6.9 The Devil's Interval

When a barbershop quartet locks a chord, something happens that is not in the score. The four voices, each on their own pitch, suddenly fuse into one. The lead, the tenor, the bass, the baritone — you stop hearing them as separate. A fifth voice appears, which no one is singing. The room rings. The hair on your arms stands up.

This is not a feeling. It is physics. When four notes are tuned into exact whole-number ratios with each other, their waveforms reinforce instead of clashing, and the air in the room carries a composite tone that wasn’t there a moment ago. The barbershop tradition calls this the “lock and ring,” and the singers who can do it have spent years learning to bend their notes away from the pitches a piano would play. They are not singing the chords on the page. They are singing the chords that ring.

The secret of the lock is the seventh. Not the major seventh of the keyboard. Not the minor seventh either. A bent seventh, lower than minor by about a quarter of a semitone, tuned to a ratio of 7:4 with the root. It is one of the most consonant intervals that exists. And it is nowhere on the piano.

The piano cannot play it because the piano cannot reach the seventh harmonic. The Western scale, as the keyboard implements it, is built from powers of 2 (octaves), powers of 3 (fifths), and powers of 5 (thirds). There is no factor of 7 anywhere in it. Whatever the keyboard plays as a “seventh” — a major seventh of 15:8, a minor seventh of 16:9 — it is an approximation built out of the materials at hand. The actual seventh harmonic, the one a string or a column of air or a human voice will produce naturally, the one the ear recognizes as a deep consonance, has been silently amputated from the framework.

This is not a small omission. Barbershop is one tradition that recovered it. Jazz is another. Blues is another. Gospel is another. All four come from the same root.

When the historians tell you that barbershop and jazz “independently discovered” the harmonic seventh, ask them what they mean by independent. Both grew out of Black American vocal and instrumental traditions of the late nineteenth and early twentieth centuries. Both inherited a harmonic sensibility from African vocal practices that had crossed the Atlantic in the worst possible conditions and survived. The intervals were kept alive in the places the framework wasn’t paying attention: the field hollers and work songs of the rural South, the church choirs and revival meetings where the voices outnumbered any keyboard, the storefront churches and after-hours clubs in Black neighborhoods of the cities the great migration filled, the porches and the streetcorners where four men could stand and start a chord and let it ring. The white concert tradition was not where this music was kept. The white concert tradition was where, eventually, this music was discovered — meaning, where its existence was finally acknowledged by the framework that had been writing it off as out of tune for a hundred years.

Jazz is the place where the bending became a vocabulary. The blue note — the bent third, the bent seventh — was systematized into compositional practice. The pianist Thelonious Monk built his harmonic language out of intervals that the equal-tempered keyboard could only gesture at, leaning on dissonance and silence to point at consonances the instrument could not name. The horn players had it easier; brass and reeds can bend. A trumpet or a saxophone can lean into the harmonic seventh directly, and the great players did, and the great singers — Billie Holiday, Sarah Vaughan, Ella Fitzgerald — moved their voices fluidly between the cartoon and the harmonics underneath, sometimes inside a single sustained note. Listeners who grew up only with the cartoon heard this as “soul” or “feeling” or “expressive deviation.” What they were actually hearing was a more accurate tuning.

What the keyboard couldn’t reach, the voice always could.

The medieval church called a different interval the devil's interval: the tritone, half an octave exactly, the maximally unstable interval that does not want to resolve. Diabolus in musica. The story most people have heard is that medieval Christians thought the tritone was Satanic and forbade it. The actual history is duller and more interesting. Plainchant avoided the tritone because it does not resolve, not because anyone thought the devil lived in it. The “diabolus” label appears in 18th-century counterpoint treatises as a mnemonic for student composers: this interval is unstable, treat it carefully. It was a warning, not an excommunication.

But the spirit of the label — that there are intervals beyond the pale, that some sounds belong outside the acceptable framework — has had a long career. Every tradition that bent its notes off the equal-tempered grid has been received, by the framework, as something close to wrong. The blue note “out of tune.” The barbershop seventh “off-key.” The Indian śruti “primitive.” The Arabic maqam “exotic.” The Mongolian throat singer “impossible to notate.” In every case, the framework’s vocabulary for what it couldn’t accommodate was a vocabulary of error.

