'aabcabcbacbabcb', 'aabcacbabacbacb', 'aabcacbacabcabc', 'aabcacbacabcacb', 'aabcacbacbacabc', 'aabcbabcabcabcb', 'aabcbabcabcbacb', 'aabcbacabacabcb', 'aabcbacabacbacb', 'aabcbacbabcabcb', 'aabcbacbabcbacb', 'aabcbacbacbabcb', 'ababacbacabcabc', 'ababacbacbacabc', 'ababcabacabcabc', 'ababcabacbacabc', 'ababcabacbacbac', 'ababcabcabacabc', 'ababcabcabacbac', 'ababcabcabcbabc', 'ababcabcbabcabc', 'ababcabcbacbabc', 'ababcbacbabcabc', 'ababcbacbacbabc', 'abacabacbabcabc', 'abacabacbacbabc', 'abacabcabacbabc', 'abacabcacbacabc']
6 6 2 1 []
6 6 3 0 ['aabcababcababcb', 'aabcababcbaabcb', 'aabcbaabcbaabcb', 'aabcbaabcbabacb', 'aabcbabacbabacb', 'ababacbabacbabc', 'ababacbabcababc', 'ababcababcababc']
7 3 3 2 []
7 4 2 2 []
7 4 3 1 []
7 4 4 0 ['aabacabacabacbc', 'aabacabacabcabc', 'aabacabacbaacbc', 'aabacabacbacabc', 'aabacabacbcaabc', 'aabacabcaabacbc', 'aabacabcaabcabc', 'aabacabcabaacbc', 'aabacabcabacabc', 'aabacabcabcaabc', 'aabacbaacbaacbc', 'aabacbaacbacabc', 'aabacbaacbcaabc', 'aabacbacabaacbc', 'aabacbacabacabc', 'aabacbacabcaabc', 'aabacbcaabcaabc', 'aabcaabcaabcabc', 'aabcaabcaabcacb', 'aabcaabcaabcbac', 'aabcaabcaacbabc', 'aabcaabcabaacbc', 'aabcaabcabacabc', 'aabcaabcabacbac', 'aabcaabcabcabac', 'aabcaabcacbaacb', 'aabcaabcbaacbac', 'aabcaabcbacabac', 'aabcaacbaabcacb', 'aabcabaacbacabc', 'aabcabacaabcabc', 'aabcabacaabcbac', 'aabcabacabaacbc', 'aabcabacabacabc', 'aabcabacabacbac', 'aabcabacabcabac', 'aabcabacbacabac', 'aabcabcabacabac', 'aabcbacabacabac', 'abacabacabacabc']
7 5 2 1 []
7 5 3 0 ['aaabcababcabacb', 'aaabcabacbaabcb', 'aaabcabacbabacb', 'aaabcabcbaaabcb', 'aaabcbaaabcbacb', 'aaabcbaabcabacb', 'aababcaabacbabc', 'aababcaabcababc', 'aababcaabcabacb', 'aababcabaacbabc', 'aababcabacababc', 'aababcabacbaabc', 'aababcbabcbabcb', 'aabacababcabacb', 'aabacbaabacbabc', 'aabacbaabcaabcb', 'aabacbaabcababc', 'aabacbaabcabacb', 'aabacbaabcbaacb', 'aabacbaacbababc', 'aabacbaacbabacb', 'aabacbabaacbabc', 'aabacbabacababc', 'aabacbabacabacb', 'aabaccabacabacc', 'aabbacabacabbac', 'aabcaabcbaabcab', 'aabcaacbacaacbc', 'aabcaacbacabcac', 'aabcabaabcababc', 'aabcabaabcabacb', 'aabcabaacbababc', 'aabcabaacbabacb', 'aabcabaacbabcab', 'aabcababacababc', 'aabcababacabacb', 'aabcabacababacb', 'aabcabacababcab', 'aabcabacbaacbab', 'aabcabacbabacab', 'aabcacaabcacabc', 'aabcacaabcacbac', 'aabcacabacabcac', 'aabcacabacacabc', 'aabcacabacacbac', 'aabcbababcbabcb', 'aabcbabcbababcb', 'aabcbabcbabcbab', 'ababacababacabc', 'ababacababacbac', 'ababacababcabac', 'ababacabacababc', 'ababacabacbabac', 'ababacbabacabac']
7 6 1 1 []
7 6 2 0 ['aabaabcbaababcb', 'aabaabcbabaabcb', 'aabababacbababc', 'aabababcabaabcb', 'aabababcabababc', 'aabababcababacb', 'aabababcbababcb', 'aababacbaababcb', 'aababacbabaabcb', 'aababacbabababc', 'aababacbababacb', 'aababacbababcab', 'aababcabaababcb', 'aababcabaabcbab', 'aababcababaabcb', 'aababcabababacb', 'aababcabababcab', 'aababcababacbab', 'aababcbaabacbab', 'aababcbabaabacb', 'aababcbabaabcab', 'aababcbabababcb', 'aababcbababcbab', 'aabacbabaabcbab', 'aabacbababaabcb', 'aabacbabababcab', 'aabacbababacbab', 'aabacbababcabab', 'aabbaabcbababcb', 'aabcababaabcbab', 'aabcabababacbab', 'aabcabababcabab', 'aabcababacbabab', 'aabcacacabcacac', 'aabcacacacbacac', 'aabcacacbacacac', 'aabcbabababcbab', 'aabcbababcbabab', 'aabcbcbcbcbcbcb', 'abababacabababc', 'abababacbababac']
7 7 1 0 ['aababababababcb', 'aabababababcbab', 'aababababcbaabb', 'aababababcbabab', 'aabababcbabaabb', 'aabababcbababab', 'aababbaababcbab', 'aababbabacababb', 'aababcbababaabb', 'aababcbabababab', 'aabbaabcbababab', 'aabcbababababab', 'abababababababc']
8 3 2 2 []
8 3 3 1 []
8 4 2 1 []
8 4 3 0 ['aaabcaabacabacb', 'aaabcaabacbaacb', 'aaabcaabcaabacb', 'aaabcaabcabaacb', 'aaabcabaacabacb', 'aaabcabaacbaacb', 'aaabcabacaabacb', 'aaabcabacabaacb', 'aabaacbaabcaabc', 'aabaacbaacbaabc', 'aababacabacaabc', 'aababacabacabac', 'aabacabaabcaabc', 'aabacabaabcaacb', 'aabacabaacbaabc', 'aabacabaacbaacb', 'aabacababacaabc', 'aabacababacabac', 'aabacabacabaabc', 'aabacabacabaacb', 'aabacabacabaacc', 'aabacabacababac', 'aabacabacabacab', 'aabacabacabacac', 'aabacabacacaabc', 'aabacabacacabac', 'aabacacabacaabc', 'aabacacabacabac', 'aabacacabacacab']
8 5 1 1 []
8 5 2 0 ['aaabaabcabaabcb', 'aaabaabcbaaabcb', 'aaababcabaabacb', 'aaabacbaabaabcb', 'aaabcabaabacbab', 'aaabcbaaabcbaab', 'aaabcbaabaabcab', 'aaabcbaabacbaab', 'aabaabacbaababc', 'aabaabcaabaabcb', 'aabaabcaababacb', 'aabaabcabaababc', 'aabaabcabaabacb', 'aabaabcababaacb', 'aabaabcbaabaacb', 'aabaacbabaababc', 'aabaacbabaabacb', 'aabaacbabaabcab', 'aababaacbaababc', 'aababaacbabaabc', 'aabababacabaabc', 'aabababacababac', 'aababacabaababc', 'aababacabaabacb', 'aababacabaabcab', 'aababacababaabc', 'aababacababaacb', 'aababacabababac', 'aababacababacab', 'aabacababaabacb', 'aabacababaabcab', 'aabacababaacbab', 'aabacabababaacb', 'aabacabababacab', 'aabacababacaabb', 'aabacababacabab', 'aabacacabacaacc', 'aabacacabacacac', 'aabacacacabaacc', 'aabacacacabacac']
8 6 1 0 ['aaabababacbabab', 'aaabababcababab', 'aabaabaababcbab', 'aabaababaababcb', 'aabaababaabcbab', 'aabaabababaabcb', 'aabaabababacbab', 'aabaabababcabab', 'aabaababacbabab', 'aabaababcababab', 'aabaababcbaabab', 'aabaabbaabaabcb', 'aabaabcbaababab', 'aabaabcbabaabab', 'aababaababaabcb', 'aababaabababacb', 'aababaabababcab', 'aababaababacbab', 'aababaababcabab', 'aababaabacbabab', 'aababaabcababab', 'aabababaabababc', 'aabababaababacb', 'aabababaababcab', 'aabababaabacbab', 'aabababaabcabab', 'aabababaacbabab', 'aababababaababc', 'aababababaabacb', 'aababababaabcab', 'aababababaacbab', 'aabababababaabc', 'aabababababaacb', 'aababababababac', 'aabababababacab', 'aababababacaabb', 'aababababacabab', 'aabababacabaabb', 'aabababacababab', 'aababacabaababb', 'aababacabaabbab', 'aababacababaabb', 'aababacabababab', 'aababbaabacabab', 'aababbabcbababb', 'aabacabababaabb', 'aabacababababab', 'aabacacacacaacc', 'aabacacacacacac']
9 2 2 2 []
9 3 2 1 []
9 3 3 0 ['aaabaacabaacabc', 'aaabaacabacaabc', 'aaabaacabcaaabc', 'aaabacaabaacabc', 'aaabacaabacaabc', 'aaabacaabcaaabc', 'aaabcaaabcaaabc', 'aaabcaaabcaaacb', 'aaabcaaabcaabac', 'aaabcaaabcabaac', 'aaabcaabacaabac', 'aaabcaabacabaac', 'aaabcabaacaabac', 'aaabcabaacabaac', 'aabaacabaacabac', 'aabaacabacaabac', 'aabacaabacaabac']
9 4 1 1 []
9 4 2 0 ['aaaaabcbaaaabcb', 'aaaabacbaaaabcb', 'aaaabacbaaabacb', 'aaaabacbaaabcab', 'aaaabcabaaaabcb', 'aaaabcabaaabacb', 'aaaabcabaaabcab', 'aaabaabacabaabc', 'aaabaabcaabaabc', 'aaabaabcaabaacb', 'aaabaacbaaabacb', 'aaabaacbaaabcab', 'aaabaacbaabaabc', 'aaabaacbaabaacb', 'aaabaacbaabacab', 'aaabaaccaabaacc', 'aaabacabaaabacb', 'aaabacabaaabcab', 'aaabacabaabaacb', 'aaabacabaabacab', 'aaabacabaabcaab', 'aaabacbaaabcaab', 'aaabbaacaabbaac', 'aaabcaabaaabcab', 'aaabcaabaabaacb', 'aaabcaabaabacab', 'aaabcaabaabcaab', 'aaabcaabaacbaab', 'aaabcaacaabcaac', 'aaabcaacaacbaac', 'aaabcaacabacaac', 'aabaabaacabaabc', 'aabaabaacababac', 'aabaabaacbaabac', 'aabaabacaabaabc', 'aabaabacaababac', 'aabaabacabaabac', 'aabaacabaabaacb', 'aabaacabaababac', 'aabaacabaabacab', 'aabaacababaabac', 'aabaacababaacab', 'aabaacacaabaacc', 'aabaacacaabacac', 'aabaacacabaacac', 'aababaacabaabac']
9 5 1 0 ['aaabaabaabaabcb', 'aaabaabaabacbab', 'aaabaabaabcabab', 'aaabaabaabcbaab', 'aaabaabacbaabab', 'aaabaabcabaabab', 'aaabaabcbaabaab', 'aaababaaababacb', 'aaababaaababcab', 'aaababaaabacbab', 'aaababaaabcabab', 'aaababaabaabacb', 'aaababaabaabcab', 'aaababaabacbaab', 'aaababaabcabaab', 'aaababacbaabaab', 'aaababcabaabaab', 'aaabacbaabaabab', 'aaabcabaabaabab', 'aaabcbaabaabaab', 'aabaabaabaababc', 'aabaabaabaabacb', 'aabaabaabaabcab', 'aabaabaabaacbab', 'aabaabaababaabc', 'aabaabaababaacb', 'aabaabaababacab', 'aabaabaabacabab', 'aabaabaabcaabab', 'aabaabaacbaabab', 'aabaababaabaabc', 'aabaababaabaacb', 'aabaababaababac', 'aabaababaabacab', 'aabaababaacabab', 'aabaabababaabac', 'aabaabababaacab', 'aabaababacaabab', 'aabaabacaabaabb', 'aabaabacaababab', 'aabaabacabaabab', 'aabaabbaabaacab', 'aabaacabaababab', 'aabaacababaabab', 'aabaacacaacacac', 'aabaacacacaacac', 'aababaababaabac', 'aababaababaacab']
10 2 2 1 []
10 3 1 1 []
10 3 2 0 ['aaaaaabcaabaacb', 'aaaaabcaaabaacb', 'aaaaabcaabaaacb', 'aaaabaaacbaaabc', 'aaaabaacabaaabc', 'aaaabaacbaaaabc', 'aaaabacaaabaacb', 'aaaabacaaabacab', 'aaaabacaabaaabc', 'aaaabacaabaaacb', 'aaaabacaabaacab', 'aaaabacabaaaabc', 'aaaabacabaaaacb', 'aaaabacabaaacab', 'aaaabcaaaacbaac', 'aaaabcaaabaaacb', 'aaaabcaaabaacab', 'aaaabcaaacaabac', 'aaaabcaaacabaac', 'aaaabcaaacbaaac', 'aaaabcaabaaacab', 'aaabaaacabaaabc', 'aaabaaacabaabac', 'aaabaaacbaaabac', 'aaabaabaacaabac', 'aaabaabacaaabac', 'aaabaacaabaaabc', 'aaabaacaabaaacb', 'aaabaacaabaaacc', 'aaabaacaabaabac', 'aaabaacaabaacab', 'aaabaacaabaacac', 'aaabaacaacaabac', 'aaabaacabaaabac', 'aaabaacabaaacab', 'aaababaacaabaac', 'aaabacaaabacaab', 'aaabacaaabacaac', 'aaabacaaacabaac', 'aaabacaabaabaac', 'aaabacaabaacaab', 'aaabacaabaacaac', 'aaabacaacaabaac', 'aabaabaacaabaac']
10 4 1 0 ['aaaabaaabaaabcb', 'aaaabaaabcbaaab', 'aaaabaabaacbaab', 'aaaabaabacabaab', 'aaaabaabcaabaab', 'aaaabcbaaabaaab', 'aaabaaabaaabacb', 'aaabaaabaaabcab', 'aaabaaabaabaacb', 'aaabaaabaabacab', 'aaabaaabaabcaab', 'aaabaaabaacbaab', 'aaabaaabacabaab', 'aaabaaabcaabaab', 'aaabaabaaabaabc', 'aaabaabaaabaacb', 'aaabaabaaabacab', 'aaabaabaaabcaab', 'aaabaabaaacbaab', 'aaabaabaabaaabc', 'aaabaabaabaaacb', 'aaabaabaabaabac', 'aaabaabaabaacab', 'aaabaabaabacaab', 'aaabaabaacaabab', 'aaabaabaacabaab', 'aaabaabacaabaab', 'aaabaacaabaaabb', 'aaabaacaabaabab', 'aaabaacaacaacac', 'aaabaacabaabaab', 'aaabaacacaaacac', 'aaababaaababaac', 'aaababaabaabaac', 'aaababaabaacaab', 'aaababaacaabaab', 'aaabacaabaabaab', 'aaabacaacaacaac', 'aabaabaabaabaac']
11 2 1 1 []
11 2 2 0 ['aaaaaaabcaaaacb', 'aaaaaabacaaaabc', 'aaaaaabcaaaaabc', 'aaaaaabcaaaaacb', 'aaaaaabcaaaabac', 'aaaaabaaacaaabc', 'aaaaabaaacaabac', 'aaaaabaacaaaabc', 'aaaaabaacaaabac', 'aaaaabacaaaaabc', 'aaaaabacaaaabac', 'aaaaabacaaaacab', 'aaaaabacaaabaac', 'aaaaabacaabaaac', 'aaaaabcaaaabaac', 'aaaaabcaaabaaac', 'aaaabaaaacaabac', 'aaaabaaaacabaac', 'aaaabaaacaaabac', 'aaaabaaacaabaac', 'aaaabaacaaaabac', 'aaaabaacaaabaac', 'aaaabaacaabaaac', 'aaaabacaaabaaac', 'aaabaaacaaabaac']
11 3 1 0 ['aaaaaabaaaaabcb', 'aaaaaabcbaaaaab', 'aaaaabaaaaabacb', 'aaaaabaaaaabcab', 'aaaaabaaaabaacb', 'aaaaabaaaabacab', 'aaaaabaaaabcaab', 'aaaaabaaabaaacb', 'aaaaabaaabaacab', 'aaaaabaaabacaab', 'aaaaabaaabcaaab', 'aaaaabaaacbaaab', 'aaaaabaacabaaab', 'aaaaabaacbaaaab', 'aaaaabacaabaaab', 'aaaaabacabaaaab', 'aaaaabcaaabaaab', 'aaaaabcaabaaaab', 'aaaabaaaabaaabc', 'aaaabaaaabaaacb', 'aaaabaaaabaacab', 'aaaabaaaabacaab', 'aaaabaaaabcaaab', 'aaaabaaaacbaaab', 'aaaabaaabaaaabc', 'aaaabaaabaaaacb', 'aaaabaaabaaabac', 'aaaabaaabaaacab', 'aaaabaaabaacaab', 'aaaabaaabacaaab', 'aaaabaaacabaaab', 'aaaabaacaabaaab', 'aaaabacaaabaaab', 'aaaabacaaacaaac', 'aaabaaabaaabaac', 'aaabaaabaaacaab', 'aaabaaabaabaaac', 'aaabaaacaaabaab']
12 1 1 1 []
12 2 1 0 ['aaaaaaaaaaaabcb', 'aaaaaaaaaaabacb', 'aaaaaaaaaaabcab', 'aaaaaaaaaabaacb', 'aaaaaaaaaabacab', 'aaaaaaaaaabcaab', 'aaaaaaaaabaaacb', 'aaaaaaaaabaacab', 'aaaaaaaaabacaab', 'aaaaaaaaabcaaab', 'aaaaaaaabaaaacb', 'aaaaaaaabaaacab', 'aaaaaaaabaacaab', 'aaaaaaaabacaaab', 'aaaaaaaabcaaaab', 'aaaaaaabaaaaacb', 'aaaaaaabaaaacab', 'aaaaaaabaaacaab', 'aaaaaaabaacaaab', 'aaaaaaabacaaaab', 'aaaaaaabcaaaaab', 'aaaaaabaaaaaabc', 'aaaaaabaaaaaacb', 'aaaaaabaaaaaacc', 'aaaaaabaaaaabac', 'aaaaaabaaaaacab', 'aaaaaabaaaaacac', 'aaaaaabaaaacaab', 'aaaaaabaaacaaab', 'aaaaaabaacaaaab', 'aaaaaababaaaaac', 'aaaaaabacaaaaab', 'aaaaaabacaaaaac', 'aaaaabaaaaabaac', 'aaaaabaaaaacaab', 'aaaaabaaaaacaac', 'aaaaabaaaabaaac', 'aaaaabaaaacaaab', 'aaaaabaaaacaaac', 'aaaaabaaabaaaac', 'aaaaabaaacaaaab', 'aaaaabaaacaaaac', 'aaaabaaaabaaaac']
13 1 1 0 ['aaaaaaaaaaaaabc', 'aaaaaaaaaaaabac', 'aaaaaaaaaaabaac', 'aaaaaaaaaabaaac', 'aaaaaaaaabaaaac', 'aaaaaaaabaaaaac', 'aaaaaaabaaaaaac']
4 4 4 4 ['abcdabcdabcdabcd']
5 4 4 3 []
5 5 3 3 []
5 5 4 2 []
5 5 5 1 []
6 4 3 3 []
6 4 4 2 ['abacbdbcabacbdbc']
6 5 3 2 []
6 5 4 1 []
6 5 5 0 ['aabacbacbacbacbc', 'aabacbacbacbcabc', 'aabacbacbcabacbc', 'aabacbacbcabcabc', 'aabacbcabacbacbc', 'aabacbcabacbcabc', 'aabacbcabcabacbc', 'aabacbcabcabcabc', 'aabcabacbacbacbc', 'aabcabacbacbcabc', 'aabcabacbcabacbc', 'aabcabacbcabcabc', 'aabcabcabacbacbc', 'aabcabcabacbcabc', 'aabcabcabcabacbc', 'aabcabcabcabcabc', 'aabcabcabcabcacb', 'aabcabcabcabcbac', 'aabcabcabcacbacb', 'aabcabcabcbacabc', 'aabcabcabcbacbac', 'aabcabcacbabcacb', 'aabcabcacbacbacb', 'aabcabcbacabcabc', 'aabcabcbacabcbac', 'aabcabcbacbacabc', 'aabcabcbacbacbac', 'aabcacbabcacbacb', 'aabcacbacbacbacb', 'aabcbacabcabcabc', 'aabcbacabcabcbac', 'aabcbacabcbacabc', 'aabcbacabcbacbac', 'aabcbacbacabcabc', 'aabcbacbacabcbac', 'aabcbacbacbacabc', 'aabcbacbacbacbac', 'abacabacbacbacbc', 'abacabacbacbcabc', 'abacabacbcabacbc', 'abacabacbcabcabc', 'abacabcabacbacbc', 'abacabcabacbcabc', 'abacabcabcabacbc', 'abacabcabcabcabc', 'abacabcabcbacabc', 'abacabcbacabcabc', 'abacabcbacbacabc', 'abacbacabcabacbc', 'abacbacabcabcabc', 'abacbacbacabcabc']
6 6 2 2 []
6 6 3 1 []
6 6 4 0 ['aabcabcababcabcb', 'aabcabcbaabcabcb', 'aabcabcbaabcbacb', 'aabcbacbaabcbacb', 'aabcbacbabacbacb', 'ababacbacbabacbc', 'ababacbacbabcabc', 'ababacbcababacbc', 'ababacbcababcabc', 'ababcabacbabacbc', 'ababcabacbabcabc', 'ababcabcababcabc', 'ababcabcabacbabc', 'abacbabcabacbabc']
7 3 3 3 []
7 4 3 2 []
7 4 4 1 []
7 5 2 2 []
7 5 3 1 []
