MAPPING MUSIC

Category: counterpoint

  • MapLab 7. Twelve-Tone Trichords in a Ternary Trio

    Schoenberg’s famous twelve-tone row technique, devised in the early 20th century, prescribed a compositional method to order choice of pitches in the dense forest of possibilities with 12 chromatic scale steps. Music composed in this manner was called “atonal,” but this Lab will create a distinctive individual tonality from a 12-tone series of four trichords.

    The sample piece, in a classic ternary form, will be scored for a chamber trio of three different instruments.

    The so-called atonal music was complex and mostly dark and dissonant. Can a technique like this also be used to methodically create coherent tonal characters of brighter, simpler sonority?

    1. Choose a model

    One of my favorite 12-tone pieces is Schoenberg’s Wind Quintet, Op. 26. Premiered in 1924 on Schoenberg’s fiftieth birthday, it was dedicated to Schoenberg’s young grandson Arnold. In four movements, it has a graceful neo-classical Viennese quality. It also is historically only the second piece ever (after his Suite for Piano, Op. 25) composed with Schoenberg’s new 12-tone row technique.

    The slow third movement, tempo marked “Etwas langsam,” displays a clear use of a 12-tone row in graceful counterpoint between the five instruments. The facile scoring of these instruments explores textures ranging from isolated solo to duos, trios, and even dense five-part counterpoint.

    2. Design a 12-tone series

    I imagine the typical way composers design a 12-tone row is by writing a line, choosing the interval from one pitch class to the next while not reusing any pitch class, until all 12 are included. That leads readily to a linear, contrapuntal approach.

    For our example in this lab, we’ll work instead with trichords, each a group of three pitch classes making a three-note array and set of three intervals. Here is such a row (A) and a slight variation of it (B):

    The A version also shows the three intervals in each trichord.

    Note that the 2 5 and 5 2 scale patterns are the same, representing a set that is inversionally symmetrical. (That is not true of 4 1 and its unique inversion, 1 4.) Its complete-octave array is 2 5 5. Think of it as a circle of the twelve semitones in an octave. Unwinding the circle through octaves looks like: 2 5 5 2 5 5 2 5 5 2 . . . etc. This is a palindrome, reading backwards the same as forwards.

    Here is a voicing into 3-voice chords:

    You can see (and hear) that the voiced-out trichords can make constellations predominantly of 5-semitone and 7-semitone intervals. This is what makes the sound of the progression coherent and somewhat more like Copland than Schoenberg in character.

    3. Twelve-tone rows

    Now we can delve into the linear potential of the series we’ve constructed. I’ll call my principal row P in keeping with standard theory, designating the transposition by the name of the starting pitch class. (Standard theory uses pitch class numbers — C = 0 — to designate the starting pitch.)

    Serial technique is just a box of 12-tone tools, which you can use in any way you wish. For the last row above, I shuffled the order of the trichords. The shuffle puts the trichords inside out, with the 1 4 array now at the beginning. This gives me more options when building counterpoint between two lines with each its own row form. Like rearranging four toy blocks, the shuffling still builds a 12-tone aggregate collection no matter the order of trichords or pitches within the trichords.

    4. Imitative lines

    As we observed earlier in Opus 26, a row is a natural material for building contrapuntal lines that are similar in interval shape. When these similar shapes are reinforced with similar or identical rhythmic patterns, the lines mimic each other in their back-and-forth conversation.

    5. Ternary form

    Of all the classic form models, ternary is one of the simplest in concept and strongest to perceive. Opening material is contrasted with a middle section of different tempo, texture, register, rhythmic pace, etc., followed by a return to the opening material: A B A. (Several familiar classic model forms are ternary, including Sonata Allegro, Da Capo Aria, and Minuet and Trio.) Often the returning opening material of the third section is embellished or varied in some way while not clouding recognition that it is a recurrence of the opening.

