# The twenty-four duple minor formations

The most common contra formation is the hands-four. There’s plenty of them — “improper,” “proper,” “reverse progression indecent Becket”. This essay is an attempt to catalog them all, and how to get from one to another.

#### Ground rules

A couple basics. For most of this post I’m considering the specific four spots on the floor where people take hands-four. I’m ignoring rotations smaller than 90 degrees, like the diamond formation, (see also Larry Jennings’ sawtooth formation), or skew translations.

I use “formation” to describe two different concepts. One is the position of the dancers at the beginning of a dance. Another is the position after every move, especially if it finishes on one of the four original spots. Discussion of other spots — like a wave of four, a diamond, a line of four — are saved for later.

When used as a general term, “Becket” means that you and your partner are on the same side of the set. Even if it’s a proper-like “Becket” where all the men are above the women.

Facing/orientation of dancers is irrelevant for this discussion.

The couple traveling down the set is always the ones, regardless of their orientation or position.

Hands fours are shown from the caller’s viewpoint on stage, so the top of the screen is the bottom of the hall.

“First diagonals” refer to first corner positions — when facing across, those on the right diagonal of a hands-four. In a regular improper set-up, they’d be the women. Then “second diagonals” would be the men. In this post, I use “corners” and “diagonals” interchangeably.

In an article this long, there’s bound to be typoes. Let me know if you suspect some.

Asides and digressions are colored in olive green.

#### Defining the problem

Here’s the standard improper formation:

W2 M2 M1 W1

Removing the labels give you four holes to fill with four unique objects (M1,W1,M2,W2):

() () () ()

Basic combinatorics tells you there are 24 (4*3*2*1) possible arrangements.

The most useful way to group these is by the identity of the person on your diagonal. In improper, this is your same-sex neighbor. In proper this is your opposite-sex neighbor. You could also have your partner on your diagonal. Once this is fixed pretty much everyone else falls into place, and you can only vary things by symmetries of rotations (improper to becket) or reflections (improper to indecent) around the central point.

#### The improperish formations family

Let’s start with your same-sex neighbor on the diagonal, and limit it to those where you’re across from your partner. (i.e. no Beckets yet.) Best-known is improper:

W2 M2 M1 W1

Applying the mirror transformation through the center of the set gives a formation best known as “indecent”, though it’s picked up other names as well:

M2|W2 W1|M1

Applying the mirror transformation across the set gives the ones below the twos:

M1 W1 ----- W2 M2

(Dashed lines are just showing where the mirror is, relative to the original improper formation.)

I call this “Progressed improper.” You get there from improper after a neighbor balance and swing. It’s used in a handful of dances, especially by Rick Mohr, like “Leave the Wine” and “The Grass Valley Glide.”

And finally, you can apply both these transformations to the base improper, and get “progressed indecent.”:

W1 M1 M2 W2

(Digression: You could number the progressed indecent couples differently by defining the top couple as ones, and the bottom couple twos. Now couples are improper, but progressing backwards. You’ve got “Galena.”

One of the side stories of this analysis is that “progressed indecent” is just reverse-progression improper. You can permute this stuff, so indecent is also reverse progressed improper, and any other confusing grouping you care to use.)

So these four formations, labelled the A-group (Across from partner) are

Improper | Indecent | Prog Impr | Prog Indecent W2 M2 | M2 W2 | M1 W1 | W1 M1 M1 W1 | W1 M1 | W2 M2 | M2 W2

Rotating them 90 degrees gives the B-group (Becket), the other four formations with a same-sex neighbor on your diagonal:

| Indecent | Indecent | Becket-cw | Becket-cw | Becket-ccw | Becket-ccwM1 W2|W1 M2|W2 M1|M2 W1W1 M2|M1 W2|M2 W1|W2 M1

(I’m using *italics* to represent the B-group. This will matter later.)

(cw and ccw are clockwise and counterclockwise, representing the direction of progression.)

Most modern contra dances have only symmetrical moves. Getting from one of these formations to another is easy, either by rotating the set (a rotational transformation), or a symmetric changing of pair(s) of dancers (a reflection transformation).

