Today, we're talking about serialism. Serialism can actually occur in
collections with less than 12 tones but you're probably most familiar
with the concept of serialism as it relates to 12-tone serialism. This
also tends to be a topic that sticks in people's heads a little bit.
If you ever took a post tonal theory course in your undergraduate
degree for example, you probably remember the matrix of 12 pitches but
I want you to be careful in thinking that you already know about
serialism because the matrix is really not that important in the grand
scheme of things. It just tends to stick with people. So before we go
any further too, I want to contrast this with our topic from last week
which was pitch class sets. So pitch class sets are unordered,
generally speaking. Both normal form and prime form are abstract
concepts. What I mean by abstract is that they don't relate to the way
the notes appear in the music so much. It's just the smallest possible
ordering of a given set of notes. It's not how the notes are actually
arranged in the score. So in almost the exact opposite way, the
12-tone series is entirely defined by the order in which the pitches
appear in the score. So it's much more literal, it's much more
directly observable in the music. Because, think about it, if it was
unordered if we just put it in the smallest possible order then every
single series would be exactly the same. It would be pitch classes
zero through 11. So the order is actually everything in serialism
whereas with sets the order is not that important. Like I said, a
serial piece can actually have any number of pitches but the most
common is to deal with 12 pitches. I only bring this up to point out
that the word, serial, refers to the fact that it occurs in an order.
So like if you have a television series, you're supposed to watch all
those episodes in order. So it's called the serialism because it's all
about the order. It's not actually about the 12 tones. But we're going
to talk mostly about 12-tone serialism. The series or the row is kind
of like the theme or the scale that a piece is based on. It's the
referential collection for the entire piece. Each composer has their
own ways of designing a 12-tone row but you should know that they are
designed. They're not just random. People think 12-tone music sounds
random but it's really the opposite of random. Rows are also usually
not shared from piece to piece, and so different serial pieces can
sound really different from one another. It doesn't necessarily have a
recognizable sound. So now that we've kind of set the stage, let's
talk about the basic operations in serialism. Transposition and
inversion are used in 12 tone rows just like we did in pc sets. But
unlike pc sets, transposing or inverting a 12-tone row results in the
same collection of pitch classes because we're using all of the pitch
classes. So like if I invert 0, 1, 4, i_0, I get 0, 11, 8, I get
different pitch classes, but if I invert a 12-tone row, I have the
same pitch classes because all the pitch classes were there in the
first place. So again, the order is everything. That's the only way
that you can tell if a row has been transposed or inverted. So you're
going to let normal form and prime form to go when you're considering
series. So operations are discerned then through the ordering of the
pitches rather than the pitch content, and one way that you can check
the ordering of the pitches is by using the intervals between each
successive pair of pitches. So the prime form of the row is somewhat
arbitrary designation, and it basically just means we're going to use
this form of the row as a reference. So if you just transpose that
prime version of the row, of course the interval succession is
remaining unchanged. That's the definition of transposition. We keep
the intervals the same, we just move it to a different pitch level.
