The idea
There are two ways to get a texture out of one small idea, and they sound nothing alike.
Imitation: state the idea, then state it again in another voice, offset in time and usually in pitch. The second voice is the same thing arriving late. Bach's Inventions run on this.
Phase: state the idea, then run a second copy at a different cycle length over a shared pulse. The two copies drift against each other and re-align by arithmetic. King Crimson's "Discipline" runs on this, and so does Reich's Piano Phase.
Both give you a rich two-part surface from almost no material. Imitation has a destination, because the imitating voice eventually stops imitating and goes free. Phase has a period, because the pattern returns to its starting alignment and could run forever. Knowing which one you are writing is the difference between a piece that ends and a piece that stops.
Watch
Learn Bach's Prelude in C Major (https://www.youtube.com/watch?v=9CVNMgyU5bI)
- The reduction that makes arpeggio writing tractable: collapse each bar to its chord and you get C, then D minor 7 over C, then G7 over B [02:32]–[03:15]. Every arpeggio figure in the piece is a chord plus an order of attack.
- The opening figure is a symmetrical shape, up and then "exactly the opposite back down" [05:50]–[06:08]. That is a cell you can rotate, which is what L13's exercises ask you to do. How to Learn Classical Music (https://www.youtube.com/watch?v=G4n7_T7sqfo)
- The rule behind interlocking guitar parts: "when you jump around like that your mind is putting together the lines of the closest notes" [15:42]–[16:12]. One player, one stream of notes, two perceived voices.
Learn Bach's Prelude in C Major
- voice-leading as minimum motion, and how an arpeggio figure is really several lines
How to Learn Classical Music
- the perceptual rule that makes interlocking arpeggios work: the ear joins the closest notes into lines
Listen
Bach, Invention No. 1 in C, BWV 772. Pass one: right hand only. Pass two: left hand only. Pass five: catch the moment the left hand stops being a copy, inside the first four bars.
Bach, Invention No. 4 in D minor, BWV 775. In 3/8, fifty-two bars, sixteenth notes six to a bar. Pass one: feel the meter in three. Pass two: count the bars the right hand plays alone before the left enters. Pass five: find the places where the two voices are a beat apart on the same falling shape.
Bach, Invention No. 8 in F, BWV 779. Thirty-four bars in 3/4. Pass one: the subject is a leap-up bar followed by a fall-down bar. Pass five: find the point where the hands swap.
King Crimson, "Discipline." Pass one: find the pulse and tap it, without counting bars. Pass two: follow one guitar for the whole track. Pass five: find where the two parts land together and estimate how many pulses apart the landings are. Tamm describes the piece as one where "beat groupings of two, three, four, five, and even seventeen jockey for the baffled listener's attention," over a pulse the players share.
King Crimson, "Frame by Frame." Tamm's description is your listening guide: a "pointillism versus slashing rhythm duel" between Fripp and Belew, and a coda "with polyrhythmic points." Character differentiation is doing the work that pitch does in Bach.
Reich, Piano Phase. Two pianos, a twelve-note pattern on five pitches (E, F sharp, B, C sharp, D). Both lock in unison, one holds tempo while the other accelerates until it is one note ahead, then they lock again, twelve times, back to unison. Listen once for the mechanism and once for what Reich cares about, the "impersonal, unattended, psycho-acoustic by-products": sub-melodies neither player is playing.
The comparison
The exact parallel. Bach's Invention and Crimson's "Discipline" both generate a whole two-part texture from a cell of under ten notes, and both make the second part hearable by offsetting it against the first rather than by giving it new material. The source course names the mechanism for Bach: the subject is "a kit of parts" whose components can be separated, recombined, inverted, sequenced and fragmented (Source course, Lesson 7). Fripp's own account of the Crimson texture is the same claim from the other side, quoted in his diary and reproduced by Tamm: "There's often good anarchism, where we all have our own parts, each worth listening to and autonomous, but played together. The listener can switch attention from one instrument to another."
Autonomous, and the listener switches attention. That is the source course's definition of contrapuntal independence, arrived at by a rock guitarist in a rehearsal room in 1981.
Where it breaks, first break. Bach's second voice imitates and then stops. In BWV 772 the right hand states the subject in the first two beats of bar one and the left hand answers with the identical seven notes an octave lower on beat three of the same bar. By bar three neither hand is imitating. The left hand is playing eighth notes against the right hand's sixteenths, and the piece has become free two-voice writing with subject fragments dropped into it. The imitation was a way in, not the substance.
