The only sizes a fifth will make

Three rungs of this ladder have described the same object without noticing it. The pentatonic is the longest run of the chain of fifths with no semitone in it. The diatonic set is seven of the twelve, and it has two sizes of every step, which is a property 448 of the other 461 seven-note selections do not have. And twelve fifths do not make seven octaves, which is a fact about the same chain and has been filed in a different field.

Put the chain on one axis and count.

What two step sizes buys #

A scale with two step sizes is one in which every generic interval — every second, every third, every fourth — comes in exactly two specific sizes. That is Myhill’s property, and the third rung of this ladder is about what it is worth: it is the condition under which a name can do useful work. A third is three semitones or four, and calling both of them thirds is allowed because there are only two to distinguish and they alternate predictably.

The two facts are the same fact. A chain of one interval whose folded step pattern uses two lengths is exactly a set whose generic intervals come in two sizes, and the proof is one line: every generic interval of a chain is itself a chain, shortened, so it inherits the two-length structure of the steps.

What is not obvious, and is the whole of this rung, is that the sizes at which this happens are a series with a rule. They are not scattered. They are not a matter of which sizes happen to sound good. They are 2, 3, 5, 7, 12, 17, 29, and they continue 41 and 53, and there is nothing between 7 and 12 and nothing between 12 and 17.

The rule is the comma’s rule #

A fifth is 701.955 cents and an octave is 1200. The chain closes when some number of fifths lands back on some number of octaves, and it never does, because log₂(3/2) is irrational — which is what a comma is, stated the other way round.

The near misses are the convergents of that irrational number’s continued fraction, and this site has been computing them since the sixth rung of the comma ladder. Written as fractions of an octave they are 1/2, 3/5, 7/12, 24/41, 31/53. The denominators are 2, 5, 12, 41, 53.

Those are five of the seven cardinalities above. The other two — 3, 7, 17, 29 — are the semiconvergents, the intermediate fractions that sit between one convergent and the next.

So the sizes at which a chain of fifths becomes a usable scale, and the sizes at which a chain of fifths nearly closes, are one list. The pentatonic and the diatonic set are on it for the same arithmetic reason that 12 and 53 are the equal divisions worth building.

The same five numbers. A generator nearly seven cents flat of pure produces the identical list of well-formed sizes, which says the sizes are a property of the ratio being close to 7/12 rather than of the fifth being pure. That is why the divisions that approximate the fifth well — 12, 29, 41, 53 — are the ones that recur: they are the ones whose best fifth lands in the same neighbourhood.

That is a genuine unification and it was not planned. The comma ladder and the scale ladder were written in different phases about different objects, and the object turns out to be the same one measured at two scales: at five and seven notes the near-miss is what makes the scale even, and at twelve and fifty-three it is what makes the tuning close.

Where the failures fail #

The rows that are not on the list are more instructive than the rows that are.

Four fifths folded into an octave give C, D, E, G — steps of 2, 2, 3, 5 semitones. Three sizes. Six give C, D, E, F♯, G, A — steps of 2, 2, 2, 1, 2, 3. Three sizes again, and the pattern has a stray semitone in it that belongs to no family.

Read as a family they are one construction stopped at five places, and the five-note and seven-note rows are the pentatonic and the diatonic set. Nothing chose them: they are where a chain of fifths happens to have exactly two step sizes, and the scales a tradition ended up with are the members of that list small enough to sing.

The reason is countable. Adding one note to a chain of k notes splits exactly one of its k steps into two, and the two pieces are the two sizes of the parent scale’s steps. So a chain grows by turning one large step into a large and a small; the number of large steps falls by one and the number of small ones rises by one, until the large steps run out and a new, smaller size appears. Two sizes exist only at the moments just before a size is exhausted, and those moments are the series.

That is a proof rather than an observation, and it is worth having because it explains why the series has to be sparse: between one exhaustion and the next there is a run of sizes at which three lengths coexist, and the run gets longer as the numbers get bigger. Between 17 and 29 there are eleven sizes and not one of them works.

