23d Examples - Serialism and Tone Rows

For many people, the terms tone row and matrix are synonymous with post-tonal theory. Luckily, your basic understanding of set theory comprehending tone rows fairly simple.

Starting small

Let’s use set classes as a way to build up to understanding tone rows. To do this, start by writing out each unique, normal-form pc set represented in the set class of (013). Use your ability to transpose and invert that pc set to find them quickly. You can also think of this as listing every pc set represented by a single prime form.

Conclusion

Because of your study of prime form, you probably realized that there are twenty-four possibilities. Because each of the pc sets will be in normal form, we know that there is only one pc set for each possible starting pitch. We can represent these using transposition notation.

• T₀[0,1,3] = [0,1,3]
• T₁[0,1,3] = [1,2,4]
• T₂[0,1,3] = [2,3,5]
• T₃[0,1,3] = [3,4,6]
• T₄[0,1,3] = [4,5,7]
• T₅[0,1,3] = [5,6,8]
• T₆[0,1,3] = [6,7,9]
• T₇[0,1,3] = [7,8,t]
• T₈[0,1,3] = [8,9,e]
• T₉[0,1,3] = [9,t,0]
• T_t[0,1,3] = [t,e,1]
• T_e[0,1,3] = [e,0,2]

There are a further twelve unique pc sets that result from inverting our prime form pc set. Of course, for each of these, you should normalize the pc set after inversion.

• T₀I[0,1,3] = [9,e,0]
• T₁I[0,1,3] = [t,0,1]
• T₂I[0,1,3] = [e,1,2]
• T₃I[0,1,3] = [0,2,3]
• T₄I[0,1,3] = [1,3,4]
• T₅I[0,1,3] = [2,4,5]
• T₆I[0,1,3] = [3,5,6]
• T₇I[0,1,3] = [4,6,7]
• T₈I[0,1,3] = [5,7,8]
• T₉I[0,1,3] = [6,8,9]
• T_tI[0,1,3] = [7,9,t]
• T_eI[0,1,3] = [8,t,e]

Each one of these twenty-four pc sets contains a unique collection of pitch classes.

Visually representing a set class

In the above example, you could describe T₀ and T₀I as the reverse order of the same intervals moving away from a central pitch. The only obvious correlation between the actual pitch classes is that both of the pc sets contain 0–which happens to be the interval of transposition for those two sets. (The interval of transposition is the “0” in T₀.) If you look at each pair of transposed and inverted pc sets, you will notice the same thing; they always center the interval of transposition. If you wanted to represent this visually, you could plot each transposition and inversion on a chart, on which this central, shared pitch for each transposition shows the inversion and transposition branching out. Look at our trichords charted this way.

Inv pc set	inv pc 1	inv pc 2	common pc	tran pc 2	tran pc 3	Tran pc set
T0I	9	e	0	1	3	T0
T1I	t	0	1	2	4	T1
T2I	e	1	2	3	5	T2
T3I	0	2	3	4	6	T3
T4I	1	3	4	5	7	T4
T1I	2	4	5	6	8	T5
T6I	3	5	6	7	9	T6
T7I	4	6	7	8	t	T7
T8I	5	7	8	9	e	T8
T9I	6	8	9	t	0	T9
TtI	7	9	t	e	1	Tt
TeI	8	t	e	0	2	Te

Here, the “common pc” column shows the pitch class that is common to both the inversion and the transposition pc sets. (Note that to create this chart, the inverted form of the pc set is written in descending form rather than ascending form, so you must read it backwards to find its normal form.) You can make a chart like this for any pc set, and this would extremely helpful if you were analyzing a piece of music that had this trichord present. Rather than only looking for intervallic patterns, you could quickly refer to your chart to identify whether a specific trichord belongs to this set class. As you begin searching pieces for set classes, you will appreciate not having to transpose and invert every time; it is much easier to have a complete list for reference, and you could use our new (013) trichord “matrix” for this purpose.

Tone rows, serialism, and 12-tone music

Before we go further, we should briefly define the genre of music most associated with set theory and matrices. (Although set theory can be used to study any type of music as long as it uses the twelve pitches from the chromatic scale.) Serialism is any music in which some aspect of the composition is based on a pre-defined repeatable pattern; this can be the melodic intervals, harmonic intervals, harmonies, rhythm, or any other aspect of music that could be described in a series.

12-tone music is a sub-genre within serialism in which a fixed series of all twelve pitches is used to generate both the melodic and harmonic content of the piece. There are a variety of ways in which composers have employed this, but in its strictest form, all twelve pitches must be used before a pitch can be repeated. The series that determines the order of all twelve pitch classes is called a tone row; this will always be a listing of all twelve pitch classes with a fixed interval order. To provide compositional variety, the tone row may be altered to any of its transpositions or inversions, and any tone row may also be played in retrograde – all pitches in reverse order. So from this, there are four forms of a tone row.

