Constructing the 12 tone row: Discussions with Mark Gould

Journal entries by composer and pianist Laurie Conrad

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Constructing the 12 tone row: Discussions with Mark Gould

Post by figaro » Wed Jun 20, 2007 9:32 pm

Constructing the 12 tone row: Discussions with Mark Gould: A Composer’s Journal Entries: May 22- 25, 2007


Tuesday, May 22

Received another e-mail from the composer/theorist Mark Gould today:

Hello Laurie!

amid my compositions I am still writing about my personal experience
of 12-note composing and music....

but perhaps we could start a new thread on how we put our rows
together in the first place - the origin of the row.


I quickly wrote him back: How we put the row together, the origin of the row - great topic - you start. I’ll try to add on.

Thursday, May 24

Found a new e-mail from Mark Gould in my in-box today. I have inserted my comments or questions to him, to further a dialogue between us:

Mark: Hello Laurie!

I thought I would put this little item forward whilst I contemplate the complexities of the Piano Trio - whose row has changed, and so in part contributed to my original idea, and maybe helps me understand what I am doing with 'rows' when composing.

Composing the Row
When I first began to write twelve note music, I believed firmly in the ideas that I had read - namely that the row is a 'thematic' shape, a 'melody' in effect. Therefore, the row seems to have little to do with harmony - only the implied harmonies of the melodic successions.

LC: Where did you read that the row is a 'thematic' shape, a 'melody', dear Mark? In my experience the row can be a melody - or a series of harmonies, or a combination of melody and harmony.

Mark: This I eventually developed into unique technique I have spoken of - dividing the row not into groups of adjacent notes but every second, third, fourth or sixth note.

These groups were my 'harmony', and I always spent many hours taking my melody and dissecting it to form the characteristic harmonic components. I quickly discovered that I was forming my rows from similar harmonies. Eventually, I realized that it was the harmonies that were deciding my rows - the result being that the melodic material was flowing from the harmonic connections. In other words my music was 'about' the chords - the row was merely a linearisation of harmony.

LC: Hmm ... I am not sure that I can separate harmonic movement and melodic movement in that way. Aren’t they intertwined? Unless we are speaking of a single, isolated tone - harmonies, no matter how approximate, will always be implied. But yes, I also see what you mean ... Have you found that over the years that your basic harmonic progressions or chords have changed?

Mark: However, my encounter with another book, Haimo's 'Schoenberg's Serial Odyssey' showed me other techniques - namely taking one row and pulling out a few notes that were the same as those from another row - that I had not previously contemplated. It seemed that there was a different approach to harmony - one based on adjacent set segments rather than my divided rows.

LC: “ - namely taking one row and pulling out a few notes that were the same as those from another row -“ Could you explain this further, Mark? I am not entirely sure of your meaning.

Mark: Compositionally they are almost incompatible - and for a time I wrote music where the row was used one way or the other exclusively.
Now, row conception is a complex and fraught process. I still divide the set into my divisions, my 'cycles', but primarily, I write a melodic line, a sensation of the character of the new work, a 'quip' or even a longer line. Harmony is still absent, but now I look at using non-adjacent notes as melody, checking and re-checking.

LC: Are you saying that now harmonic and melodic notes alternate in the construction of your row? Or that now you are considering only the melodic aspect of the row ...

Mark: One interesting outcome of this self-analysis is the 'problem of nine', as I call it. Often, when just idling on the concept of a new work, I will begin a melody, often on a blank sheet of paper - just a pair of lines, the lower being E and the upper being F of the edges of a treble stave. I write out the melody, rhythms and all, I write quickly, but then I stop, uncertain, and think - what notes are left from the chromatic? At that point I count up and discover once again that I have got to nine notes. The three remaining notes I write out in letter names. I decide that they don't form a nice end to the melody - maybe they are the harmony - so I insert them at the possible locations, evenly spread between the notes of the melody - a theme over a triad. Or sometimes they get split up - one or two notes become an anacrucis or similar.