Summarizing all the possible transformations:

The small number in the upper left-hand corner of each formation box is merely a convenient label.

Here’s tracking “The Baby Rose” by David Kaynor:

(Balances, do-si-do, and star left 1 don’t change positions of anyone.)

- It starts in Box 1, improper.
- The neighbor swing trades places with your neighbor on the side. Five steps clockwise gives Box 6, progressed improper.
- The circle left 3/4 moves you six steps counterclockwise to Box 8.
- The partner swing trades places with your neighbor on the side, five steps counterclockwise to Box 3, Becket cw.
- Ladies chain has the second diagonals trade, three steps clockwise to Box 6.
- The progression, or redefinition of hands-four moves you five steps counterclockwise back to Box 1.

Digressions: There’s one more transformation — redefinition of hands fours, or progression. This is equivalent to trading with the person on the side. As M1, you just need to assume that M2 and W2 can represent any number of potential neighbors, and W1 is either a partner or a shadow. Redefining hands fours in Becket formation takes you from a hands-four with your partner to a hands-four with your shadow.

Also, be careful with these transformations — they are not commutative — the order you apply them in matters. For instance, “circle left 3/4, then trade places with person on side of set” gives a different result from “trade places with person on side of set, then circle left 3/4.”

Four of these formations are special, and show up at some point in just about every modern contra dance. All swing partner on the side results in Becket-cw or becket-ccw. All swing neighbor on side yields indecent or progressed improper.

#### The proper formations family

So far we’ve avoided the proper formations, where your opposite-sex neighbor is on your diagonal. There’s eight, with similar transformations between them. Once again, there’s an (A)cross and a (B)ecket group.

But how do these relate to the improperish family?

Improper | Proper W2 M2 | W2 M2 M1 W1 | W1 M1

You can’t get between the two of these by any rotation or reflection around a central point of symmetry. Instead you have to use an asymmetric move, like ones half figure-eight. More on this in a bit.

#### Diagonal formations

The final eight formations have your partner on the diagonal, so I am call them the diagonal family. No dance starts here, and it’s rare to even visit.

Even though there’s no Beckets, there still is an A-group and a B-group. This will only be relevant when discussing transformations between families.

#### Other family members

There’s more formations than just the four corners of a square. People can be in a line of four, a star promenade, a diamond, a wave of four, and probably much more. While I can’t group these by the above individual formations, I can classify them into a general family. (Improperish, Properish, or Diagonal.)

The trick is to apply a rotationally symmetric transformation around the center of the hands-four, until you can make a match.

- In the case of a diamond, this is just a 45 degree rotation.
- In the case of a wave of four, this is two equal rotations in the same direction (both clockwise, for instance) around two points symmetrically arranged around the center of the hands-four. In a standard wave of four (improper formation, and do-si-do to a wave), those two rotation points are the neighbor right-hand connection. If you rotate each pair of dancers 90 degrees clockwise, you find out you’ve got something in the improper family.
- The same thing applies to a line of four, but yields some unexpected results. (A line of four is the same as a wave of four, as dancer facing is irrelevant.) Take the line of four at the beginning of “Scout House Reel.” In this case, the two rotation points are the hand connection with your neighbor. Rotate each pair of dancers 90 degrees clockwise, and you’ll find you’re in the diagonal family. (“String of Swings” uses this exact trick, as I describe in more detail later.)

#### Transforms between families

Going from one of these three families to another requires an asymmetric transformation. From a given formation in one family (say, Becket-cw), there are eight transformations to the formations in another family.

- Two involve a non-diagonal pair of dancers trading places. For instance, ones half-figure eight.
- Four involve three dancers circling left or right one position, while the fourth dancer stays put. Circles of three are the equivalent of two half-figure eights, one perpendicular to the other.
- The final two involve a transformation equivalent to a quarter double-figure-eight across or along the set. (Not quite as exotic as it sounds. One example is face down-the-hall, go down the hall, centers turn as couples while ends turn alone, bend the line. See “The Double Rainbow.”) (This is the equivalent of two transformations — one asymmetric non-diagonal pair of dancers trade places, and then everyone symmetrically trades places with the person in the perpendicular direction.)