This is kind of interesting and it will make more sense once you see a
concrete example. You would think that in a retrograde row, retrograde
is where you just read the pitches from right to left instead of left
to right. So in a retrograde row you would think, "Okay, the intervals
then are reversed," but it's interesting, the intervals are not only
reversed they're also inverted, and it makes sense more if you have a
concrete example in front of you. I just want to get through this for
now. Inverted rows have inverted intervals. Again, this makes sense
and that a retrograde inverted row has reversed intervals and that's
because it is the inversion of the retrograde row. So let's talk about
a real piece of music. In a 12-tone piece, usually the first row form
that you see is designated as the prime ordering and I use the term
designated because again it's arbitrary. The composer doesn't usually
say, this is the prime form of the row. So the rest of the row forms
then will be calculated and labeled in relation to that prime
ordering. So this schoenberg string Quartet begins with what's called
p2 in the violin part, and P stands for prime, prime meaning first. So
this is going to be our prime row, and the two indicates that the
first pitch of this row is D Pitch Class 2. So note that some people
though use what we call movable zero, nomenclature for prime rows, and
so some people would call this first row and the violin one, they
would call that P0. I and Strauss, who wrote this book that we're
referencing, find this to be kind of unnecessarily confusing. So we
use a fixed zero nomenclature where we're just going to call the prime
rows based on the first pitch that happens in that prime row. So this
is P2. So let's now determine what order the pitches would come in for
R2. R2 is named for the last pitch in the series and this may seem
confusing but it helps us to calculate it because we know that R2 is
the same as P2, just read backwards. So P2 is 2, 1, 9,10, 5, 3, 4, 0,
8, 7, 6, 11. R2 is just reading that series of notes backwards 11, 6,
7, 8, 0, 4, 3, 5, 10, 9, 1, 2. Another piece of information to be
aware of is again the interval sequence. In the previous slide I
mentioned that the integrals of an R row will be reversed and
inverted, and we can see that in action here. In blue, I'm measuring
the intervals between pitch classes as ordered pitch class intervals,
and this is the kind of interval that always moves clockwise around
the clock face. So in the P2 row, let's just take one example. We see
the first interval between D and C sharp between two and one is 11. In
the retrograde row, D and C sharp come at the end of the row. Here's D
and C sharp, and of course they come in the opposite order because
we're reading this row right to left to get our retrograde row. So the
reason that the intervals are not only backwards, but inverted becomes
a little more obvious when you think about an actual piece of music.
If I read this note D C sharp, I get the pitch class interval 11
because we're going from 2:00 all the way around to 1:00. But if I
read it the other way around, then we're going clockwise just from
1:00 to 2:00. So that's why R rows are related to P rows with inverted
and reversed intervals. Okay. So now, let's try an I row. From P2,
we're going to find I7. The seven refers to the starting pitch of the
row for an I row. So we will start with seven. In an I row, the
intervals between pitches are inverted. So each interval in blue above
the P2 row here will be inverted and turned into its complement mod
12, and that will produce the following pitch in our I7 row. So if we
know the sequence of intervals between the pitches, we can use that to
find those pitches. So that means that 11 will become one, eight will
become four, one will become 11, seven will become five, etc, going
all the way down the line. So then, we can find the pitches. So seven
plus one gives eight, eight plus four gives zero, zero plus 11 gives
11, 11 plus five gives four. We can keep going like that all the way
down and this is our I7 row. So again, the reason it's called I7 is
because the first pitch is seven and the reason it's an inverted row
is because the intervals are inverted. Next, let's make an RI row by
starting with I7. RI rows are I rows backwards, just like how R rows
are P rows backwards. So we can start from our I7 row and write it
backwards to yield RI7 because RI rows are named for their last number
just like R rows are. So here's seven and here's seven in the I and
RI7 rows. Just like with R rows, RI rows and I rows have reversed and
inverted intervals for the same reason. So here, what I'm giving you
are the ordered pitch class intervals for each row form. You can
examine and confirm the relationships between these rows this way. So
between P and R, the intervals are reversed and inverted. Between P
and I, the intervals are inverted. Between I and RI, the intervals are
reversed and inverted because the relationship between I and RI is the
same as the relationship between P and R. But interestingly, P and RI,
even though we think of them as the most distantly related, have the
same intervals, just in reverse order. So that's interesting to look
at. This information is useful for recognizing the transformations of
a row that you may come across in a piece. It may be easier to look
for certain special intervals than to look for certain pitches because
again, every row will have every pitch. So for example, in the
Schoenberg row here, I noticed that there are two interval class ones
in quick succession in each row form. So here, they come up as 11s,
here, they come up as ones, here, they're ones, and here, they are 11s
again. So I can distinguish the row forms then by looking for those
icy ones. Seeing if they're rendered as pitch class intervals one or
11, and then making note of the interval that follows them. So if I
see two 11s and then a five, I know that we're in a P row because we
don't get two 11s and a five anywhere else in these rows. So I've
noticed that if a student ever learned anything about atonal theory,
the only thing they remember is making one of these matrices, which is
really funny to me because I don't think it's that helpful. I think
what I just told you is more useful, practically speaking, than making
a matrix. But something about it just sticks with people. So the
matrix just shows you all the possible transformations of a given row.
