Acoustics & Music Theory
why it sounds like that
Scales and modes

Seven of the twelve, chosen unevenly

A major scale is a selection of seven positions out of twelve, and the selection is lopsided on purpose. The two semitones sit where they do because an even choice would destroy the thing that makes a scale usable.
15 min read 8 figures The same shape, movedSmall whole numbers

The major scale is usually introduced as a sequence: tone, tone, semitone, tone, tone, tone, semitone. That is accurate and it hides the interesting part, which is that the sequence is not symmetrical and could not usefully be.

The major scale as a cycleThe twelve semitones drawn as a cycle, with the notes of the scale filled in. The gaps between filled positions are the step pattern, and reading them round the circle is what makes the scale's asymmetry obvious.CC♯DE♭EFF♯GA♭AB♭B2 · 2 · 1 · 2 · 2 · 2 · 1steps, in semitonesthe short chords are the semitones — two of them, unevenly spaced
Fig. 1 Twelve semitone positions drawn as a cycle, with the seven of the major scale filled in and the gaps between them joined. The two short chords are the semitones. They are not opposite each other, not evenly spaced, and not adjacent — and every one of those three facts is load-bearing.

Seven filled positions out of twelve. Five gaps of two semitones and two gaps of one, arranged so that the two short gaps are separated by two steps in one direction and three in the other.

Any other arrangement of five tones and two semitones is also a scale, and several of them are in use. But this particular arrangement has properties the others do not, and they are what a scale is for.

What an even division would cost

The fastest way to see why the asymmetry matters is to remove it.

Divide the octave into six equal steps and the whole-tone scale results. It is a perfectly good set of pitches; Debussy used it extensively and it has a distinctive sound. It is also, in a specific and fatal sense, featureless.

In a whole-tone scale every note has exactly the same relationship to every other note. Start on any of the six and the pattern ahead is identical. There is no way to tell, from the intervals alone, which note is the first one — the scale has no landmarks, and therefore no home.

The same is true of the diminished scale, the augmented scale, and every other equal division. Symmetry destroys position. A listener cannot locate themselves in a pattern that looks the same from everywhere.

The circle of fifthsThe twelve pitch classes arranged so that each is a fifth above the last. Keys next to each other differ by one sharp or flat, which is why the notes of a key form a contiguous arc rather than a scattering.CaGeDbAf♯Ec♯Ba♭F♯e♭C♯b♭A♭fE♭cB♭gFdone step= one fifth= one sharpouter ring: major keys · inner ring: their relative minors
Fig. 2 The twelve pitch classes arranged by fifths, with the notes of one major scale filled. They form a contiguous run of seven — not a scattering — which is the property that makes the scale a coherent object rather than an arbitrary selection.

The major scale’s asymmetry means every degree has a unique interval signature. The note a semitone below its upper neighbour and a tone above its lower one is the seventh degree, and it is the only degree that is; a listener who has heard a few notes can work out where they are and hear the tonic as a destination rather than a convention.

The contiguity property

There is a much stronger statement available about the major scale, and it is the one that explains why this selection rather than another.

Arrange the twelve pitch classes by fifths rather than by pitch. The seven notes of any major scale form an unbroken run of seven adjacent positions. F, C, G, D, A, E, B — that is C major, and the run has no gaps.

That is a remarkable property and not a common one. Most seven-note selections out of twelve scatter across the circle of fifths. The diatonic set is one of very few that do not, and the consequences follow immediately:

  • Every note is a fifth away from another note of the scale, except the two at the ends of the run.
  • The scale contains six perfect fifths and exactly one tritone — the maximum number of fifths a seven-note set can have.
  • Two scales whose runs overlap in six positions differ by one note, which is why neighbouring keys are neighbours.
Neighbouring keys differ by one noteThe seven notes of several major keys, laid out against the chromatic scale. Keys a fifth apart share six of their seven notes, and the one that differs is the note that changes the key signature.CC♯DE♭EFF♯GA♭AB♭BE♭ majorB♭ majorF majorC majorG majorD majorA majorone note enters and one leaves for each step round the circle
Fig. 3 The seven notes of several major keys against the chromatic scale. Keys a fifth apart share six of their seven notes, and the one that differs is what a key signature records. That relationship exists because each key is a contiguous run of fifths, shifted by one.

