X. Twelve-Tone Music
Twelve-tone composers may view the notes in a tone row as ‘equal’, but they do not appear to feel the same way about different row forms. Instead, rows with certain properties have disproportionately attracted composers’ attention. This chapter surveys some of the ‘special’ types of properties and row forms to look out for. A recurring focus is on the properties of the smaller, constituent parts of a row – its internal segments. There are two ways to view these constituent parts: ‘overlapping’ and ‘discrete’.
Overlapping Segments and the ‘All-Interval’ Row
Considering every ‘overlapping’ subsegment of a row means looking at segments starting at each pitch in turn. For instance, for dyads (2 pitches, 1 interval), we look at pitches 1 and 2, then 2 and 3, followed by 3 and 4 and so on. By considering 2 pitches at a time, and by stepping forwards by 1, there’s always one pitch overlapping. That specific approach gives us the interval content of a row, and our first notable row type: the ‘all-interval’ row.
While all standard 12-tone rows include all 12 distinct pitches, only some also feature all 11 distinct intervals between neighboring pitches. There are 1,928 distinct rows forms with the property, but again, some of those have appealed to composers more than others. One true-by-definition property of these rows is that there is a tritone between each pair of notes around the central pair, i.e. between notes 1-12, 2-11, 3-10, 4-9, 5-8, and 6-7, for instance between A and D#; Bb and E; and so on in the following example:
This example of an all-interval row is a linear layout of the so-called ‘Grandmother chord’ (Nicolas Slonimsky). To produce this succession of pitches, start with a semi-tone up (interval class 1) then a tone down (i.c. 10) and continue to alternate odd and even intervals with the odd intervals getting successively larger and the even ones smaller. As a consequence, the resulting pitch succession can be view as two interleaved chromatic scales (as shown in the example) which is essentially a chromatic wedge and can therefore been seen to have precedents in tonal works such as fugues by Bach (BWV 548) and Shostakovich (24 Preludes and Fugues, Op.87 No.15).
Discrete Segments and ‘Derived’ Rows
The alternative segmentation method is to look at the ‘discrete’ (not-overlapping) parts of a row. Apart perhaps from the all-interval row property specifically, this is the more common way of thinking about row forms. Given this constraint, a 12-tone row can be divided into 6 x dyads, 4 x trichords, 3 x tetrachords, or 2 x hexachords. The fact that there are so many of these options is a property of the number 12 and one of the benefits of having a 12-based system.
In turning from overlapping to discrete segments, we also tend to turn our attention from considerations of ‘all’ to ‘only’. Specifically, it has been a preoccupation of some serial composers to find and use row forms featuring several instances of one only one pitch class set.
Here’s a classic example from Webern’s String Quartet, Op. 28. The slurs below the notes (as well as the brackets and barlines) indicate the 3 discrete tetrachords belonging to the same set class: [0,1,2,3] (i.e. a chromatic cluster). Slurs above show a similar consistency in the discrete dyads which are all semi-tones and thus instances of a semi-tone: i.e. set [0,1].
Rows with this property are sometimes called ‘derived’ rows. Minimally (and most usefully) this is used in the sense that the whole row can be considered to be made out of (‘derived’ from) one pcset. It’s worth also noting the more specific use of the term where a new row with this property derives more directly from an existing with the relevant set as one of its subsegment. For instance, the row above might have derived from another row that happened to have a [0,1,2,3] subsegment in it.
From a compositional and listener-oriented perspective, derived rows are very suggestive. Because the set-class content of a row doesn’t change when it’s transposed, inverted, etc., these set classes will circulate constantly throughout a piece, even as different row forms are used. Therefore, a derived row guarantees the regular recurrence of a set classes, which can be helpful in cultivating a particular type of unity.
‘Invariance’ refers to the preservation of something. Any musical attribute (such as a series of intervals, dynamics, rhythms, pitches, … ) may be kept the same from one context to another. While other parts of the music change, the aspect in question is not varied: hence ‘invariant’. In twelve-tone theory, we are mostly concerned with intervallic invariance and pitch class segmental invariance. The first type – intervallic invariance – is very common. Any time a row is transposed, the ordered intervallic content of the row is unchanged. (Likewise, retrograde inversion creates retrograde intervallic invariance.)
