Pearsall Chapter 4 Unordered set and their operations

Summary of important concepts and terms We have learned that when analyzing music using set theory, a new terminology is required. For example, instead of referring to perfect 5ths, we say interval class 7. Similarly, in place of of chords and scales, we have sets. These sets can be ordered - where pitches or pitch classes are placed in the same order in which they occur in a segment of music. Sets can also be unordered, where pitches are not placed in the same order they appear in a musical segment. Unordered sets occur when musical segments contain pitches that occur simultaneously, or in no particular order. In order to analyse these unordered sets, we impose an order on them known as the normal order. This is a process that equates with reducing tonal chords to their root position (stacking them in thirds). Similarly, the process of reducing a set to its normal order involves condensing the pitch classes (pcs) of a set into a single octave. Once sets are in their normal order, relationships between sets can be found. These relationships provide insights into how music, particularly atonal music is organized. You will see that through operations such transposition and inversion, sets can resemble each other, much like families. These family resemblances between sets can be expressed in one form, known as the prime form. The prime form is an abstraction of set classes that gives a unique "picture" of that particular collection of notes. If two sets have the same prime form, we can be assured that they will sound similar to one another. Sets with the same prime form contain the same number of pitches and the same collection of intervals between its pitches, hence they are in some sense aurally "equivalent. These prime forms are similar to the tonal concepts of major, minor, augmented and diminished- but there are many more of them. Prime forms of sets begin on 0 and span the smallest possible interval from bottom to top. In addition, the smallest intervals are packed most closely toward the bottom (to the left of the set). 1. Transposition Unordered sets cans be transposed shift the set up or down the required number of semitones. See Pearsall, Ex4.1 a) The pitches have been listed from the lowest note which happens to be C up to Ab. b) the pitches have been transposed down 7 semitones or a perfect 5th, and again listed from the bottom up- starting on F up to Ab.

Note: The AIS (adjacent interval series) is the same, therefor they are transpositionally equivalent. 2. Normal Order The normal order organizes a set of unordered pitch classes so that relationships between sets can be analysed. It reduces a set to the order that spans the smallest interval. To find the normal order of a pitch class set:

First reduce the set to a single octave. To do this you list the pcs in ascending numerical order.

Then rotate the set (go through all the inversions) to find the order that

spans the smallest interval (bottom to top). Pearsall, Ex 4.2 NB

a) Measure/bar 1 is depicted using tonal intervals and interval classes. b) Measure/bar 22 is depicted using pitch classes where C=0.

Placing sets in normal order has revealed the transpositional equivalence of the sets bars 1 and 22 of Schoenbergs Phantasy for Violin and Piano Accompaniment have the same AIS. Glenn Gould

https://www.youtube.com/watch?v=3RqfXRBt6XI When there is more that one rotation with the same smallest outside interval, the order with the tightest packing at the bottom is the normal order. To find the order with the tightest packing at the bottom, you compare the intervals from the bottom to the second last pitch. If you again get the same interval, continue this process- compare the intervals from the bottom to the third last pitch. See figure 4.4. Normal Order, the Short cut.

1. Write out the intervals between consecutive pitch classes and the first and last pitch class.

2. Identify the largest interval class. 3. If the largest interval class is between the first and last pitch classes the

normal order is this one the same as the original unordered set. 4. If not write out the set beginning on the top most pc of the interval the

right of the largest interval class.

NB this only works if there is only one instance of the largest interval. Where there is more than one, follow the process above, figure 4.4.

3. Mapping Transpositions and Inversions. Elements of sets can be transposed, or inverted and transposed. This operation or process is known as mapping. To ascertain the transposition level of a PC set, compare its normal order with the source set. This entails subtracting any member of the first set from the corresponding member of the second set, (the set whose transposition level you are trying to ascertain). See Pearsall, Fig 4.6. The AIS of inversionally equivalent sets are similar the reverse of each other. See Pearsall, Ex 4.3 Toru Takamitsus Rocking Mirror Daybreak, for two violins, excerpt from no. 1. Autumn. http://www.classicalarchives.com/web_player.html However, the normal orders of inversionally equivalent sets sometimes have AIS that are NOT the reverse of each other. Therefore, when trying to ascertain whether or not two unordered pc sets are inversionally equivalent, all rotations of one of the sets need to be checked. As long as one of the rotations contains an AIS that is the reverse of one the other sets AIS they are inversionally equivalent. See Pearsall, figure 4.9. Reminder unordered pitch sets (where the register of pitches- or the pitch space- is acknowledged) contain the same pitches as an ordered pitch set, but presents them from bottom to top. The method for finding the transposition of unordered pitch sets is similar to that used for pitch class sets. When transpositions occur in pitch space, the letter P is included. Tp/n (Tee pee sub-n). A negative sign denotes a downward transposition. See Pearsall Fig. 4.12. NB The notes in the first chord are represented as an unordered pitch set, so they are ordered from bottom to top, Cb, F, Bb. Using integers, If C=0, this set is: -1 5 10. Similarly, the second chord or set is Db, Gb, C or 1 6 12 in pitch space where C=0. As it is an inversion the set is reversed. So the set is 12 6 1. The first two inversionally equivalent chords map onto each other at T11.

See figure 4:13 Pitch-pace index is 11. Or T p/11 Figure 14.14 shows this process using notation note that the inversions are presented in alto clef mod 12 (D G C#) and then transposed up 11 to get T p/11 I (Db Gb C). Also at figure 14.14, The process demonstrated on the keyboard clearly shows that the AIS of these inversionally equivalent unordered pitch sets are the reverse of each other. 4. Prime Forms As mentioned above, prime forms are a method for classifying sets. The prime form begins on 0 and span the smallest possible interval from bottom to top. In addition, the smallest intervals are packed most closely toward the bottom.

We cite the prime form of a set by putting it in a normal form, most compact towards the left, that begins on pc 0.

Here are the steps for finding the prime form of a set:

1. Make sure your set is in normal form. 2. Now invert this set, and place the result also in normal form. 3. Now, transpose both normal forms to begin on 0. 4. Finally, compare the two 0-transposed sets. Which is more compact

towards the left? That one is taken as the prime form -- and the name of the class to which your original set belongs.

The tricky part can come in step 2. Normally if you invert a set, subtracting each of its integers from 0, the result will appear in "reverse-normal" order. You simply have to re-reverse this order to place your inversion in its own normal form. You can then proceed to steps 3 and 4. For example, here's the procedure for finding the prime form of set [2,3,4,7,8]:

However, as we saw earlier, the intervallic makeup of some classes of sets means that the normal form of a set and of its inversion are not always simply reverses of each other. For example, find the prime form of set [8,10,11,1,2,5]:

It turns out that the normal form of a set with pcs 4, 2, 1, 11, 10, and 7 is [10,11,1,2,4,7] In this case, if we were to carry out Step 2 just reversing the digits of the inversion, both the normal form and then the prime form would be have been incorrect.