So the scale is not an arbitrary lopsided selection. It is the seven-note set that packs in as many perfect fifths as possible, and the lopsidedness is what packing them in produces.

Where the semitones have to be

Given seven notes in a chain of fifths, the step pattern is determined — there is no further choice to make.

Take the run F–C–G–D–A–E–B and sort it by pitch: C, D, E, F, G, A, B. The gaps come out 2, 2, 1, 2, 2, 2, 1 automatically. The two semitones appear at exactly the places where the fifth-chain wraps, and their position is a consequence of the contiguity rather than an independent decision.

This is worth restating because it inverts the usual teaching order. The step pattern is normally given first, as a formula to memorise. It is better understood as an output: choose seven notes that are as connected by fifths as possible, and the tone-tone-semitone pattern falls out with nothing left to decide.

The simple ratios, and the twelve equal stepsOne octave laid out in cents. Above the line, the frequency ratios of small whole numbers, where they actually fall; below it, the twelve equal steps. The two sets almost never coincide.6/5minor third5/4major third4/3fourth3/2fifth8/5minor sixth5/3major sixthCC♯DE♭EFF♯GA♭AB♭BC+16-14-2+2+14-16the ratios of small whole numberstwelve equal steps of exactly 100 cents1200 cents to the octave
Fig. 4 One octave in cents, with the ratios of small whole numbers where they fall. The scale’s notes cluster near these ratios, which is not a coincidence — a chain of fifths generates them, and the ear was already listening for them.

The tritone, and why one is exactly right

A seven-note diatonic set contains exactly one tritone, and that number is neither an accident nor a nuisance.

Zero tritones would mean a completely consonant set with no internal tension and no way to establish a key. Two or more would mean multiple points of maximum instability, and no single one of them could function as a signpost.

One tritone means there is exactly one maximally unstable interval in the scale, it occurs between the fourth and seventh degrees, and every listener within the tradition knows where it wants to go — a resolution that is two voices moving by a semitone each. The dominant seventh chord contains it; the resolution of that chord is the strongest key-defining gesture the system has; and the whole machinery of functional harmony is built on the fact that there is precisely one of them.

Roughness across an octaveSensory dissonance between two complex tones as the upper one is swept through an octave, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The deep wells land on the fourth, the fifth and the octave; the thirds sit on shoulders rather than in wells, which is a real feature of this model and not a defect of the drawing.semitones above the lower tone6/55/44/33/25/32/1CC♯DE♭EFF♯GA♭AB♭BCroughest at about a semitonesmooth at the simple ratios
Fig. 5 Roughness across an octave. The tritone sits on high ground between the fourth’s well and the fifth’s — it is not the roughest interval available, but it is the roughest one inside a diatonic scale, and its position between two deep wells is why it is heard as needing to move.

The pentatonic case

Take the same argument and stop the chain of fifths earlier, and something equally well-formed comes out.

Five notes in a contiguous fifth-run — F, C, G, D, A — sorted by pitch give C, D, F, G, A. The step pattern is 2, 3, 2, 2, 3: no semitones at all, and no tritone.

The pentatonic scale as a cycleThe twelve semitones drawn as a cycle, with the notes of the scale filled in. The gaps between filled positions are the step pattern, and reading them round the circle is what makes the scale's asymmetry obvious.CC♯DE♭EFF♯GA♭AB♭B2 · 2 · 3 · 2 · 3steps, in semitonesthe short chords are the semitones — two of them, unevenly spaced
Fig. 6 The pentatonic scale as a cycle. Five notes, gaps of two and three semitones, and no short chords at all — there are no semitones in this set, which is why nothing in it is unstable and why any two of its notes sound acceptable together.

That is the anhemitonic pentatonic scale, and it appears independently in Chinese, Japanese, Scottish, West African, Andean and Native American traditions. The convergence is often described as mysterious. It is not: it is the five-note contiguous fifth-run, it is the set with the fewest dissonances available, and any tradition that builds a scale from stacked fifths will find it.

Its property is the opposite of the diatonic scale’s. With no semitones and no tritone, nothing in it is unstable, so any note can be played against any other without a wrong answer. That makes it forgiving — which is why it is what a beginner is given, and why it is what a blues improviser reaches for over a chord sequence that is changing underneath.