• Prime (P) - a tone row or any of its transpositions
• Retrograde (R) - any prime row played in reverse order
• Inversion (I) - the inversion of the original prime tone row and all its transpositions
• Retrograde inversion (RI) - any inverted row played in reverse order

Because any of these tone row orders can start on all twelve pitches, there are forty-eight possible arrangements of any tone row: twelve prime tone rows, twelve inverted tone rows, twelve retrograde tone rows, and 12 retrograde inversion rows. We label these using the abbreviations P, R, I, and RI followed by a subscript number to represent the transposition. (e.g. P₅ or RI₆).

Expanding the set

Now imagine that you were trying to create a visual representation for all forty-eight tone row possibilities in the same way we did above for the (013) trichord. You would have to create one of our charts for both the prime form and retrograde, and each row would twenty-three pitch classes in it. Luckily, there is a simpler, space efficient way to do this–a tone row matrix, so let’s set some ground rules. If you were to use the normal and/or prime form, there is technically only one form for an aggregate pitch set: (0123456789te). But for tone rows, we treat the intervallic pattern as a fixed part of the pc set–meaning we do not put the pc set into normal form. So you know, we can create nearly a million different distinct combinations of all twelve pitch classes if we fix the intervallic structure as well!

For illustration purposes, try creating the first two lines of our twenty-three pitch class chart, T₀/T₀I and T₁/T₁I, for the following pc set: T₀ = (0,e,6,7,t,9,4,3,5,8,1,2)

To do this, write out T₀, and then write out the inversion backwards from the first number. Then do the same for, T₁.

Conclusion

The answer to the pc set would take up too much screen for me to succinctly write in a table here, but I’ve written it in plain text here. (Remember that the inverted row must be read from the center note, the bolded note, moving backward.)

• T₀I t e 4 7 9 8 3 2 5 6 1 0 e 6 7 t 9 4 3 5 8 1 2 T₀
• T₁I e 0 5 8 t 9 4 3 6 7 2 1 0 7 8 e t 5 4 6 9 2 3 T₁

What do you notice as you work through this? Can you think of different methods for creating T₁I rather than just mirroring T₁? While the trichord’s chart above was not too unwiedly for addressing all possible variations, can you think of a way to create a chart that shows every combination (transposed and inverted) of a tone row that would take less space than the clunky chart above?

Developing a matrix

What if rather than writing T₀ and T₀I in a straight line, we rotated the inversion pc set, T₀I, downward at 90 degree angle to create a verical column? This would create the outline for a 12 by 12 grid with T₀ as the top row and T₀I as the first column.

–	T0I
T0	0	e	6	7	t	9	4	3	5	8	1	2
–	1
–	6
–	5
–	2
–	3
–	8
–	9
–	7
–	4
–	e
–	t

This grid is a called a matrix. We could then fill in each row of our matrix with the transpositions of the original pc set and each column with inversions. For example, where would the other two pc sets that we figured out in our original example, T₁ and T₁I, fit in this grid?

Filling out the matrix

Because the top row is the original, prime version of this pc set and the column is its inversion, it makes sense to place the next pc sets, T₁ and T₁I, in the row and column respectively that begin with the pitch class “1”. It would look like this:

–	T0I										T1I
T0	0	e	6	7	t	9	4	3	5	8	1	2
T1	1	0	7	8	e	t	5	4	6	9	2	3
–	6										7
–	5										6
–	2										3
–	3										4
–	8										9
–	9										t
–	7										8
–	4										5
–	e										0
–	t										e

Because we have done this correctly, you’ll notice the pitch classes T₁ and T₁I perfectly with the pitch classes in the old columns.

The great thing about this system is that once each of the transpositions (rows) are filled in correctly, each column will be a correctly transposed version of the inverted tone rows. Once filled in, a matrix such as this shows every transposition and inversion of any aggregate pc set.

Completing our tone row matrix

Using our terminology of P, R, I, and RI, complete the tone row matrix above by filling in each row and then labeling. Is it necessary to go in a particular order? What is the easiest way to fill it out for you? Knowing that we use P, I, R, and RI to label each direction of the matrix, how would you differentiate each tone row? Would retrograde tone rows be labeled by their starting pitch or their corresponding prime label?

Conclusions

The correctly completed tone row matrix for our original row would be:

–	I0	Ie	I6	I7	It	I9	I4	I3	I5	I8	I1	I2	–
P0	0	e	6	7	t	9	4	3	5	8	1	2	R0
P1	1	0	7	8	e	t	5	4	6	9	2	3	R1
P6	6	5	0	1	4	3	t	9	e	2	7	8	R6
P5	5	4	e	0	3	2	9	8	t	1	6	7	R5
P2	2	1	8	9	0	e	6	5	7	t	3	4	R2
P3	3	2	9	t	1	0	7	6	8	e	4	5	R3
P8	8	7	2	3	6	5	0	e	1	4	9	t	R8
P9	9	8	3	4	7	6	1	0	2	5	t	e	R9
P7	7	6	1	2	5	4	e	t	0	3	8	9	R7
P4	4	3	t	e	2	1	8	7	9	0	5	6	R4
Pe	e	t	5	6	9	8	3	2	4	7	0	1	Re
Pt	t	9	4	5	8	7	2	1	3	6	e	0	Rt
–	RI0	RIe	RI6	RI7	RIt	RI9	RI4	RI3	RI5	RI8	RI1	RI2	–

Read below for important notes on building and using a matrix.

