Five notes, and no semitones

The five black keys of a piano have a well-known property. Played in any order, in any combination, at any speed, they produce something that sounds deliberate. A listener with no training can improvise on them for a minute and produce nothing objectionable. The demonstration is a staple of music workshops and it always works.

The usual explanations are that the pentatonic is primordial, or universal, or built into the ear. None of those is needed. The property has a cause, the cause is arithmetic, and it says exactly how many notes the trick survives.

The derivation #

Start on C and go up by fifths. C, G, D, A, E. Fold those into one octave and sort them: C, D, E, G, A. That is the major pentatonic, and no decision was made anywhere in the process except when to stop.

Take one more fifth — B — and the collection becomes C, D, E, G, A, B. Now B and C are adjacent, one semitone apart, and the property is gone.

So the pentatonic is not five notes somebody selected for their pleasantness. It is the longest run of the chain of fifths that contains no semitone, and the number five is the answer to a question rather than a preference.

The same walk explains the diatonic set as well. Seven notes is the longest run that contains no two adjacent semitones — the two it has are as far apart as they can be — and that is the property the major scale is built on. Five and seven are consecutive answers to the same question asked with different strictness.

Which computation produced the number #

The figure walks the chain rather than being told the answer. At each length $k$ it takes the pitch classes $\{7i \bmod 12 : i < k\}$, sorts them, and measures every gap between consecutive members including the one that wraps round the octave. The smallest of those gaps is printed beside each row.

For $k = 5$ the sorted set is $\{0, 2, 4, 7, 9\}$ and the gaps are $2, 2, 3, 2, 3$ — summing to 12, as they must. The smallest is 2.

For $k = 6$ the set is $\{0, 2, 4, 7, 9, 11\}$ and the gaps are $2, 2, 3, 2, 2, 1$. The smallest is 1, and the row is marked accordingly.

Nothing in the generator knows that five is supposed to be special. It measures, and five is where the measurement changes.

What is in it, and what is not #

The absence of semitones is only half the story, and the other half is better.

A five-note set has ten pairs of notes in it. Counting what interval class each pair belongs to gives the set’s interval content, which is a complete description of what can be sounded together in it. For the pentatonic the count is: no semitones, three whole tones, two minor thirds, one major third, four fourths, and no tritones.

Two of those six numbers are zero, and they are the two that matter. The semitone is the roughest interval available and the tritone is the most unstable, and a set that contains neither of them cannot produce either, in any voicing, in any order.

Run the same count on the major scale and it comes out as two semitones, five whole tones, four minor thirds, three major thirds, six fourths and one tritone. Twenty-one pairs, of which three are the awkward ones. The pentatonic’s ten pairs contain none.

That is the whole of the workshop demonstration. It is not that the pentatonic is beautiful. It is that it has been stripped of the two intervals capable of sounding wrong, and what is left cannot be assembled badly.

And no other five notes will do #

The two zeros are a property the pentatonic has, and the more interesting question is how many other five-note sets have it. There are 792 of them, falling into 66 classes under transposition, and the count is small enough to do exhaustively.

One. The two conditions the workshop demonstration rests on pick out the pentatonic uniquely among five-note sets, and there is nothing else to choose. The other two semitone-free necklaces are 0 2 4 6 10 and 0 2 4 6 9, whose step patterns are 2 2 2 4 2 and 2 2 2 3 3 — both of them mostly whole tones with a gap thrown in, both of them carrying tritones, and neither of them a scale anybody uses.

So the derivation by fifths and the derivation by exclusion arrive at the same object from opposite directions. One says: walk the chain and stop before the semitone. The other says: of every way of choosing five notes from twelve, name the ones that cannot form a semitone or a tritone. The first is a procedure that produces one answer and the second is a filter that admits one, and both name the same set.

The seven-note case is worth putting beside it because the filter behaves completely differently there. No seven-note set of twelve is free of tritones at all — seven notes cannot avoid it, since removing five notes cannot remove all six tritone pairs — so the diatonic set’s single tritone is not a choice its designer made but the minimum available. The pentatonic avoids the interval; the diatonic set carries as little of it as anything of its size can.

That reframes the relationship between the two scales that the chain-of-fifths derivation presents as one of degree. Five and seven are consecutive stopping points on one walk, and they are on opposite sides of a threshold: below it a set can be built with no unstable interval in it, and above it every set has one and the question becomes how few.