Crimson's second guitar never stops. The two "Discipline" parts are cycles of different lengths over one pulse, so their relationship repeats when the cycle lengths coincide. Nothing is answered because nothing was asked, so the form has to come from somewhere else, which is why the piece adds and subtracts layers rather than developing them.
Where it breaks, second break. Bach differentiates his voices by rhythm and contour, since they share a timbre and often a register. Crimson's guitars share a timbre too and are differentiated by attack character and cycle length instead. Tamm's phrase for the texture is a "rock gamelan," and he is precise about the mechanism: two, three, or all four players playing "rapid-fire ostinati that interlock and counterpoint each other in a glittering pointillistic texture." He names six tracks where you can hear it, "Elephant Talk" and "Frame by Frame" among them.
A correction to the received story. It is often said that Fripp got this from Steve Reich. Tamm says otherwise, and he is the source closest to Fripp. Fripp denied any direct Reich influence, and when he made No Pussyfooting with Brian Eno in 1972 he had heard neither Reich nor Glass, though Eno had. Fripp later got to know Reich's music and liked it only so far, telling Garbarini in 1979: "It takes me to a point at which something really interesting could happen, but doesn't quite make that jump. Because it is preconceived and orchestrated." The honest statement is that Crimson's interlocking texture and Reich's phase music are convergent, both plausibly downstream of gamelan. Do not tell a student that Fripp copied Reich.
Radiohead. This band now has an analytical literature and you can use it. Brad Osborn's book gives five rhythmic techniques in chapter 3, each with his own examples: odd-cardinality meter ("2+2=5", "15 Step"), changing meter ("Sail to the Moon"), Euclidean and maximally even rhythms, grouping dissonance ("Weird Fishes", "Lotus Flower"), and polytempo ("The Gloaming", "Bloom"). Two of those bear directly on this lesson. Grouping dissonance is a two-part texture made by regrouping a fixed pattern against a fixed meter, which is the same move Fripp makes and the reason "Weird Fishes" belongs next to "Discipline." Polytempo is the case where the two parts are not sharing a pulse at all, which is where the Crimson comparison stops working.
Osborn's earlier article on math-rock grooves gives you the tool for the transitions: the pivot pulse, "the slowest temporal level preserved in a given meter change." It predicts what a meter change costs the listener. Going from 4/4 to 3/4 preserves the quarter note and barely disturbs entrainment. Going from 4/4 to 7/8, or 7/8 to 15/16, splits the primary pulse in half and does. When your own two parts change meter together, compute the pivot pulse first and you will know in advance whether the listener stays with you.
Polyphia and Chon. These belong in the lesson by ear, and no analysis of either is in front of me. Treat every specific structural claim about those two bands as to verify, and transcribe before you assert.
And the traffic in the other direction. Jonny Greenwood performed Reich's Electric Counterpoint, and Reich then wrote Radio Rewrite (premiered 5 March 2013, London Sinfonietta) on material from "Everything In Its Right Place" and "Jigsaw Falling Into Place."
At the piano
Learn one Invention's two voices as two pieces. Take BWV 772. Learn the right hand alone at 50 bpm with the metronome on the beat, using the wrist-drop and the printed fingering. Then the left hand alone to the same standard. Only then together. Slower than learning it as a piano piece, and that is the exercise.
The subject is seven notes, C, D, E, F, D, E, C, in sixteenths after a sixteenth rest, filling the first two beats of bar one. The left hand plays the same seven an octave lower from beat three of that bar. The inverted form appears early and supplies the episodes.
Build the interlocking texture yourself. Right hand, a five-note cycle in sixteenths. Left hand, a four-note cycle in sixteenths. Same pulse. Start them together and let them run twenty cycles. They realign every twenty sixteenths, which is every five beats in 4/4, which is why the bar keeps moving.
RH (5): E5 G5 B4 D5 A4 | repeat
LH (4): E3 B3 E4 B3 | repeat
At 60 bpm this is playable in a week. At 100 it is a Crimson part. Greg's metronome ladder from the Bach-in-an-hour video is the tool: 62, 70, 80, 93, 101, and drop the sheet music partway up.
Exercises
1. Invention subject autopsy
Do. Write the BWV 772 subject on staff paper and mark its two components, the rising scale fragment and the turning figure. Then write four new figures from those two components alone, in different orders and directions.
