Music Hub › Guides › Theory and harmony › Intervals: naming, recognising and building

The system everyone uses without ever naming it

One guide in this corpus explains how to stack thirds to build a seventh chord. Another explains why a vocal harmony is sung a third or a sixth above a melody rather than at any other gap. A third builds a major scale semitone by semitone, degree after degree, and a fourth transposes a whole chord chart along a circle of fifths. All of them handle intervals constantly, without ever stopping to say what an interval actually is: they assume the reader can already recognise a third or a fifth, and build their own subject on top of that assumed ability. This guide is that missing brick, the one none of the others lays because each of them already takes it for granted.

An interval measures the gap between two notes, and that name is actually built from two independent pieces of information rather than one. The first is a number, second, third, fourth, fifth, sixth, seventh or octave, which counts how many degrees of the scale separate the two notes, in the simplest way possible: by counting note letters. The second is a quality, perfect, major, minor, augmented or diminished, which pins down the exact gap in semitones once that number is fixed. The two always have to be read together, never one without the other: saying only "a third" without specifying major or minor leaves half the information missing, exactly like saying only "major" without specifying which number.

This two-tier vocabulary, number plus quality, is exactly what every neighbouring guide in this corpus uses without naming it: a third stacked on another to form a chord, a sixth chosen to harmonise a voice, a tone or a semitone to build a scale degree by degree. This guide doesn't rebuild any of those uses; it lays down the system that makes them all possible. How to count an interval in semitones, how to name it from note letters rather than from a raw semitone count alone, why certain qualities only exist for certain numbers, how to invert it, what frequency ratios and the ear itself say about its consonance, why the tritone occupies a place apart in this story, how an augmented fourth and a diminished fifth can land on the same key without ever meaning the same thing, how to recognise these gaps without thinking in numbers, and what happens once you go past the octave: each of these questions gets its own section, in that order.

Counting in semitones: the rule common to every interval

Before naming anything, you need a ruler. On a piano keyboard just as on a guitar neck, the octave splits into twelve equal steps, called semitones: C, C sharp, D, D sharp, E, F, F sharp, G, G sharp, A, A sharp, B, and back to C an octave higher. A tone, the slightly wider unit found in major and minor scales, is worth exactly two semitones. It's this division into twelve parts, and this alone, that serves as the graduated ruler for everything this guide is about to name: an interval, whatever its final name turns out to be, is first measured by counting how many of these twelve slots separate its starting note from its arrival note.

This division into twelve is nowhere near a universal given: other musical traditions, in India or the Arab world for instance, divide the octave differently, with narrower steps than the Western semitone. The system described in this guide is that of Western music as the Music Hub plays it, on a keyboard tempered into twelve equal semitones, a historical choice detailed later in this guide rather than an immutable law of physics.

Counting semitones gives a raw number, from zero, when the two notes are identical, to twelve, when they're a full octave apart. But that raw number isn't enough on its own to name an interval: two intervals that count exactly the same number of semitones can carry completely different names depending on the note letters used to write them, as the next section explains. Holding on to the semitone count alone, without the naming rule that goes with it, is the reflex that most often misleads anyone starting out in this theory, and it's the subject of the first pitfall listed at the end of this guide.

Naming an interval: the number counts letters, not semitones

The number of an interval, second, third, fourth and so on, is never worked out by counting semitones: it's worked out by counting note letters, inclusively, meaning the starting note itself counts as the first step. From C to D, two letters, C and D, are involved: that's a second. From C to E, three letters, C, D and E: that's a third. From C to G, five letters: that's a fifth. At this stage it makes no difference whether the arrival note carries a sharp or a flat: D, D♯ or D♭ each remain a second above C, because the letter stays D in all three cases. It's only once that number is fixed that any accidental steps in to pin down the interval's exact quality, a question covered in the next section.

This rule has a consequence that surprises almost everyone the first time around: two intervals that count the same number of semitones can carry different numbers, and conversely, two intervals with the same number can count different numbers of semitones. C to D♯ counts three semitones; C to E♭ also counts three semitones, exactly the same audible gap on a piano tuned in equal temperament. And yet C-D♯ is a second, since D remains the second letter after C, while C-E♭ is a third, since E is the third letter. Both are played on the same keys, but a musician reading a score never confuses them, because the name carried on the staff announces, before the first note is even played, whether the melodic line is about to cross two letters or three.

