Dissertation Diary 2016-01-28

I’ve been slogging away at chapter 4. There are inadequate and incomplete diagrams, but I’m sending the prose off for editing this morning. Hopefully that will give me a better idea of what diagrams I need. Chapter 4 stands at 23 pages and 5150 words at the moment, but I think I’m going to cut another several pages before adding more words and diagrams of better quality.

Below is the section that explains some of the weirdnesses of transformational theory that I’m using in the hybrid Chopin analysis:


Section 4.1.1

Chromaticism became more and more elaborate through the nineteenth century. Some of it can be explained with tonal relationships; for example a chord might include altered tones that resolve in a functional manner. In fact, the increasing prevalence of chords like the augmented sixth and Neapolitan (Examples 3.?? and 3.??) is a good example of this.

Additionally, chromatic mediants begin to occur both as key areas and chords, but they are fairly easily described as alterations of relative and variant third relationships discussed in section 3.1.?. [EXAMPLE] Because the T refers to the home tonic, we know the variant would normally be built on the major third mi, and that secondary functions are the opposite quality of the primary – Tv. However, if we have TV, that would be a major chord built on mi, not a closely related key at all. Other chromatic mediants are borrowed from the parallel minor, such as tR, but tr (e flat minor) is also a possibility.

Along with more chromatic chords, there is an emphasis on plagal relationships. These relationships are reflected in the theoretical treatises’ emphasis on dualism of the late nineteenth century (Section 2.?). Some composers, such as Brahms, saw many dualities such as reality/fantasy, man/woman, normal/exotic, and tried to use specific harmonies to express these distinctions, often using a plagal le-so leading tone to indicate this dichotomy.[1]

Dualism can be used as strategy to explain these types of dichotomies, and there are many examples of dualities in human experience: “Whenever we speak of consonance and dissonance, or of melody and harmony, or of homophony and polyphony, we experience the expedience of dual organization, and find it good.”[2]

The Romantic Era fascination with the fantastic and exotic provides fertile ground in which to explore the usefulness of these dualistic, unequal opposites. For this purpose, it is sometimes useful to talk about the triad built on fa as a chord that leads to tonic, particularly in minor with le-so pulling to the tonic. When this chord is not merely a cadential extension of a more traditional PAC, or the analyst wishes to highlight the plagal/dualist cadential potential, we can label fa-le-do as SubDominant – S or s – a different type of dominant on the opposite side of tonic – the original meaning of Rameau’s term.[3] This will also come in handy later in Section 4.2.1.

Other dominant alterations sometimes occur, including minor dominant (d), and dominant substitutes. Just as tonic and predominant (or now, subdominant) can have relatives and variants, dominant may also. One of the more common substitutes is dR, the major triad built on te.

Much music of the later nineteenth century is commonly described using non-tonal means; transformational theory came about in the late 1980s partially for the purpose of describing this repertoire.[4] Some of the analysis of the Chopin Prelude in Section 4.1.2 provides an example of how transformational theory might mix with Functional Analysis.

For a more thorough study of transformational theory, look at [the Neo-Riemannian SMT volume], and [the new NR reader].[5] Here is a brief overview of the concepts used in Section 4.1.3.

Transformational theory begins with the idea that chords transform one to another via steps. That is, C major (CEG) transforms to e minor (EGB) and a minor (ACE) with one step change of one voice. These are the substitute relationships described in Examples 3.? and shown on the Tonnetz of Example 3.10. Transformational theorists refer to these transformations as L (CEG-BEG) – Leitonwechsel, P (CEG-CEA) parallel, and R (CEG-CEbG) relative, based on Riemann’s original terminology.[6] These are diatonic transformations.

For more chromatic music, a concept called Parsimonious Voice Leading is often important.[7] This bases the analysis on how many half steps are required to transform a chord into another chord. The standard diatonic relations (L, R, P) would be described as -1, -1, +2, indicating half steps and direction. If one moves completely parsimoniously – only by half step – a collection of chords related by chromatic third are available: [EXAMPLE] C, c Ab, ab, E e. This is known as a hexatonic cycle, because the 6 chords composite pitch collection is a hexatonic scale.[8]

Some analyses describe pieces based on how the chords relate on a Tonnetz, but other visual representations of chordal relationships quickly proliferated. The

hexatonic cycles for triads. {EXAMPLE] Then four voice sonorities felt left out and the chicken-wire torus was designed.[9] {EXAMPLE}

For other analyses, theorists describe the pitches with numbers, as in 20th Century atonal theory.[10] These numbers can be arranged into set classes, which can help show when sonorities share interval content, even if they appear unrelated on the surface. For example, Dominant 7th chords and half diminished seventh chords share the same number of tritones and thirds, just arranged differently.[11] My Chopin analysis uses sum classes, which finds the sum of the pitch numbers (mod 12), and compares them visually that way. {EXAMPLE}. For example, a C dominant 7th chord (CEGBb) would be pitch numbers {047t} (set class 4-27, (0258)), and 0+4+7+10=21, 21-12=9, so the sum class of that chord is 9. Eb dominant 7, F# dominant 7, and Ab dom 7 all share this sum, so in some ways, these chords are related.

Even with all the transformational theory or complex chromatic relations, late Romantic music often still can be described on a large scale by function, usually as depicted on Schenker graphs. Function might be used on the large scale of a piece but not small scale, or even hypothetically, vice versa, with small scale functional cadences emphasizing a non-tonal large scale framework. That is to say, just because Functional Analysis doesn’t work all the time on a give piece or repertoire, doesn’t mean it’s not worth looking into.

[1] Notely, p ?.

[2] Harrison, 16.

[3] Rameau Traite, CHowmut?

[4] Reimaginging Riemann (?)

[5] find!


[7] Cohn, I think.

[8] Also cohn?

[9] Douthett?

[10] Straus.

[11] ?? Straus, NR stuff.

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