Segmental invariance is rarer and warrants separate comment here. Where a pitch-class segment of a row remains in place when that row is transformed, we say that the segment is “held invariant.” Let’s take another look at the row for Webern’s String Quartet, Op. 28:
The upper stave reproduces the row as we saw it before (with the discrete tetrachords shown) and the lower stave sets out the same row transposed up four semitones (or, equivalently, down 8), i.e. as P4. Notice that these two different rows (P0 and P4) comprise the same tetrachords, not only in terms of set class, but of absolute pitches:
- the first four pitches of P0 (the first ‘measure’ in the example) are the last four pitches of P4.
- the middle four pitches of P0 (the second ‘measure’ in the example) are the first four pitches of P4.
- the last four pitches of P0 (the final ‘measure’ in the example) are the middle four pitches of P4.
We say that these tetrachords are invariant segments. These segments are held invariant because they share the same relationship with one another as that shared between the rows: because the tetrachords are related by T8, when the whole row is transposed by T8, those tetrachords are “held invariant.” Put another way, when the first tetrachord is transposed up 4 semi-tones, it becomes the last tetrachord and so when the whole row is transposed by that interval, the last tetrachord ‘becomes’ the first tetrachord.
To determine when and if a pitch-class segment of a row will be held invariant:
- Find an equivalent set-class elsewhere in the row. This may be a dyad, trichord, tetrachord, etc.
- Determine the transpositional or inversional relationship between them.
- When the row is transposed or inverted by that same relationship a segment will be held invariant.
While discussion of ‘derived’ rows typically focuses on tri- and tetra-chordal sub-segments, the two discrete ‘hexachords’ of a row (i.e. the two halves, pitches 1–6 and 7–12) have attracted at least as much attention from theorists and composers over the years, not least in terms of ‘hexachordal combinatoriality’.
Let’s take this step by step. For a standard 12-tone row with each pitch stated exactly once, the first hexachord (half) of P0 obviously ‘complements’ the second half in that they make up the total chromatic together. That being the case, the first hexachord of P0 and the first hexachord of R0 are also complementary. This is a trivial step (the first hexachord of the R form has the same pitches as that of its corresponding P by definition), but it serves to introduce the relevant comparison here: the pitches of the corresponding hexachords of two row forms. Combining the first hexachords of P0and R0 gives you the total chromatic, and so does combining the second hexachord of those two forms.
By definition, this relation holds between P0 and R0, and all other P–R and I–RI pairs (e.g. I4-RI4). This ‘combinatoriality’ relation becomes meaningful when it holds between rows related by other transformations, i.e. between a P–P, a P–I, or a P-RI pair. After Babbitt, we distinguish between two degrees of this relation:
- ‘Semi-combinatorial’ pairs are related by one such transformation: transposition (P–P), inversion (P–I), or retrograde inversion (P-RI).
- ‘All-combinatorial’ pairs are related by each such serial transformation. There are only 6 distinct rows with this property where this relation holds between the row and transformations of itself. Repertoire examples include the row of Webern’s Symphonie (as explored further in this chapter).
The ‘Magic’ hexachord’
While we’re discussing hexachords, just as the properties of some rows have attracted composers, so too have some hexachords. We’ll introduce just one ‘celebrity’ hexachord here: the so-called ‘Magic’ hexachord’ (also known as the ‘hexatonic collection’, ‘hexatonic set class’, ‘Ode-to-Napoleon’ hexachord) with the following properties:
- Pitch class set 6-20 [0,1,4,5,8,9];
- Create all-combinatorial rows unusually readily, for instance, by transposition in three ways (+/-2 and 6 semi-tones). Only the whole tone scale hexachord [0,2,4,6,8,10] exceed this with combinatoriality by transposition of any odd-numbered interval;
- Related to the 1:3 ‘Distance model’ mode discussed in the Collections chapter;
- Many internal triads: this set contains within it all the pitches of the hexatonic cycle discussed in the neo-Riemannian chapter;
- Only one five-note subset: 5-21 [0,1,4,5,8];
- Only one seven-note superset: 7-21 [0,1,2,4,5,8,9], which is the complement of that 5-note subset, 5-21;
- Used in many pieces includes Webern’s Konzert (as discussed in this chapter), Schoenberg’s Ode-to-Napoleon (hence one of its names), and works by Maderna and Nono.