This distinction isn't a purist's detail: it carries the single most useful piece of information for anticipating where a musical phrase is heading. A second announces stepwise motion, the neighbouring note on the staff; a third announces a leap that skips a letter. Reading only the semitone count means losing that anticipation, whereas the full name of an interval gives it away for free, just by reading it. The international letter system, A B C D E F G, works on exactly the same counting logic as the French solfège system Do Re Mi; the Music Hub guide on letter-name notation details this crossover from one alphabet to the other for anyone who has to read both.

The five qualities, and why they don't all apply to every interval

Once the number is fixed by counting letters, the quality pins down the exact gap in semitones within that number. The system has five: perfect, major, minor, augmented and diminished. But contrary to what you might expect, these five words don't combine freely with the seven numbers: a fourth is never called major or minor, a third is never called perfect. This restriction isn't arbitrary; it comes directly from the major scale itself, the template on which the whole quality system was built.

In the C major scale, with no alterations at all, the interval between the tonic and each of the other seven degrees comes out to a single default quality: the unison, the fourth, the fifth and the octave come out perfect; the second, the third, the sixth and the seventh come out major. The minor quality is obtained by lowering one of these last four by a semitone, never one of the first four: a fourth lowered by a semitone doesn't become a "minor fourth," a term that doesn't exist in the system; it becomes a diminished fourth, the same alteration carrying a different name depending on the starting family.

This split between the two families goes back to medieval and Renaissance consonance theory, which already distinguished perfect consonances, the unison, the fourth, the fifth and the octave, from imperfect consonances, the thirds and the sixths. The word "perfect" is a direct legacy of that old vocabulary: it designates an interval whose frequency ratio, detailed later in this guide, is so simple that it traditionally tolerates no ambiguity of colour, major or minor, unlike a third or a sixth, which do change character noticeably depending on their exact size. The fourth's status, in fact, was long debated within this very classification: in the thirteenth century, the fourth and the fifth together formed what theorists called the concordantiae mediae, the middle consonances, sitting between the unison and the octave on one side and the thirds and sixths on the other; then, in the fifteenth century, the fourth came to be treated as a dissonance when it sounds on its own, a judgement set down in black and white by the Flemish theorist Johannes Tinctoris in 1473, in his Terminorum musicae diffinitorium. Modern theory has largely smoothed over this nuance by filing the fourth without reservation among the perfect intervals, but the trace of that old debate explains why it still, to this day, demands slightly separate treatment in multi-voice writing.

Augmented and diminished, finally, aren't reserved for a single family: any perfect, major or minor interval can be widened by an extra semitone to become augmented, or narrowed by a semitone to become diminished. A major third widened by a semitone becomes an augmented third; a perfect fifth narrowed by a semitone becomes a diminished fifth. Those two qualities therefore apply, in theory, to all seven numbers without exception, which is why the table below, which lists the twelve semitones of the octave over the root note C, shows two different names on one and the same central semitone: the subject of a whole section further on in this guide.

This table gives the identity card of every simple interval, from the one closest to the unison to the one closest to the octave, along with its historical consonance class inherited from the previous paragraph. It serves as a reference for the rest of this guide, exactly as the major scale formula serves as a reference for a neighbouring guide in this corpus: once this table is committed to memory, every interval encountered elsewhere, in a chord, a vocal harmony or a scale, can be identified without recalculating.

Inverting an interval: the rule of 9

Inverting an interval means moving the lower note up an octave, or the higher note down an octave, so as to flip which of the two notes sounds below the other. A third from C to E, by moving the C up to the octave above, becomes an interval from E to C: it's no longer a third but a sixth. This simple mechanism obeys a fixed arithmetic rule, known as the rule of 9: the number of an interval and the number of its inversion always add up to 9. A second inverts to a seventh, two plus seven equals nine; a third inverts to a sixth, three plus six equals nine; a fourth inverts to a fifth, four plus five equals nine; a unison inverts to an octave, one plus eight equals nine.