    For Schoenberg’s Etwas langsam third movement of the Wind Quintet, the opening material consists of two-part counterpoint between horn, marked as the Hauptstimme preeminent line, and bassoon (Fg) marked as the Nebenstimme subordinate line.

    Opus 26 middle section is faster flowing, with much more rapid, complex rhythms in a denser four-part counterpoint:

    The third section brings back but alters the opening’s two lines, moved from horn and bassoon to oboe and bassoon, with the addition of a third contrapuntal voice in the clarinet.

    For our MapLab example, the chorale-like chordal setting of the trichords we’ve already seen will be the serene opening and ending material. Imitative contrapuntal setting of the row in a faster tempo and much faster pace make the contrasting middle. (Many classic three-part forms, by the way, are the converse, fast – slow – fast.)

    Opening:

    The opening material continues with a variation of the chords, transposing them and adding rhythmic interest with single 8th-notes each arpeggiating a pitch of the chord.

    Contrasting middle:

    Note: if you’re wondering how I pulled the lower line of dotted-half-note steps from the row, I didn’t. It is free counterpoint for a simple supporting “bass line.” Those pitches are drawn from the trichords expressed above them, as unison or octave “doubling” of single pitches from the upper lines.

    The recap first brings back the opening’s single-8th-note arpeggiations then proceeds back to the still chords of the very beginning.

    6. Scoring the trio

    For this lab, choose any three different instruments. They can be in the same instrument family, like a brass trio of trumpet, horn, and trombone. Or it can mix instruments from different families, like flute, horn, and cello. You can guess from the audio above, I choose a woodwind trio of flute, oboe, and Bb clarinet.

    Get to know your chosen instruments, not only their overall range but also the particular characteristics of sub-registers. For orchestral string instruments, it is good to know their open string tunings and recognize those strings’ different qualities; the lowest string of each instrument is thicker and produces a thicker, grittier timbre. For my chosen three, flute and clarinet have wonderfully rich, dark lower registers, while the lowest few pitches of the oboe’s range are squawky. Each of the three instruments have upper ranges that thin a bit in timbre but can be elegant or powerful.

    My scoring will emphasize use of those rich low registers of the flute and clarinet. Many chords will be voiced with oboe on top, flute in the middle, and clarinet lowest. Nonetheless, the score will show them in standard score order with flute at the top. Also, the clarinet is a Bb transposing instrument. That means in the score and player’s part, a written C will sound one whole step lower as a concert Bb. When I want, say, an F to sound, the clarinet part will show a written pitch one whole step higher, G.

    For a well-balanced chamber-music trio should balance how much time each instrument gets as the active, preeminent voice in the texture (which Schoenberg would call the Hauptstimme).

    7. Finished score

    Tempo and dynamics are essential to vividly portray where the music is going — launching, growing, swelling, subsiding, approaching a cadence or climax. Articulation marks will be particular to each instrument type. Rehearsal letters or measure numbers will be helpful in rehearsal.

    Since my finished example has a childlike playful curiosity, its title follows Schoenberg’s Opus 26 dedication to his grandson. Imagine a child in a sunny garden.

    For Little Arnold

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    Continue reading Mapping the Music Universe . . .

    MapLab 8. A Small Sonata

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  • MapLab 4. Model a Metamorphosis

    As with MapLab 3, this will be multi-layer counterpoint utilizing canon in a homogenious texture. Now it will be entirely a repetitive ostinato texture — flowing, periodic rhythmic activity building a continuous texture of repeated arpeggios or melodic motives. Commonly called “minimalism,” its texture and overall rhythmic character are maximally dense.

    Multiple layers generate complex phase relationships between contrapuntal voices, with patterns of differing length repeating and changing at different times in the four layers.

    Layers of texture will change at different times to a new pattern, overlapping each other. Thus overall change of harmony unfolds gradually and continuously instead of at definite time points of harmonic rhythm, building a metamorphic form (instead of a traditional episodic sequence of chords, phrases, and sections).