LC: Interesting. I also sometimes ended up with nine notes and three left over. This was when I constructed the row melodically. To be honest: on one occasion I just left the last three notes out entirely, i.e. used a nine note row. This philosophically And musically never felt right to me, and frankly I still sometimes think about those three left out tones ...

Mark: So - the row is written out. Then comes the divisions. At this point, I often discover that several of the chords are defective - I like my harmonies to be 'even' in the sense that the dissonance level amongst all the three note chords is relatively similar, and so for the four note chords (though the dissonance level can be different) - it is good to find one set are more consonant than the other so that contrasts of harmonic tension can be drawn simply by changing the handling of the divisions.

LC: I like this idea of keeping the dissonance level of the chords relatively similar. However, as you say, it can also be helpful to establish an imbalance of dissonance within the row: then there is the possibility of a lessening (or increasing) of tension when using the retrogrades.

Mark: I may swap notes at this point, to remove a defective chord, but always checking the row has not diverged too widely from its original form. Only now will I check to see if the row is combinatorial - that the row and an inversion can be combined to form twelve notes when divided into groups of six notes.

Sometimes I find that the row is not combinatorial but the divisions into two groups of six alternate notes is. This I find interesting as it permits me to use the groups as a secondary series - for example in a slow movement or a subsidiary idea.

LC: I like this idea as well. It would lend cohesion. And I agree: any technique that use notes from another row or changes the intervals of the row in any way is a new row. We could say the rows were related, even parental or siblings - but as soon as the intervalic relationships between tones change in the row: by definition, it is a new row. So it would depend on many factors, this use of similar rows for a new piece or different movements of a larger piece. One of the advantages of twelve tone music is that a new row can be entirely different than the last one used, with entirely different and almost unlimited potential.

Mark: But I am not finished with the row yet! I then write a number of sketches utilizing the different divisions, or other ways of dividing the row. I spend a good deal of time just playing the row and its inversion through - spotting the patterns that may form the basis for extended paragraphs of music - certain chords held constant or certain transformations - sequential material. In this way I form my 'networks', whereby I can link together rows so that it is possible to connect different transformations by the common patterns.

LC: Again, would lend cohesion.

Mark: Sometimes it is that I find the row is too static - too clinical - it contains too many resonances.

So I divert myself with three variant rows:

Pull the row into a circle by joining the first note to the last, and then take every seventh note. This leads to a new row.

I Write the numbers 0 to 11 over the notes of the original row. I then sort the notes into ascending order of the chromatic scale on a lower stave, but carry these numbers to the lower stave as well. The new order of numbers is now represented as pitches, setting C being equal to 0. This set can be quite interesting to derive and can be very revealing.

LC: I am intrigued: revealing? How?

Mark: Finally, there is another set derivable. Starting with the numbered original row, on another stave I write out the chromatic scale starting on one of the notes, and number this from 0 to 11. Next, I look at the first note of the original row. Maybe we started our chromatic scale on C and the first note of the original row is G. In that case we look up G in the chromatic scale, and note the number above it - in this case 7. We then look for note number 7 in the original row, and write it down on a fresh stave. Now move to the next note in the row, and do the same again, repeating until once again you have twelve notes.

As an Example, these variants I have written out for the row of Schoenberg's Fourth String Quartet, Op.37

D C-sharp A B-flat F E-flat E C A-flat G F-sharp B

1. Every seventh note:
0 7 2 9 4 11 6 1 8 3 10 5 (0
D C A G F B E C-sharp G-sharp A-sharp F-sharp E-flat

2. Rearrange the notes into chromatic scale order and interpret the jumbled order numbers as pitches ( C == 0)
7 1 0 5 6 4 10 9 8 2 3 11
G C-sharp C F F-sharp E B-flat A G-sharp D E-flat B

3. Interpret each note as an order number, again here C == 0
2 1 9 10 5 3 4 0 8 7 6 11
A C-sharp G F-sharp E-flat B-flat F D A-flat C E B

LC: Thank you again for your clear examples.