The specifics of the transformations between families get complex. I’m leaving out the circles of three, as they’re not practical within the modern contra dance aesthetic.

To truly see the transitions, you need to break the (I)mproperish and (P)roperish families into sub-families of becket (B), and across from partner (A). In the case of the (D)iagonal family, you and your partner are never on the side of the set, though there still are the arbitrary-labelled A and B sub-families.

The necessary pair of dancers to swap is the pair that’s not on a diagonal in either formation.

The arrows represent half-figure eight/quarter double-figure eight transformations. For instance, to swap from proper becket to improper becket, either the ones must trade places with their partner, or the twos must trade places with their partner.

Circles of three represent the missing transformations, like going from Improper-A to Proper-B.

### Examples

##### The Diagonal Dilemma

(by Gene Hubert)

One of the first dances to intentionally try to get dancers in diagonal formation, as clearly indicated by the name. It starts proper.

- The A1 down-the-hall action swaps a same-sex neighbor pair to get into diagonal formation
- The A2 has a circle left and ones swing that keeps things in the diagonal family
- Then there’s the neighbor swing. There are an unequal number of rotations. (M1and W2 will swing N & 3/4 around, while W1 and M2 will swing N & 1/4 around until facing across. N is an arbitrary integer.) As far as families are concerned, this is equivalent to one opposite-sex neighbor pair turning 1/2 while the other doesn’t turn at all. This brings us back to the improperish family for the start of the B2.
- And in the B2, the one gypsy 1 & 1/2 swaps a partner pair to get back into progressed proper formation. Redefining hands fours returns things to proper, with new neighbors.

The neighbor swing here brings up a major point — an unequal interaction between one pair and another pair is the more general case of one pair trading places and the other pair doing nothing. To go from improper to proper, the ones could two-hand turn 1 & 1/2 while the twos two-hand turn just 1.

There’s a detail in the down-the-hall that I’ve swept under the rug, with two factors that cancel out. But first, let’s look at the next dance.

##### String of Swings

(by Rick Mohr and Bob Isaacs)

If you look at the hey passes (NL,2R,SSNL,1R,Nl,2R,SSNL,1R) (SSNL = Same-sex-neighbor left), you’ll see partner pairs are always meeting in the center, a sure sign of the diagonal family. They get there through an asymmetric opposite-sex neighbor interaction, just like the unequal neighbor swing rotations of “The Diagonal Dilemma.”

To do this, the ones merely step between the twos.

This is the equivalent of man one and woman two allemanding left 1/4, while woman one and man two allemande right 1/4. The net difference is equivalent to one opposite-sex neighbor pair swapping. Now the dancers are in the diagonal family.

In most dances, like “Scout House Reel“, there is a bend-the-line move following the down-the-hall, which is again the equivalent of one opposite-sex neighbor pair swapping, bringing things back into the improperish family. But not in “String of Swings.” Here, people just turn to face their neighbor, and keep using rotationally symmetric moves. They stay in diagonal formation throughout the hey, the allemande, and the twos swinging.

The neighbor swing resolves things back into the improperish family with unequal-rotation neighbor swings. Where it stays until the ones step between new neighbors.

With most down-the-hall dances, the asymmetric entry into the line of four is cancelled out by the asymmetric bend-the-line. But if there’s no bend-the-line, watch out.

##### The Diagonal Dilemma, revisited

Now we can tell the full “Diagonal Dilemma” story:

- The A1 starts proper.
- Forming the line of four puts things in diagonal family
- The same-sex neighbor swap at the bottom of the hall actually resets things back to proper family
- Bending the line puts things back into diagonal family
- The neighbor swing returns it to the improperish family
- And the ones gypsy puts things back into proper

“Equal Opportunity” by Jeff Spero is left as an exercise for the reader.

#### Andes Anomaly

(Cary Ravitz)

This is a very complex dance. But for looking at families, it’s simple.

Most of the interaction (stars, rory slides, balances) is done with the person on your diagonal. If done without the A1 rollaway, this would be your same-sex neighbor. With the rollaway of just one partner-pair, things are now in ~~diagonal~~ properish formation, and it becomes your opposite-sex neighbor. The dance stays in that family through a series of complex maneuvers. It only gets resolved back to the improperish family with the final partner swing, with unequal swing rotations for the ones and the twos.