The irony, which only became visible recently, is that those bent notes were almost never errors. They were, in most cases, more consonant than the notes the framework allowed. The 7:4 of the barbershop seventh is simpler than the 16:9 of the equal-tempered minor seventh. The blue note of the third is closer to the 6:5 or 7:6 ratios than to anything on the keyboard. The bent intervals were not approaching dissonance. They were approaching consonances older than the framework — consonances the framework had been forced to abandon when it agreed to close the circle on its own terms.

The fact that the keyboard cannot reach the seventh harmonic is not a quirk of taste. It is a mathematical necessity of equal temperament, and equal temperament was a deliberate choice — an act of suppression in exchange for a freedom. The freedom was modulation: the ability to change keys without retuning the instrument. The price was that every interval became slightly wrong, and the seventh harmonic was abandoned entirely.

To see why, you have to understand what equal temperament actually does.

The cartoon. A piano divides the octave into twelve equal semitones. “Equal” here means equal in the ratio sense: each semitone is the same multiplicative step, the twelfth root of 2, approximately 1.0595. Stack twelve of those steps and you arrive exactly at the octave. Stack seven and you arrive at the equal-tempered fifth, 2⁷ᐟ¹² ≈ 1.4983, which is just slightly less than the pure 3:2 ratio of 1.5. The fifth is close. It is not exact. And this slight wrongness propagates through every interval on the keyboard.

The major third on a piano is 2⁴ᐟ¹² ≈ 1.2599. The pure major third, ratio 5:4, is 1.25 exactly. The keyboard’s third is about 14 cents sharp of the pure third — enough to make a held a cappella chord sound subtly fuzzy when compared with one tuned to actual integer ratios. The minor seventh on a piano is 2¹⁰ᐟ¹² ≈ 1.7818. The harmonic seventh, 7:4 = 1.75, is about 31 cents flatter — well outside the range of what most listeners hear as “the same note.” The piano’s seventh is, in a precise sense, a different sound from the seventh that occurs naturally in any vibrating physical body.

The piano is not playing music as the universe makes it. It is playing a cartoon of music — a twelve-color reduction of an infinite-color landscape, simplified to fit the constraints of fixed pitch and key-change freedom. Harmony develops this in more detail: the cartoon was a deliberate trade-off, and within its limits an extraordinarily generative one. But it is still a cartoon. And generations of musicians who grew up with the cartoon have come to believe that the cartoon is what music sounds like. When they hear a choir tune its intervals to the pure ratios — the ringing, the lock, the hair standing up — they call it magical. What they are hearing is just music played without the cartoon.

Pythagoras’s two crises. The Pythagoreans staked everything on the rationality of number. All things, they said, are number — by which they meant the positive integers and their ratios. The whole cosmos was built out of ratios. And music, where the simple ratios produced the consonances, was the proof. The octave was 2:1. The fifth was 3:2. The fourth was 4:3. The simpler the ratio, the more consonant the interval. This was not metaphysics; it was empirical observation, verifiable on a stretched string.

Then they tried to close the circle.

If you stack twelve fifths, you should — by the same logic that built the system — arrive at seven octaves. You do not. You arrive at (3/2)¹² ≈ 129.746, when seven octaves are 2⁷ = 128. The ratio between them, about 1.0136, is what we now call the Pythagorean comma. It is small, but it is not zero, and no manipulation of pure fifths will close the gap. The circle does not close. You cannot stack pure fifths through twelve keys and arrive home.

This is the same problem as the diagonal of the square. The famous story — Hippasus drowned, the secret suppressed, the crisis of incommensurability — concerns √2: the diagonal of a unit square cannot be expressed as a ratio of integers, and the Pythagorean commitment that all things are number falls apart. That story is mostly legend, but the underlying intellectual catastrophe is real. What is striking, and to my knowledge under-discussed, is that the same catastrophe is sitting in their music theory the whole time, in a different form. The diagonal asks: what is the ratio between the side and the diagonal of a unit square? There is no rational answer. The closed circle of fifths asks: what is the ratio between twelve fifths and seven octaves? There is no rational answer. Both are irrationalities lurking in domains the Pythagoreans believed they had ratio-built.