Mark: As you can see these are very different to the original row of Schoenberg's.

LC: Different, yes. But in some ways very similar, especially harmonically. An interesting concept, dear Mark - and very creative. Harmonically, in the sense that we can still find the outline of 7th and 9th chords etc. within the new row. Which, as you well know - is not always the case.

Mark: This might seem mere alchemy with numbers and nothing to do with music. True to some degree, but rather these variants I find interesting in that they can spark a more interesting row than the one that generated it. The result can be a row usable for a new work - and because the technique is very abstract, there is little way of knowing what these variants are and what kind of row will turn up. I like to think of this as an almost random process - a forcing of the mind away from conscious conception to an intuitive one.

LC: Well said and well done.

Mark: And there is one last way I have uncovered a row. Once, as a - I came across, in a toy shop, dice with twelve sides. The Dodecahedron. This die is used in a type of adventure fantasy game, to determine the outcome of choices made by the players. I bought a few of them and have on occasion written aleatoric music, by throwing the die to determine the next pitch, C here is the number 12. This again I use to build a row - choosing only the next number rolled if it has not been rolled before.

Crazy though this might seem, I have had some remarkable outcomes - the row for my 13th Quartet was devised in this way (with minor polishing afterwards). And this work is commented upon for its successful incorporation of tonal components into the twelve note context.

LC: The subconscious helped you.

Mark: For a time in my late - I thought I could devise all of my work from one row - producing variants for each new work - that in some way every row I use can be traced back to a single thematic conception - one that would contain the essence of my own creative being - compositional DNA so to speak.

LC: An interesting concept.

Mark: However, I could not consciously arrive at new rows that were related, only create a new row and then see how it related to the ur-series. I abandoned this in my twenties, as I had begun to write much more complex and dissonant music that required rows not containing any tonal elements.

Of course, in my thirties, I developed my interest in microtonal music, which had been a hobby within a hobby so to speak. In 2001 I published an article which attempted to derive analogies to the diatonic scale but operating in a microtonal context. It is this work that (along with my long time study of Bartóók) led me to believe that twelve notes are inherently imbued with tonality. I rapidly altered my approach to twelve note music - and now attempt to contain tonal elements in my rows.

LC: When you say “twelve notes”, I assume you mean the twelve chromatic tones within an octave? How did microtonality lead you to this conclusion, Mark? I could see your conclusion in the sense that there are physical laws that govern the overtone series. And quarter tones, and other microtones, would set up an adjacent harmonic overtone series ... (Hmm ... is “series” both the singular and the plural? In any case, I mean plural.) We could say that each isolated tone has its own place in the harmonic system set up by the fundamental tone. However, we could also say that each isolated tone becomes its own harmonic system, carries within it its own harmonic system - which was more Schonberg’s view. In any case, please say more about this if you would.

As you know, my rows have also become more tonal as I grow older. I have often thought that J.S. Bach would have liked and perhaps used the twelve tone technique - in some ways it is a throwback to the compositional limitations and musical conundrums of the Renaissance. And not totally unlike the constraints of his fugal forms and the consequent limitations imposed on him as a composer. The joy (and challenge) facing the composer when working within such a strict, even inflexible form is figuring out how to use the technique, finding a way to break through or use the form to write beautiful or profound, meaningful music. However, in Bach’s case - his mission seems to have been to establish tonality, the tonal system.

Mark: I am 38 now. I am not sure what form my music will take next. I am discovering again that tonality often dominates my twelve note work - triads and seventh chords abound, freely juxtaposed in the context of the row. Textures are scoured for ugly dissonances and I find myself writing smaller intervals - melodic lines stepwise or constrained in shape. It as if I have become tired of jaggedness and sharpness of harmony, and am choosing softer curves and rounded chords. My Wind Quartet of 2005 is typical - almost like Brahms in places, but in some way seen through modern eyes.

