Twentieth Century Harmony – Chapter 1: Intervals

Chapter 1 in Persichetti’s text is largely rudimentary material covered in a different way that I kind of slightly prefer just a lot. It covers Intervals, along with a look at overtone influences and mediums, as well as interval vectors without the numbers.

CONSTRUCTION

To dive right in, Persichetti begins by defining the differences between consonant and dissonant intervals, which will be used in this fashion throughout the remainder of this series as well as his book. All of the intervals he defines are the same as in standard theory, but he arranges them in the order of most consonant to most dissonant intervals, which he calls a tension arrangement.

P5 and P8 – open consonances
3s and 6s – soft consonances
M2 and m7 – mild dissonances
m2 and M7 – sharp dissonances
P4 – consonant or dissonant*
tritone – ambiguous**

* The perfect fourth, as he notes, “sounds consonant in dissonant surroundings, and dissonant in consonant surroundings”
** “It sounds primarily neutral in chromatic passages and restless in diatonic passages”

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INVERSION AND SPACING

When it comes to intervals, the spacing will alter the exact nature of the interval, while the exact notes may stay the same (a M7 and a M14, while they would both contain C and B, have different qualities; the same is true of a M7 and m2).

When a m2 is inverted to a M7, the interval loses some of its sting and becomes more widespread in tone. This is why the m2 comes in before the M7 in a tension arrangement.

Other intervals, when inverted, alter their basic functions. A perfect fifth, when inverted, becomes a different entity altogether and has a markedly different function as a perfect fourth.

Compound intervals also have an altered character; as Persichetti notes, “If intervals are spaced more than an octave apart, the soft consonances become richer; open consonances and the consonant perfect fourth become stronger; dissonances become less biting, but more brilliant. The tritone, neutral in chromatic passages becomes more ambiguous and veiled; …in diatonic progressions, it becomes even less addicted to resolution.”

INTERVALS IN CHORDS

Here is where I have a distinct disagreement with Persichetti. In the opening of this section, he claims that intervals that are more than two in number become chords. I, however, take the stance that dyads are also chordal in nature, and can be used as such. Just a minor annoyance that starts an otherwise solid section. Granted, he does state “Two or more intervals occurring simultaneously form what is usually felt to be a chord.” So maybe I’m wrong there.

Here is where Persichetti starts to deviate from normal rudimentary theory courses. In this section, he does not focus simply on normal triads, rather he begins his whimsical adventure into the depths of the book by teaching of all the different chordal structures there are, such as chords of equidistant intervals, chords of similar intervals, and chords with mixed intervals. Seen below:

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This is where Persichetti, without using the numerical interval vector system, begins using interval vectors. I will be adding numerical references in combination with the visual aspect of his chordal analysis.

When picking a chord apart, Persichetti uses a four part system of analyzing the intervallic composition of a chord. They are as follows:

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It’s important to connect the lines between the two noteheads, or at least the spaces/lines used due to it becoming an absolute clusterfuck of lines after around five or so pitches. Below are examples of chords analyzed in this method:

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Now, to start with the numerics behind Persichetti’s system, let’s start with the first chord of Ex. 1-14.

If I haven’t given a primer on interval vectors, because I have no idea if I have or not, an interval vector is constructed with a series of numbers within two what-I-like-to-call carrot brackets (). The series of numbers is six digits long, with each digit representing the number of intervals of the given type within the chord or set. They follow this order:

An interval vector of a major chord, which just so happens to be the first chord in Ex. 1-14, would then be . The next chord on 1-14 is a minor 7 chord with an omitted fifth, which would be notated as , and so on so forth. I’m going to stop there, you can do the rest, I almost have enough faith in you to do so.

Persichetti notes that, while a chord containing nothing but sharp intervals is dissonant as fuck, chords containing tritones tend to be less stable, due to the nature of the tritone. He also notes on the ambiguity of the perfect fourth once more, bringing up that the surroundings of the fourth will lend it its quality in Ex. 1-17 (seen a bit above). When the other intervals are dissonant, the fourth will sound consonant, and vice versa.