This rule comes directly from the scale's seven-letter structure, with the octave closing the cycle on the eighth. Counting from the lower note up to its octave, both ends included, always comes to eight steps. The 9 comes from what happens next, and it's worth seeing once so you never have to memorise the rule blindly: the starting interval and its inversion both count the note they share, the middle one, which is the top of the first and the bottom of the second. So it gets counted twice. Eight steps, plus that double-counted note, and the total comes to nine. Subtracting the starting interval's number from nine then gives the inversion's number directly, without ever needing to recount the letters a second time. It's a calculation shortcut, not an isolated numerical coincidence.

Quality, too, inverts according to a fixed rule, and this second rule is at least as useful to remember as the one for numbers. A perfect interval stays perfect after inversion: the fourth, perfect, inverts to a fifth, also perfect. A major interval becomes minor, and a minor interval becomes major: a major third inverts to a minor sixth, a minor third inverts to a major sixth. An augmented interval becomes diminished, and a diminished interval becomes augmented: an augmented fourth inverts to a diminished fifth, which links directly to the subject of the tritone covered later in this guide.

Two mistakes almost always accompany learning this rule. The first is inverting the number without inverting the quality, or the other way round: the two operations always happen together, never one without the other, because they describe the same transformation seen from two different angles. The second is confusing inversion with transposition: inverting an interval changes neither of its two notes, only their position from one octave to another changes, whereas transposing shifts both notes together to a new pitch while keeping the same starting interval. Inversion, by contrast, literally turns the interval into a different one, with a number and a quality different from the starting point.

Why some intervals sound consonant: from Pythagorean ratios to the critical band

Western music theory very early on sought to explain in numbers why some intervals sound stable and others tense. Tradition credits the Pythagorean school, in Greece, around the sixth century BCE, with the first systematic description of the most stable consonances through frequency ratios between simple whole numbers: the octave corresponds to a ratio of two to one, the perfect fifth to a ratio of three to two, the perfect fourth to a ratio of four to three. The smaller the two numbers in the ratio, the more consonant the corresponding interval was long judged to be, a hierarchy that overlaps fairly closely, though not perfectly, with the modern distinction between perfect and imperfect consonances raised in the section on qualities.

This construction by stacking pure fifths, handled since Antiquity, nonetheless runs into an arithmetic problem that has a name, the Pythagorean comma. Stacking twelve pure fifths on top of one another, each in the exact ratio of three to two, never quite lands back on seven complete octaves: the gap, tiny but very real, works out to a ratio of 531,441 to 524,288, or around 23.5 cents, almost a quarter of a semitone. This slight mismatch is nothing like a calculation error: it's an unavoidable mathematical property of the system, one that occupied keyboard theorists for centuries before a practical solution took hold, the one the next section details under the name equal temperament.

This approach through frequency ratios still holds true today, but modern psychoacoustics has added to it an explanation that owes nothing to whole numbers alone anymore. In 1965, the Dutch researchers Reinier Plomp and Willem Levelt published a model of dissonance built on the notion of the critical band, the range of frequencies the ear processes as a single bundle rather than as two distinct sounds. Two frequencies very close to one another, inside that band, produce a rough beating that the ear perceives as dissonance; the harshness of that beating, what the two researchers call sensory roughness, peaks when the gap between the two frequencies comes close to a quarter of the critical band's width, then drops off on either side of that peak, toward zero as the frequencies draw together into a unison, and toward zero again as they move apart beyond the band. Applied to complex tones rich in harmonics, the way a voice or a real instrument is, rather than to two isolated pure frequencies, this model brings back troughs of minimal roughness exactly on the intervals with the simplest frequency ratios, the octave, the fifth, the fourth, the thirds: the physiology of the modern ear arrives, by a completely different route, at the same hierarchy of whole-number ratios that the Pythagoreans had established from nothing more than observing vibrating strings, more than two thousand years earlier.

Equal temperament against just intonation: the hidden compromise behind every key

The Pythagorean comma highlighted in the previous section poses a very concrete problem for any fixed-pitch instrument, such as a piano or a fretted guitar: if every fifth is tuned to its pure ratio of three to two, the twelfth fifth in the cycle never lands exactly back on the starting octave, and some keys sound noticeably more in tune than others depending on where in the cycle you happen to be. For centuries, different tuning systems tried to spread out this slight excess unevenly, sacrificing the tuning accuracy of a few rarely used keys to preserve that of the most common ones.