7 5 4 0 ['aabacbaacbabcabc', 'aabacbabcaabcabc', 'aabacbabcaacbabc', 'aabacbacabacbabc', 'aabacbacbaacbabc', 'aabcaabcabacbacb', 'aabcaabcabcabacb', 'aabcaabcbaacbacb', 'aabcaabcbacabacb', 'aabcaacbacabacbc', 'aabcaacbacabcabc', 'aabcaacbacbaacbc', 'aabcaacbacbacabc', 'aabcabaacbacbabc', 'aabcababcaabcabc', 'aabcababcaacbabc', 'aabcabacababcabc', 'aabcabacabacbabc', 'aabcabacabacbacb', 'aabcabacabcababc', 'aabcabacabcabacb', 'aabcabacbaabcabc', 'aabcabacbaabcacb', 'aabcabacbaacbabc', 'aabcabacbaacbacb', 'aabcabacbacababc', 'aabcabacbacabacb', 'aabcabcaabcacabc', 'aabcabcaabcacbac', 'aabcabcabaacbabc', 'aabcabcabacababc', 'aabcabcabacabacb', 'aabcabcacabacabc', 'aabcacabacabcabc', 'aabcacabacabcacb', 'aabcacabacbacabc', 'aabcacabcaabcacb', 'aabcacabcabacabc', 'aabcacbaabcacbac', 'aabcacbacabacabc', 'aabcacbacabacacb', 'aabcacbacabacbac', 'aabcbabcabcbabcb', 'aabcbabcbacbabcb', 'aabcbcbaabcbcbcb', 'ababacabacbacabc', 'ababacabcabacabc', 'ababacabcabacbac', 'ababacbacabacabc', 'ababacbacabacbac', 'ababacbacabcabac', 'ababcabacabacbac', 'ababcabacabcabac', 'ababcbabcabcbabc', 'ababcbacbabcbabc']
7 6 2 1 []
7 6 3 0 ['aaabcbaabcbaabcb', 'aababacbabacbabc', 'aababacbabcababc', 'aababcababacbabc', 'aababcababcababc', 'aababcababcabacb', 'aababcabacbaabcb', 'aababcabacbabacb', 'aabacbaabcbaabcb', 'aabacbababcababc', 'aabacbabacbaabcb', 'aabacbabacbababc', 'aabacbabacbabacb', 'aabacbabacbabcab', 'aabacbabcabaabcb', 'aabacbabcababacb', 'aabacbabcababcab', 'aabaccabaccabacc', 'aabbacabbacabbac', 'aabcabaabcbaabcb', 'aabcababacbaabcb', 'aabcababacbabacb', 'aabcababacbabcab', 'aabcababcabaabcb', 'aabcababcababacb', 'aabcababcababcab', 'aabcababcabacbab', 'aabcabacbabaabcb', 'aabcabacbabacbab', 'aabcacabcacabcac', 'aabcacabcacacbac', 'aabcacabcacbacac', 'aabcacacbacacbac', 'aabcacbacacabcac', 'aabcacbacacbacac', 'aabcbaabcbababcb', 'aabcbababcababcb', 'aabcbabacbababcb', 'ababacababacbabc', 'ababacababcababc', 'ababacbabacababc', 'ababacbabacbabac']
7 7 1 1 []
7 7 2 0 ['aabababcabababcb', 'aabababcbaababcb', 'aabababcbabaabcb', 'aababcabababcbab', 'aababcbaababcbab', 'aababcbaabbaabcb', 'aababcbabaabcbab', 'aababcbababaabcb', 'aababcbababacbab', 'aabbaabcbabaabcb', 'aabcbabaabcbabab', 'aabcbababacbabab', 'ababababacbababc', 'ababababcabababc', 'abababacbabababc']
8 3 3 2 []
8 4 2 2 ['aabaacdcaabaacdc']
8 4 3 1 []
8 4 4 0 ['aaabcacbaaabcacb', 'aabaacbaacbaacbc', 'aabaacbaacbacabc', 'aabaacbaacbcaabc', 'aabaacbacabaacbc', 'aabaacbacabacabc', 'aabaacbacabcaabc', 'aabaacbcaabaacbc', 'aabaacbcaabacabc', 'aabaacbcaabcaabc', 'aabacabaacbaacbc', 'aabacabaacbacabc', 'aabacabaacbcaabc', 'aabacabacabaacbc', 'aabacabacabacabc', 'aabacabacabcaabc', 'aabacabcaabacabc', 'aabacabcaabcaabc', 'aabcaabcaabcaabc', 'aabcaabcaabcaacb', 'aabcaabcaabcabac', 'aabcaabcaacbaacb', 'aabcaabcabacabac', 'aabcaacbaabcaacb', 'aabcabacaabcabac', 'aabcabacabacabac', 'abacabacabacabac']
8 5 2 1 []
8 5 3 0 ['aaabcabaabcabacb', 'aaabcabacbaaabcb', 'aaabcabacbaabacb', 'aabaacbabacababc', 'aababacababacabc', 'aababacabacababc', 'aababcaababcaabc', 'aababcaabacababc', 'aababcaabacbaabc', 'aababcaabcabaabc', 'aabacabaabcaabcb', 'aabacabaabcabacb', 'aabacabaabcbaacb', 'aabacababacababc', 'aabacababacabacb', 'aabacababcaabacb', 'aabacabacabaabcb', 'aabacabacababacb', 'aabacabacacabacc', 'aabacabacbaabacb', 'aabacacabacabacc', 'aabacacabacacabc', 'aabacbaabacbaabc', 'aabacbaabacbaacb', 'aabacbaabcabaabc', 'aabacbaabcabaacb', 'aabacbaacbabaacb', 'aabbacababacabac', 'aabbacabacababac', 'aabcaabcabaabcab', 'aabcaabcacaabcac', 'aabcaacbacaabcac', 'aabcaacbacaacbac', 'aabcabaabcabacab', 'aabcabaacbabacab', 'aabcababacababac', 'aabcababacabacab', 'aabcabacababacab', 'aabcabacacabacac', 'aabcacaabcacabac', 'aabcacabacabacac', 'aabcacabacacabac', 'ababacababacabac']
8 6 1 1 []
8 6 2 0 ['aaabbcbbaaabbcbb', 'aabaabcababaabcb', 'aabaabcbaabaabcb', 'aabaabcbaababacb', 'aababaacbabababc', 'aabababaacbababc', 'aabababacabababc', 'aabababcaabababc', 'aabababcaababacb', 'aabababcabaababc', 'aababacabababacb', 'aababacbaababacb', 'aababacbaababcab', 'aababacbabaabacb', 'aababcabaababcab', 'aababcabaabacbab', 'aababcababaabcab', 'aababcbabababcbb', 'aababcbbaababcbb', 'aabacabaabbacabb', 'aabacabababacabb', 'aabacabbaabacabb', 'aabacacacabacacc', 'aabacaccaabacacc', 'aabacbabaabacbab', 'aabacbabaabcabab', 'aabacbababaacbab', 'aabbacabaabbacab', 'aabbacabababacab', 'aabcababaabcabab', 'aabcababaacbabab', 'aabcabababacabab', 'aabcacacaabcacac', 'aabcacacabacacac', 'abababacabababac']
8 7 1 0 ['aabaabbaabcbaabb', 'aababaabababcbab', 'aababaababcbabab', 'aababaabbaababcb', 'aababaabcbabaabb', 'aabababaabababcb', 'aabababaababcbab', 'aabababaabcbabab', 'aababababaababcb', 'aababababaabcbab', 'aabababababaabcb', 'aabababababababc', 'aababababababacb', 'aababababababcab', 'aababababababcbb', 'aabababababacabb', 'aabababababacbab', 'aabababababcabab', 'aabababababcbabb', 'aababababacababb', 'aababababacbabab', 'aababababcababab', 'aababababcbababb', 'aabababacabababb', 'aabababacbababab', 'aabababcabababab', 'aabababcbabababb', 'aababacababababb', 'aababacbabababab', 'aababcababababab', 'aababcbababababb', 'aabacabababababb', 'aabacacacacacacc', 'aabacbababababab', 'aabbacababababab', 'aabcabababababab', 'aabcacacacacacac', 'abababababababac']
9 3 2 2 []
9 3 3 1 []
9 4 2 1 []
9 4 3 0 ['aaabacabaacbaabc', 'aaabcaaabcabaacb', 'aaabcaaacbaacabc', 'aaabcaabaacbaabc', 'aaabcaabacaabacb', 'aaabcaabacabaabc', 'aaabcaabacabaacb', 'aaabcaacabacaabc', 'aaabcaacabacaacb', 'aaabcaacbaacaabc', 'aaabcaacbaacabac', 'aaabcabaacaabacb', 'aaabcabaacabaacb', 'aabaacabaacbaabc', 'aabaacbaabcaabac', 'aababaacabacabac', 'aababacaabacabac', 'aababacaabcaabac', 'aababacabaacabac', 'aabacaabacabaabc', 'aabacaabacabaacb', 'aabacaabacabaacc', 'aabacaabacabacab', 'aabacaabbaacabac', 'aabacaabcaabacab', 'aabacaabcaabacac', 'aabacaabcaacabac', 'aabacaacabacaabc', 'aabacaacabacabac', 'aabacaacbaacabac', 'aabacabaabacabac', 'aabacabaacababac', 'aabacabaacabacab', 'aabacabaacabacac', 'aabacabacaabacac']
9 5 1 1 []
9 5 2 0 ['aaabaabcabaaabcb', 'aaabaabcabaabacb', 'aaababacbaaabacb', 'aaababcabaaabacb', 'aaababcabaaabcab', 'aaabacbaaabacbab', 'aaabacbaabaaabcb', 'aaabacbaabaabacb', 'aaabacbaabaabcab', 'aaabacbabaaabcab', 'aaabcabaaabcabab', 'aaabcabaabaabcab', 'aaabcabaabacbaab', 'aabaabaacbaababc', 'aabaabaacbabaabc', 'aabaababcaabaabc', 'aabaabacabaababc', 'aabaabacabaabacb', 'aabaabacababaabc', 'aabaabacababaacb', 'aabaabacbaabaabc', 'aabaabacbaabaacb', 'aabaabcaabaabcab', 'aabaabcaababaabc', 'aabaabcaababaacb', 'aabaabcaababacab', 'aabaabcabaabaacb', 'aabaabcabaabacab', 'aabaacbaabaacbab', 'aabaacbaababaacb', 'aabaacbaababacab', 'aabaacbabaabacab', 'aababaabcaababac', 'aababaacababaabc', 'aababaacababaacb', 'aababaacabababac', 'aababaacbaababac', 'aabababaacababac', 'aabababacaababac', 'aabababacabaabac', 'aababacaababacab', 'aababacaabbaabac', 'aababacaabbaacab', 'aababacabaabacab', 'aababacababaabac', 'aababacababaacab', 'aabacaabbaabacab', 'aabacaabbaacabab', 'aabacaacabacaacc', 'aabacaacabacacac', 'aabacaaccaabacac', 'aabacabaabacabab', 'aabacababaacabab', 'aabacacaabacacac']
9 6 1 0 ['aabaabaabaababcb', 'aabaabaabaabcbab', 'aabaabaababaabcb', 'aabaabaababacbab', 'aabaabaababcabab', 'aabaabaabcbaabab', 'aabaababaabaabcb', 'aabaababaababacb', 'aabaababaababcab', 'aabaababaabacbab', 'aabaababaabcabab', 'aabaababacbaabab', 'aabaababcabaabab', 'aabaabacbabaabab', 'aabaabcababaabab', 'aabaacacacacacac', 'aababaababaababc', 'aababaababaabacb', 'aababaababaabcab', 'aababaababaacbab', 'aababaabababaabc', 'aababaabababaacb', 'aababaabababacab', 'aababaababacabab', 'aababaabacababab', 'aababaabbaababac', 'aababaabcaababab', 'aababaacababaabb', 'aababaacbaababab', 'aabababaabababac', 'aabababaababacab', 'aabababaabacabab', 'aabababaacababab', 'aababababaababac', 'aababababaabacab', 'aababababaacabab', 'aabababababaabac', 'aabababababaacab']
10 2 2 2 []
10 3 2 1 []
10 3 3 0 ['aaaabacaabacaabc', 'aaaabacaabcaaabc', 'aaaabcaaabacaabc', 'aaaabcaaabcaaabc', 'aaaabcaaabcaaacb', 'aaaabcaaabcaabac', 'aaaabcaaacbaaacb', 'aaaabcaabacaaabc', 'aaaabcaabacaabac', 'aaabaaacbaaacabc', 'aaabaaacbaacaabc', 'aaabaaacbacaaabc', 'aaabaacaabaacabc', 'aaabaacaabacaabc', 'aaabaacaabcaaabc', 'aaabaacabaaacabc', 'aaabaacabaaacbac', 'aaabaacabaacaabc', 'aaabaacabaacabac', 'aaabaacabacaaabc', 'aaabaacabacaabac', 'aaabaacabcaaabac', 'aaabacaaabacaabc', 'aaabacaaabcaaabc', 'aaabacaaabcaabac', 'aaabacaaabcabaac', 'aaabacaabaaacabc', 'aaabacaabaaacbac', 'aaabacaabaacaabc', 'aaabacaabaacabac', 'aaabacaabacaaabc', 'aaabacaabacaabac', 'aaabacaabacabaac', 'aaabacabaacaaabc', 'aaabacabaacaabac', 'aaabacabaacabaac', 'aaabcaaabcaabaac', 'aaabcaabaacaabac', 'aaabcaabaacabaac', 'aaabcaabacaabaac', 'aaabcabaacaabaac', 'aabaacaabaacabac', 'aabaacaabacaabac', 'aabaacabaacaabac']
10 4 1 1 []
10 4 2 0 ['aaaaabcabaaaabcb', 'aaaaabcbaaaaabcb', 'aaaaabcbaaaabacb', 'aaaabacabaaabacb', 'aaaabacabaabaacb', 'aaaabacbaaaabacb', 'aaaabacbaaaabcab', 'aaaabacbaaabaacb', 'aaaabcaabaaabcab', 'aaaabcaabaabacab', 'aaaabcabaaaabcab', 'aaaabcabaaabacab', 'aaabaaacbaabaabc', 'aaabaaaccaabaacc', 'aaabaabaacabaabc', 'aaabaabacaabaabc', 'aaabaabcaaabaabc', 'aaabaabcaaabaacb', 'aaabaabcaabaaabc', 'aaabaacabaaabacb', 'aaabaacabaabaacb', 'aaabaacacaabaacc', 'aaabaacbaaabaacb', 'aaabaacbaaabacab', 'aaabaacbaaabcaab', 'aaabaacbaabaaacb', 'aaabaaccaaabaacc', 'aaabaaccaabaaacc', 'aaabacaabaaabcab', 'aaabacaabaabacab', 'aaabacabaaabacab', 'aaabacabaaabcaab', 'aaabacabaabaacab', 'aaabbaacaaabbaac', 'aaabbaacaababaac', 'aaabcaabaaabcaab', 'aaabcaabaaacbaab', 'aaabcaabaabacaab', 'aaabcaacaaabcaac', 'aaabcaacaabacaac', 'aaabcaacabaacaac', 'aabaabaacabaabac', 'aabaabacaabaabac', 'aabaabacaababaac', 'aabaacaababaacab', 'aabaacabaabaacab', 'aabaacacaabaacac']
10 5 1 0 ['aaabaaabaabcbaab', 'aaabaabaaabaabcb', 'aaabaabaaabcbaab', 'aaabaabaabaaabcb', 'aaabaabaabaabacb', 'aaabaabaabaabcab', 'aaabaabaabacbaab', 'aaabaabaabcabaab', 'aaabaabacbaaabab', 'aaabaabacbaabaab', 'aaabaabcabaaabab', 'aaabaabcabaabaab', 'aaababaaabacbaab', 'aaababaaabcabaab', 'aaabacbaabaabaab', 'aaabcabaabaabaab', 'aabaabaabaabaabc', 'aabaabaabaabaacb', 'aabaabaabaababac', 'aabaabaabaabacab', 'aabaabaabaacabab', 'aabaabaababaabac', 'aabaabaababaacab', 'aabaabaabacaabab', 'aabaabaacabaabab', 'aabaababaabaabac', 'aabaababaabaacab', 'aabaababaababaac', 'aabaababaacaabab', 'aabaacaababaabab']
11 2 2 1 []
11 3 1 1 []
11 3 2 0 ['aaaaaabcaaabaacb', 'aaaaaabcaabaaacb', 'aaaaabacaabaaacb', 'aaaaabcaaabaaacb', 'aaaaabcaaabaacab', 'aaaabaaaacbaaabc', 'aaaabaaacabaaabc', 'aaaabaaacbaaaabc', 'aaaabaacaabaaabc', 'aaaabaacaabaaacb', 'aaaabaacabaaaabc', 'aaaabaacabaaaacb', 'aaaabacaaabaaabc', 'aaaabacaaabaaacb', 'aaaabacaaabaacab', 'aaaabacaabaaaabc', 'aaaabacaabaaaacb', 'aaaabacaabaaacab', 'aaaabcaaaacbaaac', 'aaaabcaaabaaacab', 'aaaabcaaabaacaab', 'aaaabcaaacaaabac', 'aaaabcaaacaabaac', 'aaaabcaaacabaaac', 'aaabaaacabaaabac', 'aaabaabaacaaabac', 'aaabaabaacaabaac', 'aaabaacaabaaabac', 'aaabaacaabaaacab', 'aaabaacaabaaacac', 'aaabaacaabaabaac', 'aaabaacaabaacaab', 'aaabaacaabaacaac', 'aaabaacaacaaabac', 'aaabaacaacaabaac']
11 4 1 0 ['aaaabaaaabaaabcb', 'aaaabaaaabcbaaab', 'aaaabaaabaaaabcb', 'aaaabaaabaaabacb', 'aaaabaaabaaabcab', 'aaaabaaabaacbaab', 'aaaabaaabacabaab', 'aaaabaaabacbaaab', 'aaaabaaabcaabaab', 'aaaabaaabcabaaab', 'aaaabaabaaabaacb', 'aaaabaabaaabacab', 'aaaabaabaaabcaab', 'aaaabaabaacbaaab', 'aaaabaabacabaaab', 'aaaabaabcaabaaab', 'aaaabaacbaaabaab', 'aaaabacabaaabaab', 'aaaabacbaaabaaab', 'aaaabcaabaaabaab', 'aaaabcabaaabaaab', 'aaabaaabaaabaabc', 'aaabaaabaaabaacb', 'aaabaaabaaabacab', 'aaabaaabaaabcaab', 'aaabaaabaaacbaab', 'aaabaaabaabaaabc', 'aaabaaabaabaaacb', 'aaabaaabaabaacab', 'aaabaaabaabacaab', 'aaabaaabaacabaab', 'aaabaaabacaabaab', 'aaabaaabcaaabaab', 'aaabaaacbaaabaab', 'aaabaabaaabaabac', 'aaabaabaaabaacab', 'aaabaabaaabacaab', 'aaabaabaaacabaab', 'aaabaabaabaaabac', 'aaabaabaabaaacab', 'aaabaabaabaabaac', 'aaabaabaabaacaab', 'aaabaabaacaaabab', 'aaabaabaacaabaab', 'aaabaacaabaaabab', 'aaabaacaabaabaab', 'aaabaacaacaaacac', 'aaabaacaacaacaac']
12 2 1 1 []
12 2 2 0 ['aaaaaaabcaaaaacb', 'aaaaaabaacaaaabc', 'aaaaaabacaaaaabc', 'aaaaaabcaaaaaabc', 'aaaaaabcaaaaaacb', 'aaaaaabcaaaaabac', 'aaaaaabcaaaabaac', 'aaaaabaaaacaabac', 'aaaaabaaacaaabac', 'aaaaabaacaaaabac', 'aaaaabacaaaaabac', 'aaaaabacaaaabaac', 'aaaaabacaaabaaac', 'aaaaabacaabaaaac', 'aaaabaaaacaabaac', 'aaaabaaacaaabaac', 'aaaabaacaaaabaac', 'aaaabaacaaabaaac', 'aaabaaacaaabaaac']
12 3 1 0 ['aaaaaabaaaaaabcb', 'aaaaaabaaaaabacb', 'aaaaaabaaaaabcab', 'aaaaaabacbaaaaab', 'aaaaaabcabaaaaab', 'aaaaabaaaaabaacb', 'aaaaabaaaaabacab', 'aaaaabaaaaabcaab', 'aaaaabaaaabaaacb', 'aaaaabaaaabaacab', 'aaaaabaaaabacaab', 'aaaaabaaaabcaaab', 'aaaaabaaacbaaaab', 'aaaaabaacabaaaab', 'aaaaabacaabaaaab', 'aaaaabcaaabaaaab', 'aaaabaaaabaaaabc', 'aaaabaaaabaaaacb', 'aaaabaaaabaaabac', 'aaaabaaaabaaacab', 'aaaabaaaabaacaab', 'aaaabaaaabacaaab', 'aaaabaaaacabaaab', 'aaaabaaabaaaabac', 'aaaabaaabaaaacab', 'aaaabaaabaaabaac', 'aaaabaaabaaacaab', 'aaaabaaabaacaaab', 'aaaabaaacaaabaab', 'aaaabaaacaaacaac', 'aaaabaaacaabaaab', 'aaaabaabaaabaaac', 'aaaabaabaaacaaab', 'aaaabaacaaabaaab', 'aaaabaacaaacaaac', 'aaabaaabaaabaaac']
13 1 1 1 []
13 2 1 0 ['aaaaaaaaaaaaabcb', 'aaaaaaaaaaaabacb', 'aaaaaaaaaaaabcab', 'aaaaaaaaaaabaacb', 'aaaaaaaaaaabacab', 'aaaaaaaaaaabcaab', 'aaaaaaaaaabaaacb', 'aaaaaaaaaabaacab', 'aaaaaaaaaabacaab', 'aaaaaaaaaabcaaab', 'aaaaaaaaabaaaacb', 'aaaaaaaaabaaacab', 'aaaaaaaaabaacaab', 'aaaaaaaaabacaaab', 'aaaaaaaaabcaaaab', 'aaaaaaaabaaaaacb', 'aaaaaaaabaaaacab', 'aaaaaaaabaaacaab', 'aaaaaaaabaacaaab', 'aaaaaaaabacaaaab', 'aaaaaaaabcaaaaab', 'aaaaaaabaaaaaabc', 'aaaaaaabaaaaaacb', 'aaaaaaabaaaaaacc', 'aaaaaaabaaaaacab', 'aaaaaaabaaaacaab', 'aaaaaaabaaacaaab', 'aaaaaaabaacaaaab', 'aaaaaaabacaaaaab', 'aaaaaaabbaaaaaac', 'aaaaaaabcaaaaaab', 'aaaaaaabcaaaaaac', 'aaaaaabaaaaaabac', 'aaaaaabaaaaaacab', 'aaaaaabaaaaaacac', 'aaaaaabaaaaabaac', 'aaaaaabaaaaacaab', 'aaaaaabaaaaacaac', 'aaaaaabaaaacaaab', 'aaaaaabaaacaaaab', 'aaaaaabaabaaaaac', 'aaaaaabaacaaaaab', 'aaaaaabaacaaaaac', 'aaaaabaaaaabaaac', 'aaaaabaaaaacaaab', 'aaaaabaaaaacaaac', 'aaaaabaaaabaaaac', 'aaaaabaaaacaaaab', 'aaaaabaaaacaaaac']
14 1 1 0 ['aaaaaaaaaaaaaabc', 'aaaaaaaaaaaaabac', 'aaaaaaaaaaaabaac', 'aaaaaaaaaaabaaac', 'aaaaaaaaaabaaaac', 'aaaaaaaaabaaaaac', 'aaaaaaaabaaaaaac', 'aaaaaaabaaaaaaac']