Whether they noticed both at the time, we cannot say with certainty. The diagonal is the famous crisis, the one that survives in the doxography. The musical crisis is implicit in their own tuning system but does not appear to have been explicitly faced as a crisis until much later. It is tempting to suggest that the comma was the deeper crisis — the one that ran closest to their actual practice, since the Pythagoreans spent more time with monochords than with hypotenuses — and that the geometric version may have been easier to talk about because it was less embarrassing. This is speculation. But the shape of the difficulty is identical, and a tradition committed to the universality of rational ratio could not, in principle, accommodate either one.

What the centuries did, in both domains, was choose suppression. In geometry, the irrational hypotenuse was eventually absorbed by enlarging the concept of number. In music, the irrational comma was eventually absorbed by equal temperament, which adopts an irrational ratio (the twelfth root of 2) as its fundamental step and uses that irrationality to close the circle smoothly. The keyboard’s freedom of modulation is bought with the same coin Pythagoras would not accept: irrationality as the price of closure.

The label and what it has been used to label. The medieval church’s diabolus in musica was, narrowly, about the tritone — the interval at exactly half an octave, the maximally unstable point in twelve-tone equal-tempered space. Plainchant avoided it because it does not resolve. There was no claim that the devil lived in it.

But the cultural pattern of labeling unfamiliar harmonic territory as forbidden, evil, or wrong is older and more persistent than the specific medieval usage. And in the centuries since equal temperament became the global default — first in European art music in the eighteenth and nineteenth centuries, then in the global music industry of the twentieth — that pattern has had abundant material to work on.

The harmonic seventh, the bent third, the microtonal ornaments of the global vocal traditions, the modes of Arabic and Indian classical music, the polyrhythmic and polyharmonic practices of African vocal music — all of these were systematically labeled by Western musicology as either “primitive” (a tradition that had not yet developed the framework’s sophistication), “exotic” (interesting but other), or simply “out of tune” (an error against a standard the framework took to be universal). The very vocabulary of musical error — flat, sharp, off-key, wrong note, bent — assumes that the equal-tempered grid is the reference, and that departures from it require justification.

The Black American traditions that produced blues, jazz, and barbershop received this treatment for over a century. The “blue note” was an “expressive deviation.” The barbershop seventh was a “stylistic choice.” The microtonal inflections of gospel were “soulful” — a compliment that quietly preserved the assumption that what was being inflected was the framework, and the inflection was a feeling about it rather than a more accurate tuning. None of these descriptions were neutral. All of them positioned Black harmonic practice as a modification of a tradition the framework took to be standard, when the historical reality was closer to the opposite: the bent intervals carried older, more consonant ratios that the European tradition had been forced to give up when it adopted the cartoon.

This is not a claim that European music is impoverished and African music is rich. It is a claim about labels. The label “out of tune” did real work for a long time on traditions that were, by the ear’s own physical criteria, more in tune than the framework that judged them. “Devil’s interval” was never used in this way directly — the term remained attached to the tritone — but the shape of the labeling, the willingness to call unfamiliar consonance evil because the framework could not accommodate it, has run continuously through Western musical reception for centuries. Whether or not the specific term migrated, the cultural pattern it names has done its work elsewhere under other names.

The keyboard is a boundary. There is a way to see the whole picture that the chapter has been circling. The keyboard is not a musical instrument in the simple sense. It is a translation layer between two domains: the rich space of harmonic vibration as the universe offers it, and the discrete twelve-tone vocabulary that Western notation can write down.

The first domain is worth pausing on, because it is easy to picture pitch as a number line — a frequency axis — and that picture flattens what is actually there. A sounding body does not emit a single frequency. It emits a spectrum: a fundamental plus its integer harmonics, the second at twice the fundamental, the third at three times, the fourth at four times, the seventh at seven times, and so on, each one its own ringing component, the whole stack giving the sound its timbre and its place in harmony. What the ear hears as “a note” is the spectrum. What the ear hears as “a consonance between two notes” is the alignment between their spectra — the second harmonic of one voice meeting the third harmonic of another, the third meeting the fifth, the fourth meeting the seventh. The lock and ring of the barbershop chord is what alignment in this space feels like from the inside. The harmonic seventh is consonant with the root not because 7:4 is a pleasant number on a line but because the seventh harmonic of the root is, by definition, exactly there.

