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General Music Theory and Tutorials => Advanced Composition => Topic started by: RJB54 on February 08, 2019, 12:01:38 PM

Title: Expanded Serialism
Post by: RJB54 on February 08, 2019, 12:01:38 PM
The purpose of this thread is to be a general description of what has become my normal compositional approach. I have reached a point in my work as a composer that I needed to take a breath and clearly define for myself what it is that I am doing in an organized manner rather than continuing to just 'wing it' as I compose with all of these concepts just rattling around semi-randomly in my head.

To that end, (having found in the past that writing something down, having to create actual, coherent, sentences, helps to crystalize my thoughts and ideas and clear my thinking) I started creating a thesis, as it were, of the theoretical basis for what I was doing as a composer. Subsequently, I decided that if I was actually going to go to the trouble of writing this stuff down I might as well post it here with the thought that others might find it amusing or edifying (but hopefully not absurd nonsense).

One of the triggers for my deciding to go through with this documentation process was that while I keep calling my compositional approach serial, it really isn't. This is due to the fact that the manner in which I use the tone row in my compositions has drifted so far from the traditional serialist approach that there is almost no correlation between what I do and what a true serialist would do. Therefore, calling my stuff serial gives an incorrect impression of what is going on.

I have decided to start calling what I do Expanded Serialism (for lack of a better term). Referring to what I do in this manner identifies the two main aspects of this compositional approach, which is music which is based on tone rows but which makes use of techniques which expand on the traditional methods of utilizing said rows.

At bottom, my treatment/use of a tone row is, to some extent, conceptually similar to what goes on in a diatonic piece (stay with me here :)). In a diatonic piece in C Major the composer doesn't just keep repeating the C Major scale over and over or doesn't just keep presenting only the seven pitches of that scale in various configurations. No, the composer utilizes various techniques to coherently and logically incorporate into their composition in C Major pitches which are not part of the C Major scale as well as incorporating keys other than C Major. In this manner that composer can expand the content of their C Major composition.

Similarly, in my works, the tone row functions as the starting point and basis for the material which appears (as the C Major scale does in a C Major composition) even though some of (and sometimes a lot of) the material in the piece is not explicitly part of a row.

The starting point for my thinking (composing) in this manner has come from Alban Berg's mature serial works, the opera Lulu and the Violin Concerto. In these pieces he explored various ways of utilizing the tone row as a controller/generator of an expanded palette of possible pitch combinations not limited exclusively to a given tone row and not limited excusively to Schoenberg's rules for how a row should be manipulated.

I will not discuss the traditional serial techniques/concepts here, first, because there are many books which cover that subject far better than I ever could, and, second, these traditional techniques/concepts (other than the fundamental concept of the row aspects (prime, inversion, retrograde, and retrograde inversion)) have virtually nothing to do with my music.

In the past I have railed against what I have termed the mathematical serialists, that is, those composers/theorists whose concern is primarily oriented to the mathematical ramifications of Schoenberg's tone row system instead of being concerned with writing good music which someone might enjoy listening to using the row. Since I desire to write music that people would, hopefully, enjoy, I ignore most of it.

Since this whole Expanded Serialism idea is a potentially large topic and I do not know what the maximum size of an individual post can be, as well as wanting to have the info presented in bite-sized chunks, I will break the whole thing down into topics which will be posted individually.

The contents of this thread will not be fixed. In other words, I reserve the right to add, change, or delete things if my thinking changes along the way as a result of either my continuing development as a composer or comments from others.

Finally, I am in no way trying to say or imply that the things which will be discussed in this thread are all my ideas. In fact, as I will repeatedly state, most of the base concepts to be discussed here come from techniques that Berg used in his compositions. I might use them or expand them in my own ways, but the basic ideas will usually come from Berg.

Currently, the topics discussed in this thread are:
1.  Tonality vs. Atonality
2.  Multiple Rows and Row Derivation
3.  Row Rotation
4.  Interpolation
5.  Contiguous Segmentation
6.  Non-Contiguous Segmentation
7.  Sequence
8.  Multi-Directional Segmentation
9.  Set-Class Transformation
10. Doubling

Title: Re: Expanded Serialism
Post by: RJB54 on February 08, 2019, 12:16:23 PM
Tonality vs. Atonality

The first topic to be discussed is less a description of technique than philosophical musings. A lot of what I have to say in this topic has been said before in various threads but I'll cover it again it here because it describes the foundation of my thinking regarding the writing of non-diatonic, non-common practice, music.

