Five pieces built on two systems of just intonation, drawn from an analysis of the music of Catherine Christer Hennix (1948–2023) —intuitionist mathematician, a student of Pandit Pran Nath in La Monte Young's circle— and of her ensembles. The ratios between the pitches, and the way the low tones are produced, come from there.
i Two systems, one prime apart
ii The operation
iii The five pieces (with extracts)
iv How much the author does
v Who is listening
Two prisoners in neighbouring cells who talk to each other by knocking on the wall. The wall is what separates them, but it is also what lets them communicate. Every separation is a link.
— Simone Weil, Gravity and Grace
i Two systems, one prime apart
System 1 — harmonics of 36.72 Hz · threes and sevens
1/1 — 73.4 Hz
9/8 — 82.6 Hz
21/16 — 96.4 Hz
3/2 — 110.2 Hz
7/4 — 128.5 Hz
System 2 — A of 110.1 Hz · fives and sevens
1/1 — 110.1 Hz
8/7 — 125.8 Hz
5/4 — 137.6 Hz
3/2 — 165.1 Hz
7/4 — 192.7 Hz
The first is made of threes and sevens —harmonics 1, 3, 7, 9 and 21 of a fundamental of 36.72 hertz that never sounds—, so it admits no major third and no possible chord, and therefore no rest. The second has fives and sevens: over an A of 110.1 hertz, the ratios 1/1, 8/7, 5/4, 3/2 and 7/4, which form the most fused chord acoustics allows. So fused that an ear trained on the piano recognises it and at the same time cannot place it: the fifth is almost the piano's, and the others fall as much as thirty-one cents outside. These are the intervals Western tuning discarded in order to modulate.
Composing this way is not choosing pitches: it is choosing ratios. Simone Weil observes that in the Pythagorean tradition ἀριθμός —number— and λόγος —relation, word— are used interchangeably one for the other, and that the search for the mean proportional, which is the centre of Greek geometry, «is the very idea of mediation». A just intonation is, in that literal sense, a set of relations before it is a set of notes.
The two systems share an A: the 3/2 of the first falls at 110.16 hertz —the third harmonic of its fundamental— and the tonic of the second at 110.1. Less than a cent. The second system is built on a pitch the first one contains.
Hence what lies between the two systems is a piece and not a cut. Philolaus: «things that are alike and of the same kind have no need of harmony». Only those that are unlike need it.
ii The operation
None of the pieces emits its own lowest frequency. Every title names a magnitude its system manufactures and no instrument performs: the invisible harmony is greater than the manifest one.
The voices, close to sine tones and fixed in pitch, pass together through a single saturation stage —one stage, on the sum, never one per voice— where they multiply and produce the sums and differences of their frequencies: two voices 27.5 hertz apart manufacture a 27.5 hertz tone that no instrument plays. They are tuned to their system's grid within a few cents, so those new frequencies land on the series or very close to it. The low tone of each piece is the difference between two high voices: not a tone but a narrow cluster around one, and the throb comes from that minimal inexactness.
Hennix: «the basic premise of my music is that I highlight the harmonics of that drone». Here the harmonics are what sounds and the drone is what results: the series is kept and the path reversed. As Jacinta Gorriti writes, inheritance is not settled by mere iteration. To inherit is, in Althusserian terms, «to leave the circle while remaining inside it»: to do something else with what was received and to be transformed, at the same time, in that act.
The ear adds differences of its own. Two sustained, precisely tuned tones produce inside the cochlea a third pitch that is not in the air, and a set of harmonics induces the perception of the absent fundamental. Those frequencies depend on the pitches arriving summed at the same ear, so they change with the position of the body in the room: moving alters what sounds.
Weil uses for that kind of thing the Greek word that gives the record its title, metaxu: what stands in the interval. A difference tone belongs to neither of the two voices that produce it, is a harmonic of neither, and stops existing the moment one of them leaves. It is, strictly, what happens between two things. A musical interval, looked at properly, is a distance that joins: what these pieces do is draw sound from that distance — not from the pitches, which are the two sides of the wall, but from the knocking.
iii The five pieces
27.5 Hz (10:00)
Dense and consonant to the point of discomfort: twenty-three voices across five octaves, from 54 to 2,013 hertz. The voices of the low chord sit 27.5 hertz apart, which is how the piece manufactures the bass tone nobody plays. Some are shifted by half a hertz, and that is the throb: one cycle every two seconds.