The solution that eventually won out almost everywhere goes by the name equal temperament: dividing the octave into twelve rigorously identical semitones, each in a frequency ratio equal to the twelfth root of two, so that no fifth, no third, no interval except the octave itself is quite pure anymore, but the twelve keys become, in exchange, exactly as in tune or as out of tune as one another. This mathematical solution wasn't born in a single country: it was described a generation apart, and entirely independently, by the Chinese mathematician and prince Zhu Zaiyu, in a treatise published in 1584, and then by the Flemish mathematician Simon Stevin, in an unfinished manuscript, Van de Spiegheling der Singconst, written around 1605 and published only in 1884, three centuries after his death, with neither of the two aware of the other's work.

To compare just intonation, built on exact whole-number ratios, with equal temperament, built on this rigorously uniform division, music theory has used, since the late nineteenth century, a unit called the cent: a logarithmic split of the octave into 1200 equal parts, a hundred cents per semitone, introduced in 1885 by the British phonetician Alexander John Ellis in the appendix to his annotated translation of Hermann von Helmholtz's treatise on the sensations of tone. Expressed in cents, the gaps between the two systems become immediately readable, even to an ear that doesn't consciously hear them.

This table shows a fact that often surprises: the octave is the only interval that equal temperament reproduces without the slightest compromise, exactly 1200 cents in both systems, precisely because that's the one constraint the twelfth-root-of-two calculation was built to preserve to the letter. Every other interval departs from its just version by a few cents, the fifth and the fourth almost imperceptibly, barely two cents, the thirds and the sixths more noticeably, thirteen to sixteen cents, a gap that an ear trained on just intonation, a string quartet's or an a cappella vocal ensemble's for instance, can genuinely perceive on a freshly tuned piano. Neither of these two systems is "more true" than the other: just intonation maximises the purity of each interval taken on its own, at the cost of uneven keys, while equal temperament slightly sacrifices every interval to guarantee that all keys stay equally playable on a single fixed keyboard, a compromise that is precisely what makes it possible to play indifferently in any key without retuning the instrument between two pieces.

The tritone: the interval that splits the octave in two, and the legend of the diabolus in musica

Among the twelve semitones of the octave, only one sits in a perfectly central position: six semitones, exactly half the octave, a point of balance so strict that in equal temperament, the augmented fourth and its inversion, the diminished fifth, land on exactly the same value of 600 cents, with not the slightest gap between the two, a symmetry unique to this interval and this interval alone in the whole system. It's this exact split of the octave into two identical halves that earns the tritone, its usual name, its reputation as the most unstable interval in the system: neither the direction nor the destination of the movement is suggested by the symmetry of the gap alone, which explains why it almost always calls for resolution toward a more stable interval, most often the third or the sixth, rather than settling on itself.

This very real instability gave rise to a stubborn legend, still widely repeated in music teaching today: that of a tritone banned by the medieval Church, under the name diabolus in musica, the devil in music. Close examination of the sources doesn't back up that story. According to the Harvard musicologist Thomas Forrest Kelly, no known medieval reference links the tritone to the devil; the earliest written attestations of the phrase only date back to the eighteenth century. The German organist and theorist Andreas Werckmeister cites the formula as early as 1702, treating it as an already old usage in his telling; the Austrian composer Johann Joseph Fux writes it down in black and white in 1725, in his counterpoint treatise Gradus ad Parnassum, in the form "mi contra fa est diabolus in musica"; and the German composer Johann Mattheson writes in 1739 that "the old singers called this interval mi contra fa, or the devil in music." None of these three sources dates from the Middle Ages itself.

What the Middle Ages genuinely did avoid, on the other hand, is the interval itself, for reasons far more practical than theological. The Italian monk and theorist Guido d'Arezzo, at the very start of the eleventh century, built his hexachord solmisation system in a way that specifically avoided the encounter between F and B, allowing a B♭ that closes off the hexachord without ever forming this tritone, though he himself never used the word diabolus. The real difficulty was one of accuracy: an interval this unstable, with no simple consonant anchor to guide the singer's ear, was simply hard to pitch correctly a cappella, a technical constraint of choral practice rather than a formal religious ban.