So if you write the P row in the top row like I've done here, then you
write the inversion in the first column. So 11 inverted becomes one,
seven inverted becomes five, eight inverted becomes four, and so on.
So this yields our P0 row and a I0 row. It's nice to start with zero,
but you actually don't have to. It works either way. So then in each
new row, you write the version of the row transposed by the interval
that you see in the first column. So this starts out with a nice
little zero one. So we're going to write each pitch class transposed
by one. So 11 will become zero, seven will become eight, eight will
become nine, etc. Then for five, we'll do 11 transposed by five, which
would give us four, zero, etc, and so on. So that's how you fill out
an entire matrix for you. But this would yield all the P rows, of
course, but also all the other row forms. So you would read it left to
right for the P rows, you'd read it top to bottom for the I rows.
You'd read it right to left for your R rows and you'd read it bottom
to top for your RI rows. You can find matrix generators online that
will build these for you. I really don't mind if you use them because
this is just like a mechanical calculator. Why not use a calculator if
you have it? Just know that creating the matrix is not really an
analysis, it's not a big deal. That doesn't impress me if you made a
matrix. It's just a tool that will help you analyze and it's nothing
more. So it's just a first step. Another first step in a serial
analysis is to do what's called a 12 count of the piece. This means
finding each series of 12 notes and labeling them with the number of
the pitch that they are within the row, not their PC number. So here
in this example, you can see that each pitch has been labeled 1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11. So what this is showing you is that the RI7
row is deployed in the piece with the pitches of our RI7 in this
particular order. Wait, where did 12 go? Here it is. Twelve is
overlapping with one. So again, this isn't really an analysis because
you're just labeling things in a very true false sort of way, but it
gets you an idea. It gets you a starting point and entry into the
piece. So for example, now that we have the 12th count of this piece,
we can see some interesting things about it, like every row here,
every row statement includes a four-note chord that takes four
different pitches of the row and sounds same as a simultaneity all at
once. Also, that sometimes the rows have overlapping pitches, where
pitch 12 of RI7 becomes pitch one of P7 and other things like that. So
we can use this as a place to start to make interesting statements,
but again, it's not really interesting on its own. All right. Let's
zoom back out again away from particular operations and just talk
about how you can pose with a row. Serialism can seem like a very
mechanical way of composing, but it really isn't. Because even though
the PC's are decided in advance, the order of them is decided in
advance, literally nothing else is. So you still have to choose the
intervals you're going to use, if they're going to be the small
versions, the big versions, harmonic, melodic, you still have to
choose your phrasing where the cadences are, you still have to choose
your dynamics, your articulation, your text, your instrumentation. So
there's a lot of ways that things can go even after you have the row.
I could give you all the same row and you could all compose very
different pieces with it. But beyond that, the composer designs the
row in a very intentional way and so we're going to talk about that a
little bit in the upcoming slides. So this is a good place to take a
break. I know the video is a little long today. All right. Let's talk
about subset structure. A 12-tone row can be effectively described in
terms of its subset structure. Different subsets yield different
intervallic characteristics which in turn affects the sounds of the
row. So for example, Webern seems to like rows that emphasize interval
class one, whereas Berg likes rows that emphasizes ICs three, four,
and five. Composers like to embed their series with smaller sets that
they're interested in exploiting. So when investigating subset
structure, you want to consider dyads, trichords, tetrarecords, and
hexachords because these are the numbers that divide evenly into 12.