At 07:13 AM 7/28/2011, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>> The reason is that the JI major scale is small - its block is
>> only one unit tall in the 5-wardly direction. As a result,
>> vectors that fit in it can only take a few different angles,
>> none of which produce the (v, v+a, v+b, v+a+b) case (I think).
>
>This is part of my conjecture about N=3 DE scales. The conjecture says
>that any N=3 DE scale is either:
>
>* A scale with 5 notes per period of the pattern "aabcb", the shape of
>whose block is therefore a square around a central point (a.k.a. "plus sign")
>
>* A scale with 7 notes per period of the pattern "aabacab", the shape
>of whose block is therefore a hexagon around a central point, or
>
>* An arbitrarily large scale that lies entirely on two adjacent lines
>in the lattice (i.e. "one unit tall" in some direction).

You bet! -C.

Great work! I have a method in mind to produce the block-based
version of this, which I hope to code up over the next day or two.

-Carl

At 07:19 AM 7/28/2011, you wrote:
>Carl asked for some results for N=4 DE scale patterns, so here they
>are up to 16 notes:
>
>1 1 1 0 ['abc']
>1 1 1 1 ['abcd']
>2 1 1 0 ['aabc', 'abac']
>2 1 1 1 ['abacd']
>2 2 1 0 ['aabcb', 'ababc']
>3 1 1 0 ['aaabc', 'aabac']
>2 2 1 1 ['abacdc']
>2 2 2 0 ['aabccb', 'abacbc', 'abcabc']
>3 1 1 1 []
>3 2 1 0 ['aaabcb', 'aababc', 'aabacb', 'aabacc', 'aabbac', 'aabcab',
>'aabcac', 'ababac']
>4 1 1 0 ['aaaabc', 'aaabac', 'aabaac']
>2 2 2 1 ['aabcdcb']
>3 2 1 1 ['abacabd', 'abacadc']
>3 2 2 0 ['aaabccb', 'aabacbc', 'aabcabc', 'aabcacb', 'aabcbac',
>'aabcbcb', 'abacabc']
>3 3 1 0 ['aababcb', 'aabcbab', 'abababc']
>4 1 1 1 []
>4 2 1 0 ['aaaabcb', 'aaabacb', 'aaabcab', 'aabaabc', 'aabaacb',
>'aabaacc', 'aababac', 'aabacab', 'aabacac']
>5 1 1 0 ['aaaaabc', 'aaaabac', 'aaabaac']
>2 2 2 2 ['abcdabcd']
>3 2 2 1 ['abacbdbc']
>3 3 1 1 []
>3 3 2 0 ['aabcabcb', 'aabcbacb', 'ababacbc', 'ababcabc', 'abacbabc']
>4 2 1 1 ['aabaacdc']
>4 2 2 0 ['aaabcacb', 'aabaacbc', 'aabacabc', 'aabcaabc', 'aabcaacb',
>'aabcabac', 'abacabac']
>4 3 1 0 ['aaabbcbb', 'aabaabcb', 'aabababc', 'aababacb', 'aababcab',
>'aababcbb', 'aabacabb', 'aabacacc', 'aabacbab', 'aabbacab',
>'aabcabab', 'aabcacac', 'abababac']
>5 1 1 1 []
>5 2 1 0 ['aaaaabcb', 'aaaabacb', 'aaaabcab', 'aaabaabc', 'aaabaacb',
>'aaabaacc', 'aaabacab', 'aaabbaac', 'aaabcaab', 'aaabcaac',
>'aabaabac', 'aabaacab', 'aabaacac']
>6 1 1 0 ['aaaaaabc', 'aaaaabac', 'aaaabaac', 'aaabaaac']
>3 2 2 2 []
>3 3 2 1 []
>3 3 3 0 ['abacbacbc', 'abcabcabc']
>4 2 2 1 []
>4 3 1 1 []
>4 3 2 0 ['aaabcabcb', 'aaabcbacb', 'aabacbabc', 'aabcaabcb',
>'aabcaacbc', 'aabcababc', 'aabcabacb', 'aabcacabc', 'aabcacacb',
>'aabcacbac', 'aabcbabcb', 'aabcbcbcb', 'ababacabc', 'ababacbac',
>'ababcabac', 'ababcbabc']
>4 4 1 0 ['aabababcb', 'aababcbab', 'aabbaabcb', 'aabcbabab', 'ababababc']
>5 2 1 1 []
>5 2 2 0 ['aaaabcacb', 'aaabacabc', 'aaabaccab', 'aaabcaabc',
>'aaabcaacb', 'aaabcabac', 'aabaacabc', 'aabaacbac', 'aababaacc',
>'aabacaabc', 'aabacabac', 'aabacacab']
>5 3 1 0 ['aaabaabcb', 'aaababacb', 'aaababcab', 'aaabacbab',
>'aaabcabab', 'aaabcbaab', 'aabaababc', 'aabaabacb', 'aabaabcab',
>'aabaacbab', 'aababaabc', 'aababaacb', 'aabababac', 'aababacab',
>'aabacaabb', 'aabacaacc', 'aabacabab', 'aabacacac']
>6 1 1 1 []
>6 2 1 0 ['aaaaaabcb', 'aaaaabacb', 'aaaaabcab', 'aaaabaacb',
>'aaaabacab', 'aaaabcaab', 'aaabaaabc', 'aaabaaacb', 'aaabaaacc',
>'aaabaabac', 'aaabaacab', 'aaabaacac', 'aaababaac', 'aaabacaab',
>'aaabacaac', 'aabaabaac']
>7 1 1 0 ['aaaaaaabc', 'aaaaaabac', 'aaaaabaac', 'aaaabaaac']
>3 3 2 2 []
>3 3 3 1 []
>4 2 2 2 ['abacdabacd', 'abacdabadc']
>4 3 2 1 ['abacbadabc']
>4 3 3 0 ['aabacbacbc', 'aabacbcabc', 'aabcabacbc', 'aabcabcabc',
>'aabcabcacb', 'aabcabcbac', 'aabcacbacb', 'aabcbacabc', 'aabcbacbac',
>'abacabacbc', 'abacabcabc', 'abacbacabc']
>4 4 1 1 []
>4 4 2 0 ['aabcababcb', 'aabcbaabcb', 'aabcbabacb', 'ababacbabc', 'ababcababc']
>5 2 2 1 []
>5 3 1 1 []
>5 3 2 0 ['aaabcabacb', 'aabaacbabc', 'aababacabc', 'aababcaabc',
>'aabacababc', 'aabacabacb', 'aabacabacc', 'aabacacabc', 'aabacbaabc',
>'aabacbaacb', 'aabbacabac', 'aabcaabcab', 'aabcaabcac', 'aabcaacbac',
>'aabcababac', 'aabcabacab', 'aabcabacac', 'aabcacabac', 'ababacabac']
>5 4 1 0 ['aabaababcb', 'aabaabcbab', 'aababaabcb', 'aababababc',
>'aabababacb', 'aabababcab', 'aabababcbb', 'aababacabb', 'aababacbab',
>'aababcabab', 'aababcbabb', 'aabacababb', 'aabacacacc', 'aabacbabab',
>'aabaccaacc', 'aabbaabbac', 'aabbacabab', 'aabcababab', 'aabcacacac',
>'ababababac']
>6 2 1 1 ['aabaacadac']
>6 2 2 0 ['aaaabcaacb', 'aaabaacabc', 'aaabacaabc', 'aaabcaaabc',
>'aaabcaaacb', 'aaabcaabac', 'aaabcabaac', 'aabaacabac', 'aabacaabac']
>6 3 1 0 ['aaabaaabcb', 'aaabaabacb', 'aaabaabcab', 'aaabacbaab',
>'aaabcabaab', 'aabaabaabc', 'aabaabaacb', 'aabaababac', 'aabaabacab',
>'aabaacabab', 'aabaacacac', 'aababaabac', 'aababaacab']
>7 1 1 1 []
>7 2 1 0 ['aaaaaaabcb', 'aaaaaabacb', 'aaaaaabcab', 'aaaaabaacb',
>'aaaaabacab', 'aaaaabcaab', 'aaaabaaabc', 'aaaabaaacb', 'aaaabaaacc',
>'aaaabaacab', 'aaaabacaab', 'aaaabbaaac', 'aaaabcaaab', 'aaaabcaaac',
>'aaabaaabac', 'aaabaaacab', 'aaabaaacac', 'aaabaabaac', 'aaabaacaab',
>'aaabaacaac']
>8 1 1 0 ['aaaaaaaabc', 'aaaaaaabac', 'aaaaaabaac', 'aaaaabaaac', 'aaaabaaaac']
>3 3 3 2 []
>4 3 2 2 []
>4 3 3 1 []
>4 4 2 1 []
>4 4 3 0 ['aabcabcabcb', 'aabcabcbacb', 'aabcbacabcb', 'aabcbacbacb',
>'ababacbacbc', 'ababacbcabc', 'ababcabacbc', 'ababcabcabc',
>'ababcacbabc', 'abacbabcabc']
>5 2 2 2 []
>5 3 2 1 []
>5 3 3 0 ['aabacabacbc', 'aabacabcabc', 'aabacbaacbc', 'aabacbacabc',
>'aabacbcaabc', 'aabcaabcabc', 'aabcaabcacb', 'aabcaabcbac',
>'aabcaacbabc', 'aabcabaacbc', 'aabcabacabc', 'aabcabacbac',
>'aabcabcabac', 'aabcbacabac', 'abacabacabc']
>5 4 1 1 []
>5 4 2 0 ['aaabcbaabcb', 'aababacbabc', 'aababcababc', 'aababcabacb',
>'aababcbabcb', 'aabacbaabcb', 'aabacbababc', 'aabacbabacb',
>'aabacbabcab', 'aabaccabacc', 'aabbacabbac', 'aabcabaabcb',
>'aabcababacb', 'aabcababcab', 'aabcabacbab', 'aabcacabcac',
>'aabcacacbac', 'aabcacbacac', 'aabcbababcb', 'aabcbabcbab',
>'aabcbcbcbcb', 'ababacababc', 'ababacbabac']
>5 5 1 0 ['aababababcb', 'aabababcbab', 'aababcbaabb', 'aababcbabab',
>'aabbaabcbab', 'aabcbababab', 'abababababc']
>6 2 2 1 []
>6 3 1 1 []
>6 3 2 0 ['aaaabcabacb', 'aaabcaabacb', 'aaabcabaacb', 'aabaacbaabc',
>'aababacaabc', 'aababacabac', 'aabacabaabc', 'aabacabaacb',
>'aabacabaacc', 'aabacababac', 'aabacabacab', 'aabacabacac',
>'aabacacaabc', 'aabacacabac']
>6 4 1 0 ['aaababacbab', 'aaababcabab', 'aabaabaabcb', 'aabaababacb',
>'aabaababcab', 'aabaabacbab', 'aabaabcabab', 'aababaababc',
>'aababaabacb', 'aababaabcab', 'aababaacbab', 'aabababaabc',
>'aabababaacb', 'aababababac', 'aabababacab', 'aababacaabb',
>'aababacabab', 'aabacabaabb', 'aabacababab', 'aabacacaacc', 'aabacacacac']
>7 2 1 1 []
>7 2 2 0 ['aaaaabcaacb', 'aaaabacaabc', 'aaaabcaaabc', 'aaaabcaaacb',
>'aaaabcaabac', 'aaabaaacabc', 'aaabaaacbac', 'aaabaacaabc',
>'aaabaacabac', 'aaabacaaabc', 'aaabacaabac', 'aaabacaacab',
>'aaabacabaac', 'aaabcaabaac', 'aabaacaabac']
>7 3 1 0 ['aaaabaaabcb', 'aaaabcbaaab', 'aaabaaabacb', 'aaabaaabcab',
>'aaabaabaabc', 'aaabaabaacb', 'aaabaabacab', 'aaabaabcaab',
>'aaabaacbaab', 'aaabacabaab', 'aaabcaabaab', 'aaabcaacaac',
>'aabaabaabac', 'aabaabaacab', 'aabaababaac', 'aabaacaabab']
>8 1 1 1 []
>8 2 1 0 ['aaaaaaaabcb', 'aaaaaaabacb', 'aaaaaaabcab', 'aaaaaabaacb',
>'aaaaaabacab', 'aaaaaabcaab', 'aaaaabaaacb', 'aaaaabaacab',
>'aaaaabacaab', 'aaaaabcaaab', 'aaaabaaaabc', 'aaaabaaaacb',
>'aaaabaaaacc', 'aaaabaaabac', 'aaaabaaacab', 'aaaabaaacac',
>'aaaabaacaab', 'aaaababaaac', 'aaaabacaaab', 'aaaabacaaac',
>'aaabaaabaac', 'aaabaaacaab', 'aaabaaacaac']
>9 1 1 0 ['aaaaaaaaabc', 'aaaaaaaabac', 'aaaaaaabaac', 'aaaaaabaaac',
>'aaaaabaaaac']
>3 3 3 3 ['abcdabcdabcd']
>4 3 3 2 []
>4 4 2 2 ['abacdcabacdc']
>4 4 3 1 []
>4 4 4 0 ['aabccbaabccb', 'abacbacbacbc', 'abacbacbcabc',
>'abacbcabacbc', 'abcabcabcabc']
>5 3 2 2 []
>5 3 3 1 []
>5 4 2 1 []
>5 4 3 0 ['aaabcabcbacb', 'aabacbabcabc', 'aabacbacbabc',
>'aabcaabcbacb', 'aabcaacbacbc', 'aabcababcabc', 'aabcababcacb',
>'aabcabacabcb', 'aabcabacbabc', 'aabcabacbacb', 'aabcabcaabcb',
>'aabcabcaacbc', 'aabcabcababc', 'aabcabcabacb', 'aabcabcacabc',
>'aabcabcacbac', 'aabcabcbabcb', 'aabcacabcabc', 'aabcacabcacb',
>'aabcacbaabcb', 'aabcacbabacb', 'aabcacbacabc', 'aabcacbacacb',
>'aabcacbacbac', 'aabcbaabcbcb', 'aabcbabcabcb', 'aabcbabcbacb',
>'aabcbacabacb', 'aabcbacbabcb', 'ababacbacabc', 'ababcabacabc',
>'ababcabacbac', 'ababcabcabac', 'ababcabcbabc', 'ababcbabcabc',
>'ababcbacbabc', 'abacabacbabc']
>5 5 1 1 []
>5 5 2 0 ['aababcababcb', 'aababcbaabcb', 'aababcbabacb',
>'aabcababcbab', 'aabcbaabcbab', 'aabcbabacbab', 'abababacbabc',
>'abababcababc', 'ababacbababc']
>6 2 2 2 []
>6 3 2 1 []
>6 3 3 0 ['aabaacbaacbc', 'aabaacbacabc', 'aabaacbcaabc',
>'aabacabaacbc', 'aabacabacabc', 'aabacabcaabc', 'aabcaabcaabc',
>'aabcaabcaacb', 'aabcaabcabac', 'aabcabacabac', 'abacabacabac']
>6 4 1 1 []
>6 4 2 0 ['aaabcabaabcb', 'aaabcababacb', 'aaabcbaaabcb',
>'aaabcbaabacb', 'aabaacbababc', 'aabaaccabacc', 'aababaacbabc',
>'aababacababc', 'aababcaababc', 'aababcaabacb', 'aababcabaabc',
>'aabacabaabcb', 'aabacababacb', 'aabacacabacc', 'aabacbaabacb',
>'aabacbaabcab', 'aabacbabaacb', 'aabaccaabacc', 'aabaccabaacc',
>'aabbacaabbac', 'aabbacababac', 'aabcabaabcab', 'aabcabaacbab',
>'aabcababacab', 'aabcacaabcac', 'aabcacabacac', 'ababacababac']
>6 5 1 0 ['aabaababcbab', 'aabaabbaabcb', 'aabaabbacabb',
>'aabaabcbaabb', 'aababaababcb', 'aababaabcbab', 'aabababaabcb',
>'aabababababc', 'aababababacb', 'aababababcab', 'aababababcbb',
>'aabababacabb', 'aabababacbab', 'aabababcabab', 'aabababcbabb',
>'aababacababb', 'aababacbabab', 'aababcababab', 'aababcbababb',
>'aabacabababb', 'aabacacacacc', 'aabacbababab', 'aabbacababab',
>'aabcabababab', 'aabcacacacac', 'abababababac']
>7 2 2 1 []
>7 3 1 1 []
>7 3 2 0 ['aaaabcaabacb', 'aaaabcabaacb', 'aaabaacbaabc',
>'aaabacaabacb', 'aaabacabaabc', 'aaabacabaacb', 'aaabacbaaabc',
>'aaabacbaaacb', 'aaabcaaabcab', 'aaabcaaacbac', 'aaabcaabaabc',
>'aaabcaabaacb', 'aaabcaabacab', 'aaabcaacaabc', 'aaabcaacabac',
>'aaabcaacbaac', 'aaabcabaacab', 'aabaabcaabac', 'aabaacabaabc',
>'aabaacabaacb', 'aabaacabaacc', 'aabaacababac', 'aabaacabacac',
>'aabaacacabac', 'aabaacbaabac', 'aabaaccaabac', 'aababaacabac',
>'aababacaabac', 'aabacaabacab', 'aabacaabacac', 'aabacaacabac']
>7 4 1 0 ['aaabaabaabcb', 'aaabaabacbab', 'aaabaabcabab',
>'aaabaabcbaab', 'aaababaabacb', 'aaababaabcab', 'aaababacbaab',
>'aaababcabaab', 'aaabacbaabab', 'aaabbaacaabb', 'aaabcabaabab',
>'aaabcbaabaab', 'aabaabaababc', 'aabaabaabacb', 'aabaabaabcab',
>'aabaabaacbab', 'aabaababaabc', 'aabaababaacb', 'aabaababacab',
>'aabaabacabab', 'aabaabbaabac', 'aabaabcaabab', 'aabaacabaabb',
>'aabaacacacac', 'aabaacbaabab', 'aababaababac', 'aababaabacab',
>'aababaacabab', 'aabababaabac', 'aabababaacab']