Doubling is an important factor when it comes to chords, as the specific doubling will alter the timbral aspects of the chord, and can be used to strengthen specific pitches or achieve specific timbral qualities. These are most clearly seen with a group containing instruments of various timbres, such as strings, woods, brass, and percussion. He notes that the spacing is also of paramount importance to the voicing of the chord, and that, while proper doubling ensure the use of the wide intervals at the bottom of the chord and narrow intervals at the top of the chord, these can be altered for various effects. For example, placing the spacings in reverse of the normal voicings achieves a feeling of tautness, due to the stretching at the top of the register.

OVERTONE INFLUENCE

The overtone series is an important factor of deciding chord spacings. The tone that creates it, the fundamental, generates a series of overtones above it, known as partials. These are most commonly heard in the timbral differences between different instruments; for example, a flute has a very strong second and fourth partial, where as a clarinet has a very strong third and fifth partial. These are further exaggerated in the performance of the two instruments; the flute’s register key transposes by octave (partial 2), whereas the clarinet’s register key transposes by perfect twelfth (partial 3). Here’s an overtone series on C2:

persichetti 5

Any tone has implications of this series in both horizontal and vertical applications, and these implications may be used simultaneously in harmonic and melodic uses. Furthermore, instruments with the capability of generating high overtones will sound incredibly resonant, yet very dissonant due to the crowding of the overtone series. The same tone played on instruments of middle and low overtone accentuation will sound consonant, but lack the resonance of the high-overtone instrument. These differences, as stated above, give instruments their specific character.

Basic harmonic materials may also be traced to this series. I’m just going to put a picture of this one because it’s a bit wordy and he does it far better than I can:

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Chords tend to resonant more when the tones of the given chord closely resemble the overtone series than other orders, which is why the normal voicings for chords place the wider intervals at the bottom of the chord.

 

MEDIUM

Oh, this is fabulous topic to broach within the first chapter of a book on theory.

The medium of the material used in performance will also have a critical impact on the quality of what is being presented, as well as other aspects like dynamics, articulation, spacings, and tension. In the event that the same idea is presented in two different types of ensemble (ie: piano solo vs. women’s chorus), the two may have drastically different qualities to them. An example:

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Awareness of timbre is essential for good harmonic and melodic craft, and will dictate the effectiveness of the presentation of any material in a given performance. I feel that a fantastic example of this would be the use of a tenor tuba in Holst’s The Planets, “1. Mars the Bringer of War.” In this, he uses the tenor tuba to present a quite heroic melodic line, tempered by the mellowness of the sound, which additionally gives it contrast to the trumpets that enter soon after. This gives the tenor tuba’s solo a far darker weight when compared to the historic context of the time, which was during a small war in Europe commonly referred to as “World War I.” Thomas Goss, the head of Orchestration Online, has a reference to this point in the work, and talks about it in the context of functional timbre, as well as the contrast it brings. If I can ever find the video again, I will post it here, along with credits. If you can find anything by him analysis wise, I *highly* recommend it.

 

APPLICATIONS

This will be a separate blog post, as I haven’t actually done any of them yet. At the time of writing, and unless I change it, I will be doing around 25% of the problems presented in each chapter. The methods of number selection fall to a distinct panel of people known as “Anybody Who Happens to Have Me as a Friend on Facebook or Happens to Have my Phone Number,” of which two will choose random numbers between x and y to ultimately decide my fate. This chapter has been decided by my colleague and fellow Dr. Theory Slap’d writer, Djesso (check his music out here: The Mighty Djesso), and our mutual student, Buttons (when stuff gets posted, I’ll do a thing on it because advertisement). They have each chosen the problems I will do, those numbers being 3, 6, 11, and 12. I will, however, post the problems from the chapter below. I strongly urge you to do them, because practice is good, and this book has fantastic problems for learning the applications of the different concepts within each chapter:

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