Some rows are used more by composers than others. Often this is because of the row's properties.

This chapter explains some row properties that are especially common:

all interval

derived rows

Invariance

Hexachordal combinatoriality

Partially-ordered sets

The Twelve-Tone Anthology has more detail on this topic.

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" segment of a row means looking at segments starting at each pitch in turn. For instance, for dyads (two pitches, one interval), we look at pitches 1 and 2, then 2 and 3, followed by 3 and 4, and so on. By considering two pitches at a time and stepping forward by one, there’s always one pitch overlapping. That specific approach gives us the interval content of a row and allows us to identify our first notable row type: the all-interval row.

While all standard twelve-tone rows include all twelve distinct pitches, only some also feature all eleven distinct intervals between neighboring pitches (Example 1). There are 1,928 distinct row forms with this all interval property, but again, some of them 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 and 12, 2 and 11, 3 and 10, 4 and 9, 5 and 8, and 6 and 7. In Example 1, for instance, the row starts with A and ends with D♯, and so on.

Example 1. An all-interval row from Luigi Nono, Il Canto Sospeso.

This example of an all-interval row is a linear layout of the so-called "Grandmother chord" (credited to Nicolas Slonimsky). To produce this succession of pitches, start with a semitone up (interval class 1), then a tone down (interval class 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 viewed as two interleaved chromatic scales (as shown in Example 1) which is essentially a chromatic wedge and can therefore be seen to have precedents in tonal works such as fugues by Bach (BWV 548) and Shostakovich (24 Preludes and Fugues, Op. 87, no. 15).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.

Discrete Segments and "Derived" Rows

The alternative segmentation method is to look at the discrete (non-overlapping) parts of a row. This is perhaps the most common way of thinking about row forms. Given this constraint, a twelve-tone row can be divided into six dyads, four trichords, three tetrachords, or two 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 only one pitch class set.

Example 2 is a classic example from Anton Webern’s String Quartet, Op. 28. The brackets below the notes indicate the three discrete tetrachords belonging to the same set class: (0123), a chromatic tetrachord. Slurs above show a similar consistency in the discrete dyads, which are all semitones.



Example 2. A row from Anton Webern's String Quartet Op. 28, divided into discrete segments. Notice the limited number of set classes and intervals.

Rows with this property are sometimes called derived rows, meaning that the whole row can be considered to be made out of ("derived" from) one pitch-class set. It’s worth also noting the more specific use of the term, where a new row with this property is derived more directly from an existing one with the relevant set as one of its subsegments. For instance, the row above might have been derived from another row that happened to have a (0123) 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 set classes, which can be helpful in cultivating a particular type of unity.

(Segmental) Invariance

Invariance refers to the preservation of something. Any musical attribute (such as a series of intervals, dynamics, rhythms, or 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 the term "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” (Example 3). The upper staff reproduces the row as we saw it in Example 2 (with the discrete tetrachords shown), and the lower staff sets out the same row transposed up