Counting what is available

A small piece of arithmetic makes the scale’s distinctiveness concrete rather than rhetorical.

There are (127)=792\binom{12}{7} = 792 ways to choose seven pitch classes out of twelve. Allowing for rotation — since a scale started on a different degree is the same necklace — that collapses to 66 distinct seven-note patterns.

Of those 66, exactly one is a contiguous run on the circle of fifths. Exactly one has six perfect fifths and one tritone. Exactly one has the property that every interval size appears in exactly two varieties — every second is either major or minor, every third either major or minor, and never three sizes of anything.

That last property has a name, Myhill’s property, and the sets that have it are called well-formed. The diatonic set has it, the pentatonic set has it, and among seven-note sets in twelve-tone space the diatonic set is alone.

So when the major scale is described as special, that is not a cultural loyalty dressed up as mathematics. It is one out of sixty-six, distinguished by several independent criteria that all pick out the same one.

Two sizes of everything

The two-varieties property deserves unpacking, because it is what makes the scale learnable.

In a major scale there are seconds, and they come in two sizes — the tone and the semitone. There are thirds, in two sizes: major and minor. Fourths come in two sizes, perfect and augmented; fifths in two, perfect and diminished; sixths and sevenths in two each.

Never three. A musician learning the scale has to learn, for each interval class, which degrees give the large version and which give the small — a manageable amount of information, and the same amount for every interval class.

The seven modes, brightest firstThe same seven pitch classes started on each of its degrees in turn, ordered by how many of their notes are raised. Each row differs from the one below it by exactly one note, and that note moves down one semitone each time.CC♯DE♭EFF♯GA♭AB♭BCLydianIonianMixolydianDorianAeolianPhrygianLocrianbrightest at the top, darkest at the bottomeach row drops exactly one note by a semitone from the row above
Fig. 7 The same seven pitch classes started on each of its degrees in turn. Every row has the same two-sizes property, because rotating a necklace does not change which sizes it contains — only which degree each size falls on.

A set without the property would need three or four sizes of third, and the names would multiply. That is exactly what happens in the octatonic scale, and it is part of why octatonic music is harder to hear one’s way around in.

The scale is a filter as much as a palette

The last thing worth saying about a seven-note selection is what it does to the five notes it leaves out.

A chromatic note in a diatonic context is not merely an additional colour. It is heard against the scale — as a deviation, an inflection, a note borrowed from somewhere. That reading is only available because the scale established an expectation for it to depart from.

The result is that a seven-note scale gives twelve usable notes rather than seven: seven that are inside and five that are audibly outside, and the outside ones carry meaning precisely because they are outside. A twelve-note scale gives twelve notes and no outside.

That is the deepest reason not to use all twelve. Not that seven sound better, but that a subset creates a context, and context is what makes a note mean something beyond its pitch. Modes exploit exactly this: the same seven notes, a different tonic, and every degree’s meaning reassigned.

The diatonic set is old and widespread, and the claims made about it are routinely too strong.

It is present in ancient Greek theory as the diatonic genus, alongside chromatic and enharmonic genera that used quarter-tones and that European practice abandoned. It appears in Mesopotamian tuning instructions from the second millennium BC, on tablets that describe tuning a lyre through a chain of fifths and fourths and which produce diatonic sets. It is the basis of medieval European modes and of a great deal of folk music across Europe and Asia.

It is not universal. Indonesian sléndro is close to five equal steps and is not a subset of twelve. Arabic maqam systems use intervals between the semitone and the tone that no twelve-note keyboard can produce. Indian classical music works with twenty-two śruti and treats certain notes as continuously variable within a range.

The safe claim is this: a chain of fifths generates the diatonic set, and traditions that tune by fifths tend to find it. That is a claim about a method, not about human nature, and it explains the convergences without asserting anything about what music must be.

Notes on the keyboardA piano keyboard with the notes under discussion marked. The keyboard is used throughout this site because it shows distance rather than name, and distance is what the theory is about.CEG3 notes sounding
Fig. 8 The keyboard, which encodes one particular answer. The white keys are a diatonic set and the black keys are a pentatonic one, which is a physical accident of an instrument designed around one repertoire — and which has shaped how the entire subject is taught ever since.