>8 2 1 1 []
>8 2 2 0 ['aaaaabcaaacb', 'aaaabaacaabc', 'aaaabacaaabc',
>'aaaabcaaaabc', 'aaaabcaaaacb', 'aaaabcaaabac', 'aaaabcaabaac',
>'aaabaaacabac', 'aaabaacaabac', 'aaabacaaabac', 'aaabacaabaac', 'aabaacaabaac']
>8 3 1 0 ['aaaabaaaabcb', 'aaaabaaabacb', 'aaaabaaabcab',
>'aaaabaabaacb', 'aaaabaabacab', 'aaaabaabcaab', 'aaaabaacbaab',
>'aaaabacabaab', 'aaaabacbaaab', 'aaaabcaabaab', 'aaaabcabaaab',
>'aaabaaabaabc', 'aaabaaabaacb', 'aaabaaabacab', 'aaabaaabcaab',
>'aaabaaacbaab', 'aaabaabaaabc', 'aaabaabaaacb', 'aaabaabaabac',
>'aaabaabaacab', 'aaabaabacaab', 'aaabaacaabab', 'aaabaacaacac',
>'aaabaacabaab', 'aaababaabaac', 'aaababaacaab', 'aaabacaabaab',
>'aaabacaacaac', 'aabaabaabaac']
>9 1 1 1 []
>9 2 1 0 ['aaaaaaaaabcb', 'aaaaaaaabacb', 'aaaaaaaabcab',
>'aaaaaaabaacb', 'aaaaaaabacab', 'aaaaaaabcaab', 'aaaaaabaaacb',
>'aaaaaabaacab', 'aaaaaabacaab', 'aaaaaabcaaab', 'aaaaabaaaabc',
>'aaaaabaaaacb', 'aaaaabaaaacc', 'aaaaabaaacab', 'aaaaabaacaab',
>'aaaaabacaaab', 'aaaaabbaaaac', 'aaaaabcaaaab', 'aaaaabcaaaac',
>'aaaabaaaabac', 'aaaabaaaacab', 'aaaabaaaacac', 'aaaabaaabaac',
>'aaaabaaacaab', 'aaaabaaacaac', 'aaaabaabaaac', 'aaaabaacaaab',
>'aaaabaacaaac', 'aaabaaabaaac']
>10 1 1 0 ['aaaaaaaaaabc', 'aaaaaaaaabac', 'aaaaaaaabaac',
>'aaaaaaabaaac', 'aaaaaabaaaac', 'aaaaabaaaaac']
>4 3 3 3 []
>4 4 3 2 []
>4 4 4 1 []
>5 3 3 2 []
>5 4 2 2 []
>5 4 3 1 []
>5 4 4 0 ['aabacbacbacbc', 'aabacbacbcabc', 'aabacbcabacbc',
>'aabacbcabcabc', 'aabcabacbacbc', 'aabcabacbcabc', 'aabcabcabacbc',
>'aabcabcabcabc', 'aabcabcabcacb', 'aabcabcabcbac', 'aabcabcacbacb',
>'aabcabcbacabc', 'aabcabcbacbac', 'aabcacbabcacb', 'aabcacbacbacb',
>'aabcbacabcabc', 'aabcbacabcbac', 'aabcbacbacabc', 'aabcbacbacbac',
>'abacabacbacbc', 'abacabacbcabc', 'abacabcabacbc', 'abacabcabcabc',
>'abacabcbacabc', 'abacbacabcabc']
>5 5 2 1 []
>5 5 3 0 ['aabcababcabcb', 'aabcababcbacb', 'aabcabcbaabcb',
>'aabcabcbabacb', 'aabcbaabcbacb', 'aabcbacbabacb', 'ababacbabacbc',
>'ababacbabcabc', 'ababacbacbabc', 'ababacbcababc', 'ababcababcabc',
>'ababcabacbabc']
>6 3 2 2 []
>6 3 3 1 []
>6 4 2 1 []
>6 4 3 0 ['aabacbaacbabc', 'aabcaabcabacb', 'aabcaabcbaacb',
>'aabcaacbacabc', 'aabcabaacbabc', 'aabcababacabc', 'aabcabacababc',
>'aabcabacabacb', 'aabcabacacabc', 'aabcacabacabc', 'aabcacabacacb',
>'aabcbabcbabcb', 'ababacabacabc', 'ababacabacbac', 'ababacabcabac',
>'ababacbacabac', 'ababcabacabac', 'ababcbabcbabc']
>6 5 1 1 []
>6 5 2 0 ['aabaabcbaabcb', 'aababacbababc', 'aababcabaabcb',
>'aababcabababc', 'aababcababacb', 'aababcbaabacb', 'aababcbababcb',
>'aabacbabaabcb', 'aabacbabababc', 'aabacbababacb', 'aabacbababcab',
>'aabcabaabcbab', 'aabcabababacb', 'aabcabababcab', 'aabcababacbab',
>'aabcacacabcac', 'aabcacacacbac', 'aabcacacbacac', 'aabcbabababcb',
>'aabcbababcbab', 'aabcbcbcbcbcb', 'abababacababc', 'abababacbabac',
>'abababcababac', 'abababcbababc']
>6 6 1 0 ['aabababababcb', 'aababababcbab', 'aabababcbaabb',
>'aabababcbabab', 'aababcbabaabb', 'aababcbababab', 'aabbaabcbabab',
>'aabcbabababab', 'ababababababc']
>7 2 2 2 []
>7 3 2 1 []
>7 3 3 0 ['aaabacabacabc', 'aaabacabcaabc', 'aaabcaabacabc',
>'aaabcaabcaabc', 'aaabcaabcaacb', 'aaabcaabcabac', 'aaabcaacbaacb',
>'aaabcabacaabc', 'aaabcabacabac', 'aabaacabaacbc', 'aabaacabacabc',
>'aabaacabcaabc', 'aabaacbaacabc', 'aabaacbaacbac', 'aabaacbacaabc',
>'aabaacbacabac', 'aabaacbcaabac', 'aabacaabacabc', 'aabacaabcaabc',
>'aabacaabcabac', 'aabacabaacabc', 'aabacabaacbac', 'aabacabacaabc',
>'aabacabacabac']
>7 4 1 1 []
>7 4 2 0 ['aaaabcbaaabcb', 'aaabacbaaabcb', 'aaabacbaabacb',
>'aaabacbaabcab', 'aaabcabaaabcb', 'aaabcabaabacb', 'aaabcabaabcab',
>'aabaabacababc', 'aabaabacbaabc', 'aabaabcaababc', 'aabaabcaabacb',
>'aabaabcabaabc', 'aabaabcabaacb', 'aabaacbaababc', 'aabaacbaabacb',
>'aabaacbaabcab', 'aabaacbabaabc', 'aabaacbabaacb', 'aabaacbabacab',
>'aabaaccaabacc', 'aabaaccabaacc', 'aababaacbaabc', 'aababacabaabc',
>'aababacababac', 'aabacabaabacb', 'aabacabaabcab', 'aabacababaacb',
>'aabacabababac', 'aabacababacab', 'aabacacabaacc', 'aabacacabacac',
>'aabacacacabac']
>7 5 1 0 ['aaabababacbab', 'aaabababcabab', 'aaababacbabab',
>'aaababcababab', 'aabaabaababcb', 'aabaabaabcbab', 'aabaababaabcb',
>'aabaababacbab', 'aabaababcabab', 'aabaabcbaabab', 'aababaabababc',
>'aababaababacb', 'aababaababcab', 'aababaabacbab', 'aababaabcabab',
>'aababaacbabab', 'aabababaababc', 'aabababaabacb', 'aabababaabcab',
>'aabababaacbab', 'aababababaabc', 'aababababaacb', 'aabababababac',
>'aababababacab', 'aabababacaabb', 'aabababacabab', 'aababacabaabb',
>'aababacababab', 'aababbaabacab', 'aababbabcbabb', 'aabacababaabb',
>'aabacabababab', 'aabacacacaacc', 'aabacacacacac']
>8 2 2 1 []
>8 3 1 1 []
>8 3 2 0 ['aaaaabcaabacb', 'aaaaabcabaacb', 'aaaabcaabaacb',
>'aaabaaacbaabc', 'aaabaacabaabc', 'aaabaacbaaabc', 'aaabacaabaabc',
>'aaabacaabaacb', 'aaabacaabacab', 'aaabacabaaabc', 'aaabacabaaacb',
>'aaabacabaacab', 'aaabcaaacbaac', 'aaabcaabaacab', 'aaabcaacaabac',
>'aaabcaacabaac', 'aabaabaacabac', 'aabaabacaabac', 'aabaacabaabac',
>'aabaacabaacab', 'aabaacabaacac', 'aabaacacaabac']
>8 4 1 0 ['aaabaaabaabcb', 'aaabaaabcbaab', 'aaabaabaaabcb',
>'aaabaabaabacb', 'aaabaabaabcab', 'aaabaabacbaab', 'aaabaabcabaab',
>'aaababaaabacb', 'aaababaaabcab', 'aaabacbaabaab', 'aaabcabaabaab',
>'aabaabaabaabc', 'aabaabaabaacb', 'aabaabaababac', 'aabaabaabacab',
>'aabaabaacabab', 'aabaababaabac', 'aabaababaacab', 'aabaabacaabab',
>'aabaacabaabab', 'aabaacacaacac']
>9 2 1 1 []
>9 2 2 0 ['aaaaaabcaaacb', 'aaaaabacaaabc', 'aaaaabcaaaabc',
>'aaaaabcaaaacb', 'aaaaabcaaabac', 'aaaabaaacaabc', 'aaaabaaacabac',
>'aaaabaacaaabc', 'aaaabaacaabac', 'aaaabacaaaabc', 'aaaabacaaabac',
>'aaaabacaaacab', 'aaaabacaabaac', 'aaaabacabaaac', 'aaaabcaaabaac',
>'aaaabcaabaaac', 'aaabaaacaabac', 'aaabaaacabaac', 'aaabaacaaabac',
>'aaabaacaabaac']
>9 3 1 0 ['aaaaabaaaabcb', 'aaaaabcbaaaab', 'aaaabaaaabacb',
>'aaaabaaaabcab', 'aaaabaaabaacb', 'aaaabaaabacab', 'aaaabaaabcaab',
>'aaaabaacbaaab', 'aaaabacabaaab', 'aaaabcaabaaab', 'aaabaaabaaabc',
>'aaabaaabaaacb', 'aaabaaabaabac', 'aaabaaabaacab', 'aaabaaabacaab',
>'aaabaaacabaab', 'aaabaabaaabac', 'aaabaabaaacab', 'aaabaabaabaac',
>'aaabaabaacaab', 'aaabaacaaabab', 'aaabaacaaacac', 'aaabaacaabaab',
>'aaabaacaacaac']
>10 1 1 1 []
>10 2 1 0 ['aaaaaaaaaabcb', 'aaaaaaaaabacb', 'aaaaaaaaabcab',
>'aaaaaaaabaacb', 'aaaaaaaabacab', 'aaaaaaaabcaab', 'aaaaaaabaaacb',
>'aaaaaaabaacab', 'aaaaaaabacaab', 'aaaaaaabcaaab', 'aaaaaabaaaacb',
>'aaaaaabaaacab', 'aaaaaabaacaab', 'aaaaaabacaaab', 'aaaaaabcaaaab',
>'aaaaabaaaaabc', 'aaaaabaaaaacb', 'aaaaabaaaaacc', 'aaaaabaaaabac',
>'aaaaabaaaacab', 'aaaaabaaaacac', 'aaaaabaaacaab', 'aaaaabaacaaab',
>'aaaaababaaaac', 'aaaaabacaaaab', 'aaaaabacaaaac', 'aaaabaaaabaac',
>'aaaabaaaacaab', 'aaaabaaaacaac', 'aaaabaaabaaac', 'aaaabaaacaaab',
>'aaaabaaacaaac']
>11 1 1 0 ['aaaaaaaaaaabc', 'aaaaaaaaaabac', 'aaaaaaaaabaac',
>'aaaaaaaabaaac', 'aaaaaaabaaaac', 'aaaaaabaaaaac']
>4 4 3 3 []
>4 4 4 2 ['aabcdcbaabcdcb']
>5 3 3 3 []
>5 4 3 2 []
>5 4 4 1 []
>5 5 2 2 []
>5 5 3 1 []
>5 5 4 0 ['aabcabcabcabcb', 'aabcabcabcbacb', 'aabcabcbacabcb',
>'aabcabcbacbacb', 'aabcbacabcbacb', 'aabcbacbacbacb',
>'ababacbacbacbc', 'ababacbacbcabc', 'ababacbcabacbc',
>'ababacbcabcabc', 'ababcabacbacbc', 'ababcabacbcabc',
>'ababcabcabacbc', 'ababcabcabcabc', 'ababcabcacbabc',
>'ababcacbabcabc', 'ababcacbacbabc', 'abacbabcabacbc',
>'abacbabcabcabc', 'abacbacbabcabc']
>6 3 3 2 []
>6 4 2 2 ['abacabadbacabd', 'abacabdabacabd', 'abacadcabacadc']
>6 4 3 1 []
>6 4 4 0 ['aaabccbaaabccb', 'aabacbaacbacbc', 'aabacbaacbcabc',
>'aabacbacabacbc', 'aabacbacabcabc', 'aabacbcaabacbc',
>'aabacbcaabcabc', 'aabacbcabaacbc', 'aabcabaacbacbc',
>'aabcabaacbcabc', 'aabcabacabacbc', 'aabcabacabcabc',
>'aabcabcaabcabc', 'aabcabcaabcacb', 'aabcabcaabcbac',
>'aabcabcaacbabc', 'aabcabcabacabc', 'aabcabcabacbac',
>'aabcacbaabcacb', 'aabcbacaabcbac', 'aabcbacabacabc',
>'aabcbacabacbac', 'aabcbcbaabcbcb', 'abacabacbacabc', 'abacabcabacabc']
>6 5 2 1 []
>6 5 3 0 ['aabacbabacbabc', 'aabacbabcababc', 'aabcaabcbaabcb',
>'aabcaacbcaacbc', 'aabcababcababc', 'aabcababcabacb',
>'aabcabacbaabcb', 'aabcabacbababc', 'aabcabacbabacb',
>'aabcacabcacabc', 'aabcacabcacacb', 'aabcacabcacbac',
>'aabcacacbacabc', 'aabcacacbacacb', 'aabcacbacacbac',
>'aabcbaabcbabcb', 'aabcbababcabcb', 'aabcbabacbabcb',
>'aabcbabcababcb', 'aabcbacbababcb', 'ababacabcababc',
>'ababacbabacabc', 'ababacbabacbac', 'ababacbabcbabc',
>'ababacbacababc', 'ababcababcabac', 'ababcababcbabc', 'ababcabacbabac']
>6 6 1 1 []
>6 6 2 0 ['aababcabababcb', 'aababcbaababcb', 'aababcbabaabcb',
>'aabcbabaabcbab', 'aabcbababacbab', 'abababacbababc', 'abababcabababc']
>7 3 2 2 []
>7 3 3 1 []
>7 4 2 1 []
>7 4 3 0 ['aaabcaabcabacb', 'aaabcabacabacb', 'aaabcabacbaacb',
>'aababacabacabc', 'aababcaabcaabc', 'aabacabaacbabc',
>'aabacababacabc', 'aabacababcaabc', 'aabacabacababc',
>'aabacabacabacb', 'aabacabacabacc', 'aabacabacacabc',
>'aabacabacbaabc', 'aabacabacbaacb', 'aabacacabacabc',
>'aabacbaabcaabc', 'aabacbaacbaabc', 'aabacbaacbaacb',
>'aabbacabacabac', 'aabcaabcaabcab', 'aabcaabcaabcac',
>'aabcaabcaacbac', 'aabcaabcabaacb', 'aabcaabcabacab',
>'aabcaabcacabac', 'aabcaacbacabac', 'aabcababacabac',
>'aabcabacababac', 'aabcabacabacab', 'aabcabacabacac',
>'aabcabacacabac', 'aabcacabacabac', 'ababacabacabac']
>7 5 1 1 []
>7 5 2 0 ['aaababcababacb', 'aaabcabababacb', 'aaabcababacbab',
>'aaabcbaabaabcb', 'aabaabcabaabcb', 'aabaabcababacb',
>'aabaabcbaabacb', 'aababaacbababc', 'aabababacababc',
>'aabababcaababc', 'aababacabababc', 'aababacababacb',
>'aababacbaababc', 'aababacbaabacb', 'aababcaababcab',
>'aababcabaabacb', 'aababcabaabcab', 'aababcbababcbb',
>'aabacabababacb', 'aabacababacabb', 'aabacacabacacc',
>'aabacacacabacc', 'aabacbaabacbab', 'aabacbabaabcab',
>'aabacbabaacbab', 'aabbacabababac', 'aabbacababacab',
>'aabcabaabcabab', 'aabcababaacbab', 'aabcabababacab',
>'aabcababacabab', 'aabcacaabcacac', 'aabcacabacacac',
>'aabcacacabacac', 'abababacababac']
>7 6 1 0 ['aababaabababcb', 'aababaababcbab', 'aababaabcbabab',
>'aabababaababcb', 'aabababaabcbab', 'aababababaabcb',
>'aababababababc', 'aabababababacb', 'aabababababcab',
>'aabababababcbb', 'aababababacabb', 'aababababacbab',
>'aababababcabab', 'aababababcbabb', 'aabababacababb',
>'aabababacbabab', 'aabababcababab', 'aabababcbababb',
>'aababacabababb', 'aababacbababab', 'aababcabababab',
>'aababcbabababb', 'aabacababababb', 'aabacacacacacc',
>'aabacbabababab', 'aabaccaaccaacc', 'aabbaabbaabbac',
>'aabbacabababab', 'aabcababababab', 'aabcacacacacac', 'ababababababac']
>8 2 2 2 []
>8 3 2 1 []
>8 3 3 0 ['aaabaacbaacabc', 'aaabaacbacaabc', 'aaabacaabacabc',
>'aaabacaabcaabc', 'aaabacabaacabc', 'aaabacabacaabc',
>'aaabacabcaaabc', 'aaabcaaabcaabc', 'aaabcaaabcaacb',
>'aaabcaaabcabac', 'aaabcaaacbaabc', 'aaabcaabaacabc',
>'aaabcaabacaabc', 'aaabcaabacabac', 'aaabcaabacbaac',
>'aaabcaabcaabac', 'aaabcabaacaabc', 'aaabcabaacabac',
>'aaabcabaacbaac', 'aaabcabacaabac', 'aabaacabaacabc',