Properties of the matrix

Finally let’s touch on some general important concepts and common mistakes that you will encounter when creating tone row matrices.

Important concepts

• The most often used labeling method for tone row matrices is fixed-zero notation, but movable-zero is preferred by some. Either is possible, but make sure that you are consistent in your application and clearly state which method you are using in any analysis.
  • In fixed-zero, 0 will always be the pitch class that includes C.
  • In movable-zero, the analyst chooses a primary pitch to designate as 0 based on the important features of the piece (e.g. it is often the first pitch class of the piece) and then determines each other pitch class’s number based on the intervallic relationship to the designated 0 pitch class.
• In a tone row matrix, label each prime and inversion tone row by the abbreviation P or I respectively, followed by a subscript integer of the first pitch class for that row.
• However, label retrograde (R) and retrograde inversion (RI) rows by their corresponding prime (P) and inversion (I) rows. If you don’t have a completed matrix available, but you know that a tone row is a R or RI form, you can just label it using the last pitch of the tone row, because that is the first pitch class of the corresponding P or I row. Therefore, R and RI rows are always labeled using their last pitch.
• When building a matrix, there are two ways to arrange the prime and inversion tone rows of the matrix depending on which transposition of the tone row that you choose to place in the top level of the matrix.
  • One possible layout prioritizes a specific iteration of the tone row by placing it in the top row of the matrix in its non-transposed iteration. Typically, your original tone row will be taken from a piece of music, so many theorists prefer to format the matrix in the top position. You must then determine each prime row based on this original row.
    • For example, if the first iteration of the row started on A-flat, the first row in the matrix would be that row. If you are using fixed-zero notation this would be labeled P₈, but if you are using movable-zero, this would be labeled P₀.
  • On the other hand, a matrix is meant to be a useful tool for composers and analysts, so it’s primary role is to help you track all possible uses of the row. It is easier to create a matrix that starts on 0, because creating the first inverted row (I₀) can be created using your standard fixed-zero inversions. (i.e. 0 becomes 0, 1 becomes e, 2 becomes t, etc.) If using fixed-zero, though, it is rare that the piece will begin on pitch class 0 (C). Instead, you can simply transpose your entire tone row to start on 0, write this as the top left row of your matrix, and then create the matrix as we did above. Even though your rows will be in a different order, every row will be labeled identically to the other method because you are using fixed-zero notation.
    • For example, if the first iteration of the row started on A-flat (pc 8 in fixed-zero), you would subtract 8 (or add 4) to transpose the entire row to begin on pitch class 0. You can then place this as the top row of your matrix and build the matrix normally. All rows will be labeled correctly according to fixed-zero notation. To make sure that you remember which row is the original row, you will probably want to mark it in some way. I do this by circling the row label (e.g. P₈) of the row that begins the piece.
• When done correctly, any tone row matrix will have a pitch class that extends diagonally from the top left cell of the matrix to the bottom right cell. In the correct chart above, you can see that the pitch class 0 is in the first cell and then connects diagonally across the entire matrix.

Common mistakes

• The most common mistake when creating a matrix, other than simply mis-transposing, is the mislabeling of the tone rows. Remember that the labels for R and RI rows are based on their corresponding prime and inverted rows, NOT the starting pitch of the R or RI rows.
• When using a matrix to analyze a 12-tone piece of music, you will occasionally encounter tone row elision. Two rows are said to be elighted if the ending pitch class(es) of of one row are actually shared to become the beginning pitch class(es) of the next row. If you encounter a tone row that seems to have less than twelve pitch classes, this is likely the reason.
• If you use movable-zero, you will need to be careful in creating your first inverted row because you will not be inverting around C=0. Proponents of this method usually suggest inverting interval by interval away from the first pitch class. (e.g. If the first interval of the row is an ascending M3, then the first interval of your inverted row will be a descending M3. You will continue this process for each interval.)

Technology is great…to a point

As with most things, some enterprising people have created a shortcut to all of this work. There are now free-to-use matrix calculators such as the 12-tone assistant at this excellent website. These are great to speed up your analysis, but as a student, make sure that you understand the principles of why and how these work before becoming entirely reliant on them. Even when trying to use this website, you must understand exactly what output you are trying to achieve before you can choose the correct option.

Admittedly, it is unlikely that you will ever be required to create a tone row matrix without access to a calculator (unless you are taking a music theory exam.) But the tedium of transposing each of the rows by hand will help you notice patterns in the tone row and help you to remember the nuances of the tone row as you analyze. This often provides insight into the analysis that would be missed by those relying on a calculator to create the matrix.