VIII. 20th- and 21st-Century Techniques

Collections

Mark Gotham; Megan Lavengood; Brian Moseley; and Kris Shaffer

Key Takeaways

This chapter introduces a number of pitch collections that appealed to many composers in the 20th century:

  • collection, as separate from major/minor scales or diatonic modes
  • : a five-note collection that corresponds to the black keys of the piano; can also be generated as a stack of five perfect fifths or through the pattern 2–2–3–2–3
  • : a six-note collection that is made up entirely of notes separated by whole steps
  • : an eight-note collection that is formed by alternating whole- and half steps
  • : a six-note collection generated with the pitch interval pattern 1–3–1–3–1–3
  • : a seven-note collection similar to the mixolydian mode but with a [latex]\uparrow\hat{4}[/latex]; corresponds roughly to the lowest partials of the harmonic series

Other pitch collections introduced include:

  • Olivier Messiaen’s “modes of limited transposition”: a group of scales that cannot be transposed in 12 unique ways
  • The “distance model” of generating collections with a repeating pattern of pitch intervals

Chapter Playlist

Beyond the use of major/minor and the diatonic modes, there are four new(ish) collections that occupy a special place in the 20th century: the pentatonic, whole-tone, octatonic, and acoustic collections. This chapter discusses these collections along with some important questions of modal properties and extra-musical meaning.

Diatonic Collection and Pandiatonicism

You have probably encountered the concept of the diatonic collection many times already, especially in the chapters on modes in pop and in 20th-/21st-century music, or even when you first learned about major and minor keys. The diatonic collection is the basis of much Western music.

In the 20th and 21st centuries, composers sometimes used the diatonic collection, but without making any attempt to make a specific pitch sound like the the pitch center. Such examples are not tonal, nor are they modal; instead, they are considered . Igor Stravinsky often wrote pandiatonic passages; many can be heard throughout the opening of his ballet Petrushka.

Pentatonic Collection

The is prevalent in music across the globe. It is a collection built with the interval pattern ma2–ma2–mi3–ma2–mi3. The pentatonic collection can be described with reference to the diatonic modes in multiple different ways (Example 1).

  1. Using the scale degrees of a major scale, a pentatonic scale can be formed with [latex]\hat{1}\text{, }\hat{2}\text{, }\hat{3}\text{, }\hat{5}\text{, and }\hat{6}[/latex].
  2. If you think of the diatonic collection as a stack of perfect fifths (F, C, G, D, A, E, B), then the pentatonic is a narrower form of the same collection: C, G, D, A, E.
  3. Finally, if you visualize the piano, note that the complement of C, D, E, F, G, A, B, C (the white notes) is G♭, A♭, B♭, D♭, E♭ (the black notes). The black-note collection is, once again, the pentatonic.

Example 1. (a–c) Three ways of generating a pentatonic collection, and (d) a rotation of the pentatonic collection.

What the pentatonic collection removes from the diatonic is the two notes that create half steps ([latex]\hat{4}[/latex] and [latex]\hat{7}[/latex], if you are thinking in terms of major scale degrees). The absence of semitones arguably makes the pentatonic more readily rotatable and considered as a collection, without a strong emphasis on a particular note as tonic. Any member of the collection easily functions as a tonal center. For example, there are five unique modes of the collection given in (a) of  Example 1, formed by rotating the collection so that each pitch class becomes tonic: C pentatonic (C, D, E, G, A), D pentatonic (D, E, G, A, C), E pentatonic (E, G, A, C, D), and so on. Letter (d) of Example 1 gives an especially common rotation of the pentatonic collection that is referred to as the .

The pentatonic scale is especially common as a basis for melodic composition. One example can be heard during the first movement of Chen Yi’s percussion concerto, at the entrance of the strings at 2:13. Notice that the pentatonic melody is sometimes accompanied by atonal-sounding chords in the bras. Pentatonic melodies don’t need to be harmonized within the pentatonic collection, and in fact often aren’t; after all, there are only two possible triads in the pentatonic collection (C major and A minor, for example, in the collection C–D–E–G–A).

