Theory:

The Complete Musician(https://tinyurl.com/5b6prtrx)

Chapter 31 of 32: Analysis with Sets

Building on our brief study of Chapter 30, we looked at sets. This chapter builds on that to begin talking about analyzing non-tonal music that cannot be easily handled through other means.

Pitch Space and Pitch Class Space

Pitch space: each pitch's register is important.

pitch-class space: There are just the twelve pitch classes regardless of octave transpositiong.

Octave equivalence: the understanding that the basic pitch of a tone doesn't change regardless of octave

Enharmonic equivalence: the understanding that any single pitch can have several different names, i.e. C = Dbb = B#. All three spellings refer to the same sounding pitch. 

Combining all these concepts together enable us to represent the sum total musical space with the integers 0-11.

Intervals in Pitch Space

In this system, there is a concept known as an ‘ordered pitch interval’ or ‘OPI’. This measures both the distance and the direction from one pitch to another pitch. It also has an integer sign to show if the distance is in ascending motion or descending one, i.e. a plus sign for ascent and a minus sign for descent. So the OPI would look something like this: +4. This would mean ‘this note is four half steps higher than the other note.’

There is another concept known as the ‘unordered pitch interval', which measures just distance. This is helpful when pitches sound simultaneously, or when we're more interested in the distance between them and not the ordering. 

Intervals in Pitch-Class Space

Unlike pitch-space, pitch-class space has no ‘up’ or ‘down’, because pitch classes have no register. Distance is measured using something called a pitch-class clock.

Ordered pitch-class interval (or OPCE): restricted to clock wise motion. ‘How many hours (or rising semitones in pitch-class space) elapse from the first pitch class to the second pitch class?’. Because 12 hours (clock face) pitches ‘travel’ from 0 to 11.

Unordered pitch-c.ass interval (or UPCI): allows both clockwise and counterclockwise motion and take the ‘shortest path’. Allowing for both directions of travel, no two spots on the are more than 6 hours away, as UPCI's range from 0 to 6. Another name for an unordered pitch-class interval is an Interval class (IC).

Pitch-Class Sets

Pitch-Class Set: a group of two or more pitch classes. The cardinality of the set is the unique number of members it contains. Two-member set: dyad, three-member: trichord; four-member = tetrachord; five-member = pentachord; six-member = hexachord. Sets larger than this exist, but are less common. The creation of a pitch class is referred to as segmentation. From here these is the need to utilize theoretical tools to allow us to recognize these sets.

Arranging a Pitch-Class set in normal order

normal: the pitch class compressed into its most compact arrangement in order to facilitate comparisons between sets. To find the normal order, construct the pitch class on an interval clock with arrs that can encompass all of its pitch classes. Whichever is shortest shows the normal order. The pitchclass is then notated in clockwise order as they appear within that shortest arc. 

Sets Related by Transposition

There are different types of transposition techniques to create sets closely related to the original but not exact.

Inversions in Pitch Space

Transposed sets retain the same interval structure, every note in the set moving up or down the same number of half steps. Inversion does something different: it reflects the set around some imagined line. The imagined line is referred to as the axis of inversion. Of the line is a specific pitch class, so the inversions are built on the understanding of ‘how many half steps’ in either direction. When reflected in this way, the type of inversion is called ‘mirror reflection’. 

Inversion in Pitch-Class Space

 When inversion takes place in pitch-class spacer rather than in pitch space, the reflection takes place within the pitch-class clock. Referencing this, it will become apparent that each pair of pitch classes adds up to the same number mod-12. This sum is how the inversion is labelled. 

What is Invariance?

Using transposition and inversion, most pitch-class sets can be transformed into 23 different pitch-class sets without any exact duplicates resulting in 24 unique sets. Many sets have fewer than 12 distinct transpositions and/or fewer than 12 distinct inversions. This is where invariance plays a role. The meaning of invariance is that any given pitches in a set will become members of other sets.

When inverting a set produces no pitch-class sets beyond what can be produced by transposing the set, it is because the set is inversionally symmetrical. In other words, these sets look the same when read forward or backwards. Invariance is helpful to composers, since it permits varying degrees of pitch-class similarity when a set is transformed.

Interval-Class Vector

This is a way of determining how many pitches will be invariant. It summarizes how may times each interval class occurs among the dyads within a set. To construct an IC vector, locate each pair of pitch classes that can be found within the set. Trichords have 3 different dyads, tetrachords 6, pentachords 10, and hexachords 15 different dyads. We determine which interval class each dyad belongs to, then tally instances of each of the 6 interval classes in order, enclosed within angle brackets without any commas.

Combining Transposition and Inversion: Set Class and Prime Form

Unlike a normal order, a prime form will no longer captuer the specific pitch clases or our original set. Instead, it will represent a whole group of 24 pitch-class set. Before finding a set's prime form, we need it in normal form. The prime form is the form with the smallest second digit. The prime form is also called the set class.