Theory:

The Complete Musician(https://tinyurl.com/3an5zcen)

Chapter 30 of 32: Centricity, Extended and Non-Tertian Sonorities, and Collections

The discussion in this chapter is focused on post-tonal music. Basically, as the 19th century drew to a close, the main underpinnings of tonality had been wrenched loose from their moorings, and composers, in their constant search for new ways of organizing sound, started pushing past the strictures imposed by traditional tonal expression, and ended up creating the body of music that has come to be called post-tonal. For this chapter, the focus is on two basic techniques of post-tonal organization: centricism and atonal music. 

There are three questions posed at the beginning of this chapter that sets the stage for the rest of the chapter:

1. How can centric music emphasize a particular pitch class and create a sense of closure, even in the absence of tonality
2. What are some sonoroties, including ones constructed of dissonant intervals rather than just htirds, that play roles in post-tonal music?
3. What are some pitch-class collections other than major and minor scales that can serve as color palettes for the pitch content of post-tonal music?

What is centricity?

In centric music, a pich class - called the centric pitch class, or just the center - stands out as the primary one, and even the most stable one, but it is not a tonic, because there are not functional progressions or tonal cadences. Other factors come into play here, such as repetition, symmetry, register, and/or gesture serve to highlight the centric pitch class.

One important technique in centric music for emphasizing a specific pitch class is ostinato. Ostinato is basically continuous repetition. This approach is especially effective where there is no conclusive harmonic motion to give formal coherence to the unfolding-in-time.

Achieving Closure in Centric Music

Centric music does not feature the same cadential formulas of tonal music. There are other means for creating this sense of closure, other techniques that composers have at their disposal. Through artful manipulation of rhythm, pitch meter, and any other parameters that may be open to articulation, the sense of tonal resolution can be emulated. 

Adding Colors to Teritan Chords

It is a feature of some twentieth-century music that tertian chords extend beyond seventh chords now, to include ninths, elevenths, and thirteenths.

Added Sixths

These are not the same as thirteenths. This refers to a chord that is basically a triad with the sixth added, so CMAJ6 would be C-e-G-a, C Major Sixth. The sixth is always major, bu the way. 

Adding Sharp-Ninth Chords

The sharp nine is often represented enharmonically as a minor third. THinking of this as a chord with a ‘split’ third, meaning both major and minor thirds are available.

Non-Tertian Sonorities: Stacking by Fourths and Fifths

Quartal Harmony: Stacking chords by 4ths

Quintal Harmony: Stacking chords by 5ths

Collections:

A collection is a musical color palette, a palette of pitches which is a repository of pitch classes from which a composer can draw any number of melodic and harmonic shapes. They can be any size and contain any number of pitches. 

The largest collection is the collection of all twelve half steps which is called the chromatic collection or the aggregate.

Seven-Note Collections: Diatonic Modes

Major and natural minor scales are examples of diatonic collections. Diatonic collections:

• contains seven pitch classes
• It does not divide the octave evenly
• each pitch class has one letter name
• Contains five whole steps and two half steps when arranged as a scale
• are referred to by their key signatures i.e.: Eb/c = ‘3-flat collection’
• each diatonic collection contains seven diatonic modes

A Five-Note Collection: Pentatonic

The pentatonic collection contains five pitches. The one focused on in this chapter takes the diatonic collection and removes the two pitches that participate in the tritone. This collection contains no half-steps; adjacent ptiches are separated by either a major second or a  minor third. It is also asymmetrical. No two diatonic collections are identical

Six-Note Collections:

• whole-tone collection: partitions octave symmetrically; there are two whole-cone collections, each consisting of six tones; it cannot be rotated because it has not half-steps; it is not possible to construct an odd-numbered interval; the huge degree of symmetry in the whole-tone scale makes it difficult for any single pitch to stand out as a center
• the hexatonic collection: there are only four distinct augmented triads; are generally created by combining two augmented triads that are either a whole of half-step apart; there are four different pairs; this is called hexatonic; each collection contains three different major triads and three different minor triads

An Eight-Note Collection: Octatonic:

Now that we have looked at building two kinds of six-note collections by combining pairs of augmented triads. Now we can use the same concept to create octatonic (8 note) collections by combining pairs of diminished 7ths. There are only three distinct diminished seventh chords. Therefore there are only three ocatonic collections. The octatonic collection repeats itself every three half-steps. The octatonic collection has many interesting subsets, smaller collections found within larger ones. The harmonic possibilities are rich.