So let's look at an example. This is from the string quartet we looked
at earlier, and this is the row form and it's discreet trchords. It's
kind of a fancy way of saying this is how the row divides up into
trichords that don't share notes with one another. So we've got a 015,
we've got a 027, a 048, and a 015. So that can tell us something
interesting about the properties of the row. We can find some other
015si by looking in other places within the row. So we've got a 015
here, DC Sharp A, a 015 here, AB flat F, a 015 here, CA flat G and a
015, here G F sharp B. Then this example, this score is showing us all
the ways that 015 ends up showing up as a consequence of the
prevalence of 015 within this row. So that's the kind of thing that is
actually interesting for you to point out. "Hey, this row has a lot of
015s, and look, it seems like Schoenberg did this on purpose because
the 015s are all very prominent in the string quartet." Next, let's
look at 0148. This set class shows up, again, in a couple of different
ways in the row as a tetrachord. It's not in the discrete tetrachords,
so if you only looked at the tetrachords that way, you would not find
it. But if you look at other groupings, there's a bunch of 0148s that
are present. Again, you can see that 0148 seems to be a set-class that
Schoenberg wanted to exploit in this string quartet. So the moral of
the story there is, don't limit yourself to the discrete tetrachords,
you can look at some other tetrachords as well. Another thing that can
be exploited in a row is the property of invariants. An invariant is
something that is preserved where the series is transformed. Because
repetition is important in music in general, not just in serialism,
invariants are also very important to serial music. So for example,
invariants under inversion is when pitches are kept intact even if the
series is inverted. You probably still don't know what I mean, so
let's talk about something specific. In the Schoenberg row from
before, if we perform I5 on this row, it will preserve the two dyads
that I've marked with blue here. So 011 is here and 65 is here. If we
invert by five, we still get to keep five and six, and zero and 11. So
this is something you could exploit compositionally just to give the
most basic example. Let's say that when you get to these pitches in
your p0 row, you play them harmonically and melodically. Then when you
do your I5 row, you do the same thing; you play the first two pitches
harmonically instead of melodically and you do the same over here when
you get to 11, 0. That's something that a listener could pretty easily
pick up on and hear a connection between the p0 and the I5 row even
without being aware of the entire serial background. So how do you
figure out that you can do this? You just ask yourself which inversion
maps 11, 0 onto five, six and the answer is five for reasons you
learned last week when you studied set-classes. The index number is
what you get with this crosswise addition. Sorry, this crosswise. This
addition here, zero and five add up to five, 11 and six also add up to
five. So I5 is the inversion that will map zero, 11 onto five, six. We
can see that this is more or less what Schoenberg did with these two
pitches. So in this example on the right here, in the boxes, we've got
zero, 11 played in unison at the outset of a movement and five, six
here again played in unison at kind of an important cadence point.
Then as we move from unison playing to breaking things up, it seems
that zero, 11 and five, six are used as important phrase inceptions
for each of these rows. We can also do an invariance under inversion
with larger groups of pitches. So if you look at this row, you can see
with the teal brackets, different trichords that have been bracketed
and preserved under inversion. So 10, two, six is still there in I7
just in a slightly different order, but here's 10, two, six. Zero,
four, eight comes over here, becoming four, zero, eight and 11, one,
three comes down here becoming three, one, 11. This is from Berg's
row. So we can see that Berg has exploited these invariant tetrachords
by writing these similar lines that emphasize the sameness between
them. So in conclusion, we will definitely learn about some other ways
of handling a 12-tone series, these are really only some of the
possibilities. I will have you take a look at some of those other ways
of doing things. Making a matrix and doing a 12-count, it's just a
first step in analyzing a 12-tone piece and by itself it's not
terribly interesting, but it can lead you to uncover more about how
the piece is constructed which then can lead you to find more
interesting things. Composers do a lot of different things to create
interest in their 12-tone pieces and that's really what the goal is of
analyzing a 12-tone piece. Kind of figuring out what makes this piece
interesting. The 12-tone process is its genesis, but what makes it
unique beyond that? All right.