The fundamental basis of my music is the idea that operating in a non-diatonic environment does not mean that entities which were used in common practice should be avoided. The idea that they should be avoided primarily comes from Schoenberg and had a strong influence on Webern and his followers (Babbet, Boulez, Rochberg, etc.).

Schoenberg always had a dichotomous relationship with common practice. On the one hand, he always strongly argued that his forays into atonality (a term he hated, but that we're stuck with) and then serialism were natural, and needed, developments from common practice and that there were historical and creative continuities between common practice and what he was doing. On the other hand, he very much wanted to be viewed as a progressive thinker kicking over the traces, as it were, and wiping out the past. This side of him lead to the 'rules' of atonality and serialism which I feel had a very deleterious effect on the development of classical music in the 20th century.

I think that part of the reason Schoenberg placed such an emphasis on his 'progressive' side was due to the fact that Schoenberg and Webern spent the first four decades of the 20th century constantly jostling each other for the title of the Most Progressive Composer. They were always vying with each other over who could come up with the newest, most progressive, idea. This challenge/counter-challenge certainly occurred in private as letters of Schoenberg, Webern, Berg, and others have shown. Publicly, they kept trying to outdo each other in their compositions. This being the case, any references to common practice elements had to be suppressed/ignored as such elements would get in the way of claiming to be the most progressive.

One result of this was Schoenberg's insistence that the most fundamental of common practice entities were to be avoided. The entities being the consonant intervals (thirds, fourths, fifths, and sixths) both horizontally and vertically; thus, the prominence given by the Second Viennese School to augmented and diminished intervals, as well as seconds, sevenths, and ninths. Fourths and Fifths were acceptable if they appeared in a context of quartal/quintal harmonies which, of course, are not common practice.

One of the biggest problems with this idea is that it contradicts nature as represented by the laws of acoustical physics. It is nature that declares the strong feeling of rightness in the relationships between the major third, the perfect fifth, and the perfect octave. This is why so many people don't like atonal/serial music. It is not, as the serialists always argued, that people don't like it because its too new, or too different, or that they're just not used to it yet. While, to some extent, this is true, the real reason for the dislike is that so much of atonal/serial music goes out of its way to deny the third and fifth relationship defined by nature and this results in the music sounding off and unpleasant to most people. The fault is not with the people who feel this way, but rather, the fault is with the theorists and composers for denying a fundamental fact of nature.

Is this denial of the third and fifth relationship a requirement for writing serial music? No, it is not. It was a choice which Schoenberg made for his own reasons and purposes and it was a choice which Webern made to follow Schoenberg down that path, while Berg chose to not completely follow that path. Despite Berg's frequent paying of fealty to Schoenberg in public, when it can time to write his music Berg did not find it within himself to completely abrogate the realities of nature. This is why Berg has the reputation of being a regressive, backward looking, composer, while Webern has the reputation of being a progressive.

In my view Berg was the most progressive of the three because he operated in a creative environment which incorporated all of the possibilities of composing with 12 tones while Schoenberg and Webern operated in a more regressive manner by deliberately limiting their compositional possibilities to a subset of all those which were available to them due to their limiting themselves to the rules Schoenberg established for strict serialism. To me this is little different from a common practice composer who limited themselves to a subset of the available diatonic possibilities by limiting themselves to only writing pieces in C Major, G Major, and F Major and ignoring every other possibility. The history of contemporary music could have possibly gone down a different, better, path had Berg been willing to openly challenge Schoenberg on the issues he disagreed with Schoenberg on rather than openly supporting Schoenberg and covertly challenging him by writing music which in many ways did not adhere to all of Schoenberg's rules.

We now arrive at the crux of the matter. Do tonality and atonality have to be viewed as opposing concepts? I think that this is probably one of the reasons why both Schoenberg and Berg hated the term atonality because it implied this dualist conflict between the tonal and the not tonal. Berg, in particular, in most of his compositions, and certainly in Lulu and the Violin Concerto, clearly did not subscribe to this dichotomy. His approach was to do whatever was needed, to use whatever materials were at hand, to express what he needed to express. The result is the inclusion of tonal, or at least tonalistic, elements in his atonal/serial compositions. It is clear in his compositions that Berg did not subscribe to the dualistic opposition of tonal and atonal, but rather, viewed them as being differing aspects of the same thing, the world of sound defined by nature.