That same tritone, today, is no longer a bogeyman: it's actually one of the most heavily used engines of tonal harmony, present between the third and the seventh of every dominant seventh chord, where its tension mechanically calls for resolution to the following chord, a mechanism detailed by another guide in this corpus devoted to seventh chords. The legend of its medieval banishment dies hard precisely because the instability it describes is, itself, very real: only the religious explanation for that instability is myth, not the instability itself.

An augmented fourth, a diminished fifth: same keys, different functions

The letter-naming rule laid out earlier in this guide produces a result that throws almost everyone the first time they meet it: two intervals that sound identical on a piano or a guitar can carry two completely different names depending on which note is chosen to write them. The tritone offers the clearest example of this in the whole system. From C, going up an augmented fourth leads to F♯: C, D, E, F form the four letters of a fourth, and the sharp adds the semitone that widens it. From that same C, going up a diminished fifth leads to G♭: C, D, E, F, G form the five letters of a fifth, and the flat removes the semitone that narrows it. F♯ and G♭ land on exactly the same black key of the piano, six semitones from C in both cases, and yet one is called an augmented fourth, the other a diminished fifth: two intervals different in name, identical in sound, a phenomenon theory calls enharmonicity.

This difference in name isn't just a notational choice left to chance or to a composer's habit: it carries information about the expected direction of the melodic movement, a convention inherited directly from classical multi-voice writing. An augmented interval naturally tends to widen further, to resolve outward: F♯, as an augmented fourth above C, pulls upward, toward the G that follows it a semitone above, like a local leading tone drawing the ear toward the next note of the scale. A diminished interval, conversely, naturally tends to narrow further, to resolve inward: G♭, as a diminished fifth above C, pulls downward, toward the F that precedes it a semitone below. The same black key on the piano thus carries two strictly opposite directions of resolution depending on the name given to it, and it's precisely this directional information, invisible to the ear alone but explicit on a score, that the choice of name is meant to convey.

This same logic reappears, in negative, in the single strangest property of the diminished seventh detailed in this corpus's guide on suspended, diminished and augmented chords: because that chord divides the octave into four rigorously equal minor thirds, each of its four notes can, depending on context, be notated and function as the root, which multiplies its possibilities for resolving into different keys. The tritone does exactly the same thing on a more modest scale, with two notes rather than four: one and the same pair of keys, two names, two possible directions of resolution, and it's this functional ambiguity, not just an acoustic one, that explains why composers have exploited it as a pivot for modulation for centuries, long before jazz harmony generalised its use under the name tritone substitution.

Recognising an interval by ear, without a chord chart or a song

Naming an interval on paper isn't enough to recognise it in the heat of a song: the theory laid out in the previous sections only becomes a usable tool once it's paired with a sound landmark memorised once and for all, without having to recount semitones every time. The principle is always the same: pair each interval with a familiar tune that opens with exactly that gap, then recall that tune from memory as soon as an unfamiliar interval turns up, rather than trying to hear it in the abstract.

The perfect fourth and the perfect fifth, the two most commonly used perfect intervals after the octave, are often recognised thanks to an instrument that imposes them physically: a natural horn or bugle, with no valves or slide, can only produce the notes of its tube's harmonic series, and the very first gaps available in that series, before the degrees narrow down into seconds, are precisely an octave, then a fifth, then a fourth. It's directly for this acoustic reason that military bugle calls and traditional horn calls, built on these available notes alone, almost always leap by a fourth or a fifth rather than moving in steps: the horn leaves them no other choice in its low register. A two-tone siren, of the kind some emergency vehicles use, generally alternates between two fairly close notes instead, often somewhere around a second or a third depending on the model, a good landmark for the ear once you know it's a small, repeated interval rather than a big leap.

The major second, the smallest interval that moves stepwise within a major scale, is heard most simply in the scale itself played note by note, ascending or descending: every step of that ladder, with two exceptions detailed in this corpus's guide on major and minor scales, is precisely a major or minor second. A public-domain anthem like Beethoven's Ode to Joy, which became a European emblem long after its composition in 1824, opens with a melody that moves almost entirely through these small stepwise motions, an excellent training ground for the ear before tackling wider leaps.

For the third and the sixth, a simple two-note doorbell chime, very common on front doors, most often descends a third, major or minor depending on the model: a domestic landmark, available without an instrument, for fixing the colour of that gap. The ascending major sixth, wider and less spontaneous to sing accurately on the first try, trains well on a very old traditional tune popularised in summer camps and English-speaking scout movements, which opens on exactly that leap before coming back down in stepwise motion.