What the thirteenth fifth is #

The clearest place to watch a size get exhausted is the step from twelve notes to thirteen, because the new length that appears has a name.

Twelve pure fifths folded into an octave have steps of 113.69 and 90.22 cents — the Pythagorean chromatic scale, seven large steps and five small ones. Add the thirteenth fifth and it splits one of the large steps into 90.22 and 23.46, and 23.46 cents is the Pythagorean comma. The comma is not an error that turns up at the end of the chain; it is the step size the chain acquires at its thirteenth note, and it is present as a step in every larger chain from there on.

At seventeen notes the large steps have all been used up: the pattern is twelve steps of 90.22 and five of 23.46, two sizes again, and the scale is back on the list. At twenty-nine the same thing happens one level down, at forty-one again, at fifty-three again — and the two sizes there are 23.46 and 19.84, which differ by three and a half cents. That is the point at which a chain of fifths has become, for any practical purpose, an equal division, and it is why fifty-three is on everybody’s short list.

So the series is one process seen at five magnifications. Each time the larger step runs out, the two sizes left are the smaller step and the difference — and the difference is a comma of the previous level.

The pentatonic is not a smaller diatonic #

Because the two are neighbours in the series, it is tempting to say the pentatonic is the diatonic set with two notes taken out. It is, in the sense that the diatonic set contains it; and the arithmetic says something better.

The pentatonic’s step pattern is 2-2-3-2-3: two sizes, two and three semitones. The diatonic’s is 2-2-1-2-2-2-1: two sizes, two and one. Growing from five to seven splits each of the two three-semitone steps into 2 + 1. The three-semitone step is the parent of the semitone, which is why the pentatonic has no semitones at all — not because its builders avoided them, but because a five-note chain has not yet produced one.

The tempered fifth makes a different list #

Everything above uses the pure 3:2. Substitute the tempered fifth of exactly 700 cents and the arithmetic changes in a way that is easy to state and surprisingly rarely stated.

Three things follow, and the third is the one worth carrying.

The tempered fifth is more permissive, not less. Eight, nine, ten and eleven notes of the tempered chain each have two step sizes, of 100 and 200 cents. The pure chain has three sizes at every one of those. Equal temperament does not merely close the circle; it makes four extra scale sizes well formed on the way there.

And then it stops. The pure chain runs forever, so it has a 17-note scale and a 29-note scale and a 41. The tempered chain has twelve notes and then repeats itself, so those sizes do not exist in it — not as bad scales, as no scales.

So the two agree exactly on 2, 3, 5 and 7. The pentatonic and the diatonic set are the sizes at which just and equal intonation cannot be told apart by this test. Every argument this ladder has made about the shape of the diatonic set is therefore an argument that survives temperament, which is not true of the arguments the comma ladder makes about the same chain. That is why a rung about set structure can be written without saying which tuning it is in, and a rung about thirds cannot.

The same question of another generator #

Nothing above is about fifths in particular. Any generator has a series, and the series is the denominators of its continued fraction.

A chain of pure thirds gives usable scales at three, four, seven and ten notes. A chain of the 7:4 gives them at 2, 3, 4, 5, 6, 11 and 16. Neither list contains twelve. Whether a generator produces a familiar scale is a property of where its size sits between the whole numbers, and there is no sense in which the fifth is privileged by this argument — it is privileged by consonance, which is a different argument on a different rung and about the ear rather than about the arithmetic.

A count that says nothing about sound #

It is worth being blunt about what this rung is and is not.

Every number here comes from folding a chain into an octave and counting how many distinct gaps result. No listener appears in it. No spectrum appears in it. The word consonance has not been used, and the fifth’s only role is that it is 701.955 cents; substitute any other irrational fraction of an octave and the machinery runs unchanged and produces a different list.

That is a strength for what it establishes and a limit on what it can be asked. It establishes that five and seven are not arbitrary and not separately motivated — that a tradition which starts from the fifth and keeps adding notes will find those two sizes and skip the ones between them, whatever it thinks it is doing. It cannot establish that the fifth is where anyone should have started. That argument is about roughness, it is made on a different ladder, and it is the only part of this account that involves an ear at all.