Whole-Tone Collection

The is exactly what it sounds like: a scale made up entirely of six whole steps. Similar to the pentatonic scale, the whole-tone scale is rotationally ambiguous, since there is only one size of step. Composers often exploit this ambiguity by using the whole-tone collection to produce an unsettled feeling in the listener (in film and TV, one famous trope is to use the whole-tone scale to accompany a dream sequence).

There are only two unique whole-tone collections: one that contains the even-numbered [0, 2, 4, 6, 8, 10] and one that contains the odd-numbered pitch classes [1, 3, 5, 7, 9, 11]. If you need to distinguish between these two collections in an analysis, you can use the abbreviations WT0 and WT1: “WT” stands for “whole tone,” and the subscript number indicates the pitch C (0) or C♯ (1) (Example 2).

Example 2. There are only two whole-tone collections. Transposing WT1 up a half step would yield WT0 again.

Octatonic Collection

The is built with an alternation of whole steps and half steps, leading to a total of eight distinct pitches (Example 3). Jazz musicians refer to this as the diminished scale, as it fits well with a fully diminished seventh chord. (Of course, there can be other scales that have eight distinct pitches, but this is the one called the octatonic scale.)

Example 3. There are only three octatonic collections. 

The interval content of this collection is very homogeneous, and this intervallic consistency leads to one of its most interesting properties. As shown in Example 3, there are only three possible octatonic collections. When we transpose the first collection above (OCT0,1) by 3—adding 3 to each of the integers in the collection—<0, 1, 3, 4, 6, 7, 9, 10> becomes <3, 4, 6, 7, 9, 10, 0, 1>, which is the same as the first collection, just starting on a different pitch. If you need to distinguish between these two collections in an analysis, you can use the abbreviations OCT0,1, OCT1,2, and OCT2,3: “OCT” is an abbreviation of “octatonic,” and the subscript numbers represent pitches that create a half step unique to that scale (C,C♯; C♯,D; and D,E♭, respectively).

Joan Tower frequently uses the octatonic collection, and it is particularly audible in the opening of her piece Silver Ladders.

Unlike the other collections discussed here, the octatonic collection appears with some frequency prior to the 20th century, especially in Russia.[1] The octatonic collection can produce several familiar triadic harmonies, as shown in Example 4: eight major/minor triads (and four diminished triads, not shown), and four each of every type of seventh chord except the major seventh chord. However, there are no chords related by by fifth, so no tonic/dominant motion is possible. Instead, there is a plethora of root motion by third. Frédéric Chopin uses the octatonic over a B♭7 chord in his Ballade in G minor (Example 5). Thanks to all those triadic harmonies, it’s a versatile mode that can imply tonal associations while also inviting a freer movement among tonalities not traditionally regarded as being closely related.

Example 4. The octatonic collection contains many triadic harmonies. 

Example 5. Octatonic collections in Chopin’s Ballade in G minor (1836).

Hexatonic Collection

The hexatonic collection is a six-note collection that is formed by alternating minor seconds and minor thirds. The name “hexatonic” refers to its six notes, and while there are other possible scales with six notes (for example, the blues scale), the name “hexatonic collection” always refers to this particular group of notes. Like octatonic collections, hexatonic collections can only be transposed four times before returning to the same group of notes again (Example 6), and they are similarly named according to their lowest semitone (e.g., HEX0,1 is the hexatonic collection containing C–C♯).

Example 6. Four possible hexatonic collections.

Again like octatonic collections, hexatonic collections contain triads but do not suggest a particular tonic chord or home key. As shown in Example 7, each collection contains three major triads, three minor triads, and two augmented triads. Juxtaposing two augmented triads that are one semitone apart is another way of generating the hexatonic scale.

Example 7. Triads present in a hexatonic collection.

Acoustic Collection

The acoustic collection is based on the lowest intervals of the overtone series. This is significant because there has long been an association between those lowest intervals and the notion of musical consonance. The result is a mode that resembles the major scale but with a [latex]\uparrow\hat{4}[/latex] and [latex]\downarrow\hat{7}[/latex] (Example 8).

Example 8. The acoustic scale is derived from the notes of the overtone series

 

New Ways of Organizing Pitch

Many composers of the 20th and 21st centuries have looked for new ways of generating pitch collections for their music.