I, of course, clearly follow Berg in this regard. I don't consider myself as writing tonal or atonal music, but, rather, music which merges the two with emphasis placed on one side or the other based upon the expressive needs of the moment. This could be viewed as being somewhat self-contradictory (after all, I am basing my music on one of the most formulistic of music theories), but I feel very strongly, that, in all situations, "it's never the tool's fault", it is always how the tool is used that matters. Just because Schoenberg, Webern, and their ilk used the tool of the tone row in a poor manner (in my opinion) doesn't mean that the tone row itself is inherently a bad tool. It's just that some composers chose to use the tool in a way which resulted in music that most people are not interested in listening to. As a composer one doesn't have to use the row in that manner though and I don't.
Title: Re: Expanded Serialism
Post by: RJB54 on February 08, 2019, 12:31:20 PM
Multiple Rows and Row Derivation

The next topic to be discussed can be a fairly complex one, Multiple Rows and Row Derivation.

The rules as laid down by Schoenberg state that there should only be a single row used for a composition. However, in Lulu Berg ended up making use of more than one tone row. He began composing Lulu following Schoenberg's rule by initially only using a single row, but, he found part way through composing the first act that relying on a single row was too limiting and he needed additional base material. One of the ways he added such additional material was by using more than one tone row.

Berg included more that one row in Lulu by using the technique of row derivation (which I will discuss below).

For my compositions, I expand on Berg's approach by using multiple more or less independently created rows which have been constructed in various ways in order to provide different musical capabilities. For example, I have one row constructed in a way which provides useful melodic shapes while another row is constructed in a way which provides useful vertical coincidences (chords), etc. In my works I often make use of multiple rows operating simultaneously in various instruments. The ebb and flow of the interactions of the various harmonic and melodic possibilities of the multiple rows can result in music which can be either quite dissonant or almost tonal depending on how the material is handled.

As said above, Berg used the technique of row derivation to create and justify his use of multiple rows in Lulu claiming that he actually was following Schoenberg's rule of one row per composition.

The technique of row derivation means creating additional rows by extracting pitches from a single base row in a given numerical pattern. In this way Berg made available to himself additional rows which he found to be useful. He convinced himself that since these rows were all derived from the same base row he wasn't really breaking Schoenberg's rule about using only a single row.

The process of row derivation is performed as follows. You choose a numerical pattern and then, starting at the first ordinal position of your base row, you extract the pitches from the base row following the numerical pattern. When you reach the end of the base row you wrap back around to the begining of the row again. If the pattern causes you to land on an already extracted pitch, you slide to the right until you land on a pitch which has not been extracted as yet and then move on from there following the pattern.

Using a very boring row as the base row (which is really just the semitonal scale) we start with a row of 0-1-2-3-4-5-6-7-8-9-10-11 (using pitch numbers rather than pitch class names).

The simplest example of derivation would be deriving a row based on a pattern of 2, that is, extract every other pitch. In doing so you would end up with a derived row of 0-2-4-6-8-10-1-3-5-7-9-11. Once pitch 10 has been extracted moving forward 2 would take you back to pitch 0 which has already been extracted. Therefore, you slide to the right to pitch 1 which has not been extracted yet. You extract pitch 1 and move forward 2 from there.

When deriving a row based on a pattern of 3, that is, every third pitch, you would end up with a derived row of 0-3-6-9-1-4-7-10-2-5-8-11. Moving to the right 3 after extracting pitch 9 causes you to land on pitch 0 which has already been extracted. Therefore, you slide to the right to pitch 1 which has not been extracted yet. You extract pitch 1 and move forward 3 from there. Once pitch 10 is extracted, moving forward 3 lands you on pitch 1 which has already been extracted; therefore, you slide to pitch 2, extract it, and go on from there.

You can get as complicated as you want with the extraction pattern. In fact, Berg used a fairly complex pattern to extract what turned out to be one of the most important rows in Lulu, 'Dr. Schoen's Row'. The pattern used was 2-3-4-4-3-2. When using this pattern to derive a row from our base row you would end up with a row of 0-2-5-9-1-4-6-8-11-3-7-10.