The honest joint statement is a division of labour. Consonance chooses the generator; arithmetic chooses the sizes. Neither half does the other’s work, and a great deal of writing about why the diatonic scale is what it is runs the two together and ends up claiming that seven notes are consonant, which is not a thing seven notes can be.

Two families on one list #

The nine entries of the fifth’s series are not nine of the same thing, and the difference is visible the moment the two step sizes are printed beside each other rather than merely counted.

Every semiconvergent has a large step more than twice its small one, and every convergent has one less than twice. Nine for nine, with no near misses on either side of the line — the closest are 1.85 and 2.26.

The reason is the splitting argument run one step further. A chain grows by splitting a large step into a small one and a remainder, so a size whose large step is still more than twice its small one has another split of the same kind left in it, and one whose large step is under twice does not: the next note has to break the small step and introduce a third length. A convergent is a size that has just run out of splits, which is exactly what a near-closure of the chain is, and the two descriptions the section above unified are here separated again into which of them a given row is.

That sorts the list into the two things it has been treating as one. The convergents — 2, 5, 12, 41, 53 — are the sizes at which the chain is nearly an equal division and its two steps nearly merge. The semiconvergents — 3, 7, 17, 29 — are the sizes at which the two steps are strongly distinct.

And it answers the objection the closing section makes to itself. The twelve-note Pythagorean chain has two sizes at a ratio of 1.26, which is why nobody hears it as a scale of two step kinds; the diatonic set has two sizes at a ratio of 2.26, which is why everybody does. Tone and semitone are more different from each other than the two sizes of anything else on the list except the seventeen and the twenty-nine, and the property that makes a name do useful work is not having two sizes but having two that are far apart. The diatonic set is on the list twice over: once for being on it at all, and once for being the largest ratio among the sizes small enough for anyone to sing.

What the picture cannot show #

Two step sizes is not the same as two useful step sizes, and the section above sorts the list on exactly that. What the property cannot say is whether a listener uses the distinction; the ratio says how large it is, and no rung here measures the first from the second.

Nothing here says which rotation is the scale. A chain of seven fifths is a set. Whether it is heard as major or as Dorian is not decided by anything in this figure, and cannot be, because rotating a set does not change its step sizes — only their order.

And the series is about one generator at a time. Real tuning systems have used two: a chain of fifths with a pure third dropped in, which is what meantone is a repair to. Two generators make a two-dimensional lattice rather than a chain, the counting argument above does not apply to it, and what happens there is the second comma’s subject.

Whose music this is a claim about #

The five-note and seven-note scales appear in traditions with no contact between them, and the usual explanation is that both come from the fifth. This rung sharpens that into something checkable: if a tradition builds from a chain of fifths, its scale sizes should be on the list, and if it builds some other way they should not be.

The prediction survives contact with the awkward cases better than expected. Slendro’s five near-equal steps are not a chain of fifths and its steps are near enough equal that it has one step size rather than two — which is off the list at both ends, and consistent with a tradition that does not tune to a chain at all. The Thai seven-note near-equal tuning is the same case at seven. Both are excluded by this criterion and both are excluded by the historical account too, which is a small piece of corroboration for a criterion that is otherwise pure arithmetic.

Where it says nothing is on the six-note and eight-note scales that European music does use — the whole-tone and the octatonic. Neither is generated by anything and neither has two step sizes; the whole-tone has one and the octatonic’s thirds are all the same size. They are built on a different principle, which is symmetry, and this ladder has no rung about it.

The ladder from here #

The series answers how many notes and the census on the previous rung answers which seven. What neither answers is why the universe has twelve positions in it, and the next rung asks exactly that: run the whole four-property census in a universe of nineteen, or twenty-four, or fifty-three, and see whether twelve is special or whether every universe has its own diatonic set.

The answer turns out to be completely regular, and it is not the one this ladder has been implying.