Messiaen’s modes of limited transposition

Olivier Messiaen was interested in composing with collections that can only be transposed a few times before they repeat themselves, such as the octatonic collection. Messiaen’s “modes of limited transposition,” as he called them, are shown in the table in Example 9 and in notation in Example 10. The numbers in parentheses refer to  in semitones.

Messiaen Mode Number Interval pattern (in half-steps) Number of unique transpositions Number of unique rotations Misc.
1 2, 2, 2, 2, 2, 2 2 1 Corresponds to the whole-tone scale
2 (1, 2), (1, 2), (1, 2), (1, 2) 3 2 Corresponds to the octatonic scale
3 (2, 1, 1), (2, 1, 1), (2, 1, 1) 4 3
4 (1, 1, 3, 1), (1, 1, 3, 1) 6 4
5 (1, 4, 1), (1, 4, 1) 6 3
6 (2, 2, 1, 1), (2, 2, 1, 1) 6 4
7 (1, 1, 1, 2, 1), (1, 1, 1 , 2, 1) 6 5

Example 9. Messiaen’s Modes of Limited Transposition.

Example 10. Messiaen’s modes of limited transposition, beginning on C.

These are not the only collections that have limited transpositions: other / of these collections may also have this property. Even within this set, Mode 1 is a subset of Modes 3 and 6. [2]

Distance model modes

Distance model modes involve a more restricted set of modes of limited transposition. These modes are formed with an alternation between two intervals, such as:

  • 1:2 = alternating semi and whole tones (the octatonic mode again)
  • 1:3 = semitones and minor thirds, sometimes called the hexatonic mode/collection, or in parlance, the “magic” hexachord
  • 1:5 = semitones and perfect fourths

This style of organization is strongly associated with Béla Bartók, as we’ll see in the next chapter.

Other modes

This is just the tip of the iceberg, even for 20th-century classical music. Among the other main areas to explore are:

  • Microtonal modes, beloved of Ligeti and the spectral school, which focus on alternate tuning systems and avoid any assumption of the equal-tempered 12.
  • Synthetic modes, derived by alteration of diatonic, as in the non-standard key signatures of certain movements in Bartók’s Mikrokosmos.
  • One-off cases like the “scala enigmatica” of Verdi’s Ave Maria (which is hardly used as a scale/mode in any other case).

Important Considerations with Collections

Why do some of these modes keep cropping up in different contexts? Again, that’s a big issue that has attracted a great deal of theoretical attention. Here are some highlights:

  • Correspondence to the natural overtone series. One hypothesis is that people like modes that see the important pitches align with those low down in the harmonic series, and thus with what are conventionally called (octaves, fifths, etc.). Clearly the is a particularly literal implementation of this idea.
  • Symmetry is a key preoccupation of 20th-century composers. One reason for this is the desire to create a new kind of order not rooted (pun intended…) in the “from-the-bass-up” world of the overtone series and fundamental bass harmony. Symmetry can be internal to scale, as in the rotational symmetry of the modes of limited transposition, or else between scales, as in Bartók’s Cantata Profana.
  • Maximal evenness. A prominent theory of modal construction emphasizes the even distribution of pitches in the space.[3] Think of the diatonic modes again. The diatonic collection is made up of mostly whole tones, with only two semitones that are as far away from each other as possible. This maximized spacing between the semitones means that the pitches of the diatonic collection are maximally even.
Further reading
  • Agmon, Eytan. 1990. “Equal Divisions of the Octave in a Scarlatti Sonata.” In Theory Only 11 (5): 1–8.
  • Clough, John, and Jack Douthett. 1991. “Maximally Even Sets.” Journal of Music Theory 35, no. 1–2 (Spring–Autumn): 93–173.
Assignments
  1. Worksheet on collections (.pdf, .mscz). Asks students to spell one example of each of the collections from this chapter.
  2. Analyze Lili Boulanger’s resplendent Hymne au Soleil. Identify modes and collections used, along with related techniques. Scores can be found on IMSLP and MuseScore. Both include the original French text and an English translation in the underlay.

  1. Agmon (1990) even sees it in a Scarlatti sonata (K. 319 b.62ff).
  2. The acoustic scale does not have this property, so it does not appear.
  3. This emerges primarily from Clough and Douthett 1991.

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OPEN MUSIC THEORY by Mark Gotham; Megan Lavengood; Brian Moseley; and Kris Shaffer is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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