Cary Ravitz has since explored the diagonal formation deliberately, presumably when submitting for the “Greenfield Formation” contest.

##### An Other, Whirled

(by me, written based on all of the above)

I wanted to write a same-sex star promenade dance. That meant I couldn’t lead into the figure with the standard same-gender allemande in the middle. My choices were either a partnership (the ones) allemanding, or the first corners. I went with the ones allemanding, assuming it’d be easier for them to self-identify.

So I needed diagonal formation. The quickest way to get an opposite-sex neighbor pair to swap was with a long lines, selective neighbor roll away.

Then I needed a resolution to get out of diagonal formation. The easiest unequal opposite-sex neighbor interaction was a neighbor swing. So I had to pack in two swings, making the rest of the dance choreography very tight.

Had I gone with first corners doing the allemande, I could have just gotten into the proper-Becket family, and gotten out with a partner swing, skipping the neighbor swing and allowing for more fun stuff out of the star-promenade, like a hey.

##### Hull’s Victory

(Traditional)

It starts proper. Then things get odd.

There’s an asymmetric partner-pair exchange, as the ones allemande right halfway. Simultaneously the twos shift up until they’re outside the ones to form the wave of four. This is an asymmetric opposite-sex neighbor interaction. (More technically an asymmetric interaction with the place where the opposite-sex neighbor will be.) These two transformations resolve to the diagonal family.

Most of the B1 is ones lead down, turn-as-a-couple and come back until between the twos. This changes formation groups from diagonal-B to diagonal-A, so their same-sex neighbor is on the side. (They are now directly between their neighbors, facing up.) This lets them interact asymmetrically with their same-sex neighbor during the cast, returning things to the properish family.

##### The Women’s Wall

(Al Olson)

Contra corners is a quirky figure, a triple minor figure shoehorned into a duple-minor set. It has a different symmetry. In most figures, the center of rotational symmetry is the middle of a hands-four. In contra corners, the center of rotational symmetry is directly between the ones. (There’s another directly between the twos.)

The dance starts proper, and at the end of the A1 is progressed proper. The contra corners, plus turning the second contra corner enough, has the ones on the sidelines, facing their first contra corner. This means the ones have swapped places, putting things in the improperish family. (Progressed improper.)

Then things almost get weird. All the men are facing down, and the women are facing up, looking for a neighbor to swing. If the swing finished faced across, this would be an unequal rotation between neighbor swings, putting things in the diagonal family. Except that Al Olson makes them have an equal rotation by all facing the “women’s wall”, so things stay progressed improper.

The final move that changes formation is the ones swing, and end facing up, which returns the formation to progressed proper.

##### Equinox

(by Jim Kitch)

The first half of this dance is unremarkable. After the second circle, things are:

W1 M2 M1 W2

After the small shift left, you have

W1 M1 M2 WS W2

(WS stands for a shadow woman)

Two things have happened here: the shift left half a position, and redefinition of hands-four. Combined, those are the equivalent of swapping the ones, putting things back in the properish family. (Here we’re treating woman one as equivalent to her shadow.)

(Note we have to start with the improperish-B family to get to the properish-B family, with everyone across from a same-sex neighbor.)

The resetting of hands four is very unusual in this dance. In most dances, resetting of hands-fours keeps two dancers together from the original hands four. The tiny shift left keeps three dancers together from the original hands-four. Normal transformation rules no longer apply.

The end of the dance resolves with a partner swing, returning things to the improperish family.

(I’m not completely satisfied with the above explanation. I may revisit this.)

##### Fiddleheads

(Ted Sannella)

The dance starts improper. Then the ones do something while the twos stand still, putting things in the diagonal family, which seems to violate my entire analysis system.

The catch is my rules above only apply when couples are trading places by a 180 degree rotation between themselves, or one couple is rotating 180 degrees relative to the other couple. But all bets can be off in exotic asymmetric moves. Like the first move of Fiddleheads.

Sometimes we can break it down.