George Perle in his book on Lulu quite vehemently derided the idea of Berg using such a method to arrive at his rows because it is a purely mechanical process with no creativity. Perle insists on this even though Berg himself stated that that is how he did it. One aspect of Perle's argument is that Berg couldn't possibly have known that deriving a row based upon the 2-3-4-4-3-2 pattern, for example, would result in a usable row. However, Perle is misstating what Berg was actually doing. Berg did not say something like "I need a row for Dr. Schoen so I'm going to derive a row following a pattern of 2-3-4-4-3-2 because I know it will be a good row for that purpose". No, what Berg did was use this derivation technique to derive from the base row many, if not all, of the rows which can possibly be derived from the base row using many different numerical patterns. Berg then chose from the results those rows which suited his creative purposes.
Title: Re: Expanded Serialism
Post by: RJB54 on February 08, 2019, 12:36:02 PM
Row Rotation

The next topic to be discussed is a simple one, Row Rotation.

Row rotation was used extensively by Berg, particularly in Lulu. In this technique you continue to present the pitches of the row in proper order; however, you rotate the row to begin on a pitch other than the first. For example, instead of presenting the pitches of the row in their normal ordinal positions of 0-1-2-3-4-5-6-7-8-9-10-11, you could present the pitches rotated 4, that is, in an ordinal order of 4-5-6-7-8-9-10-11-0-1-2-3.

Of course, as far as Schoenberg was concerned, this technique is invalid because, officially, the rotated row should be viewed as a new, different, row and not as a permutation of the original row. However, that issue didn't seem to bother Berg so it doesn't bother me.
Title: Re: Expanded Serialism
Post by: RJB54 on February 09, 2019, 07:32:33 AM

The next topic to be discussed is Interpolation.

Interpolation is where non-serial material is inserted between serial pitches. In my work an interpolation will almost always consist of interval cycles of either fixed or mixed type.

A fixed interval cycle is one where, as might be expected, all of the intervals are the same. For example, the semitonal scale is an interval-1 cycle (1-1-1...), while the wholetone scale is an interval-2 cycle (2-2-2...), etc.

A mixed cycle is one where there is more than one interval involved. In Berg's usage, and mine, a mixed cycle must always follow a numerical pattern. Cycles of '1-2-3-4', '1-1-1-2-2-3', '1-2-3-4-3-2-1', or '2-4-2' are all valid mixed cycles, while a random pattern such as '1-5-3-7-2-6' is invalid.

While the vast majority of interpolations in my music will be of the linear type described above, I will also, on occasion, make use of a circular type of interpolation. These circular types are the Neighbor Note, the Double Neighbor Note, and the Turn.

While these interpolations are similar to their common practice equivalents there are two main differences in my usage.

The first difference is that the same interval must be used throughout the interpolation. This differs from common practice where the intervals which appear for double neighbors and turns are typically defined by the current key. In my usage, both neighbors must be either minor seconds or major seconds.

The second difference is that in my usage incomplete neighbors (such as echappees) are not allowed. The primary pitch must always appear at the beginning and end of the interpolation to confirm/re-enforce the serial pitch.

The purpose of an interpolation is to (in Schenkerian terminology) prolong a given row pitch. This results in the creation of a new prolongational level. Additional prolongational levels can be created by inserting an interpolation into an interpolation.

Continuing in the Schenkerian vein, the row can be viewed as the background level. An interpolation based upon a row pitch (which is a background pitch) creates a new prolongational level, the first middleground level. If an interpolation appears prolonging a pitch in the first middleground level this creates another prolongational level, the second middleground level. This prolongational approach, that is, the adding of additional content to the piece by prolonging pitches of the next lower prolongational level, continues until you arrive at the surface of the music (the foreground level) which is what you hear when the music is performed.

A real world example of this can be seen in the first fourteen bars of my posted composition, Meditation #2. These fourteen bars present a single tone row. How can this be true since, obviously, there's a lot more than twelve notes in these bars? It was accomplished via interpolation.

