Permutations

How many ways are there to order two couples? or three? or four? Each of these orderings is a permutation (or, in older parlance, a change). And each implies a certain kind of progression. By looking at all permutations we can examine the various possible styles of progression.

I exclude from this analysis progressions which go outside the minor set, I do not have a good notation for this behavior. So I don't talk about double progression in a duple minor, nor triple progression in a triple minor.

Notation: I write a permutation by showing where everyone ends. Everyone starts in 1 2 3 4 ... order. If I write a permutation as (1 2 3 4) then no one has moved at all, everyone ends where they started. This is called an identity progression. But if I write (4 3 2 1) then everyone has moved to the other end of the set from where they started.

Changes of 2

There are only two permutations of two couples: The identity permutation, which leaves everyone where they started, and the permutation which mixes people up.

(1 2), Identity

In the identity permutation the 1s remain at the top of the set and the 2s at the bottom. This won't work in a duple minor format (you are doomed to have the same neighbors forever), but it works fine for a two couple dance where no progression is expected.

(2 1)

Everyone ends up where their neighbor started. This is the standard duple minor progression.

Changes of 3

There are six permutations of three couples

(1 2 3), Identity

Everyone ends where they started. There are lots of three couple dances with no form of progression.

(3 1 2) and (2 3 1)

Both of these permutatios end with everyone in a different place, and they both have period three (that is if you dance a dance with one of these permutations then you will end where you started on the third time but not before). These are exactly the requirements for a standard three couple dance with progression.

If you want a mixer, then you have the men progress with one of these permutations, while the women progress with the other.

(3 1 2) - An unexpected triple minor

This permutation also works as a triple minor. I've never seen any triple minor that actually used it — they all use the simpler (2 1 3) permutation instead. However there are lots of 3 couple dances which do use it and they can all be danced as triple minors if anyone wanted to.

For those used to normal triple minors it has some unexpected behaviors: the 2s+3s retain their roles to the ends of the set, just as the 1s do. This might make the dance easier to learn for the 2s+3s. Also, everyone progresses: the 1s and 2s move down (unexpected for 2s) and the 3s move up two places.

This permutation has an unfortunate effect: If you want to insure that the 2s get to dance as 1s then you may need to be tricky.

If the number of couples in your set is divisible by six, then when there are two couples out at the top you must have them change places (and they do not change when there are two out at the bottom)
If the number of couples in your set is equal to 1 mod 6 (7,13...) then you do not change when there are two couples out at top, nor at bottom. Equally you may change at both top and bottom.
If the number of couples in your set is equal to 2 mod 6 (8,14...) then you change when there are two couples out at bottom but not at top.
If the number of couples in your set is equal to 3 mod 6 (9,15...) then you do not change when there are two couples out at top, nor at bottom. Equally you may change at both top and bottom.
If the number of couples in your set is equal to 4 mod 6 (10,16...) then you change when there are two couples out at bottom but not at top.
If the number of couples in your set is equal to 5 mod 6 (11,17...) then you do not change when there are two couples out at top, nor at bottom. Equally you may change at both top and bottom.

In other words how you behave when you are out depends on how many couples were in the set when you started, which is unexpected. If you always have 2 mod 6 or 4 mod 6 couples in a set then behavior is similar to that in a normal triple minor dance so it might be best to stick with that size.

Similarly any Scottish 3 couple dance with this progression can be danced as an unusual 3 couple dance in a 4 couple set.

As examples I present:

Joseph Pimentel's Earth Adored.
Anthea Macdonald's Mistress Mary

(2 3 1) - The rather useless double progression triple minor

This permutation could also be used in a double progression triple minor (indeed, it has been, look at Step Stately). Usually in a triple minor we expect the 2s and 3s to alternate roles, but this doesn't happen here, so that may be a disadvantage. Double progression triple minors have awkward end effects. It is possible to make them work, but it isn't obvious and probably only should be played with in a workshop situation.

However, this permutation works very nicely when the line only has 5 couples (what the Scottish would call a three couple dance in a five couple set). There the dance happens first with the 1 2 3 couples (the 4s and 5s being out at bottom), then again with the 1 4 5 couples (the 2s and 3s being out at top). After the second iteration the 1s are at the bottom. Then the 2nd couple starts and dances twice, and so on.

