Exercise 5-2. Set inversion and inversional equivalence of sets

A set can be inverted around axis 0 -- the conventional axis for inversion -- by subtracting each of its pcs from 0 (= 12). In such an inversion, pcs map onto each other in consistent pairs: 0/0, 1/11, 2/10, 3/9, 4/8, 5/7, and 6/6. The inversion operation is held to be coupled with a transposition in a compound procedure called TnI. (An inversion with no subsequent transposition is called T0I: that is, the transposition level is 0).

The TnI operation can be broken down into four stages:

1. Make sure the given set is in normal form.
2. Invert the pcs of the set around 0, using the consistent mapping given above.
3. Place the resulting set in normal form.
4. Transpose the new set by number "n".

Here's an example. Let's find T5I of set [3,5,8].
1. Place the set in normal form: [3,5,8]
2. Invert its pcs around 0:
 3 5 8 9 7 4
3. Place the resulting set in normal form: [4,7,9]
4. Transpose the new set by 5:
 4 7 9 + 5 5 5 = 9 0 2 set [9,0,2]
So T5I of [3,5,8] = [9,0,2]

The tricky part can come in step 3. Normally if we invert a set, the result will appear in "reverse-normal" order, for instance, 9 7 4 above. We simply have to "re-reverse" this order to place the new set in its correct normal form: [4,7,9]. Not always, however! The intervallic makeup of some classes of sets means that the normal form of such a set and of its inversion are not simple retrogrades of each other. For example, look what happens if we try to find T0I of set [8,10,11,1,2,5]:
1. Place the set in normal form: [8,10,11,1,2,5]
2. Invert its pcs around 0:
 8 10 11 1 2 5 4 2 1 11 10 7
3. Place the resulting set in normal form:

Not [7,10,11,1,2,4] but [10,11,1,2,4,7]!

Carrying out the steps for finding this set's normal form, we discover that [10,11,1,2,4,7] is actually more compact towards the left than [7,10,11,1,2,4].

4. Transpose the new set by 0: [10,11,1,2,4,7]
So T0I of [8,10,11,1,2,5] = [10,11,1,2,4,7]. You can never tell when you'll run across a set of this type, so be careful in step 3!

1. On a printout of this page or on a separate sheet of paper, invert and transpose the following sets (which are already in normal form) by the values indicated. Write the resultant sets in normal form.

 1 T0I of [2,6,9] Comment 2 T9I of [0,1,2,3,4,5] Comment 3 T7I of [6,7,8,10,2] Comment 4 T5I of [5,6,11,0] Comment 5 T11I of [5,6,11,0] Comment

If you can invert and transpose a set to map onto another set, then the two sets are said to be inversionally equivalent to each other. As with transpositional equivalence, such sets are considered to be of the same type, or members of the same set class.

1. Below are some pairs of sets. On a printout of this page or on separate sheet of paper, tell whether or not the two sets in each pair are inversionally equivalent. (Invert the first set and place the result in normal order. Find the Tn level at which the initial pc of this new set maps to the initial pc of the second given set. Then see whether the rest of the pcs map.)