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Triple minor sets

Q: In English country dancing a "triple minor" set consists of a line of n couples, numbered 1, 2, 3, 1, 2, 3, ... from the top: each block of three couples 1, 2, 3 is called a "minor set". At each turn of the dance the 1s move down one place and the 2s move up one place, and the 2s and 3s then swap numbers to form new minor sets numbered 1, 2, 3 (omitting the new top couple). During any turn only those in complete minor sets dance (experienced dancers sometimes do a fudged version in an incomplete minor set, but ignore this for now).

The 1s move down the set: when they reach the bottom - 1 position they immediately move to the bottom, and wait two turns to restart as 3s. The others move (more slowly) up the set, alternately dancing as 2s and 3s: when they reach the top - 1 position they wait one turn, dance once as 2s to get to top place, and wait two turns to restart as 1s. Normally the dance has t = 3u + 1 turns, for some u.

Many people believe that couples spend a greater proportion of time dancing if n is not a multiple of 3. Is this true?
David Barnert, ECD discussion list, 19 Jun 2001

Q: Generalise to minor sets of m couples, as follows: after the first turn the 1s have moved down one place and as before become the top of new minor sets, so the couple below them change their number to 2, the next couple to 3, etc. End effects in this case are left as an exercise for the reader. Let n = pm + q, and while we're at it generalise the number of turns to t = um + v. How does the number of couples affect the efficiency?

Notes: There are many "duple minor" English country dances, with m = 2. There are quite a few "triple minors", with m = 3 as above, which were particularly popular in the 17th and 18th centuries because they tend to give the 2s and 3s more flirting time. Many Scottish country dances also have this type of progression, almost always with n = 4, m = 3 and t = 8 (although with such a small n the two sets of end effects meet...). Hampstead Manor is a "quadruple minor", with m = 4, but I don't know details, or any other such dances. A dance with m = 3 or 4 should not be confused with a duple minor that has a triple or quadruple progression. Nor is it the same as a double contra, such as Jim Kitch's Double Chocolate, which is different again.


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