Done when: you can play all four at 60 bpm and say which component each one starts from.
2. Imitation at three distances
Do. Take one of your four figures into the right hand. Write it again in the left hand an octave below, entering after two beats, then after one beat, then after four.
Done when: you can play all three versions and state which distance produced the most usable harmony between the voices, and why.
3. Invertible counterpoint
Do. Write a four-bar subject and a four-bar countersubject that work with the subject on top. Then swap them so the countersubject is on top and the subject is underneath, without changing a note of either.
Done when: both arrangements are playable with no unintended strong-beat dissonance. If a fifth became a fourth and broke, rewrite the pair until it survives the swap.
4. Phase against imitation, same cell
Do. Take one five-note cell. Build version A as canonic imitation, left hand entering two beats after the right, both in 4/4. Build version B as phase, right hand in cycles of five sixteenths and left hand in cycles of four, over the same pulse.
Done when: you can play thirty seconds of each and describe out loud how each one would have to end.
5. The Crimson cycle, transcribed
Do. Transcribe eight bars of one "Discipline" guitar part by ear into Sibelius and count its cycle length in sixteenths. Then do the other guitar.
Done when: you have two cycle lengths written down, you can state how many pulses pass before the two cycles realign, and you can play the shorter of the two at the piano.
6. Compound melody
Do. Write a single line of sixteenth notes for one hand that alternates between two registers a tenth apart, so the ear hears two voices. Use the rule from the video: the ear joins the closest notes.
Done when: a listener who cannot see your hands reports hearing two lines, and you can notate what they heard on two staves.
Compose
Write a two-voice piece of sixty-four bars that changes engine once.
Bars 1 to 32 are imitative. One subject, stated in one voice and answered in the other, developed by inversion, sequence and fragmentation, with at least one passage of invertible counterpoint.
Bars 33 to 64 are phase. Take the same subject, cut it to a cycle of five or seven notes, and run it against a copy of itself at a different cycle length over an unbroken pulse. The transition between the two halves must be audible and must take no more than two bars.
Instrumentation: piano for the writing, then arranged for two guitars and bass. Both guitar parts must be playable by one player each at tempo, which means you write them on the instrument, not only on the page.
Checkpoint
- Play both voices of BWV 772 hands together at 60 bpm from memory, and play each hand alone as a melody.
- State the BWV 772 subject by note name, say where the left hand answers it, and say what the left hand is doing by bar three.
- Play your five-against-four interlocking pattern at 80 bpm for twenty cycles without drift.
- Hand over the two transcribed "Discipline" cycles with their lengths in sixteenths and the realignment number.
- Explain the difference between imitation and phase in two sentences, including what each one implies about how the piece ends.
- Say what is documented and what is not about Fripp and Reich.
Read
- Source course, Lesson 7 (/library/garnett-course.pdf). The subject as a kit of parts, imitation, countersubject, inversion, sequence, fragmentation, invertible counterpoint, and rhythmic differentiation as the guarantor of independence. Read it after you have learned BWV 772 by ear, not before.
- Eric Tamm, Robert Fripp: From Crimson King to Crafty Master. "The Rock Gamelan" and "Metrical Complications" are three pages and the best short account of how that band's texture works. The chapter on Fripp and minimalism settles the Reich question.
- Brad Osborn, Everything in its Right Place: Analyzing Radiohead (2016), chapter 3, https://doi.org/10.1093/acprof:oso/9780190629229.001.0001, and his earlier article "Beats that Commute," Gamut 3/1 (2010): 43-67, https://doi.org/10.7290/gamut6nma. The book gives you the five rhythmic techniques and the grouping-dissonance examples. The article gives you the pivot pulse, which is the one tool here that predicts what a meter change will cost the listener.
- Steve Reich, "Music as a Gradual Process" (1968), https://ccrma.stanford.edu/courses/220b-winter-2024/readings/Reich_Gradual-Process.pdf. Three pages. Read it for two sentences: "once the process is set up and loaded, it runs by itself," and "One can't improvise in a musical process." Both are constraints the second half of your composition accepts.
- Scores of BWV 772, 775 and 779. Free editions are adequate, and you check the source course against the score, not the reverse. The course calls the D minor subject a descending scale in even eighth notes. The score has it rising, in sixteenths, in 3/8: D, E, F, G, A, B flat in bar one, then C sharp, B flat, A, G, F, E in bar two, left hand entering an octave below in bar three.