    1. Choose a model

    The classic granddaddy of this whole genre is Terry Riley’s monumental 60-to-90-minute improvisatory piece, In C. My own 1984 homage to that classic, EFFULGENCE, models with Riley’s many innovative techniques.

    2. Select a source scale

    While any scale can work, those most commonly used are diatonic scales. In the TC example, we’ll go with the same as In C, a C-major/A-minor no-sharps-or-flats key signature. (We’ll see later, however, that a motive can be transposed into another diatonic scale and key signature.)

    3. Make motives

    First, design two or three motives, basic shapes of 3 to 7 pitches from the source scale.

    TC example:

    Motive R gets extended by the addition of two pitches, F and E. The last example shows motive T’s shape shifted to a different level of the diatonic scale (what Sibelius calls a diatonic transposition). A motive can also be truncated to as few as two notes:

    4. Plan a stream of motive variants

    Motive patterns can and should vary in length, especially when rhythmic values are mostly all 8th-notes, providing a changing landscape of rhythmic vitality. In the TC example, however, most patterns are 5 8th-notes long. Since 5 is a prime number, and set in a 3 4 meter, the overlaps of these 5-patterns in the competing lines fulfills that energetic complexity of rhythmic fabric.

    TC example

    For the pitch motives, a process of adding or abandoning pitches to make the next pattern creates the metamorphic unfolding process that is the true magic of this lab. In the TC example below, this add/abandon process is color coded:

    • GREEN for newly added pitches
    • BLUE for pitches appearing in a different octave than in the previous pattern
    • PURPLE for pitches that will appear next in a different octave
    • RED for pitches that will be abandoned in the next pattern

    You can see that by letter K the original C-major diatonic is modulating to a new diatonic, Bb major. These two keys have in common 5 pitch classes, and the patterns capitalize on the F, G, and C common tones to connect smoothly. (Riley’s In C also modulates, eventually adding F# and Bb in much the same way Bach inflects the C-major tonality toward the end of his famous C Major Prelude that launches Book I of the Well-Tempered Klavier.)

    Here is the lead voice of the ostinato canon:

    You can see that the number of repetitions of a pattern and the overall duration of its presence in the texture vary throughout. Patterns E, F, K, and P run for five full measures in the lead line alone (plus delayed answers in the whole texture), while the simple transitional pattern N runs for only five beats in the lead line.

    5. Spin the canonic counterpoint

    The time delays of canonic answer should be chosen not to match the length of the typical pattern. Otherwise, the answers would lock into fixed duplications of each other, making a rigid, uninteresting periodicity. Each new motive-pattern entry is highlighted below with a new dynamic marking. Here is a sample excerpt starting around pattern H:

    The answers all enter at unison or octave, with timings determined by a mostly trial-and-error method as follows:

    • PP – 9 beats later at unison, then 9 more beats down an octave
    • P – same
    • MF – almost same, but shortened last answer comes one 8th-note early
    • MP – (for a 3-8th-note pattern) 5 8th-notes later then 7 8th-notes after that
    • PP – top voice leads, answers are 2 beats later then 3 8th-notes after that

    This last is what we described in MapLab 3 as a stretto, answers coming in with very short time delay.

    6. Interrupt with an interlude

    As with In C, the ostinato texture can blast through from beginning to end in a continuous monolithic stream. Another form scheme, which I will invoke in the TC example, breaks the stream with an interrupting interlude before a coda to come. Of course, it’s another canon, a stretto of cascading downward dotted quarter-notes.

    7. Ending an ostinato stream

    Several considerations . . .

    First, since you’ve built a canon with staggered entrances, the last notes will be staggered as well. To make any kind of cadential closure, however, you’ll want to have them stop at the same time, right? That is accomplished simply by truncating the answering lines and/or adding repetitions of the final pattern in the lead voice.