The problem with double progression triple minors in a longer set is that at some point you have four couples out at the top. If the top couple starts then the fourth couple down is not in any minor set. Or, perhaps it is better to look at as a quadruple minor dance, with the fourth couple happening to be inactive (after all lots of 18th century triple minors had an inactive third couple, so why not an inactive fourth? In fact in the 19th century there was usually an inactive couple between every active triple in a triple minor)

The other option for when you have 4 couples out at the top, is that the second couple starts the dance. This actually works well, but will be quite unexpected for your dancers.

An Example.

The Playford dance Jenny Come Tie My Cravat is a multipart triple minor, which Colin Hume interpreted as a three couple longways dance with this progression. And I decided to play with it as an example of this style.

Similarly any Scottish 3 couple dance with this progression can be danced as an 3 couple dance in a 5 couple set.

Chris Brooks' Blue Butterfly

Which means that any three couple dance can be danced as a triple minor without changing the figures of the dance.

(1 3 2), (3 2 1) and (2 1 3)

These permutations all have period two and only mix up two couples. They don't work in a three couple set.

(2 1 3)

This one mixes up the top two couples, and is exactly the permutation needed for a standard triple minor.

There are two permutations which work for a triple minor (2 1 3) and (3 1 2) so it should be possible to write a mixer where the 2s and 3s exchange partners (the 1s need to keep theirs), but having one sex use one permutation and the other, the other.

(3 2 1)

This permutation also leads to a double progression triple minor, and here the 2s and 3s do alternate roles as expected. It has the same disturbing end effects as other double progression triple minors. Back in the 1970s&80s Al Olsen experimented with this format; see: Baskets of Brew.

(1 3 2)

This dance mixes up the bottom two couples. It's sort of a reverse triple minor. The 3s move up and the 1s+2s move down, alternating roles. I know of no dance with this format, but if you wanted to play, you could take any normal triple minor and reverse everything (3s for 1s, up for down, etc.)

Changes of 4

There are twenty-four permutations of four couples

(1 2 3 4), Identity

Everyone ends where they started. There are four couple dances with no form of progression.

(4 1 2 3), (3 1 4 2), (2 4 1 3), (4 3 1 2), (2 3 4 1), and (3 4 2 1)

All of these permutations have period 4 and will work for a 4 couple set either in a square or longways formation.

You might think that with so many to choose from you would be able to get a mixer as we did in the three couple case (where everyone is in a different position with a different partner every time). But it turns out that no matter which two permutations you choose, at some point at least one couple will be dancing with their original partner before the end of the set. Gary Roodman has proved this fact mathematically, while I have proved it more prosaically by getting a computer to iterated over every possibility - none worked.

(2 1 4 3), (4 3 2 1), and (3 4 1 2)

These three permutations all have period 2 but they also mix up all the couples each time.

In his search for a four couple mixer where every dancer dances in every position available to his/her sex and always has a different partner, Gary Roodman realized that if you alternate between two permutations you can sometimes find a mixer.

Now obviously any such permutation must mix up all the dancers. It turns out that all the 6 permutations of period 4 will mix everyone up, as will the three permutations of period 2 listed above.

But oddly the period 4 permutations do not work in combo to produce a mixer, only the period 2 permutations (this rather surprised me).

Example of one mixer for a square where each dancer has a different partner and different position each time through the music.
I use a square as my example, but the same set of permutations will work in a 4 couple longways dance.

Men	Women
(2 1 4 3) (3 4 1 2)	(3 4 1 2) (4 3 2 1)
(2 1 4 3) (4 3 2 1)	(3 4 1 2) (2 1 4 3)
(2 1 4 3) (3 4 1 2)	(4 3 2 1) (2 1 4 3)
(2 1 4 3) (4 3 2 1)	(4 3 2 1) (3 4 1 2)
(3 4 1 2) (2 1 4 3)	(2 1 4 3) (4 3 2 1)
(3 4 1 2) (4 3 2 1)	(2 1 4 3) (3 4 1 2)
(3 4 1 2) (2 1 4 3)	(4 3 2 1) (3 4 1 2)
(3 4 1 2) (4 3 2 1)	(4 3 2 1) (2 1 4 3)
(4 3 2 1) (2 1 4 3)	(2 1 4 3) (3 4 1 2)
(4 3 2 1) (3 4 1 2)	(2 1 4 3) (4 3 2 1)
(4 3 2 1) (2 1 4 3)	(3 4 1 2) (4 3 2 1)
(4 3 2 1) (3 4 1 2)	(3 4 1 2) (2 1 4 3)

I list twelve combinations for completeness, but really there are only six, as any combinantion of two permutations on the men with two on the woman is matched by applying the men's permutations to the women and vice versa.

