Seven of the twelve, chosen unevenly

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.

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 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. Counting rather than saying so: of the 66 distinct seven-note shapes in the twelve, exactly one is a contiguous run on the circle of fifths, and it is this one. Not one of a few — the only one, because a contiguous run is what “shifted along a cycle” means and every shift of it is the same shape. 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.

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.

That claim is also a count rather than an assertion, and the distribution is steep:

Six is the maximum and one shape reaches it. The reason is the same reason it is the only contiguous run: the fifths make a twelve-cycle, choosing seven vertices from a twelve-cycle gives at most six adjacent pairs, and six is achieved only by taking seven in a row. So “the most fifths” and “an unbroken run” are two descriptions of one thing, and the diatonic set is the unique object that satisfies either.

The median shape has four. So a seven-note set drawn at random from the twelve carries two thirds of the fifths this one does, and the gap between the best and the typical is not enormous — what is unusual is not the number but that exactly one shape achieves it, which is what makes the property a definition rather than a ranking.

The two counts also settle a question the section on the tritone raises and cannot answer from inside itself: whether packing fifths and having one tritone are the same requirement or two. They are not the same. Sixteen shapes have a single tritone and only one has six fifths, so the tritone condition is fifteen times looser and the fifths condition implies it rather than the other way round — a contiguous run of seven on the fifths cycle spans an interval of six fifths, which is one tritone, and a set with fewer fifths has its notes spread over more of the cycle and picks up more of them. One property is doing all the work and the other is its consequence, which is worth knowing before either is offered as a reason the scale is what it is.

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.

Against the simple ratios the twelve equal steps land unevenly: the fifth misses by 2 cents and the fourth by 2, and the thirds and sixths miss by 14 to 16 — so a division that gets the strong coincidences almost exactly right pays for it on the weaker ones. Where the semitones fall inside the seven is what decides which degrees are near which, and it is fixed by the chain rather than chosen.

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 is not an option, and saying so is worth a paragraph because the usual framing offers it as one. The six tritone pairs partition the twelve pitch classes, so any seven notes must complete at least one of them — a set of seven cannot fit inside six pairs while taking at most one from each. Every seven-note scale in the twelve has a tritone, whatever anybody wanted. The counting confirms the pigeonhole: over the 66 shapes, sixteen have one tritone, forty have two and ten have three, and none has none.

So the choice was never between having a tritone and not; it was between one and more. 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.

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.

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 $\binom{12}{7} = 792$ 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.

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.

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.