Every act of playing a note on a keyboard performs a transduction from this rich harmonic space onto a flat grid of twelve fundamentals per octave, with the harmonic relationships between fundamentals approximated by the nearest grid points. Once the music is on the grid, it can be transmitted reliably — copied, taught, modulated through twenty-four keys without retuning. That is the freedom equal temperament bought. What it cost is the very alignments that were the point. The seventh harmonic is not on the grid because no power of 2 times no power of 3 times no power of 5 reaches a factor of 7. The bent third sits in a place the grid has no name for. The ornaments and inflections of the global vocal traditions track alignments between harmonics that the grid was built to forget.

The voice does not have to go through that boundary. A singer can target any frequency the ear is hearing, including the frequency that would make their voice’s harmonic spectrum lock with another voice’s spectrum. A horn player, with embouchure and breath, can do the same. A string player without frets — a violinist, a cellist — can place a finger anywhere on the harmonic series. The traditions that kept the bent intervals alive were not coincidentally vocal and string-based. They were vocal and string-based because the voice and the unfretted string are channels that do not pass through the keyboard’s transduction. What the keyboard had to discard, they could keep.

This is exactly the boundary-stack argument of The Boundary Is a Stack applied to music. Communication between two interiorities passes through a layered apparatus of translations. Each layer is a transduction, and each transduction risks loss. A wide channel passes more through; a narrow channel passes less. The keyboard is a narrow channel for the harmonics of physical vibration. The voice is a wider channel for the same thing. When the framework that grew up around the keyboard came to mistake the keyboard’s vocabulary for the full range of musical possibility, it was confusing the channel with the capacity. What was not playable on a piano was not absent from music. It was being kept alive in channels the framework had stopped listening to.

The traditions that kept the bent intervals alive were never producing what the framework took them to be producing — a “modification” of a “standard.” They were producing what the framework had narrowed down from. The cartoon was the narrowing. The bending was the source returning.

What the bending actually does. There is one further observation that closes the chapter.

When a singer bends a note off the equal-tempered grid, they are moving in some direction. Sometimes they bend toward a more consonant ratio — the 7:4 of the harmonic seventh, the 6:5 of the pure minor third, the simple integer relationships that produce the ringing lock. Sometimes they bend away from any clean ratio at all, into deliberate dissonance — the scream, the wail, the held suspension that refuses to resolve.

These are opposite directions. One leads toward what the ear hears as deep consonance, more inevitable than the cartoon could produce. The other leads toward what the ear hears as suspended tension, the deliberate refusal of rest.

Heaven and hell. Same act of bending. Opposite destinations.

The cartoon vocabulary cannot tell them apart. Both look, from inside equal temperament, like “out of tune.” The ear that grew up only with the cartoon hears them as the same kind of wrongness. The ear that learns to hear the harmonics underneath hears them as opposites — one returning home to ratios the framework abandoned, the other leaving home for territories the framework cannot map.

The traditions that learned to bend learned both directions, and learned to use them together. Jazz, blues, gospel, barbershop, sean-nós, qawwali, the maqam improvisations of the Arab world, the raga elaborations of Hindustani classical music — every one of them uses bent notes as a vocabulary that distinguishes heaven from hell, that knows when a held suspension is approaching rest and when it is choosing not to. The framework that called all bending “out of tune” missed the most important thing about it: the bending was where the music was deciding what kind of world it was offering.

The devil’s interval, in the medieval sense, was about a sound that wouldn’t resolve. In the deeper sense the title is gesturing at here, it is about every interval that the framework couldn’t hold — most of which were not the devil at all, but the door the framework had quietly shut behind it.

The cartoon and what it suppresses is also the shape of We Used to Think. Now We Know. and The Word "Just" Is a Confession: the framework discards the mechanism and discards the phenomenon with it, the small word does all the philosophical work, and the loss is invisible to the framework that took it. Harmony develops the equal-tempered trade-off in more detail. The Simplest Numbers Come First shows why the simplest ratios appear first whenever a system bothers to count them. And Love takes up what happens when the bending leads, as it sometimes does, all the way home.