LC: As you know, my story is similar, although I more tend towards French Impressionism. When I first began writing 12 tone music, my rows contained many half step intervals, which naturally often led to jagged dissonance. However, I found that over time, the ear began to hear the dissonant intervals, for example the 7ths and 9ths, as consonant. That sort of honest twelve tone music - if delicately written - is like a rare wine, an acquired taste. And a rarified air to breathe.

Mark: But I wander from my topic - the row and its origin - now I find myself appraising the row for its subtleties and its melodic concision, forming as I said, stepwise themes. Harmonically, I haven't diverged much from my original techniques - and still stare at the row to find the inner elements.

LC: That is the joy of writing twelve tone music, isn’t it ... Finding the endless patterns and possibilities, melodic and harmonic, hidden within the row. When working with a new row, I always feel like a child let loose in a toy store or candy shop. And I will usually continue to use the same row, often for many pieces - until I feel I have explored it sufficiently. Sometimes I will go back to them later, to make sure I have exhausted my ideas.

Mark: But one thing is certain - a row has never come to me complete, finished in all its details - I have had to drag it from the creative fires, note by note, and then hammer it into shape. Often, when I am at last happy with the row I almost feel that my work is done - that I have fashioned the row like a sword, and would rather appreciated it for what it is than sully it by using it in composing.

LC: What a wonderful image. However, for me - the row is used more like a new mode, a new scale, even though it is not constructed stepwise. And in my case, the row is very much only the means to a greater end - i.e. writing the music. But I do agree: the row is everything. All depends on it, its construction. The set order of notes demands that they are very carefully chosen.

Mark: But, to continue my metaphor - but in applying the sword to the toil of writing music, I find it has become a ploughshare, bent, beaten and sorry looking.

LC: Actually, the way you describe the finished, decided-upon row you have constructed: you should probably have it inscribed on a T shirt. I would buy one.
There, that's about it for the row. Sometimes free, sometimes obscure! And far too long - words words words. I sometimes wish I could just say - String Quartet in F-minor or some such thing. But a row does not condescend to simple labels. I did once think of enumerating all the different unique rows (not transpositions or inversions or retrogrades etc of each other), but I could not determine the how, and then wondered why, so I gave up.

LC: There would be too many of them. The number of possible rows is almost limitless: 12 taken to the 12th power. However, to analyze a finite number of rows would be fascinating, and useful to theorists, composers - and interpreters.

Mark: My piano trio is at last in motion, but the Piano sonata is still stuck on the starting blocks - it lacks a row, and no matter how I bash the materials I cannot get a row out of the music. So I may just bin the material and think of something else...


LC: Personally, I would never even consider extracting a row from a passage of music already written. In my view, one must start with the row. Those carefully chosen 12 tones, in their set order, can outline the melodies and harmonies one inwardly hears and wishes to express musically - or at least can make them possible. But to try to construct a row from an already existing passage of written music - seems like an impossible and futile task. Working in this way would also, in my opinion, remove most of the joy and exhilaration of the element of discovery as a composer. Just as composers throughout history worked within the framework of an existing mode or scale have found that same pure joy and inspiration of discovery and possibility within a set framework.

I say these words in deep respect for you and your fine method of constructing a row - and for other composers who might draw the row from initial melodic and harmonic ideas, from already written musical passages. My viewpoint is merely my opinion, and hopefully will help bring about more thought and probing to this discussion.

Thank you Mark, for all your effort and deep thought and writing on this subject of constructing a row - and for your clarity and detail. I find these discussions most worthwhile and interesting - and sincerely hope they continue.

Today it is the perfect temperature, in the low seventies F. Later we will go to Windgarth, our house on the lake. I will think of you. Best wishes dear Mark.


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