On beat 1 of bar 1, the first pitch of a given row (which is a background pitch) appears. Then from beat 1.3 through beat 2.1 this pitch is prolonged via a wholetone interval cycle: A (beat 1.1), G (beat 1.3), F (beat 1.4), and Eb (beat 2.1). This is the first middleground level. Each of the pitches in the first middleground level are also prolonged creating another prolongational level which in this case is the foreground level. The A is prolonged (a second time) by a mixed interval cycle, in this case a cycle with the pattern 1-2-3. The G is prolonged by a mixed interval cycle also with a pattern of 1-2-3. The F is prolonged (actually delayed) by a mixed cycle of 2-4-2. The Eb is prolonged by a mixed cycle of 1-2-3-4-5-7. Only on the upbeat of beat 4 in bar 2 does the second pitch of the row (D) appear. In bar 4 the third pitch of the row (F#) appears and is prolonged through bar 5 by a mixed cycle of 1-2-3. In bar 6 the fourth pitch of the row (B) appears and is also prolonged by a mixed cycle of 1-2-3. The last pitch of bar 6 (E) is the fifth pitch of the row. The music continues in this manner until the last pitch of the row (F) appears on the downbeat of bar 14. This pitch is then prolonged via a mixed cycle of 1-2-3-4-3-2-1. The F appears again (viewed as an echo tone, a technique often used by Berg) which is itself prolonged via a wholetone cycle leading to the pitch on the downbeat of bar 15 (B) which is the first pitch of a different row.

As another example, later on in the piece there is an entire discrete passage (bars 72 through 79) where not a single row pitch appears. This is because the entire passage encompasses a wholetone cycle prolongation of the G in bar 71 which is the last tone of a given row. The cycle is G (beat 71.2), F (beat 72.1), Eb (beat 72.4), C# (beat 73.2), B (beat 74.1), A (beat 75.1), G (beat 76.1), F (beat 77.1), and D# (beat 79.1). Each of these pitches are prolonged via additional interpolations of various types. It should be noted that while all of the other pitches in the cycle are prolonged, the C# is both delayed and prolonged. It is delayed via a mixed cycle interpolation and prolonged via a semitone cycle interpolation. The prolongation of the D# (via a wholetone cycle) leads to the G# on beat 1 of bar 79 which is the first pitch of the next row to be presented.

Depending on how you handle it you can generate quite a bit of content from a single interpolation. A simple example can be seen in bars 16 and 17. There is a single mixed cycle interpolation through bars 16 and 17 which is prolonging the B on beat 1 of bar 16 (which is a row tone). However, it feels like there are two events, one in bar 16 and one in bar 17, because the D# on the upbeat of beat 16.4 jumps up an octave and seems to be the starting point of another diminution, but, in reality, this is just a continuation of the same mixed interval cycle which began at the B in bar 16.

The possibilities for composition of this technique are boundless.
Title: Re: Expanded Serialism
Post by: whitebark on February 09, 2019, 10:07:32 AM
Thanks for posting this detailed description of the theory behind your musical compositions!

Title: Re: Expanded Serialism
Post by: RJB54 on February 09, 2019, 01:22:12 PM
Jay, thanks.
Title: Re: Expanded Serialism
Post by: RJB54 on February 10, 2019, 11:26:39 AM
Contiguous Segmentation

The next topic to be discussed is Contiguous Segmentation.

This is another technique used by Berg in Lulu. This technique involves treating the pitches of a given row as if the row is a trope instead of a row.

Trope theory was defined by Josef Hauer as a way of controlling and organizing the writing of twelve tone music.

Briefly, Hauer's approach was to organize the twelve pitches based upon pitch segmentation which differs from Schoenberg's approach where organization was based on pitch order. In trope theory, the twelve tones would be separated into segments (pitch groups). Each group of segments should be unique across all tropes (except for transpositional equivalence of course). Since in trope theory pitch order doesn't matter, the pitches of a given segment can appear in any order and can be repeated any number times so long as the pitches of that segment are presented as a clearly identifiable group of notes.

What Berg did in Lulu was to take a given tone row and temporarily treat it as if it were a trope by segmenting contiguous pitches of the row and treating them in the manner of a trope rather than a row.

For example, you can segment the twelve pitches of a given row into three four-note groups (segments) and then treat the pitches in each segment freely, that is, present the four pitches in any order, repeat them as many times as desired, etc. Of course, each of the three segments must be presented in proper sequence. It is not acceptable to extract one segment from the row and only present that segment. All segments must be presented in relatively close proximity.