The pattern is the object

One last property, and it is the one that makes the scale portable.

A scale is a pattern of intervals, not a set of pitches. Move all seven notes up by the same amount and the pattern is unchanged, and every listener hears the same scale. That is why there are twelve major scales rather than one, and why they are all the same scale.

The independence is not quite free. In any tuning except equal temperament, transposing a scale changes the actual interval sizes slightly, because the twelve semitones are not all identical. In meantone, C major and F-sharp major are genuinely different scales with different internal proportions. Equal temperament is precisely the arrangement that makes transposition exact — and that is what it was adopted for.

Why not six, or eight

Seven is not the only well-formed size, and the alternatives explain why seven won.

Five works — the pentatonic set is contiguous on the circle of fifths and has the two-sizes property. It is forgiving and it is limited: no semitones means no leading note, so a key can be suggested and not asserted.

Six does not work at all as a contiguous run, because six consecutive fifths produce a set with a step pattern that includes a three-semitone gap and no tritone, and the resulting scale has weak internal landmarks.

Eight produces a run containing two tritones and two adjacent semitones. It is usable — the octatonic and bebop scales are eight-note sets — and it is unstable: two tritones means two competing points of maximum tension, and no single one can act as the signpost that defines a key.

Seven is the largest set that keeps exactly one tritone, and that is the property the whole tonal system is built on. It is also, not incidentally, close to the number of items that can be held in working memory, which may be why it is a comfortable size to think in.

Where the model stops

Twelve is assumed. The whole framing selects seven from twelve. Traditions that do not divide the octave into twelve have scales this analysis cannot describe at all, not approximately.

Equal steps are assumed. The necklace figure draws twelve equal positions, which is true only in equal temperament. In any historical tuning the beads are unevenly spaced and the picture is a schematic rather than a measurement.

A scale is not a set. Real scales have hierarchy — a tonic, a dominant, notes that are stable and notes that are passing — and none of that is visible in a picture of which positions are filled. Two traditions can use the same seven pitches and different hierarchies, and a maqam or a raga is defined by far more than its pitch content: characteristic phrases, ascent and descent patterns, ornaments attached to particular degrees. The necklace shows the least interesting thing about them.

Melodic behaviour is invisible. The figures here show which notes exist and say nothing about which follow which, and that is most of what a scale is in practice.

The ladder from here

Later rungs: the modes, and what changes when the same seven start elsewhere. Well-formedness and Myhill’s property, which formalise what makes this set special. The pentatonic in detail. Harmonic and melodic minor, and what a raised seventh is for. Maqam, raga and the scales that are not sets. Ancient Greek genera. Scales derived from spectra rather than from fifths. The blues scale, which is not a scale. And the whole-tone and octatonic sets, which Debussy and Stravinsky used precisely because they have no home.

The Mesopotamian tuning tablets are three and a half thousand years old, and they describe tuning a nine-stringed lyre through a cycle of fifths and fourths. The instructions produce a diatonic set, and they are the earliest evidence anywhere that somebody worked out this scale rather than inherited it.

Where this goes next

CC♯DE♭EFF♯GA♭AB♭BCLydianIonianMixolydianDorianAeolianPhrygianLocrianbrightest at the top, darkest at the bottomeach row drops exactly one note by a semitone from the row above The same seven, started later A mode is not a new scale. It is the same seven notes with a different one treated as home, and the entire change in character comes from reassigning which degree the semitones fall next to. CaGeDbAf♯Ec♯Ba♭F♯e♭C♯b♭A♭fE♭cB♭gFdone step= one fifth= one sharpouter ring: major keys · inner ring: their relative minors Keys are neighbours, and the map is computed Two keys a fifth apart share six of their seven notes. That single fact generates the circle of fifths, the key signatures, the shape of nearly every modulation, and the sense that some keys are far away. one frame = one repeat of the combined wavelower — 2 per frameupper — 3 per frametheir sumthe pattern repeats after 2 cycles of the lower tone Two notes and a ratio, which is the whole of consonance Sound two tones together and the pair either settles or does not. What decides it is the ratio of their frequencies, and the rule is that simpler ratios settle — which is two and a half thousand years old and still not quite an explanation.