>'aabaacabaacbac', 'aabaacabacaabc', 'aabaacabacabac',
>'aabaacabcaabac', 'aabaacbaacabac', 'aabaacbacaabac',
>'aabacaabacaabc', 'aabacaabacabac', 'aabacabaacabac']
>8 4 1 1 []
>8 4 2 0 ['aaaabcabaaabcb', 'aaaabcbaaaabcb', 'aaaabcbaaabacb',
>'aaabacabaabacb', 'aaabacbaaabacb', 'aaabacbaaabcab',
>'aaabacbaabaacb', 'aaabcaabaabcab', 'aaabcabaaabcab',
>'aaabcabaabacab', 'aabaabaacbaabc', 'aabaabacabaabc',
>'aabaabcaabaabc', 'aabaabcaabaacb', 'aabaabcaababac',
>'aabaacababaacb', 'aabaacacabaacc', 'aabaacacabacac',
>'aabaacbaabaacb', 'aabaacbaabacab', 'aabaaccaabaacc',
>'aabaaccaabacac', 'aababaacababac', 'aababacaababac',
>'aababacabaabac', 'aabacaabbaacab', 'aabacaacabacac',
>'aabacabaabacab', 'aabacacaabacac']
>8 5 1 0 ['aaababaabacbab', 'aaababaabcabab', 'aaababacbaabab',
>'aaababcabaabab', 'aabaabaabaabcb', 'aabaabaababacb',
>'aabaabaababcab', 'aabaabaabacbab', 'aabaabaabcabab',
>'aabaababaababc', 'aabaababaabacb', 'aabaababaabcab',
>'aabaababaacbab', 'aabaababacabab', 'aabaababcaabab',
>'aabaabacbaabab', 'aabaabcabaabab', 'aabaacacacacac',
>'aabaacbabaabab', 'aababaababaabc', 'aababaababaacb',
>'aababaabababac', 'aababaababacab', 'aababaabacabab',
>'aababaacababab', 'aabababaababac', 'aabababaabacab',
>'aabababaacabab', 'aababababaabac', 'aababababaacab']
>9 2 2 1 []
>9 3 1 1 []
>9 3 2 0 ['aaaaabcaabaacb', 'aaaabacaabaacb', 'aaaabacabaaacb',
>'aaaabcaaabaacb', 'aaaabcaaabacab', 'aaaabcaabaaacb',
>'aaaabcaabaacab', 'aaabaaacbaaabc', 'aaabaacabaaabc',
>'aaabaacabaaacb', 'aaabaacabaabac', 'aaabacaabaaabc',
>'aaabacaabaaacb', 'aaabacaabaabac', 'aaabacaabaacab',
>'aaabacaacaabac', 'aaabacaacabaac', 'aabaabaacaabac',
>'aabaabaacabaac', 'aabaabacaabaac', 'aabaacaabaacab', 'aabaacaabaacac']
>9 4 1 0 ['aaabaaabaaabcb', 'aaabaaabaabacb', 'aaabaaabaabcab',
>'aaabaaabacbaab', 'aaabaaabcabaab', 'aaabaabaaabacb',
>'aaabaabaaabcab', 'aaabaabaabaabc', 'aaabaabaabaacb',
>'aaabaabaabacab', 'aaabaabaabcaab', 'aaabaabaacbaab',
>'aaabaabacabaab', 'aaabaabcaabaab', 'aaabaacbaabaab',
>'aaababaacaabab', 'aaabacabaabaab', 'aaabcaabaabaab',
>'aaabcaacaacaac', 'aabaabaabaabac', 'aabaabaabaacab',
>'aabaabaababaac', 'aabaabaacaabab', 'aabaababaabaac']
>10 2 1 1 []
>10 2 2 0 ['aaaaaabcaaaacb', 'aaaaabaacaaabc', 'aaaaabacaaaabc',
>'aaaaabcaaaaabc', 'aaaaabcaaaaacb', 'aaaaabcaaaabac',
>'aaaaabcaaabaac', 'aaaabaaaacabac', 'aaaabaaacaabac',
>'aaaabaacaaabac', 'aaaabacaaaabac', 'aaaabacaaabaac',
>'aaaabacaabaaac', 'aaabaaacaabaac', 'aaabaacaaabaac']
>10 3 1 0 ['aaaaabaaaaabcb', 'aaaaabaaaabacb', 'aaaaabaaaabcab',
>'aaaaabacbaaaab', 'aaaaabcabaaaab', 'aaaabaaaabaacb',
>'aaaabaaaabacab', 'aaaabaaaabcaab', 'aaaabaaabaaabc',
>'aaaabaaabaaacb', 'aaaabaaabaacab', 'aaaabaaabacaab',
>'aaaabaaabcaaab', 'aaaabaaacbaaab', 'aaaabaacabaaab',
>'aaaabacaabaaab', 'aaaabcaaabaaab', 'aaaabcaaacaaac',
>'aaabaaabaaabac', 'aaabaaabaaacab', 'aaabaaabaabaac',
>'aaabaaabaacaab', 'aaabaaacaabaab', 'aaabaaacaacaac',
>'aaabaabaaabaac', 'aaabaabaaacaab']
>11 1 1 1 []
>11 2 1 0 ['aaaaaaaaaaabcb', 'aaaaaaaaaabacb', 'aaaaaaaaaabcab',
>'aaaaaaaaabaacb', 'aaaaaaaaabacab', 'aaaaaaaaabcaab',
>'aaaaaaaabaaacb', 'aaaaaaaabaacab', 'aaaaaaaabacaab',
>'aaaaaaaabcaaab', 'aaaaaaabaaaacb', 'aaaaaaabaaacab',
>'aaaaaaabaacaab', 'aaaaaaabacaaab', 'aaaaaaabcaaaab',
>'aaaaaabaaaaabc', 'aaaaaabaaaaacb', 'aaaaaabaaaaacc',
>'aaaaaabaaaacab', 'aaaaaabaaacaab', 'aaaaaabaacaaab',
>'aaaaaabacaaaab', 'aaaaaabbaaaaac', 'aaaaaabcaaaaab',
>'aaaaaabcaaaaac', 'aaaaabaaaaabac', 'aaaaabaaaaacab',
>'aaaaabaaaaacac', 'aaaaabaaaabaac', 'aaaaabaaaacaab',
>'aaaaabaaaacaac', 'aaaaabaaacaaab', 'aaaaabaabaaaac',
>'aaaaabaacaaaab', 'aaaaabaacaaaac', 'aaaabaaaabaaac',
>'aaaabaaaacaaab', 'aaaabaaaacaaac']
>12 1 1 0 ['aaaaaaaaaaaabc', 'aaaaaaaaaaabac', 'aaaaaaaaaabaac',
>'aaaaaaaaabaaac', 'aaaaaaaabaaaac', 'aaaaaaabaaaaac', 'aaaaaabaaaaaac']
>4 4 4 3 []
>5 4 3 3 []
>5 4 4 2 []
>5 5 3 2 []
>5 5 4 1 []
>5 5 5 0 ['abacbacbacbacbc', 'abacbacbacbcabc', 'abacbacbcabacbc',
>'abcabcabcabcabc']
>6 3 3 3 ['abacdabacdabacd', 'abacdabacdabadc']
>6 4 3 2 []
>6 4 4 1 []
>6 5 2 2 []
>6 5 3 1 []
>6 5 4 0 ['aabacbacbabcabc', 'aabcabacbabcabc', 'aabcabacbacbabc',
>'aabcabcaabcbacb', 'aabcabcaacbacbc', 'aabcabcababcabc',
>'aabcabcababcacb', 'aabcabcabacabcb', 'aabcabcabacbabc',
>'aabcabcabacbacb', 'aabcabcacabcabc', 'aabcabcacbacabc',
>'aabcabcacbacbac', 'aabcabcbabcabcb', 'aabcabcbabcbacb',
>'aabcabcbacbabcb', 'aabcacbabacbacb', 'aabcacbacabcabc',
>'aabcacbacabcacb', 'aabcacbacbacabc', 'aabcbabcabcabcb',
>'aabcbabcabcbacb', 'aabcbacabacabcb', 'aabcbacabacbacb',
>'aabcbacbabcabcb', 'aabcbacbabcbacb', 'aabcbacbacbabcb',
>'ababacbacabcabc', 'ababacbacbacabc', 'ababcabacabcabc',
>'ababcabacbacabc', 'ababcabacbacbac', 'ababcabcabacabc',
>'ababcabcabacbac', 'ababcabcabcbabc', 'ababcabcbabcabc',
>'ababcabcbacbabc', 'ababcbacbabcabc', 'ababcbacbacbabc',
>'abacabacbabcabc', 'abacabacbacbabc', 'abacabcabacbabc', 'abacabcacbacabc']
>6 6 2 1 []
>6 6 3 0 ['aabcababcababcb', 'aabcababcbaabcb', 'aabcbaabcbaabcb',
>'aabcbaabcbabacb', 'aabcbabacbabacb', 'ababacbabacbabc',
>'ababacbabcababc', 'ababcababcababc']
>7 3 3 2 []
>7 4 2 2 []
>7 4 3 1 []
>7 4 4 0 ['aabacabacabacbc', 'aabacabacabcabc', 'aabacabacbaacbc',
>'aabacabacbacabc', 'aabacabacbcaabc', 'aabacabcaabacbc',
>'aabacabcaabcabc', 'aabacabcabaacbc', 'aabacabcabacabc',
>'aabacabcabcaabc', 'aabacbaacbaacbc', 'aabacbaacbacabc',
>'aabacbaacbcaabc', 'aabacbacabaacbc', 'aabacbacabacabc',
>'aabacbacabcaabc', 'aabacbcaabcaabc', 'aabcaabcaabcabc',
>'aabcaabcaabcacb', 'aabcaabcaabcbac', 'aabcaabcaacbabc',
>'aabcaabcabaacbc', 'aabcaabcabacabc', 'aabcaabcabacbac',
>'aabcaabcabcabac', 'aabcaabcacbaacb', 'aabcaabcbaacbac',
>'aabcaabcbacabac', 'aabcaacbaabcacb', 'aabcabaacbacabc',
>'aabcabacaabcabc', 'aabcabacaabcbac', 'aabcabacabaacbc',
>'aabcabacabacabc', 'aabcabacabacbac', 'aabcabacabcabac',
>'aabcabacbacabac', 'aabcabcabacabac', 'aabcbacabacabac', 'abacabacabacabc']
>7 5 2 1 []
>7 5 3 0 ['aaabcababcabacb', 'aaabcabacbaabcb', 'aaabcabacbabacb',
>'aaabcabcbaaabcb', 'aaabcbaaabcbacb', 'aaabcbaabcabacb',
>'aababcaabacbabc', 'aababcaabcababc', 'aababcaabcabacb',
>'aababcabaacbabc', 'aababcabacababc', 'aababcabacbaabc',
>'aababcbabcbabcb', 'aabacababcabacb', 'aabacbaabacbabc',
>'aabacbaabcaabcb', 'aabacbaabcababc', 'aabacbaabcabacb',
>'aabacbaabcbaacb', 'aabacbaacbababc', 'aabacbaacbabacb',
>'aabacbabaacbabc', 'aabacbabacababc', 'aabacbabacabacb',
>'aabaccabacabacc', 'aabbacabacabbac', 'aabcaabcbaabcab',
>'aabcaacbacaacbc', 'aabcaacbacabcac', 'aabcabaabcababc',
>'aabcabaabcabacb', 'aabcabaacbababc', 'aabcabaacbabacb',
>'aabcabaacbabcab', 'aabcababacababc', 'aabcababacabacb',
>'aabcabacababacb', 'aabcabacababcab', 'aabcabacbaacbab',
>'aabcabacbabacab', 'aabcacaabcacabc', 'aabcacaabcacbac',
>'aabcacabacabcac', 'aabcacabacacabc', 'aabcacabacacbac',
>'aabcbababcbabcb', 'aabcbabcbababcb', 'aabcbabcbabcbab',
>'ababacababacabc', 'ababacababacbac', 'ababacababcabac',
>'ababacabacababc', 'ababacabacbabac', 'ababacbabacabac']
>7 6 1 1 []
>7 6 2 0 ['aabaabcbaababcb', 'aabaabcbabaabcb', 'aabababacbababc',
>'aabababcabaabcb', 'aabababcabababc', 'aabababcababacb',
>'aabababcbababcb', 'aababacbaababcb', 'aababacbabaabcb',
>'aababacbabababc', 'aababacbababacb', 'aababacbababcab',
>'aababcabaababcb', 'aababcabaabcbab', 'aababcababaabcb',
>'aababcabababacb', 'aababcabababcab', 'aababcababacbab',
>'aababcbaabacbab', 'aababcbabaabacb', 'aababcbabaabcab',
>'aababcbabababcb', 'aababcbababcbab', 'aabacbabaabcbab',
>'aabacbababaabcb', 'aabacbabababcab', 'aabacbababacbab',
>'aabacbababcabab', 'aabbaabcbababcb', 'aabcababaabcbab',
>'aabcabababacbab', 'aabcabababcabab', 'aabcababacbabab',
>'aabcacacabcacac', 'aabcacacacbacac', 'aabcacacbacacac',
>'aabcbabababcbab', 'aabcbababcbabab', 'aabcbcbcbcbcbcb',
>'abababacabababc', 'abababacbababac']
>7 7 1 0 ['aababababababcb', 'aabababababcbab', 'aababababcbaabb',
>'aababababcbabab', 'aabababcbabaabb', 'aabababcbababab',
>'aababbaababcbab', 'aababbabacababb', 'aababcbababaabb',
>'aababcbabababab', 'aabbaabcbababab', 'aabcbababababab', 'abababababababc']
>8 3 2 2 []
>8 3 3 1 []
>8 4 2 1 []
>8 4 3 0 ['aaabcaabacabacb', 'aaabcaabacbaacb', 'aaabcaabcaabacb',
>'aaabcaabcabaacb', 'aaabcabaacabacb', 'aaabcabaacbaacb',
>'aaabcabacaabacb', 'aaabcabacabaacb', 'aabaacbaabcaabc',
>'aabaacbaacbaabc', 'aababacabacaabc', 'aababacabacabac',
>'aabacabaabcaabc', 'aabacabaabcaacb', 'aabacabaacbaabc',
>'aabacabaacbaacb', 'aabacababacaabc', 'aabacababacabac',
>'aabacabacabaabc', 'aabacabacabaacb', 'aabacabacabaacc',
>'aabacabacababac', 'aabacabacabacab', 'aabacabacabacac',
>'aabacabacacaabc', 'aabacabacacabac', 'aabacacabacaabc',
>'aabacacabacabac', 'aabacacabacacab']
>8 5 1 1 []
>8 5 2 0 ['aaabaabcabaabcb', 'aaabaabcbaaabcb', 'aaababcabaabacb',
>'aaabacbaabaabcb', 'aaabcabaabacbab', 'aaabcbaaabcbaab',
>'aaabcbaabaabcab', 'aaabcbaabacbaab', 'aabaabacbaababc',
>'aabaabcaabaabcb', 'aabaabcaababacb', 'aabaabcabaababc',
>'aabaabcabaabacb', 'aabaabcababaacb', 'aabaabcbaabaacb',
>'aabaacbabaababc', 'aabaacbabaabacb', 'aabaacbabaabcab',
>'aababaacbaababc', 'aababaacbabaabc', 'aabababacabaabc',
>'aabababacababac', 'aababacabaababc', 'aababacabaabacb',
>'aababacabaabcab', 'aababacababaabc', 'aababacababaacb',
>'aababacabababac', 'aababacababacab', 'aabacababaabacb',
>'aabacababaabcab', 'aabacababaacbab', 'aabacabababaacb',
>'aabacabababacab', 'aabacababacaabb', 'aabacababacabab',
>'aabacacabacaacc', 'aabacacabacacac', 'aabacacacabaacc', 'aabacacacabacac']
>8 6 1 0 ['aaabababacbabab', 'aaabababcababab', 'aabaabaababcbab',
>'aabaababaababcb', 'aabaababaabcbab', 'aabaabababaabcb',
>'aabaabababacbab', 'aabaabababcabab', 'aabaababacbabab',
>'aabaababcababab', 'aabaababcbaabab', 'aabaabbaabaabcb',
>'aabaabcbaababab', 'aabaabcbabaabab', 'aababaababaabcb',
>'aababaabababacb', 'aababaabababcab', 'aababaababacbab',
>'aababaababcabab', 'aababaabacbabab', 'aababaabcababab',
>'aabababaabababc', 'aabababaababacb', 'aabababaababcab',
>'aabababaabacbab', 'aabababaabcabab', 'aabababaacbabab',
>'aababababaababc', 'aababababaabacb', 'aababababaabcab',
>'aababababaacbab', 'aabababababaabc', 'aabababababaacb',
>'aababababababac', 'aabababababacab', 'aababababacaabb',
>'aababababacabab', 'aabababacabaabb', 'aabababacababab',
>'aababacabaababb', 'aababacabaabbab', 'aababacababaabb',
>'aababacabababab', 'aababbaabacabab', 'aababbabcbababb',
>'aabacabababaabb', 'aabacababababab', 'aabacacacacaacc', 'aabacacacacacac']
>9 2 2 2 []
>9 3 2 1 []
>9 3 3 0 ['aaabaacabaacabc', 'aaabaacabacaabc', 'aaabaacabcaaabc',
>'aaabacaabaacabc', 'aaabacaabacaabc', 'aaabacaabcaaabc',
>'aaabcaaabcaaabc', 'aaabcaaabcaaacb', 'aaabcaaabcaabac',
>'aaabcaaabcabaac', 'aaabcaabacaabac', 'aaabcaabacabaac',
>'aaabcabaacaabac', 'aaabcabaacabaac', 'aabaacabaacabac',
>'aabaacabacaabac', 'aabacaabacaabac']
>9 4 1 1 []
>9 4 2 0 ['aaaaabcbaaaabcb', 'aaaabacbaaaabcb', 'aaaabacbaaabacb',
>'aaaabacbaaabcab', 'aaaabcabaaaabcb', 'aaaabcabaaabacb',