Every rotation works #

There is a second property, and it is the one that makes the pentatonic useful rather than merely safe.

Rotate the set to start on each of its five notes in turn and the step patterns are $2\,2\,3\,2\,3$, $2\,3\,2\,3\,2$, $3\,2\,3\,2\,2$, $2\,3\,2\,2\,3$ and $3\,2\,2\,3\,2$. Every one of them still has a smallest gap of 2. So every rotation is itself an anhemitonic pentatonic, and every note of the set can serve as a tonic without the set acquiring a semitone.

That is not true of the diatonic set. Its seventh rotation — Locrian — has a diminished fifth above its tonic, which is the tritone, and the resulting tonic triad will not hold still. Locrian is the mode nobody writes in, and the reason is structural rather than aesthetic.

The consequence for a player is direct: on a pentatonic set, the choice of tonic is free. A melody can settle anywhere and the settling will sound like a decision. On a diatonic set it cannot, which is why modal writing has a literature about which modes work and pentatonic writing does not.

The same set, arrived at from somewhere else #

There is a second construction that produces the pentatonic, and it has nothing to do with fifths.

Ask for the five positions out of twelve that are as evenly spread as twelve allows. Twelve does not divide by five, so perfect spacing is impossible and the best available is $\operatorname{round}(12k/5)$ for $k = 0 \ldots 4$, which gives $\{0, 2, 5, 7, 10\}$. Sorted, its steps are $2\,3\,2\,3\,2$ — a rotation of the pentatonic.

Do the same for seven positions and $\operatorname{round}(12k/7)$ gives $\{0, 2, 3, 5, 7, 9, 10\}$, whose steps are $2\,1\,2\,2\,2\,1\,2$: a rotation of the diatonic set.

Two entirely unrelated procedures — stop a chain of pure fifths, and spread points as evenly as possible round a circle — produce the same two collections. One is an acoustic construction and the other is a purely combinatorial one, and neither knows about the other.

That agreement is worth being suspicious of, so it is worth checking where it fails, and the check is cheap enough to run at every size rather than at one. Comparing the k-note prefix of the chain with the k-note maximally even set, up to rotation, for every k from two to eleven: they agree at five, at seven and at eleven, and disagree at the other seven sizes.

Six is the clearest failure. The maximally even six-note set is $\{0, 2, 4, 6, 8, 10\}$, the whole-tone scale, and the six-note prefix is $\{0, 2, 4, 7, 9, 11\}$, which is something else entirely. Eleven agrees for a reason that is not interesting — both are the twelve with one note missing, and there is only one such necklace — so the substantive coincidence is exactly five and seven.

Two of ten, and the two are the two scales the world uses. That is a real property of those two numbers rather than a general fact, and it makes the agreement interesting rather than inevitable.

This dual origin — maximal evenness on one side, generation by fifths on the other — is the beginning of a considerable body of work in mathematical music theory. The result that a set generated by a single interval is maximally even exactly when the numbers work out is due to Clough and Douthett, in 1991, and five and seven in twelve are its two headline cases.

No small steps, and no leading note #

The absence of semitones has a melodic consequence as well as a harmonic one, and it is the more audible of the two.

Every adjacent pair in a pentatonic set is a whole tone or a minor third. There are no small steps at all, which means a melody moving from one note to the next always moves a noticeable distance. Pentatonic melody is therefore closer to being made of leaps than diatonic melody is, and the characteristic open, unhurried quality that gets described as “folk-like” is largely that.

The second consequence is the one a harmony textbook would notice. A leading note is a note a semitone below the tonic, and its pull is a matter of roughness and resolution rather than of convention. A pentatonic set has no note a semitone below anything, so it has no leading note, and cadence in it cannot work by semitone approach.

What it works by instead is the fourth and the fifth — the intervals the set has four of. A pentatonic melody arrives home by descending a fourth or a fifth onto the tonic far more often than by any other means, which is a pattern visible in Chinese, Scottish and West African repertoire alike, and it is what is left when the semitone is not available.

Where it comes from, physically #

The chain of fifths is not an arbitrary starting point, and it is worth saying why.

