Imagine that you wish to write a serial piece. Aside from such important considerations as performance medium, rhythm, texture, form, and overall structure of the piece (remember this is music), you would want to devote a great deal of time and thought to your Po, since the pitch structure of the entire work will be derived from it. Composers devoted a great deal of time to this task, and came up with some interesting types of Po's, as described below.
All-interval Row: a tone row containing one example of each interval from the m2-M7. The distance between the first and last note is always a tritone. Between adjacent notes of the row there will be two examples of each interval class except 6, which will be represented only once. Once a Po of this variety is set up, it's possible to write a 12-tone melody which contains the maximum of intervallic variety--all eleven intervals from the m2 to the M7. Luigi Dallapiccola and Alban Berg are known for using all-interval rows.
Derived Row: this type of row is constructed from transformations of a smaller set. For example, one might write a 3-pitch set (say, 0,1,4) and then state it in its P, I, R, and RI forms. Thus there's maximal pitch-class unity here: the tone row Po and any transposition or transformation of it will automatically have (014) sets as well. Anton Webern was particularly fond of derived rows, using them in several of his works.
Symmetrical Row: a tone row constructed with internal symmetries (some R form will be identical to some P form, or perhaps some R form will be identical to some I form; in other words, R=Pt or R=It). This type of tone row reduces the number of distinct row forms from 48 down to 24. Anton Webern was fond of symmetrical rows.
Embedded Subsets: this denotes a tone row which contains recurring subsets (e.g. 0,1,6 appears more than once). Nore carefully the difference between a derived row and a row which contains embedded subsets: a derived row is created exclusively from a smaller set (usually a trichord or tetrachord), while a row with embedded subsets simply shows the presence of several identical sets (typically a trichord or tetrachord). In a sense, a derived row is a very special type of row with embedded subsets. Virtually all serial composers write tone rows with embedded subsets, as it provides a built-in motivic unity.
Tonal Implications in a Tone Row: some composers, in particular Dallapiccola and Berg, write their Po's in such a way that they contain tonal implications--perhaps a triad or two, or a seventh chord, or a scale fragment, and so on. Berg's Violin Concerto contains the ultimate in tinal tone rows: it's made up of broken thirds with a bit of whole tone at the end.
G Bb D F# A C E G# B C# D# F
Source Sets: tone rows where the hexachordal content is determined but the hexachordal ordering is not. The composer Josef Hauer was experimenting with unordered hexachords, the combination of which created 12-tone rows, at about the same time that Schoenberg was developing the 12-tone system. In essence, Schoenberg "won out" because his system was more completely and rigorously developed than Hauer's.