The ascending perfect fifth, beyond the horn already mentioned, can also be recognised in the opening of a symphonic poem by Richard Strauss composed in 1896, popularised for a completely different audience in 1968 when a film director used it as the opening theme for his best-known science-fiction film: two notes a fifth apart, followed by a fourth that closes the octave, a progression that helps fix both the fifth and the fourth in memory at once. The tritone, finally, is recognised by its characteristic instability more than by any one precise tune: a film composer built, in 1975, one of the most recognisable motifs in cinema on a neighbouring interval, two notes alternating a single semitone apart, the minor second, one of the most dissonant gaps in the system alongside the tritone, for a tension effect that feels almost physical from the first two notes.

None of these landmarks replaces repeated practice: perfect pitch, the ability to name an interval with no point of comparison at all, remains rare, but relative pitch, the ability to recognise a gap by mentally comparing it to a memorised tune, can be trained at any age and is more than enough to accompany, harmonise or transpose a song by ear.

Beyond the octave: compound intervals

Nothing forces an interval to stay within a single octave: two notes can just as well be an octave and a half apart, two octaves apart, or separated by any distance wider still. Beyond the octave, the naming system carries on with exactly the same letter-counting logic, without changing the rule at all: you simply keep counting degrees past the eighth. A ninth picks up the second's number and adds a full octave to it; a tenth picks up the third's; an eleventh the fourth's; a thirteenth the sixth's. The general rule boils down to a single arithmetic operation, adding seven to the number of the equivalent simple interval, or, which amounts to the same thing, adding a full octave to its pitch.

These compound intervals share the quality of the simple interval they derive from: a major ninth has the same quality as a major second, simply transposed an octave higher; a minor thirteenth has the same quality as a minor sixth. What changes is therefore never the gap's intrinsic colour, only its span: a ninth sounds more open, more spread out than a second squeezed inside a single octave, exactly as a widely spread chord sounds different from the same chord voiced in a tight cluster, even though the two contain rigorously the same notes.

This numbering beyond seven isn't just an arithmetic exercise reserved for theorists: it's exactly the vocabulary that extended-chord notation borrows to name the notes it adds beyond the seventh, ninth, eleventh and thirteenth, without ever renumbering those notes as if they were merely plain seconds, fourths or sixths folded back inside the octave. This guide doesn't get into how these extended chords are built, a subject in its own right covered by this corpus's guides on seventh chords and on suspended, diminished and augmented chords; it only lays down the brick that makes their names legible: why a ninth is called a ninth and not a second, even though the note actually played, once folded back inside an octave, is precisely a second.

The invisible thread: where this system resurfaces throughout the hub

Once this vocabulary is fixed, number, quality, semitones, inversion, it becomes visible everywhere else in this corpus, even though no other guide stops to name it. The major scale and its three minor forms, described in this corpus's guide on scales for beginners, are nothing more than a fixed run of major and minor seconds stacked degree by degree: the tone-tone-semitone formula that defines them is an interval formula, expressed in the vocabulary laid out here. A seventh chord, whether dominant, major or minor, stacks three thirds on top of one another above a root, as the guide devoted to those chords details; suspended, diminished and augmented chords replace, widen or narrow one of those stacked thirds to get different colours. A vocal harmony for two or three voices chooses, note after note, between a third and a sixth above or below an already-sung melody, exactly the two intervals this guide has just named and measured.

The circle of fifths, for its part, is never anything more than one single operation repeated twelve times over, going up or down a perfect fifth, which makes it probably the purest illustration of what a simple interval can generate once repeated systematically: all twelve major keys and their key signatures follow from it entirely, with not a single note chosen at random. Letter-name notation, finally, names the same seven letters as French solfège, in the same order, with the same inclusive counting logic detailed earlier in this guide for fixing an interval's number, whichever alphabet is used to write the starting note.

None of these guides lingers on this shared mechanism, and that's precisely why none of them duplicates this one: each assumes the reader already knows what a third or a fifth is, so it can focus on what gets done with it, a chord, a harmony, a scale, a cycle of keys. This guide doesn't do what they do: it names, measures and explains the brick itself, the one all the others borrow without ever unwrapping it.

Common pitfalls to avoid

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