Anything that resonates produces partials at whole-number multiples of its fundamental, and the first two intervals in that series are the octave and the fifth: the second partial is an octave above the fundamental and the third an octave and a fifth. They are the intervals with the most coinciding partials and therefore the least roughness, and they are what a tuning procedure gets built out of because they are the easiest two to set by ear — a pure fifth beats at nothing at all.

Anything that resonates produces partials at whole-number multiples of its fundamental, and the first two intervals in that series are the octave and the fifth. They are the intervals with the most coinciding partials and therefore the least roughness, and they are what a tuning procedure gets built out of because they are the easiest two intervals to set by ear — a pure fifth beats at nothing at all.

So a chain of fifths is what any culture with strings, pipes or bells will find first, and stopping that chain somewhere is the only decision. Five and seven are the two places where stopping produces something with a useful structural property, and both of those places have been found repeatedly and independently.

Where the model stops #

“Pentatonic” does not mean one scale. Everything above concerns anhemitonic pentatonics — the ones with no semitone. There are pentatonic scales with semitones in them, and they are in wide use: the Japanese in scale runs roughly 0, 1, 5, 7, 8, which has two semitones and is not reachable by any prefix of the chain of fifths. The workshop demonstration does not work on it, and the scale is none the worse for that.

Slendro is not built this way at all. Javanese slendro has five notes to the octave that are close to equally spaced — steps of around 240 cents each, varying from gamelan to gamelan. That is not a chain of fifths cut short; it is a different construction that arrives at five notes for different reasons, and its intervals are not approximations of anything in this essay.

Nothing here says the pentatonic sounds good. It says the pentatonic cannot form the two intervals that most reliably sound bad. Those are different claims, and the second is much weaker. A set can be free of roughness and still be dull, and the reason the pentatonic is not dull has to do with what melodies do in it, which this argument does not address.

The interval count assumes octave equivalence. Counting pairs by pitch class treats a major sixth and a minor third as the same interval. That is standard and it is an assumption; a voicing where two notes are two octaves apart is not the same event as one where they are adjacent, as the critical band makes very clear.

The fifths are assumed pure, and then folded as though they were not. The chain that generates the set is a chain of 3:2 fifths, and folding those into one octave and calling the results pitch classes quietly assumes the comma has already been dealt with. In a system that has not dealt with it, the five notes are not quite the five notes drawn here.

Whose music, and when #

The chain-of-fifths derivation is not a modern reconstruction. It is written down.

Chinese theory has the 三分损益 method — sanfen sunyi, “subtract and add a third” — recorded in the Guanzi and the Lüshi Chunqiu by the third century BC. A string length is reduced by a third and then increased by a third alternately, which generates exactly a chain of fifths and fourths, and the first five notes it produces are the wu sheng: gong, shang, jue, zhi, yu. The procedure is the one in the figure above, described two and a half thousand years ago, and the stopping point at five is explicit.

The scale is then extended to seven and eventually to twelve lü by the same method, which runs into the same comma that European theory met independently.

Anhemitonic pentatonics are widespread beyond that: in Scottish and Irish traditional melody, across much of West Africa, in Andean music, in Hungarian folk song as Bartók and Kodály catalogued it, and in the blues, whose minor pentatonic is the fourth rotation of the set above with the blue notes added on top. The frequency of the pattern across unconnected traditions is real and is the source of the “universal” claim.

The claim overreaches, though. What is common is not a scale but a procedure — tune by fifths, stop before it gets awkward — and cultures that tune by other procedures arrive elsewhere.

Slendro is the standing counterexample, and it is worth taking seriously rather than filing as an exotic variant. It has five notes; it is not the pentatonic; its steps are near-equal where the pentatonic’s are two sizes; and it is tuned to the ensemble rather than to a standard, so the notion of a scale as a set of fixed pitches does not survive contact with it. Five notes to the octave is a much weaker coincidence than it looks, because five is a convenient number of things to hit.

The ladder from here #

Later rungs on this anchor: the diatonic set’s own derivation and the properties it has that the pentatonic lacks — Myhill’s property, the fact that every interval comes in exactly two sizes, and why that produces a sense of key. Hexatonic and octatonic sets, and what changes when the chain stops in the wrong place. Maximal evenness as a general construction, and which of the useful scales it produces. The blues scale, and what the blue notes are doing to a set that had no semitones by design. And the five-note systems built by other procedures, of which slendro is the best-documented.

Five black keys, and the reason they behave is that there was no sixth fifth to take.