All of my rows are always conceptually segmentable into the following segments with each segment labeled by a lowercase letter:

a: the first six row pitches.
b: the last six row pitches.
c: the first four row pitches.
d: the middle four row pitches.
e: the last four row pitches.
f: the first three row pitches.
g: the second three row pitches.
h: the third three row pitches.
i: the last three row pitches.

In addition to the nine standard segments which I apply to all rows (segments a through i). Individual rows can have additional row-specific segments identified as segments j through z.

One of my rows can be segmented into segments j and k, with segment j being the first eight row pitches and k being the last four row pitches.

Another of my rows can also be segmented into segments j and k, with segment j being the first seven row pitches and segment k being the last five row pitches.

Another of my rows can be segmented into segments j, k, and l, with segment j being the first two row pitches, segment k being the next five row pitches, and segment l being the last five row pitches.

An example of the use of this technique can be found in bars 83 and 84 of my posted composition, Meditation #2. A given row has been divided into three four-note segments (segments c, d, and e) and the pitches of each of these segments are then presented freely (within each segment) to achieve the arpeggiated figuration seen in those bars. Another example can be found in bars 88 - 90. In this case the row in question has been divided into segments j, k, and l as described above and the segments are presented in retrograde order with Segment l in bar 88, Segment k in bar 89, and Segment j in bar 90.

I have deliberately constructed rows with an eye towards the possibilities of various row segments to provide various useful entities. Some examples are:
(a) One row has been constructed so that the first four pitches present (depending on how they are voiced) either a minor seventh chord or a major chord with an added sixth.
(b) This same row is also constructed in such a way that, when rotated two notes, the first six pitches of the rotated row are the pitches for one wholetone scale, while the last six pitches present the other wholetone scale.
(c) When segmented into three note segments, that same row presents a major triad in the first three note segment while the third segment presents a diminished triad. When inverting that row, the first three note segment would present a minor triad while the third segment would also present a diminished triad.
(d) Another row was constructed so that the first seven pitches present a diatonic scale while the last five pitches present the complementary pentatonic scale (or a quartal chord).
(e)When that same row is segmented into three note segments, the first three note segment of the row presents a major triad while the second and fourth segments present minor triads. When that row is inverted, the first segment presents a minor triad while segments two and four present major triads.
(f) Another row was constructed so that its first eight pitches present a given octatonic scale while the last four pitches present a diminished seventh chord.

Row segments don't, of course, have to refer to diatonic/tonal configurations. A segment containing an atonal pitch set could provide pitches which would be just as useful in a given situation as a tonal pitch set.

A variant of this troping technique which Berg used was the reverse process, that is, by treating a trope in a serial manner. In Lulu Berg associates tropes with certain characters rather than tone rows. For example, the character Schigolch has a trope associated with him rather than a row. George Perle has named this trope Schigolch's Serial Trope. This trope is segmented into three four note segments with each segment being comprised of, ordered, contiguous, pitches of the semitonal scale. Berg often treats these segments in the free manner typical of a trope; however, it is a serial trope because at times Berg treats each four note segment of the trope as if they were four tone, mini, tone rows. Each four note segment has a prime order and they have serial aspect transformation processes applied to them such as inversion, retrograde, and retrograde inversion independent of what is being done to/with the other segments at the moment.

An extension of this serial trope concept would be to apply it to segments extracted from a given tone row rather than segments of a trope. In other words, take a tone row and segment the contents as described above. Then treat the contents of a row segment as a mini tone row, as Berg did to some trope segments, and apply serial aspect transformations to a given segment independent of the other row segments.

For example, posit a row segment of C, C#, Eb, D. This would be the prime form. One can than present the segment inverted (C, B, G#, A), or retrograde (D, Eb, Db, C), or retrograde inversion (C, Db, Eb, D). In this manner you get additional pitch groups which could perhaps be useful for the music at that moment.
Title: Re: Expanded Serialism
Post by: whitebark on February 11, 2019, 06:52:12 PM
In the words of the immortal Monte Python, "My Brain Hurts!".   

Seriously, this is interesting stuff to a budding composer to read, although I haven't attempted to explore tone rows in my music.  Thanks for posting it.