>'aaaabcabaaabcab', 'aaabaabacabaabc', 'aaabaabcaabaabc',
>'aaabaabcaabaacb', 'aaabaacbaaabacb', 'aaabaacbaaabcab',
>'aaabaacbaabaabc', 'aaabaacbaabaacb', 'aaabaacbaabacab',
>'aaabaaccaabaacc', 'aaabacabaaabacb', 'aaabacabaaabcab',
>'aaabacabaabaacb', 'aaabacabaabacab', 'aaabacabaabcaab',
>'aaabacbaaabcaab', 'aaabbaacaabbaac', 'aaabcaabaaabcab',
>'aaabcaabaabaacb', 'aaabcaabaabacab', 'aaabcaabaabcaab',
>'aaabcaabaacbaab', 'aaabcaacaabcaac', 'aaabcaacaacbaac',
>'aaabcaacabacaac', 'aabaabaacabaabc', 'aabaabaacababac',
>'aabaabaacbaabac', 'aabaabacaabaabc', 'aabaabacaababac',
>'aabaabacabaabac', 'aabaacabaabaacb', 'aabaacabaababac',
>'aabaacabaabacab', 'aabaacababaabac', 'aabaacababaacab',
>'aabaacacaabaacc', 'aabaacacaabacac', 'aabaacacabaacac', 'aababaacabaabac']
>9 5 1 0 ['aaabaabaabaabcb', 'aaabaabaabacbab', 'aaabaabaabcabab',
>'aaabaabaabcbaab', 'aaabaabacbaabab', 'aaabaabcabaabab',
>'aaabaabcbaabaab', 'aaababaaababacb', 'aaababaaababcab',
>'aaababaaabacbab', 'aaababaaabcabab', 'aaababaabaabacb',
>'aaababaabaabcab', 'aaababaabacbaab', 'aaababaabcabaab',
>'aaababacbaabaab', 'aaababcabaabaab', 'aaabacbaabaabab',
>'aaabcabaabaabab', 'aaabcbaabaabaab', 'aabaabaabaababc',
>'aabaabaabaabacb', 'aabaabaabaabcab', 'aabaabaabaacbab',
>'aabaabaababaabc', 'aabaabaababaacb', 'aabaabaababacab',
>'aabaabaabacabab', 'aabaabaabcaabab', 'aabaabaacbaabab',
>'aabaababaabaabc', 'aabaababaabaacb', 'aabaababaababac',
>'aabaababaabacab', 'aabaababaacabab', 'aabaabababaabac',
>'aabaabababaacab', 'aabaababacaabab', 'aabaabacaabaabb',
>'aabaabacaababab', 'aabaabacabaabab', 'aabaabbaabaacab',
>'aabaacabaababab', 'aabaacababaabab', 'aabaacacaacacac',
>'aabaacacacaacac', 'aababaababaabac', 'aababaababaacab']
>10 2 2 1 []
>10 3 1 1 []
>10 3 2 0 ['aaaaaabcaabaacb', 'aaaaabcaaabaacb', 'aaaaabcaabaaacb',
>'aaaabaaacbaaabc', 'aaaabaacabaaabc', 'aaaabaacbaaaabc',
>'aaaabacaaabaacb', 'aaaabacaaabacab', 'aaaabacaabaaabc',
>'aaaabacaabaaacb', 'aaaabacaabaacab', 'aaaabacabaaaabc',
>'aaaabacabaaaacb', 'aaaabacabaaacab', 'aaaabcaaaacbaac',
>'aaaabcaaabaaacb', 'aaaabcaaabaacab', 'aaaabcaaacaabac',
>'aaaabcaaacabaac', 'aaaabcaaacbaaac', 'aaaabcaabaaacab',
>'aaabaaacabaaabc', 'aaabaaacabaabac', 'aaabaaacbaaabac',
>'aaabaabaacaabac', 'aaabaabacaaabac', 'aaabaacaabaaabc',
>'aaabaacaabaaacb', 'aaabaacaabaaacc', 'aaabaacaabaabac',
>'aaabaacaabaacab', 'aaabaacaabaacac', 'aaabaacaacaabac',
>'aaabaacabaaabac', 'aaabaacabaaacab', 'aaababaacaabaac',
>'aaabacaaabacaab', 'aaabacaaabacaac', 'aaabacaaacabaac',
>'aaabacaabaabaac', 'aaabacaabaacaab', 'aaabacaabaacaac',
>'aaabacaacaabaac', 'aabaabaacaabaac']
>10 4 1 0 ['aaaabaaabaaabcb', 'aaaabaaabcbaaab', 'aaaabaabaacbaab',
>'aaaabaabacabaab', 'aaaabaabcaabaab', 'aaaabcbaaabaaab',
>'aaabaaabaaabacb', 'aaabaaabaaabcab', 'aaabaaabaabaacb',
>'aaabaaabaabacab', 'aaabaaabaabcaab', 'aaabaaabaacbaab',
>'aaabaaabacabaab', 'aaabaaabcaabaab', 'aaabaabaaabaabc',
>'aaabaabaaabaacb', 'aaabaabaaabacab', 'aaabaabaaabcaab',
>'aaabaabaaacbaab', 'aaabaabaabaaabc', 'aaabaabaabaaacb',
>'aaabaabaabaabac', 'aaabaabaabaacab', 'aaabaabaabacaab',
>'aaabaabaacaabab', 'aaabaabaacabaab', 'aaabaabacaabaab',
>'aaabaacaabaaabb', 'aaabaacaabaabab', 'aaabaacaacaacac',
>'aaabaacabaabaab', 'aaabaacacaaacac', 'aaababaaababaac',
>'aaababaabaabaac', 'aaababaabaacaab', 'aaababaacaabaab',
>'aaabacaabaabaab', 'aaabacaacaacaac', 'aabaabaabaabaac']
>11 2 1 1 []
>11 2 2 0 ['aaaaaaabcaaaacb', 'aaaaaabacaaaabc', 'aaaaaabcaaaaabc',
>'aaaaaabcaaaaacb', 'aaaaaabcaaaabac', 'aaaaabaaacaaabc',
>'aaaaabaaacaabac', 'aaaaabaacaaaabc', 'aaaaabaacaaabac',
>'aaaaabacaaaaabc', 'aaaaabacaaaabac', 'aaaaabacaaaacab',
>'aaaaabacaaabaac', 'aaaaabacaabaaac', 'aaaaabcaaaabaac',
>'aaaaabcaaabaaac', 'aaaabaaaacaabac', 'aaaabaaaacabaac',
>'aaaabaaacaaabac', 'aaaabaaacaabaac', 'aaaabaacaaaabac',
>'aaaabaacaaabaac', 'aaaabaacaabaaac', 'aaaabacaaabaaac', 'aaabaaacaaabaac']
>11 3 1 0 ['aaaaaabaaaaabcb', 'aaaaaabcbaaaaab', 'aaaaabaaaaabacb',
>'aaaaabaaaaabcab', 'aaaaabaaaabaacb', 'aaaaabaaaabacab',
>'aaaaabaaaabcaab', 'aaaaabaaabaaacb', 'aaaaabaaabaacab',
>'aaaaabaaabacaab', 'aaaaabaaabcaaab', 'aaaaabaaacbaaab',
>'aaaaabaacabaaab', 'aaaaabaacbaaaab', 'aaaaabacaabaaab',
>'aaaaabacabaaaab', 'aaaaabcaaabaaab', 'aaaaabcaabaaaab',
>'aaaabaaaabaaabc', 'aaaabaaaabaaacb', 'aaaabaaaabaacab',
>'aaaabaaaabacaab', 'aaaabaaaabcaaab', 'aaaabaaaacbaaab',
>'aaaabaaabaaaabc', 'aaaabaaabaaaacb', 'aaaabaaabaaabac',
>'aaaabaaabaaacab', 'aaaabaaabaacaab', 'aaaabaaabacaaab',
>'aaaabaaacabaaab', 'aaaabaacaabaaab', 'aaaabacaaabaaab',
>'aaaabacaaacaaac', 'aaabaaabaaabaac', 'aaabaaabaaacaab',
>'aaabaaabaabaaac', 'aaabaaacaaabaab']
>12 1 1 1 []
>12 2 1 0 ['aaaaaaaaaaaabcb', 'aaaaaaaaaaabacb', 'aaaaaaaaaaabcab',
>'aaaaaaaaaabaacb', 'aaaaaaaaaabacab', 'aaaaaaaaaabcaab',
>'aaaaaaaaabaaacb', 'aaaaaaaaabaacab', 'aaaaaaaaabacaab',
>'aaaaaaaaabcaaab', 'aaaaaaaabaaaacb', 'aaaaaaaabaaacab',
>'aaaaaaaabaacaab', 'aaaaaaaabacaaab', 'aaaaaaaabcaaaab',
>'aaaaaaabaaaaacb', 'aaaaaaabaaaacab', 'aaaaaaabaaacaab',
>'aaaaaaabaacaaab', 'aaaaaaabacaaaab', 'aaaaaaabcaaaaab',
>'aaaaaabaaaaaabc', 'aaaaaabaaaaaacb', 'aaaaaabaaaaaacc',
>'aaaaaabaaaaabac', 'aaaaaabaaaaacab', 'aaaaaabaaaaacac',
>'aaaaaabaaaacaab', 'aaaaaabaaacaaab', 'aaaaaabaacaaaab',
>'aaaaaababaaaaac', 'aaaaaabacaaaaab', 'aaaaaabacaaaaac',
>'aaaaabaaaaabaac', 'aaaaabaaaaacaab', 'aaaaabaaaaacaac',
>'aaaaabaaaabaaac', 'aaaaabaaaacaaab', 'aaaaabaaaacaaac',
>'aaaaabaaabaaaac', 'aaaaabaaacaaaab', 'aaaaabaaacaaaac', 'aaaabaaaabaaaac']
>13 1 1 0 ['aaaaaaaaaaaaabc', 'aaaaaaaaaaaabac', 'aaaaaaaaaaabaac',
>'aaaaaaaaaabaaac', 'aaaaaaaaabaaaac', 'aaaaaaaabaaaaac', 'aaaaaaabaaaaaac']
>4 4 4 4 ['abcdabcdabcdabcd']
>5 4 4 3 []
>5 5 3 3 []
>5 5 4 2 []
>5 5 5 1 []
>6 4 3 3 []
>6 4 4 2 ['abacbdbcabacbdbc']
>6 5 3 2 []
>6 5 4 1 []
>6 5 5 0 ['aabacbacbacbacbc', 'aabacbacbacbcabc', 'aabacbacbcabacbc',
>'aabacbacbcabcabc', 'aabacbcabacbacbc', 'aabacbcabacbcabc',
>'aabacbcabcabacbc', 'aabacbcabcabcabc', 'aabcabacbacbacbc',
>'aabcabacbacbcabc', 'aabcabacbcabacbc', 'aabcabacbcabcabc',
>'aabcabcabacbacbc', 'aabcabcabacbcabc', 'aabcabcabcabacbc',
>'aabcabcabcabcabc', 'aabcabcabcabcacb', 'aabcabcabcabcbac',
>'aabcabcabcacbacb', 'aabcabcabcbacabc', 'aabcabcabcbacbac',
>'aabcabcacbabcacb', 'aabcabcacbacbacb', 'aabcabcbacabcabc',
>'aabcabcbacabcbac', 'aabcabcbacbacabc', 'aabcabcbacbacbac',
>'aabcacbabcacbacb', 'aabcacbacbacbacb', 'aabcbacabcabcabc',
>'aabcbacabcabcbac', 'aabcbacabcbacabc', 'aabcbacabcbacbac',
>'aabcbacbacabcabc', 'aabcbacbacabcbac', 'aabcbacbacbacabc',
>'aabcbacbacbacbac', 'abacabacbacbacbc', 'abacabacbacbcabc',
>'abacabacbcabacbc', 'abacabacbcabcabc', 'abacabcabacbacbc',
>'abacabcabacbcabc', 'abacabcabcabacbc', 'abacabcabcabcabc',
>'abacabcabcbacabc', 'abacabcbacabcabc', 'abacabcbacbacabc',
>'abacbacabcabacbc', 'abacbacabcabcabc', 'abacbacbacabcabc']
>6 6 2 2 []
>6 6 3 1 []
>6 6 4 0 ['aabcabcababcabcb', 'aabcabcbaabcabcb', 'aabcabcbaabcbacb',
>'aabcbacbaabcbacb', 'aabcbacbabacbacb', 'ababacbacbabacbc',
>'ababacbacbabcabc', 'ababacbcababacbc', 'ababacbcababcabc',
>'ababcabacbabacbc', 'ababcabacbabcabc', 'ababcabcababcabc',
>'ababcabcabacbabc', 'abacbabcabacbabc']
>7 3 3 3 []
>7 4 3 2 []
>7 4 4 1 []
>7 5 2 2 []
>7 5 3 1 []
>7 5 4 0 ['aabacbaacbabcabc', 'aabacbabcaabcabc', 'aabacbabcaacbabc',
>'aabacbacabacbabc', 'aabacbacbaacbabc', 'aabcaabcabacbacb',
>'aabcaabcabcabacb', 'aabcaabcbaacbacb', 'aabcaabcbacabacb',
>'aabcaacbacabacbc', 'aabcaacbacabcabc', 'aabcaacbacbaacbc',
>'aabcaacbacbacabc', 'aabcabaacbacbabc', 'aabcababcaabcabc',
>'aabcababcaacbabc', 'aabcabacababcabc', 'aabcabacabacbabc',
>'aabcabacabacbacb', 'aabcabacabcababc', 'aabcabacabcabacb',
>'aabcabacbaabcabc', 'aabcabacbaabcacb', 'aabcabacbaacbabc',
>'aabcabacbaacbacb', 'aabcabacbacababc', 'aabcabacbacabacb',
>'aabcabcaabcacabc', 'aabcabcaabcacbac', 'aabcabcabaacbabc',
>'aabcabcabacababc', 'aabcabcabacabacb', 'aabcabcacabacabc',
>'aabcacabacabcabc', 'aabcacabacabcacb', 'aabcacabacbacabc',
>'aabcacabcaabcacb', 'aabcacabcabacabc', 'aabcacbaabcacbac',
>'aabcacbacabacabc', 'aabcacbacabacacb', 'aabcacbacabacbac',
>'aabcbabcabcbabcb', 'aabcbabcbacbabcb', 'aabcbcbaabcbcbcb',
>'ababacabacbacabc', 'ababacabcabacabc', 'ababacabcabacbac',
>'ababacbacabacabc', 'ababacbacabacbac', 'ababacbacabcabac',
>'ababcabacabacbac', 'ababcabacabcabac', 'ababcbabcabcbabc', 'ababcbacbabcbabc']
>7 6 2 1 []
>7 6 3 0 ['aaabcbaabcbaabcb', 'aababacbabacbabc', 'aababacbabcababc',
>'aababcababacbabc', 'aababcababcababc', 'aababcababcabacb',
>'aababcabacbaabcb', 'aababcabacbabacb', 'aabacbaabcbaabcb',
>'aabacbababcababc', 'aabacbabacbaabcb', 'aabacbabacbababc',
>'aabacbabacbabacb', 'aabacbabacbabcab', 'aabacbabcabaabcb',
>'aabacbabcababacb', 'aabacbabcababcab', 'aabaccabaccabacc',
>'aabbacabbacabbac', 'aabcabaabcbaabcb', 'aabcababacbaabcb',
>'aabcababacbabacb', 'aabcababacbabcab', 'aabcababcabaabcb',
>'aabcababcababacb', 'aabcababcababcab', 'aabcababcabacbab',
>'aabcabacbabaabcb', 'aabcabacbabacbab', 'aabcacabcacabcac',
>'aabcacabcacacbac', 'aabcacabcacbacac', 'aabcacacbacacbac',
>'aabcacbacacabcac', 'aabcacbacacbacac', 'aabcbaabcbababcb',
>'aabcbababcababcb', 'aabcbabacbababcb', 'ababacababacbabc',
>'ababacababcababc', 'ababacbabacababc', 'ababacbabacbabac']
>7 7 1 1 []
>7 7 2 0 ['aabababcabababcb', 'aabababcbaababcb', 'aabababcbabaabcb',
>'aababcabababcbab', 'aababcbaababcbab', 'aababcbaabbaabcb',
>'aababcbabaabcbab', 'aababcbababaabcb', 'aababcbababacbab',
>'aabbaabcbabaabcb', 'aabcbabaabcbabab', 'aabcbababacbabab',
>'ababababacbababc', 'ababababcabababc', 'abababacbabababc']
>8 3 3 2 []
>8 4 2 2 ['aabaacdcaabaacdc']
>8 4 3 1 []
>8 4 4 0 ['aaabcacbaaabcacb', 'aabaacbaacbaacbc', 'aabaacbaacbacabc',
>'aabaacbaacbcaabc', 'aabaacbacabaacbc', 'aabaacbacabacabc',
>'aabaacbacabcaabc', 'aabaacbcaabaacbc', 'aabaacbcaabacabc',
>'aabaacbcaabcaabc', 'aabacabaacbaacbc', 'aabacabaacbacabc',
>'aabacabaacbcaabc', 'aabacabacabaacbc', 'aabacabacabacabc',
>'aabacabacabcaabc', 'aabacabcaabacabc', 'aabacabcaabcaabc',
>'aabcaabcaabcaabc', 'aabcaabcaabcaacb', 'aabcaabcaabcabac',
>'aabcaabcaacbaacb', 'aabcaabcabacabac', 'aabcaacbaabcaacb',
>'aabcabacaabcabac', 'aabcabacabacabac', 'abacabacabacabac']
>8 5 2 1 []
>8 5 3 0 ['aaabcabaabcabacb', 'aaabcabacbaaabcb', 'aaabcabacbaabacb',
>'aabaacbabacababc', 'aababacababacabc', 'aababacabacababc',
>'aababcaababcaabc', 'aababcaabacababc', 'aababcaabacbaabc',
>'aababcaabcabaabc', 'aabacabaabcaabcb', 'aabacabaabcabacb',
>'aabacabaabcbaacb', 'aabacababacababc', 'aabacababacabacb',
>'aabacababcaabacb', 'aabacabacabaabcb', 'aabacabacababacb',
>'aabacabacacabacc', 'aabacabacbaabacb', 'aabacacabacabacc',
>'aabacacabacacabc', 'aabacbaabacbaabc', 'aabacbaabacbaacb',
>'aabacbaabcabaabc', 'aabacbaabcabaacb', 'aabacbaacbabaacb',
>'aabbacababacabac', 'aabbacabacababac', 'aabcaabcabaabcab',
>'aabcaabcacaabcac', 'aabcaacbacaabcac', 'aabcaacbacaacbac',
>'aabcabaabcabacab', 'aabcabaacbabacab', 'aabcababacababac',