Fiddleheads starts improper.

W2 M2 M1 W1

After the ones cross, it’s proper.

W2 M2 W1 M1

Then the ones individually go around one, and step into the middle to form a diamond. To make things clearer to see, I’m rotating this diamond to the left 1/8.

W2 M1 WS M2

First, WS stands for woman one’s shadow. As far as families are considered, a person can just as well be represented by their same-sex shadow. So the net effect of the first move of the A1 is to change the position of M1 and M2. This is an asymmetric same-sex neighbor swap, putting things in diagonal family.

There’s something else weird going on here. M2 moved from being next to his partner to being diagonal from his partner, without either physically moving. This happens from redefining the square of the hands-four. The twos start on the sideline of a square, and end up on the diagonal of a new, smaller diamond. This is only half the area of the original square, but in a dance situation it doesn’t matter, because dancers will (hopefully) naturally expand or contract the hands-four to the needs of the dance. And the next need is a ring balance.

After the petronella turns and the redefining of hands fours, however, you have a truly odd formation:

M2 W2 M1 W1

The resolution cannot be neatly explained in terms of pairs of dancers swapping. The ones swing is the equivalent of ones allemande left 1/4, and face down. This happens to put things in proper formation, but when you go off the charts and work outside of 180 degree pair swappings, things can get a lot more interesting, and a lot less predictable.

The remaining analysis of the dance is straightforward.

That’s enough about duple minors for now. At some point I may revisit symmetries. But hopefully some of this was comprehensible, and will let you better understand how to cope with — and use — these odd set-ups.

Here are few notes on your very interesting writeup of the 24 possibilities for two couple minor sets.

If you look at the set of four as four dancers, then as you noted, it is apparent that there are three basic formations that can be designated by the diagonal dancer. If you look at the set of four as a set of two couples, diagonal never occurs, and there are just two basic formations. Perhaps that’s why diagonal formation has not been developed.

Here is my quick overview of what is important here –

A set of four may be characterized by the diagonally opposite person

– same sex neighbor – improperish, includes improper, Becket, indecent

– opposite sex neighbor – properish, includes proper

– partner – diagonal, includes Cross Contra (we’ll see if my name sticks).

To get from/to improperish to/from properish

– ones trade places OR

– twos trade places.

To get from/to improperish to/from diagonal

– one man and two woman trade places OR

– one woman and two man trade places.

To get from/to properish to/from diagonal –

– men trade places OR

– women trade places.

You can get from improperish to diagonal without calling “one man and two woman …” – go from improperish to properish, then properish to diagonal – from improper, twos half figure eight above, women gypsy 1/2 or balance, twos trade places, balance, women trade places.

No symmetric contra figure can move from one formation to another because the diagonal person stays the same – chain, right and left through, hey, circle, star, etc.

The USUAL simultaneous swings end facing the other swinging pair with the woman on the right. So at the end of any such swing the set is improperish.

What makes diagonal formation useful is the meeting of partners in the middle in figures such as ricochet hey, chain, and gypsy for four. Where improperish gives you same sex middle meetings, properish gives you opposite sex neighbor meetings, diagonal gives you a partner meeting – much more satisfying.

The roll away in Andes Anomaly makes the dance properish, not diagonalish – ones trade places so that the diagonal person is an opposite sex neighbor.

Keep writing – it’s great to read analyses of contra choreography.

Cary Ravitz

http://ravitz.us/dance/ – with three dances that use diagonal formation (as of 11-06-2013)

Thank you for catching the Andes goof.

One very good point in your summary is _why_ to use them.

I wonder if diagonal/cross formation was never developed because it was a hard target to hit, especially if you’re trying to get there by random walk. Look at how inefficient something like “Equal Opportunity” or “Diagonal Dilemma” is. And you’ve only got so much space to pack in setting up diagonal/cross formation, the main figure, the resolution, and two swings.

Thank you very much for writing! (Especially for a summary of a post whose tale grew far too much in the telling. I’m guessing you’ve been doing the same sort of research.) I was interested to see your three dances pop up the same day I was doing the final edit on this post, but I was only able to quick slap them in as links under Andes.