IX. Twelve-Tone Music
Row Properties
Mark Gotham and Brian Moseley
Key Takeaways
- 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 (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 , for instance, the row starts with A and ends with D♯, and so on.
). There are 1,928 distinct row forms with this
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 [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).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