>'aabcababacabacab', 'aabcabacababacab', 'aabcabacacabacac',
>'aabcacaabcacabac', 'aabcacabacabacac', 'aabcacabacacabac', 'ababacababacabac']
>8 6 1 1 []
>8 6 2 0 ['aaabbcbbaaabbcbb', 'aabaabcababaabcb', 'aabaabcbaabaabcb',
>'aabaabcbaababacb', 'aababaacbabababc', 'aabababaacbababc',
>'aabababacabababc', 'aabababcaabababc', 'aabababcaababacb',
>'aabababcabaababc', 'aababacabababacb', 'aababacbaababacb',
>'aababacbaababcab', 'aababacbabaabacb', 'aababcabaababcab',
>'aababcabaabacbab', 'aababcababaabcab', 'aababcbabababcbb',
>'aababcbbaababcbb', 'aabacabaabbacabb', 'aabacabababacabb',
>'aabacabbaabacabb', 'aabacacacabacacc', 'aabacaccaabacacc',
>'aabacbabaabacbab', 'aabacbabaabcabab', 'aabacbababaacbab',
>'aabbacabaabbacab', 'aabbacabababacab', 'aabcababaabcabab',
>'aabcababaacbabab', 'aabcabababacabab', 'aabcacacaabcacac',
>'aabcacacabacacac', 'abababacabababac']
>8 7 1 0 ['aabaabbaabcbaabb', 'aababaabababcbab', 'aababaababcbabab',
>'aababaabbaababcb', 'aababaabcbabaabb', 'aabababaabababcb',
>'aabababaababcbab', 'aabababaabcbabab', 'aababababaababcb',
>'aababababaabcbab', 'aabababababaabcb', 'aabababababababc',
>'aababababababacb', 'aababababababcab', 'aababababababcbb',
>'aabababababacabb', 'aabababababacbab', 'aabababababcabab',
>'aabababababcbabb', 'aababababacababb', 'aababababacbabab',
>'aababababcababab', 'aababababcbababb', 'aabababacabababb',
>'aabababacbababab', 'aabababcabababab', 'aabababcbabababb',
>'aababacababababb', 'aababacbabababab', 'aababcababababab',
>'aababcbababababb', 'aabacabababababb', 'aabacacacacacacc',
>'aabacbababababab', 'aabbacababababab', 'aabcabababababab',
>'aabcacacacacacac', 'abababababababac']
>9 3 2 2 []
>9 3 3 1 []
>9 4 2 1 []
>9 4 3 0 ['aaabacabaacbaabc', 'aaabcaaabcabaacb', 'aaabcaaacbaacabc',
>'aaabcaabaacbaabc', 'aaabcaabacaabacb', 'aaabcaabacabaabc',
>'aaabcaabacabaacb', 'aaabcaacabacaabc', 'aaabcaacabacaacb',
>'aaabcaacbaacaabc', 'aaabcaacbaacabac', 'aaabcabaacaabacb',
>'aaabcabaacabaacb', 'aabaacabaacbaabc', 'aabaacbaabcaabac',
>'aababaacabacabac', 'aababacaabacabac', 'aababacaabcaabac',
>'aababacabaacabac', 'aabacaabacabaabc', 'aabacaabacabaacb',
>'aabacaabacabaacc', 'aabacaabacabacab', 'aabacaabbaacabac',
>'aabacaabcaabacab', 'aabacaabcaabacac', 'aabacaabcaacabac',
>'aabacaacabacaabc', 'aabacaacabacabac', 'aabacaacbaacabac',
>'aabacabaabacabac', 'aabacabaacababac', 'aabacabaacabacab',
>'aabacabaacabacac', 'aabacabacaabacac']
>9 5 1 1 []
>9 5 2 0 ['aaabaabcabaaabcb', 'aaabaabcabaabacb', 'aaababacbaaabacb',
>'aaababcabaaabacb', 'aaababcabaaabcab', 'aaabacbaaabacbab',
>'aaabacbaabaaabcb', 'aaabacbaabaabacb', 'aaabacbaabaabcab',
>'aaabacbabaaabcab', 'aaabcabaaabcabab', 'aaabcabaabaabcab',
>'aaabcabaabacbaab', 'aabaabaacbaababc', 'aabaabaacbabaabc',
>'aabaababcaabaabc', 'aabaabacabaababc', 'aabaabacabaabacb',
>'aabaabacababaabc', 'aabaabacababaacb', 'aabaabacbaabaabc',
>'aabaabacbaabaacb', 'aabaabcaabaabcab', 'aabaabcaababaabc',
>'aabaabcaababaacb', 'aabaabcaababacab', 'aabaabcabaabaacb',
>'aabaabcabaabacab', 'aabaacbaabaacbab', 'aabaacbaababaacb',
>'aabaacbaababacab', 'aabaacbabaabacab', 'aababaabcaababac',
>'aababaacababaabc', 'aababaacababaacb', 'aababaacabababac',
>'aababaacbaababac', 'aabababaacababac', 'aabababacaababac',
>'aabababacabaabac', 'aababacaababacab', 'aababacaabbaabac',
>'aababacaabbaacab', 'aababacabaabacab', 'aababacababaabac',
>'aababacababaacab', 'aabacaabbaabacab', 'aabacaabbaacabab',
>'aabacaacabacaacc', 'aabacaacabacacac', 'aabacaaccaabacac',
>'aabacabaabacabab', 'aabacababaacabab', 'aabacacaabacacac']
>9 6 1 0 ['aabaabaabaababcb', 'aabaabaabaabcbab', 'aabaabaababaabcb',
>'aabaabaababacbab', 'aabaabaababcabab', 'aabaabaabcbaabab',
>'aabaababaabaabcb', 'aabaababaababacb', 'aabaababaababcab',
>'aabaababaabacbab', 'aabaababaabcabab', 'aabaababacbaabab',
>'aabaababcabaabab', 'aabaabacbabaabab', 'aabaabcababaabab',
>'aabaacacacacacac', 'aababaababaababc', 'aababaababaabacb',
>'aababaababaabcab', 'aababaababaacbab', 'aababaabababaabc',
>'aababaabababaacb', 'aababaabababacab', 'aababaababacabab',
>'aababaabacababab', 'aababaabbaababac', 'aababaabcaababab',
>'aababaacababaabb', 'aababaacbaababab', 'aabababaabababac',
>'aabababaababacab', 'aabababaabacabab', 'aabababaacababab',
>'aababababaababac', 'aababababaabacab', 'aababababaacabab',
>'aabababababaabac', 'aabababababaacab']
>10 2 2 2 []
>10 3 2 1 []
>10 3 3 0 ['aaaabacaabacaabc', 'aaaabacaabcaaabc', 'aaaabcaaabacaabc',
>'aaaabcaaabcaaabc', 'aaaabcaaabcaaacb', 'aaaabcaaabcaabac',
>'aaaabcaaacbaaacb', 'aaaabcaabacaaabc', 'aaaabcaabacaabac',
>'aaabaaacbaaacabc', 'aaabaaacbaacaabc', 'aaabaaacbacaaabc',
>'aaabaacaabaacabc', 'aaabaacaabacaabc', 'aaabaacaabcaaabc',
>'aaabaacabaaacabc', 'aaabaacabaaacbac', 'aaabaacabaacaabc',
>'aaabaacabaacabac', 'aaabaacabacaaabc', 'aaabaacabacaabac',
>'aaabaacabcaaabac', 'aaabacaaabacaabc', 'aaabacaaabcaaabc',
>'aaabacaaabcaabac', 'aaabacaaabcabaac', 'aaabacaabaaacabc',
>'aaabacaabaaacbac', 'aaabacaabaacaabc', 'aaabacaabaacabac',
>'aaabacaabacaaabc', 'aaabacaabacaabac', 'aaabacaabacabaac',
>'aaabacabaacaaabc', 'aaabacabaacaabac', 'aaabacabaacabaac',
>'aaabcaaabcaabaac', 'aaabcaabaacaabac', 'aaabcaabaacabaac',
>'aaabcaabacaabaac', 'aaabcabaacaabaac', 'aabaacaabaacabac',
>'aabaacaabacaabac', 'aabaacabaacaabac']
>10 4 1 1 []
>10 4 2 0 ['aaaaabcabaaaabcb', 'aaaaabcbaaaaabcb', 'aaaaabcbaaaabacb',
>'aaaabacabaaabacb', 'aaaabacabaabaacb', 'aaaabacbaaaabacb',
>'aaaabacbaaaabcab', 'aaaabacbaaabaacb', 'aaaabcaabaaabcab',
>'aaaabcaabaabacab', 'aaaabcabaaaabcab', 'aaaabcabaaabacab',
>'aaabaaacbaabaabc', 'aaabaaaccaabaacc', 'aaabaabaacabaabc',
>'aaabaabacaabaabc', 'aaabaabcaaabaabc', 'aaabaabcaaabaacb',
>'aaabaabcaabaaabc', 'aaabaacabaaabacb', 'aaabaacabaabaacb',
>'aaabaacacaabaacc', 'aaabaacbaaabaacb', 'aaabaacbaaabacab',
>'aaabaacbaaabcaab', 'aaabaacbaabaaacb', 'aaabaaccaaabaacc',
>'aaabaaccaabaaacc', 'aaabacaabaaabcab', 'aaabacaabaabacab',
>'aaabacabaaabacab', 'aaabacabaaabcaab', 'aaabacabaabaacab',
>'aaabbaacaaabbaac', 'aaabbaacaababaac', 'aaabcaabaaabcaab',
>'aaabcaabaaacbaab', 'aaabcaabaabacaab', 'aaabcaacaaabcaac',
>'aaabcaacaabacaac', 'aaabcaacabaacaac', 'aabaabaacabaabac',
>'aabaabacaabaabac', 'aabaabacaababaac', 'aabaacaababaacab',
>'aabaacabaabaacab', 'aabaacacaabaacac']
>10 5 1 0 ['aaabaaabaabcbaab', 'aaabaabaaabaabcb', 'aaabaabaaabcbaab',
>'aaabaabaabaaabcb', 'aaabaabaabaabacb', 'aaabaabaabaabcab',
>'aaabaabaabacbaab', 'aaabaabaabcabaab', 'aaabaabacbaaabab',
>'aaabaabacbaabaab', 'aaabaabcabaaabab', 'aaabaabcabaabaab',
>'aaababaaabacbaab', 'aaababaaabcabaab', 'aaabacbaabaabaab',
>'aaabcabaabaabaab', 'aabaabaabaabaabc', 'aabaabaabaabaacb',
>'aabaabaabaababac', 'aabaabaabaabacab', 'aabaabaabaacabab',
>'aabaabaababaabac', 'aabaabaababaacab', 'aabaabaabacaabab',
>'aabaabaacabaabab', 'aabaababaabaabac', 'aabaababaabaacab',
>'aabaababaababaac', 'aabaababaacaabab', 'aabaacaababaabab']
>11 2 2 1 []
>11 3 1 1 []
>11 3 2 0 ['aaaaaabcaaabaacb', 'aaaaaabcaabaaacb', 'aaaaabacaabaaacb',
>'aaaaabcaaabaaacb', 'aaaaabcaaabaacab', 'aaaabaaaacbaaabc',
>'aaaabaaacabaaabc', 'aaaabaaacbaaaabc', 'aaaabaacaabaaabc',
>'aaaabaacaabaaacb', 'aaaabaacabaaaabc', 'aaaabaacabaaaacb',
>'aaaabacaaabaaabc', 'aaaabacaaabaaacb', 'aaaabacaaabaacab',
>'aaaabacaabaaaabc', 'aaaabacaabaaaacb', 'aaaabacaabaaacab',
>'aaaabcaaaacbaaac', 'aaaabcaaabaaacab', 'aaaabcaaabaacaab',
>'aaaabcaaacaaabac', 'aaaabcaaacaabaac', 'aaaabcaaacabaaac',
>'aaabaaacabaaabac', 'aaabaabaacaaabac', 'aaabaabaacaabaac',
>'aaabaacaabaaabac', 'aaabaacaabaaacab', 'aaabaacaabaaacac',
>'aaabaacaabaabaac', 'aaabaacaabaacaab', 'aaabaacaabaacaac',
>'aaabaacaacaaabac', 'aaabaacaacaabaac']
>11 4 1 0 ['aaaabaaaabaaabcb', 'aaaabaaaabcbaaab', 'aaaabaaabaaaabcb',
>'aaaabaaabaaabacb', 'aaaabaaabaaabcab', 'aaaabaaabaacbaab',
>'aaaabaaabacabaab', 'aaaabaaabacbaaab', 'aaaabaaabcaabaab',
>'aaaabaaabcabaaab', 'aaaabaabaaabaacb', 'aaaabaabaaabacab',
>'aaaabaabaaabcaab', 'aaaabaabaacbaaab', 'aaaabaabacabaaab',
>'aaaabaabcaabaaab', 'aaaabaacbaaabaab', 'aaaabacabaaabaab',
>'aaaabacbaaabaaab', 'aaaabcaabaaabaab', 'aaaabcabaaabaaab',
>'aaabaaabaaabaabc', 'aaabaaabaaabaacb', 'aaabaaabaaabacab',
>'aaabaaabaaabcaab', 'aaabaaabaaacbaab', 'aaabaaabaabaaabc',
>'aaabaaabaabaaacb', 'aaabaaabaabaacab', 'aaabaaabaabacaab',
>'aaabaaabaacabaab', 'aaabaaabacaabaab', 'aaabaaabcaaabaab',
>'aaabaaacbaaabaab', 'aaabaabaaabaabac', 'aaabaabaaabaacab',
>'aaabaabaaabacaab', 'aaabaabaaacabaab', 'aaabaabaabaaabac',
>'aaabaabaabaaacab', 'aaabaabaabaabaac', 'aaabaabaabaacaab',
>'aaabaabaacaaabab', 'aaabaabaacaabaab', 'aaabaacaabaaabab',
>'aaabaacaabaabaab', 'aaabaacaacaaacac', 'aaabaacaacaacaac']
>12 2 1 1 []
>12 2 2 0 ['aaaaaaabcaaaaacb', 'aaaaaabaacaaaabc', 'aaaaaabacaaaaabc',
>'aaaaaabcaaaaaabc', 'aaaaaabcaaaaaacb', 'aaaaaabcaaaaabac',
>'aaaaaabcaaaabaac', 'aaaaabaaaacaabac', 'aaaaabaaacaaabac',
>'aaaaabaacaaaabac', 'aaaaabacaaaaabac', 'aaaaabacaaaabaac',
>'aaaaabacaaabaaac', 'aaaaabacaabaaaac', 'aaaabaaaacaabaac',
>'aaaabaaacaaabaac', 'aaaabaacaaaabaac', 'aaaabaacaaabaaac', 'aaabaaacaaabaaac']
>12 3 1 0 ['aaaaaabaaaaaabcb', 'aaaaaabaaaaabacb', 'aaaaaabaaaaabcab',
>'aaaaaabacbaaaaab', 'aaaaaabcabaaaaab', 'aaaaabaaaaabaacb',
>'aaaaabaaaaabacab', 'aaaaabaaaaabcaab', 'aaaaabaaaabaaacb',
>'aaaaabaaaabaacab', 'aaaaabaaaabacaab', 'aaaaabaaaabcaaab',
>'aaaaabaaacbaaaab', 'aaaaabaacabaaaab', 'aaaaabacaabaaaab',
>'aaaaabcaaabaaaab', 'aaaabaaaabaaaabc', 'aaaabaaaabaaaacb',
>'aaaabaaaabaaabac', 'aaaabaaaabaaacab', 'aaaabaaaabaacaab',
>'aaaabaaaabacaaab', 'aaaabaaaacabaaab', 'aaaabaaabaaaabac',
>'aaaabaaabaaaacab', 'aaaabaaabaaabaac', 'aaaabaaabaaacaab',
>'aaaabaaabaacaaab', 'aaaabaaacaaabaab', 'aaaabaaacaaacaac',
>'aaaabaaacaabaaab', 'aaaabaabaaabaaac', 'aaaabaabaaacaaab',
>'aaaabaacaaabaaab', 'aaaabaacaaacaaac', 'aaabaaabaaabaaac']
>13 1 1 1 []
>13 2 1 0 ['aaaaaaaaaaaaabcb', 'aaaaaaaaaaaabacb', 'aaaaaaaaaaaabcab',
>'aaaaaaaaaaabaacb', 'aaaaaaaaaaabacab', 'aaaaaaaaaaabcaab',
>'aaaaaaaaaabaaacb', 'aaaaaaaaaabaacab', 'aaaaaaaaaabacaab',
>'aaaaaaaaaabcaaab', 'aaaaaaaaabaaaacb', 'aaaaaaaaabaaacab',
>'aaaaaaaaabaacaab', 'aaaaaaaaabacaaab', 'aaaaaaaaabcaaaab',
>'aaaaaaaabaaaaacb', 'aaaaaaaabaaaacab', 'aaaaaaaabaaacaab',
>'aaaaaaaabaacaaab', 'aaaaaaaabacaaaab', 'aaaaaaaabcaaaaab',
>'aaaaaaabaaaaaabc', 'aaaaaaabaaaaaacb', 'aaaaaaabaaaaaacc',
>'aaaaaaabaaaaacab', 'aaaaaaabaaaacaab', 'aaaaaaabaaacaaab',
>'aaaaaaabaacaaaab', 'aaaaaaabacaaaaab', 'aaaaaaabbaaaaaac',
>'aaaaaaabcaaaaaab', 'aaaaaaabcaaaaaac', 'aaaaaabaaaaaabac',
>'aaaaaabaaaaaacab', 'aaaaaabaaaaaacac', 'aaaaaabaaaaabaac',
>'aaaaaabaaaaacaab', 'aaaaaabaaaaacaac', 'aaaaaabaaaacaaab',
>'aaaaaabaaacaaaab', 'aaaaaabaabaaaaac', 'aaaaaabaacaaaaab',
>'aaaaaabaacaaaaac', 'aaaaabaaaaabaaac', 'aaaaabaaaaacaaab',
>'aaaaabaaaaacaaac', 'aaaaabaaaabaaaac', 'aaaaabaaaacaaaab', 'aaaaabaaaacaaaac']
>14 1 1 0 ['aaaaaaaaaaaaaabc', 'aaaaaaaaaaaaabac', 'aaaaaaaaaaaabaac',
>'aaaaaaaaaaabaaac', 'aaaaaaaaaabaaaac', 'aaaaaaaaabaaaaac',
>'aaaaaaaabaaaaaac', 'aaaaaaabaaaaaaac']
>
>