‘Partially ordered’ sets
When is ‘serial’ music not really serial? Surprising as it might seem, the answer is ‘often’. In the strict sense that 12-tone music is supposed to be consistently based on a fixed succession of the 12 pitches in a simple (transposed and / or inverted) version a referential row, much of what we call ‘serial’ music doesn’t actually work like that.
A key example of this is the idea of a ‘partially ordered’ set (another term borrowed from mathematics) which qualifies the idea of all 12 being in a fixed order to a more flexible constraint. Derived rows provide a good introduction to this situation. As we’ve seen, derived rows comprise one context in which we focus on the constituent segments (tetrachords, for instance) of a row and their set class. It’s a small step from this to a ‘row’ which is defined more by the content of those segments than any specific ordering.
For instance, in the example below, we reprise Webern’s Op.28 row one more time (upper stave) and create a new row by reversing each pair of pitches (lower stave). The second case has the same dyadic and tetrachordal segments as the first, but the internal order of those segments has changed. By definition, we still have a row with all 12 pitches, but a ‘different’ one.
We provide the Webern example for illustration; really this is a concept introduced later, by Milton Babbitt, and the kinds of re-ordering can get a lot more complex. But is it so different? While this practice clearly deviates from ‘strict’ definition of 12 pitches in a fixed order, what they share is a constant turning over of the total chromatic / aggregate. Moreover, even in ‘strict’ serialism, composers still use chords (several pitches sounding at once) which necessitates thinking more in the partially ordered than the strictly serial sense. In short, at least some form of this thinking is very, very common indeed.
Finally, how significant is it for rows to be related by partial ordering? First recall that are hundreds of millions of different orderings of the 12 pitches. Well there are fewer when you fix the hexachords in place (and fewer still for smaller sub-segments), but still a LOT. Here are some numbers:
- Any ordered of the 12 pitches: 12! = 490,001,600
- Hexachords fixed: 6!x6! = 7202 = 518,400
- Tetrachords fixed: 4!x4!x4! = 243 = 13,824
- Trichords fixed: 3!x3!x3!x3! = 64 = 1,296
- Dyads fixed: 2!x2!x2!x2!x2!x2! = 26 = 64
So this alone might be considered weak grounds for meaningful relation between row forms especially as the interval content changes with each pitch permutation. As so often in analysis, it all comes down to how the composer actually uses the property in practice.
Other ‘Special’ types of row forms
This already long chapter surveys only a few of the many row properties that composers have explored. We end simply with a short list of some others, as a nod to the huge range out there.
Symmetrical interval progressions: while a strictly symmetrical succession of pitches is obviously not possible (given that we don’t repeat pitches within the row) a symmetrical succession of intervals is, and rows with this property are popular among some composers.
‘Cyclic series’ rows (George Perle): this term and property refer to rows in which every pairs of notes separated by one all form the same interval, i.e. the interval between pitches 1 and 3 is the same as that between 2 and 4, and so on.
Rows with more or fewer than 12 notes. Some composers have explored serial processes with tone rows longer or shorter than 12 notes. Clearly this leads to slightly different properties (e.g. repeating notes in the case of rows with more than 12 notes) but the compositional processes and priorities can be the same. And as we noted at the start of this section, ‘serial’ thought is no limited to 12-tone thinking anyway. Examples include:
- Ruth Crawford Seeger, Diaphonic Suite No. 1 (1930): [7,9,8,11,0,5,1,7] (9 notes);
- Ruth Crawford Seeger, String Quartet (1931): [2,4,5,3,6,9,8,7,1,0] (10 notes);
- Elizabeth Lutyens, Requiescat In Memoriam Igor Stravinsky: [6,8,4,2,11,0,3,1,5,7,10] (11 notes);
- Elizabeth Lutyens, Chamber Concerto, Op. 8, No. 1, Mvmt. I, [3,2,5,6,11,0,7,1,9,10,2,6,8,5,4] (15 notes).
- That the wedge is the subject of a fugue in both these cases, and therefore a central focus, recurring frequently throughout, perhaps strengthens the connection. ↵