Gary Roodman has gone so far as to produce a mixer for trios in a square (rather than couples). I haven't bothered to work this one out.

(2 1 3 4)

This is the permutation for a standard single progression quadruple minor. There aren't very many quadruple minors, but both Playford and Kynaston published a few around 1720, and Pat Shaw published at least one in the 1970s.

Changes of 5

There are 120 permutations of five couples. When we have this many dancers we get a new style of permutation, we get some with period 6 which is more than the number of couples. Unfortunately the permutation isn't very interesting. Two of the dancers alternate positions, and the other 3 do too. (if the number of couples, n can be written as the sum of distinct numbers, say n=p1+p2+p3+... where the greatest common divisor of an pair of these is 1, then there will be permutations of period p1*p2*p3*... In the case of 5 p1 is 2 and p2 is 3. But I don't think this would make a very interesting dance. You'd have a pair of couples doing a 2 couple dance and a trio of couples doing a 3 couple at the same time.

However, with Gary Roodman's trick of alternating permutations we can find two period 6 permutations which, when applied alternately lead to a permutation of period 5, so it will feel to the dancers as if they are dancing 10 times. There are 120 of these, too many to list all, but I give one here:
(2 1 4 5 3) (3 4 1 5 2) -> (4 3 5 2 1) period: 5

(2 3 4 5 1), (2 3 5 1 4), (2 4 1 5 3), (2 4 5 3 1), (2 5 1 3 4), (2 5 4 1 3), (3 1 4 5 2), (3 1 5 2 4), (3 4 2 5 1), (3 4 5 1 2), (3 5 2 1 4), (3 5 4 2 1), (4 1 2 5 3), (4 1 5 3 2), (4 3 1 5 2), (4 3 5 2 1), (4 5 1 2 3), (4 5 2 3 1), (5 1 2 3 4), (5 1 4 2 3), (5 3 1 2 4), (5 3 4 1 2), (5 4 1 3 2), and (5 4 2 1 3)

There are 24 different permutations of period 5 for 5 couples. Any of these will make a reasonable permutation for a five couple longways (or circular) set.

Since there are an odd number of couples in the set, mixers are trivial. There are 72 combinations of two period 5 permutations which generate mixers. That's too many to list but the obvious one where the men move round the circle clockwise and the women counter-clockwise works just as it does for three couples.

(2 1 3 4 5)

I don't know of any quintuple minor dances, but if there were any they'd use this permutation.

Changes of 6

As might be expected there are 120 permutations of period 6 which mix up all the couples. But 6 is the first number where there are permutation of period N which do not mix up all the couples (there are 120 of these too). I'm not going to list them all because 120 is too many.

As in the four couple case there are no simple mixers (where the women move with one permutation and the men another and you never dance with anyone twice nor in the same place twice).

Again I tried the brute force approach. This is becoming difficult with 6 couples. There are 720 different permutations and if I want to check all combinations of four permutations, then I must check 720⁴ possibilities, which is almost a trillion, and that takes significant time, even on a modern computer. Luckily we can restrict our searh to permutations which mix up all elements and there are only 265 of those, which reduces our search space to 5 billion and took a little more than a minute on my laptop.

There are no solutions where each sex alternates between two permutations (at first I thoguht there were, but Gary Roodman kindly found the error in my program). Nor are there any solutions of the form (p₁, p₁, p₂, p₁, p₁, p₂) (that is each sex has two permutations but instead of alternating them, the first is applied twice, then the second is applied, then the first twice and the second). Nor are there any solutions like (p₁, p₂, p₂, p₁, p₂, p₂).

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