Title: Re: Expanded Serialism
Post by: RJB54 on February 12, 2019, 06:20:31 AM
Again, Jay, thanks.
Title: Re: Expanded Serialism
Post by: sandalwood on February 12, 2019, 12:43:29 PM
Thanks for studiously putting together this map of your compositional technique. I believe this is a serious and valuable read for those who don't take an interest in any shade of serialism as well as  who do.
Title: Re: Expanded Serialism
Post by: RJB54 on February 12, 2019, 03:28:56 PM
You're welcome. I appreciate that you appreciate it. :)
Title: Re: Expanded Serialism
Post by: RJB54 on February 14, 2019, 01:03:00 PM
Non-Contiguous Segmentation

The next topic to be discussed is Non-Contiguous Segmentation.

This is a technique which Berg made frequent use of and can either be simple or quite complicated depending on how it gets used.

In this technique you divide the pitches of the row between multiple instruments in order to create a desired melodic shape using pitches which are not continguous in the row. For example, posit that you could create a desirable melody by extracting pitches 0, 3, 5, 6, 8, 10, and 11 of the row. Then you would present the remaining pitches (1, 2, 4, 7, and 9) as accompaniment (counter-melody, background chord, or whatever). An example of this can be seen at the very beginning Berg's Violin Concerto where the solo violin presents the extracted pitches of the row which happen to coincide with the open strings of the violin with other instruments presenting the rest. Actually, what's going on is a little more complicated than that, but that analysis is outside the scope of this topic.

One aspect of Non-Contiguous Segmentation which differs from Contiguous Segmentation is that while the segments created via Contiguous Segmentation can, if desired, be treated freely as in a trope, the segments created via Non-Contiguous Segmentation cannot. This is due to the fact that they occur by the musical treatment of the pitches of the row with some non-contiguous pitches being treated as a main melody (for example) while others might be treated as an accompanying chord, a counter-melody, etc. This means that the pitches being presented must always appear in proper row order.

Use of this technique can get pretty complicated making it somewhat difficult to ensure row integrity depending on how elaborate the segmentation gets, especially if multiple rows are involved. Some passages in Lulu are extremely difficult to analyze since they are a mare's nest of extracted pitches scattered all over the place between instruments as Berg torturously attempts to create the melodies, counter-melodies, and chords that he wants while attempting to stay within the bounds of the serial framework (that is, without reverting back to free atonality).
Title: Re: Expanded Serialism
Post by: RJB54 on February 14, 2019, 01:18:20 PM

The next topic to be discussed is Sequence.

In diatonic music a sequence is a group of notes which are initially part of the currently prevailing key and are then repeated and transposed one or more times. Each iteration of the sequence results in a group of notes which are not necessarily part of the original key and/or are not necessarily participating in a 'proper' modulation through, or into, another key. The pitches are just repeated at various transpositional levels.

In my expanded serial environment, a similar concept applies. A group of row pitches can be repeated at various transpositional levels independent of the row they were originally part of, or of any other row for that matter. This means that as the group of pitches moves through the various transpositional levels, the remaining row pitches for a given transposition level (that is, those row pitches which are not participating in the sequence) do not need to appear. The pitches participating in the sequence are viewed, temporarily, as a group of pitches which are independent of any row.

Other expansion entities (such as interpolations, etc.) could, if desired, participate in the sequence. For example, if a row pitch has a wholetone cycle associated with it, that wholetone cycle could, and in my practice almost always would, be transposed through the sequence along with the row pitch.

Of course, this technique breaks a number of Schoenberg's rules, but I won't tell if you don't  :).
Title: Re: Expanded Serialism
Post by: RJB54 on August 27, 2019, 10:00:03 AM
I have updated the Interpolation topic to include a description of circular interpolations.
Title: Re: Expanded Serialism
Post by: RJB54 on June 24, 2020, 06:18:03 PM
I have updated the Non-Contiguous Segmentation topic to clarify the manner in which the pitches which result from the segmentation can be used.
Title: Re: Expanded Serialism
Post by: RJB54 on June 24, 2020, 06:21:21 PM
Multi-Directional Segmentation

The next topic to be discussed is Multi-Directional Segmentation.

Simply stated, Multi-Directional Segmentation means the row is divided into two contiguous segments with one segment starting at the first pitch of the row and moving forwards while the other segment starts at the last pitch of the row and moves backwards with the two meeting somewhere in the middle of the row.