    Think about the lead line and its answers leading to a point of harmonic stability and finality — somewhere that feels like tonic home base.

    More repetitions help slow and stop the harmonic momentum.

    In the TC example, an ostinato coda after interruption settles into and prolongs what will sound like a dominant chord in C major, then crash lands on a tonic C-major stinger.

    8. Title and listen

    The picturesque metaphor of a babbling creek made me reminisce about a favorite adventure on days off from working at the National Music Camp in Interlochen, Michigan back in the ’70s and early ’80s. We would canoe down the Platte River to its end flowing into Platte Bay on Lake Michigan. There was also a nearby spot where tiny Otter Creek trickled out onto a more secluded sandy Lake Michigan beach offering northward a spectacular view of Empire Bluff.

    Otter Creek

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    Continue reading Mapping the Music Universe . . .

    MapLab 5. Spin a solo

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  • MapLab 3. Construct a Canon

    Canon is a venerable, centuries-old compositional device, building counterpoint between a melodic line and one or more delayed and possibly transposed echoes of itself. Like a magic trick, it makes a strongly cohesive contrapuntal texture of rhythmically independent lines that are like clones of each other. Canon is more intense than a fugue, which formalizes the echo cloning technique, interspersed with free counterpoint.

    1. Study historical models

    There are many great models to study. Many 16th-century composers (notably Josquin and di Lasso) wrote canonic choral mass movements. Known more for his fugues, the great 18th-century contrapuntal master, Bach, also wrote several intriguing canons in his late work The Musical Offering. No more elegant model exists than the first movement of Anton Webern’s Symphonie Op. 21 (1928), in which four voices are spun out by successions of instruments each in turn differently coloring two to four notes of the same 12-tone line.

    Like every fine magic trick, there are several basic techniques we can learn to construct a canon. I’ll cover three, which I will call Zigzag technique, Trial-and-error technique, Rhythmic alternation, and Stretto echo.

    In 1610, Venetian composer Diruta wrote Il Transilvano analyzed Renaissance polyphonic style by codifying five species of rhythmic relationships between contrapuntal lines. Johann Joseph Fux, in his monumental 1725 pedagogy, Gradus ad Parnassum, explicated 16th-century counterpoint using these rhythmic species, of which the following are of special importance for us in this lab:

    • FIRST Species – note against note
    • FOURTH Species – lines alternating, seldom moving simultaneously
    • FIFTH Species – a mixture of rhythmic values in all lines

    2. Zigzag

    My name for it says it simply, like laying bricks one at a time but staggered to overlap.

    • Compose a few notes of the lead line. (In the example below, it is just three notes in two measures.)
    • ZIG: Establish a time delay. (in the example, one measure of two half-note beats). Duplicate the first notes (rhythm and melodic interval shape) in the following line, starting on a chosen pitch that makes the kind of vertical contrapuntal interval you desire to emphasize.
    • ZAG: Select new notes for the lead line that overlap with the ZIG notes, again making your desired vertical contrapuntal intervals. These ZAG notes need not match one-to-one the rhythms of the ZIG notes, providing the opportunity if desired to establish a Fifth-species rhythmic mixture.
    • The notes of this ZAG now ZIG into the following line, preserving the same transpositional level you established in the first ZIG.
    • Keep going as long as you wish or have stamina for. When ready to cadence, arrive at a longer note of stable pitch-sense in the lead line.

    The canonic material you just contructed can be reused transposed. Just be sure you transpose all lines together by the same transpositional interval.

    In the example below, my seven zigzag-composed measures are transposed down one semitone.

    Starting on Eb might be useful to follow the first statement of the material, which ended on D in the lead (lower) line. Or I could transpose the whole thing up 8 semitones to start on C, eliding with the middle C (bass clef) that ended the following line.

    Adding the third part enables this stair-step sequential transposition of the two-voice canon to go on and on . . .