Mike wrote:

>> The reason is that the JI major scale is small - its block is
>> only one unit tall in the 5-wardly direction. As a result,
>> vectors that fit in it can only take a few different angles,
>> none of which produce the (v, v+a, v+b, v+a+b) case (I think).
>
>That's interesting. Not sure what it means. Are there periodicity
>blocks where only one of the vectors produces 4 step sizes and the
>rest produce 3?

Only two, yes (the one that does, and its inversion).

See my original post where I define the (v,...) notation for
more details.

>> >> Paul's won nothing of the kind. Distributional evenness is a
>> >> property Keenan's scales have and that's what he's been calling
>> >> them. It's not completely clear what rank-3 MOS are yet or
>> >> what they should be called when they're understood.
>> >
>> >Are you saying that an imprint like xL+ym+zs can lead to more than
>> >one DE scale, but that there might exist one that's more symmetric
>> >than the others in an as of yet undefined way, and scales with that
>> >property will get the MOS title?
>>
>> Not exactly...
>
>So what are you saying then?

That Paul already lost that argument because of a Wilson
paper Graham found?

-Carl

Some of these have 4 step sizes, but they aren't rank 3. For example,
consider the trivial case "abcd" - if you tune it so that a is 5/4, b
is 6/5, c is 7/6, and d is 8/7, you end up with a scale that has 4
step sizes in each case, but is rank 4.

If the smallest interval is d, then we have three possible chromatic
vectors - the thing that turns d into c (49/48), the thing that turns
d into b (21/20), and the thing that turns d into a (35/32). However,
if this were a rank-3 Fokker block, we'd want the vector turning d
into a to be the sum of the vectors turning d into c and d into b,
respectively, meaning we'd want 35/32 ~= 49/48 * 21/20. This means
that 50/49 vanishes, which is the only way to turn this into a block
with 4 step sizes (unless you want to temper out 9/8 or something, but
I believe that would cause one of the unison vectors to be negative).
However, the "seconds" now end up only having three step sizes - 3/2,
4/3, and 7/5~10/7.

This scale is interesting in that there doesn't seem to be any way to
turn it into a rank-3 periodicity block without reducing at least one
of the generic interval classes to having only 3 specific sizes. I'm
not sure how widespread this behavior is. If you don't like that, your
remaining options are to get rid of 25/24, 36/35, or 49/48, but then
"abcd" doesn't effectively describe this scale anymore.

-Mike

On Thu, Jul 28, 2011 at 10:19 AM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
>
>
> Carl asked for some results for N=4 DE scale patterns, so here they are up to 16 notes:
>
> 1 1 1 0 ['abc']
> 1 1 1 1 ['abcd']
> 2 1 1 0 ['aabc', 'abac']

There can also be non-convex rank-3 epimorphic blocks (MODMOS's) that
still obey the 2^(n-1) rule. A trivial example is the scale 1/1 16/15
5/4 3/2 (2/1), taken from 5-limit JI, which looks like a square with
the top right corner cut out and shifted one unit down and one to the
left (so it looks kind of like ".L"). This has four step sizes in
every generic interval class, but is not a parallelotope. I'm not sure
if any exist like this that have more than four intervals per period.
Thanks to Graham for helping work this out as well.

This is analogous to how the porcupine-tempered JI major scale has 3
interval sizes in each class. It's also a MODMOS of the Lssssss mode
of porcupine[7] with a flattened fourth and sharpened seventh; both
intervals are altered by L-s.

-Mike

On Thu, Jul 28, 2011 at 3:00 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
> Some of these have 4 step sizes, but they aren't rank 3. For example,
> consider the trivial case "abcd" - if you tune it so that a is 5/4, b
> is 6/5, c is 7/6, and d is 8/7, you end up with a scale that has 4
> step sizes in each case, but is rank 4.
>
> If the smallest interval is d, then we have three possible chromatic
> vectors - the thing that turns d into c (49/48), the thing that turns
> d into b (21/20), and the thing that turns d into a (35/32). However,
> if this were a rank-3 Fokker block, we'd want the vector turning d
> into a to be the sum of the vectors turning d into c and d into b,
> respectively, meaning we'd want 35/32 ~= 49/48 * 21/20. This means
> that 50/49 vanishes, which is the only way to turn this into a block
> with 4 step sizes (unless you want to temper out 9/8 or something, but
> I believe that would cause one of the unison vectors to be negative).
> However, the "seconds" now end up only having three step sizes - 3/2,
> 4/3, and 7/5~10/7.
>
> This scale is interesting in that there doesn't seem to be any way to
> turn it into a rank-3 periodicity block without reducing at least one
> of the generic interval classes to having only 3 specific sizes. I'm
> not sure how widespread this behavior is. If you don't like that, your
> remaining options are to get rid of 25/24, 36/35, or 49/48, but then
> "abcd" doesn't effectively describe this scale anymore.
>
> -Mike
>
>
> On Thu, Jul 28, 2011 at 10:19 AM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>>
>>
>>
>> Carl asked for some results for N=4 DE scale patterns, so here they are up to 16 notes:
>>
>> 1 1 1 0 ['abc']
>> 1 1 1 1 ['abcd']
>> 2 1 1 0 ['aabc', 'abac']
>

On Thu, Jul 28, 2011 at 9:56 AM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > > Interesting. Keenan's 3DE scales are rank 3, but only a small
> > > subset of possible rank 3 scales, where one comma subdivides
> > > another or something... producing an effect akin to rank 2
> > > temperaments where the period subdivides the octave...
> >
> > If one comma subdivides another, wouldn't we be dealing with torsion
> > or something? I guess I'll figure it out when I read through the
> > thread.
>
> I don't think it's related to torsion at all. I think I would describe it like this: Although there are three chromata*,

LOL, chromata, chroma, I can't win. I say chromata, Carl says chroma,
I say chroma, you say chromata. Gene says don't call it a chromatic
vector, call it a chroma, Graham says it's a chromatic unison vector.
Oh well.

> there are never four different intervals in a class because the chromata always appear in a triangle pattern. One chroma is equal to the sum of the other two, and it always appears subdivided by the other two.

What do you mean "subdivided by" the other two?

> * (plural of "chroma")
>
> > > I think it's best to avoid the term "3GMP" for now.
> >
> > OK, so 3DE is the term now? Looks like Paul's finally won this battle...
>
> Haha, I didn't know there was a battle. He just told me that Myhill was defined as "exactly two" where as DE was "at most two". The thing I want to talk about is "at most N", so it seems natural to call it a generalization of DE rather than a generalization of Myhill.

The Great MOS War is over whether or not MOS means DE or Myhill. We've
been using it to be synonymous with DE (so that diminished[8] is an
MOS of diminished temperament), and Paul insists that it's synonymous
with Myhill. I've resolved to call the whole thing MOS and just stop
thinking about it.

-Mike

On Thu, Jul 28, 2011 at 10:07 AM, Keenan Pepper <keenanpepper@gmail.com> wrote:
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > Yes. Intervals in the same interval class can differ by L-m, m-s, and
> > L-s. That gives you three possible chromatic alterations, bringing you
> > to four interval sizes total. However, there are only two orthogonal
> > chroma, because c_o = c_u + c_l. I suppose we could call c_u and c_l
> > orthonormal, in a sense.
>
> Both "orthogonal" and "orthonormal" only make sense in an inner product space, which is a vector space with an inner product and hence also a metric. This would mean we have not only a way to measure the "size" or "length" of a vector, but also a way to measure the "angle" between two of them.

How can we talk sensibly about the convexity of the blocks that are
tiling the lattice if we aren't treating this as an inner product
space? The convexity of a shape has to do with the size of its
interior angles, correct?

> But interval space doesn't come with any natural metric. You can invent different metrics for it, as Gene did and used in the definition of hobbit scales, but there's no one obvious choice of metric. I haven't seen anyone ever define an inner product.

When Carl said this, I assumed everyone was treating everything as a
Euclidean space:

Carl wrote:
> Only a few linear combinations are possible, because it must
> be possible to fit v (the vector representing the generic interval
> in question) inside the block without without crossing any
> boundaries. IOW, the a-component of v must be < a itself, and
> likewise for b. I think this just means only coefficients of 1
> are possible.

For us to be talking about the "a-component" of v at all, there'd have
to exist of some inner product between v and a, correct? And does the
concept of convexity have an analogue in a taxicab space? But maybe
you could work it out with the L1 norm if you wanted to.

-Mike

On Thu, Jul 28, 2011 at 12:34 PM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
>
> >> The reason is that the JI major scale is small - its block is
> >> only one unit tall in the 5-wardly direction. As a result,
> >> vectors that fit in it can only take a few different angles,
> >> none of which produce the (v, v+a, v+b, v+a+b) case (I think).
> >
> >That's interesting. Not sure what it means. Are there periodicity
> >blocks where only one of the vectors produces 4 step sizes and the
> >rest produce 3?
>
> Only two, yes (the one that does, and its inversion).
>
> See my original post where I define the (v,...) notation for
> more details.

I can't find the original post defining the notation, only the one
where you first laid out the 4 step sizes per interval idea, but I'll
take your word for it. There do seem to be an awful lot of blocks that
have all interval sizes at 4 except for 1 at three though.

> >> >Are you saying that an imprint like xL+ym+zs can lead to more than
> >> >one DE scale, but that there might exist one that's more symmetric
> >> >than the others in an as of yet undefined way, and scales with that
> >> >property will get the MOS title?
> >>
> >> Not exactly...
> >
> >So what are you saying then?
>
> That Paul already lost that argument because of a Wilson
> paper Graham found?

How are you distinguishing between "distributional evenness" and "MOS"
in this comment?

Carl wrote:
> Paul's won nothing of the kind. Distributional evenness is a
> property Keenan's scales have and that's what he's been calling
> them. It's not completely clear what rank-3 MOS are yet or
> what they should be called when they're understood.

If Paul lost, then distributional evenness and MOS are synonymous, yes?

-Mike

Mike wrote:

>Some of these have 4 step sizes, but they aren't rank 3. For example,
>consider the trivial case "abcd" - if you tune it so that a is 5/4, b
>is 6/5, c is 7/6, and d is 8/7, you end up with a scale that has 4
>step sizes in each case, but is rank 4.

They're just scale patterns. What rank etc. they are depends how
they're mapped.

>If Paul lost, then distributional evenness and MOS are synonymous, yes?

There are all kinds of caveats with these things, and so many
variants I've forgotten. I'd have to review the Clough paper
to be sure, but I think the answer is yes.

-Carl

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> LOL, chromata, chroma, I can't win. I say chromata, Carl says chroma,
> I say chroma, you say chromata. Gene says don't call it a chromatic
> vector, call it a chroma, Graham says it's a chromatic unison vector.
> Oh well.

Well, it is an actual Greek word, and the Greek plural is "chromata". The other option is to use the English plural "chromas", which I would also be fine with. I don't see why anyone would use "chroma" as the plural.

> > there are never four different intervals in a class because the chromata always appear in a triangle pattern. One chroma is equal to the sum of the other two, and it always appears subdivided by the other two.
>
> What do you mean "subdivided by" the other two?

I mean if you make a list of all the intervals that appear in the scale, then whenever you have two intervals that differ by the large chroma, you also have one of intermediate size that differs from the other two by the two smaller chromas.

I can see why that's a confusing thing to say because it's not "subdivided" in the scale - because none of the chromas appear in the scale at all.

Keenan

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> How can we talk sensibly about the convexity of the blocks that are
> tiling the lattice if we aren't treating this as an inner product
> space? The convexity of a shape has to do with the size of its
> interior angles, correct?

Excellent question! I actually had to think for a good minute about this when Carl used the term, but a convex set is perfectly well defined in any vector space.

A "convex combination" of two vectors x and y is a linear combination of the form ax + by where a >= 0, b >= 0, and a + b = 1. Geometrically it's just the line segment between them. (A convex combintation of more than two vectors is defined similarly.)

A set of vectors in a vector space is a "convex set" if, for any two vectors in the set, the set also contains all convex combinations of those two vectors.

This definition makes no references to lengths or angles, so it works for any vector space, regardless of any metric or inner product, and (exercise for the reader) it agrees with the "interior angles" definition whenever there happens to be an inner product.

> When Carl said this, I assumed everyone was treating everything as a
> Euclidean space:
>
> Carl wrote:
> > Only a few linear combinations are possible, because it must
> > be possible to fit v (the vector representing the generic interval
> > in question) inside the block without without crossing any
> > boundaries. IOW, the a-component of v must be < a itself, and
> > likewise for b. I think this just means only coefficients of 1
> > are possible.
>
> For us to be talking about the "a-component" of v at all, there'd have
> to exist of some inner product between v and a, correct? And does the
> concept of convexity have an analogue in a taxicab space? But maybe
> you could work it out with the L1 norm if you wanted to.

Well, there are (at least) two things you could mean when you say "component". You could mean the component of some vector along a single specified unit vector direction, which is simply a dot product and obviously is only defined in an inner product space.

But you could also mean the coefficient of a certain basis vector in some basis that's already under consideration. This makes sense in any vector space because for any vector v, given a basis [e1,e2,e3...], there is a unique ordered list of coefficients [c1,c2,c3...] such that c1e1 + c2e2 + c3e3 + ... = v. The "component" or "coordinate" of v along e1 in this basis is simply c1. (Unlike the first meaning, this kind of "component" depends on all the other vectors in the basis too.)

I think Carl may have intended it in the latter sense.

Keenan

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> They're just scale patterns. What rank etc. they are depends how
> they're mapped.

Exactly. Each has a maximum rank, but no minimum rank (because you can map it to a subset of an equal temperament and make it rank-1 if you want).

Keenan

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >> Joint publishing efforts around here have failed, I think,
> >> largely because they tried to disclose the entire regular
> >> mapping paradigm in a single paper.
> >
> >I've been finding the Xenwiki a pretty good outlet.
>
> You have indeed and it will stand a long time.

Thanks. I think distributed hyperlinks and the Internet make for a new publication paradigm, if publish or perish is not a concern.

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> How can we talk sensibly about the convexity of the blocks that are
> tiling the lattice if we aren't treating this as an inner product
> space? The convexity of a shape has to do with the size of its
> interior angles, correct?

Convexity is an affine property and doesn't require angles.

> > But interval space doesn't come with any natural metric. You can invent different metrics for it, as Gene did and used in the definition of hobbit scales, but there's no one obvious choice of metric. I haven't seen anyone ever define an inner product.

Well, sure we have, for instance:

http://xenharmonic.wikispaces.com/Monzos+and+Interval+Space
http://xenharmonic.wikispaces.com/Vals+and+Tuning+Space
http://xenharmonic.wikispaces.com/Tenney-Euclidean+metrics
http://xenharmonic.wikispaces.com/Tenney-Euclidean+temperament+measures

Surprising conjecture about rank-3 distributionally even scales

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