Maximal evenness: Difference between revisions

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# ME(n, m) where n < m/2. This is a maximally even subset of Z/nZ with step sizes L > s > 1, which determines the locations of step sizes of 2 in the complement. The rest of the complement's step sizes are 1. The sizes of the chunks of 1 are L - 2 and s - 2 (0 is a valid chunk size), and the sizes form a maximally even MOS.

# ME(n, m) where n > m/2. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since n > m/2) occupy a maximally even subset of the slots of ME(n, m) (*). Now replace each 1 with "| |" and each 2 with "| . |"~~, and then~~ collapse "||" to "|". The "|"s form chunks of sizes that differ by 1 and are distributed in a MOS way by (*).

# ME(n, m) where n > m/2. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since n > m/2) occupy a maximally even subset of the slots of ME(n, m) (*). Now do the following steps:

# replace each 1 with "|.|" and each 2 with "|$|"

# collapse "||" to "|"

# remove all "."

Consider the resulting binary word of "|" and "$". The "|"s form chunks of sizes that differ by 1 and are distributed in a MOS way by (*).

(e.g.) 21211 -> |$||.||$||.||.| -> |$|.|$|.|. -> |$||$||

== Concoctic scales ==

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 # ME(n, m) where n < m/2. This is a maximally even subset of Z/nZ with step sizes L > s > 1, which determines the locations of step sizes of 2 in the complement. The rest of the complement's step sizes are 1. The sizes of the chunks of 1 are L - 2 and s - 2 (0 is a valid chunk size), and the sizes form a maximally even MOS.
-# ME(n, m) where n > m/2. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since n > m/2) occupy a maximally even subset of the slots of ME(n, m) (*). Now replace each 1 with "| |" and each 2 with "| . |", and then collapse "||" to "|". The "|"s form chunks of sizes that differ by 1 and are distributed in a MOS way by (*).
+# ME(n, m) where n > m/2. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since n > m/2) occupy a maximally even subset of the slots of ME(n, m) (*). Now do the following steps:
+# replace each 1 with "|.|" and each 2 with "|$|"
+# collapse "||" to "|"
+# remove all "."
+Consider the resulting binary word of "|" and "$". The "|"s form chunks of sizes that differ by 1 and are distributed in a MOS way by (*).
+(e.g.) 21211 -> |$||.||$||.||.| -> |$|.|$|.|. -> |$||$||
 == Concoctic scales ==