For example, one segment contains the pitches at ordinal positions 0 through 7 while the other segment contains pitches 8 through 11. These two segments would be presented more or less simultaneously with one presentation moving from ordinal 0 to 7 and the other moving from ordinal 11 to 8.

The opposite can also occur with the two segments starting in the middle with one segment going backwards to ordinal position 0, while the other goes forwards to ordinal position 11.

Either of these processes can also be being applied to a rotated row.

For example, rotate a row by 4 which results in a row of 4,5,6,7,8,9,10,11,0,1,2,3.

In this case, the first segment could present ordinal positions 4 through 9 while the second segment would present ordinal position 3 through 0 then 11 through 10.
Title: Re: Expanded Serialism
Post by: RJB54 on June 24, 2020, 06:35:52 PM
Pitch-Set Transformation

The next topic to discuss is Pitch-Set Transformation.

This topic starts as a follow-on from the Contiguous and Non-Contiguous Segmentation topics. Once a segment has been extracted from the row using either of these segmentation techniques, the resultant segment can be manipulated using pitch-set transformations.

The general idea is that (a) you extract a segment (pitch-set) from a row, then (b) apply a pitch-set transformation to that segment (pitch-set) transforming it into a different pitch-set, then, optionally, (c) apply another transformation to the second pitch-set to create a third pitch-set, etc. This allows for the creation of more content.

In my usage, Pitch-Set Transformations will typically be used in a manner analogous to Sequence (described above) in that it provides a method of generating content based on only some of the pitches of a given row independently of the remaining pitches of that row. The main difference between a Sequence and a Pitch-Set Transformation is that a Sequence will repeat the same relative pitches in the same order on differing transpositional levels, while a Pitch-Set Transformation will modify the pitches using the various transformational processes.

For my purposes I will not make use of all of the possibilities of pitch-set transformations. This is primarily due to the fact that making use of the full possibilities of this technique will take one into the realm of free atonality. For my purposes, I do not wish to go that far.
The pitch-set transformations which I will make use of are:

The complementation transformation involves transforming a pitch-set into its literal complement.

The inversion transformation involves transforming a pitch-set into its literal inversion.

The pitch transformation involves modifying one of more pitches of pitch-set-1 resulting in pitch-set-2.

The subset transformation involves extracting some of the pitches from pitch-set-1 resulting in pitch-set-2 which has a smaller cardinality than pitch-set-1.

The superset transformation involves adding one or more pitchs to pitch-set-1 resulting in pitch-set-2 which has a larger cardinality than pitch-set-1.

The vector transformation involves transforming pitch-set-1 into pitch-set-2 which will be a pitch-set whose interval vector is similar to that of pitch-set-1. In my usage there are three levels of vector relationships. From strongest to weakest they are: (1) when the two vectors are exactly the same (Forte called this the Z relationship), for example, the vectors for both sets are [013202], (2) when there is only one vector different between the two vectors, for example, [124310] and [114310], or (3) when the difference is that two of the vector values have been switched, for example, [231041] and [131042].
Title: Re: Expanded Serialism
Post by: RJB54 on September 12, 2020, 09:34:35 AM

This topic is a fairly simple one dealing with the idea of two or more row pitches being doubled by non-row pitches. This technique should be applied to two or more row pitches because applying it to a single pitch is inherently ambiguous.

Conceptually, the doubling should relate to the row pitches at either a single interval or at different intervals following a given intervallic pattern.

Doubling at a single interval is simple. The row pitches are all doubled by pitches the same interval away (third, sixth, second, etc.). For example, doubling at interval 3 means that the row pitches being doubled are all doubled at the minor third.

Doubling by intervallic pattern means each row pitch is doubled by pitches whose difference changes from row pitch to row pitch by a given interval pattern similar to that of Linear Interval Cycles. For example, for a single interval cycle of 2, the first row pitch (C) is doubled at interval 2 (Bb), the second row pitch (Eb) is doubled at an interval of 4 (B), the third row pitch (F) would be doubled at an interval of 6 (B), etc. For an interval cycle using a mixed interval pattern (say a cycle of 1-2-3)  the first row pitch (C) is doubled at an interval of 1 (B), the second row pitch (Eb) is doubled at an interval of 2 (Db), the third row pitch (F) is doubled at an interval of 3 (D), etc.