    3. Trial and error

    Let’s try a different technique to add a canonic answer, one that is facilitated by notation software such as Finale or Sibelius. This way involves

    • copy the whole lead line, not just a head motive
    • choose a time delay or maintain one already established. Paste into the new answering voice the lead line
    • Playback the synthesized audio to test aurally for contrapuntal viability.
    • If it sounds bad, analyze the vertical intervals to discover why.
    • Make a strategic choice of a transposition of the pasted-in answer, then test it aurally.
    • Keep trying different transpositions until you find one you really like.

    For traditional diatonic tonal subjects, common transpositional choices are: unison; octave; Perfect 5th (7 semitones); Perfect 4th (5 semitones).

    In the following examples, I show in the first system a trial of an added third voice in the middle, starting on E (alto clef) transposed an octave up from the lead. For the second system, I tried adding a third voice on top, transposed up a Major 9th (14 semitones) from the lead’s start on Eb to start on F (treble clef).

    Horrible, yes? Why? What vertical contrapuntal invervals are the sour ones to your ear?

    I’ll jump to a better trial that succeeds in both places.

    In this successful trial, the first system’s added middle voice transposes from the lead’s E up 13 semitones (minor 9th), and later in the second system the added upper voice transposes up also 13 semitones from the lead’s Eb to a second answer starting on E. The minor 9th is unusual, unorthodox, chromatic, not a solution we might predict . . . but it works!

    4. Rhythmic alternation

    This will be like Fux’s Fourth Species. The lead subject is best with some long note values, leaving ample time for answering voices to present pitches when it is not moving. Transpositional choices for entering answers become fixed as predominant vertical intervals throughout the canon. In this example, the first answer chooses down 11 semitones plus an octave, and the second answer enters up 7 semitones (Perfect 5th) from the first answer, which is down a Major 10th (14 semitones) from the lead line. Thus vertical (harmonic) intervals of 11, 7, and 14 semitones end up projecting harmonies based on the 7 4 array: G up 7 to D up 4 to F#, which is up 11 from G. That sets the harmonic character of pitch constellations throughout the canon.

    5. Stretto echo

    Stretto is the term used in fugue structure for when an answer to the subject happens before the subject is finished, sometimes with a delay as short as only one or two beats. For a canon, this offers an interesting strategy for choosing pitches to shape a subject that makes its own arpeggiated harmony as it goes. The answers at unison (not transposed) are literally echoes. Even with answers octave-transposed, the effect is a multivoice arpeggiation. The fascinating wrinkle, however, is that the “chord” being arpeggiated is constantly evolving, dropping one pitch and adding one new at each note of the lead line.

    Though this setup can work with other rhythmic “species” of lines, it is particularly interesting in the note-against-note conforming rhythms of “First Species.”

    Here is how it can work, using the canon above as a straightforward example.

    This analysis sounds as a rather nice progression of arpeggiated chords and simple flute line! The important point, though, is that this progression did not come first. It was built by the canonic subject line as each new pitch was chosen to make a certain array with the previous two pitches in an ongoing, evolving flow. Magic!

    6. Spin a piece

    For our example, we’ll follow the order of the example techniques:

    • two-voice zigzag canon
    • add a third voice by trial and error
    • stretto echo of the same subject
    • Rhythmically spacious subject allowing non-synchronous timing of answers
    • Recapitulation of the stretto echo canon

    The result is a fuller working out of No. 10 of the 14 specimens in my Book of Canons:

    Black Canyon

    The title comes from my photographic memories of the Black Canyon of the Gunnison River, named for the ever-present shadows the narrow canyon’s steep, sheer, tall rock walls cast on the river flowing far below. The sheer cliffs of the Black Canyon are metamorphic Precambrian gneiss and schist, streaked with thin, brighter-colored layers of pegmatite. These streaks sketched on the darker rock look like maps of ancient contrapuntal lines.

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    Continue reading Mapping the Music Universe . . .

    MapLab 4. Model a metamorphosis

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