9.01 Hz (27:00)
The crossing: the tuning migrates from one system to the other. Some voices shift so slightly that what you hear is not a change of note but of colour; others move to a different pitch. Around minute sixteen the drone sounds both of its pitches at once, and the distance between them —9.01 hertz— is the pulse. There the series comes apart: what saturation manufactures stops landing on the grid.
2115 s (4:00)
The window: four minutes cut from a long realisation with no arc, chosen because it is the stretch that most seems to have form inside a process that has none. The title is the second at which the cut begins, so it names a place rather than a frequency: it points to a piece that is not on the record.
36.72 (20:00)
Sparse, without a third, the opposite pole from the first. Seventy per cent of the energy lives in a drone at 72.8 hertz; everything else is colour in three registers that do not blend: nine voices over the drone, four of body in the middle and five high and metallic above.
504 s (10:30)
The only closed form on the record and the only one without chance. Each pitch class enters and leaves on a cycle of 24, 72 or 168 seconds, whole-number periods of the system itself. The three coincide once, at 504 seconds, and the voices still sounding stop together. None of them has that period: the coincidence produces it. A difference tone made of time.
iv How much the author does
There is no arc and no development. The pieces do not go anywhere and do not close: they end where they are left. Form is not in the duration but in the cross-section, in the configuration of distances between the parts at each moment — and in just intonation that stops being an analogy, because a chord is exactly a set of distances between parts.
What there is not is metre. Rhythm there is: the slow oscillation the voices share, the tremor of beating between neighbouring pitches, and in the last piece the explicit cycles.
For Weil, arithmetic, geometry, astronomy and music —the four branches of ancient mathematics— have no value in themselves: they are intermediaries. The analytical work that preceded this record was not, in that sense, the task before the important one. It was another way of crossing from one side to the other, and the measuring apparatus turned out to be the same one used to compose.
The method of production is a negative way: nothing sampled, no tape and no added warmth, no pitch modulation —the whole identity of the record depends on the intervals being exact—, no hand-drawn curves. An invocation without a relic. When a piece needs more life, it is asked of the system and not of the mix.
Teresa of Ávila describes the degrees of prayer with an image of a garden: it is watered by drawing water from a well, «at great labour of ours»; with a waterwheel and buckets, that is, with a device one builds and turns; with running water from a river, where «almost He is the gardener and the one who does it all»; and finally by rain, «with no labour of ours at all». It is not a scale of merit but of the progressive withdrawal of the one who works, and it is exactly what this record does: the synthesiser played pitch by pitch, the generator, the realisation chosen among many without any of its events having been decided, and 504 s, where the phases are fixed in advance and nothing is decided while the piece runs.
v Who is listening
The system can run indefinitely and never repeats a configuration. Every realisation is unique; the album is a selection, not an original. What is fixed in advance is not the order of events but which pitches can occur, and how often.
The difference tones of these pieces have two distinct origins: one happens in the air, in the saturation stage, and a machine records it with nobody listening; the other is assembled in the cochlea, and there a body is needed.
That second origin cannot be composed. The system is composed; the rest happens or does not happen in each ear, and no technique guarantees it.
John Drever asks, apropos of the old slogan about the primacy of the ear: whose ear? The cochlear non-linearity these phenomena depend on varies between people and degrades with age and with hearing damage. The correct formulation is not that the work is completed in whoever listens, as if there were a normative listener waiting on the other side. It is that it is a different work in every ear, and in some it does not happen.
What these pieces ask for, then, is not following but staying. Weil defines attention as an emptying: suspending thought, leaving it available and penetrable, waiting without seeking anything. She calls it «an effort, the greatest of efforts perhaps, but a negative effort».
Seventy-one and a half minutes. For a room, extended-response headphones, or any system that reaches below sixty hertz.
Cover: the angel of the second trumpet, f. 166 of the Beatus of Facundus (1047). Biblioteca Nacional de España, Madrid. Public domain.
