A scale built downward from a fourth

The previous rung ended by naming two things this ladder did not have, and one of them was the tetrachord as a unit of construction. It is owed for a reason: every scale result on this site so far has been produced by one of two accounts, and the tetrachord is neither.

The first account is a generator. Take a fifth and iterate it: five links give the pentatonic, seven give the diatonic set, and the sizes at which the chain produces exactly two step sizes are 2, 3, 5, 7, 12, 17 and 29 and no others.

The second is a census. Take an equal division of the octave, enumerate the subsets, and score them — which produced the four-property census over all 349 shapes and the sweep over every universe from four to thirty.

Both begin with a fixed grid of positions. The tetrachord does not, and it is the older account by two thousand years.

The unit #

A tetrachord is a span of a fourth — 498 cents, the ratio 4:3 — with two notes inside it. The bounding notes never move. Everything a tradition names is interior.

The three Greek genera are the three families of filling. The diatonic puts the two interior notes so that the steps are roughly a semitone and two tones. The chromatic puts them close together at the bottom, leaving a leap of nearly a minor third at the top. The enharmonic puts them closer still — steps of 63 and 49 cents in Archytas’ tuning — leaving a pure major third above them.

Each genus has several tunings, and the tunings are precise. Ptolemy’s Harmonics tabulates them as ratios, and this site draws them from those ratios rather than from any modern approximation.

Two of them, and an octave #

The unit becomes a scale by being used twice.

That is the disjunct arrangement. The conjunct one shares a note between the two tetrachords, so the pair spans a minor seventh and the octave is closed at one end instead; the Greek Greater Perfect System is a chain of four tetrachords, alternately conjunct and disjunct, spanning two octaves, and the modern word scale is a poor translation of it.

Now read the disjunct scale the way the tradition did.

So the two accounts agree on this one scale. The chain of fifths reaches the diatonic set at seven links; two tetrachords reach it in one step; and Ptolemy’s intense diatonic is the just tuning of it. The diatonic set is where the two constructions intersect, which is a good reason it is the scale two very different theoretical traditions both ended up with.

And then they part #

The agreement stops immediately outside that case, and the clearest demonstration is a jins that is in constant use.

Build an octave from hijaz and hand the result to the census.

Read as a census the octave built from two hijaz tetrachords has two step sizes for some of its generic intervals and three for others, where the diatonic set has exactly two throughout — so a twelve-note structural test scores it as a near miss. That is the test’s verdict and not the tradition’s: the hijaz tetrachord is a unit with a name, a repertoire and a place in a theory, and its position in a census of seven-note subsets of twelve is a fact about the census.

Hijaz is one of the six the census can see, which makes the failure sharper rather than milder: the mode is inside twelve, the enumeration reaches it, and every property the enumeration computes comes back saying it is malformed. The invisible cases are a separate problem; this one is visible and unaccounted for.

So the whole apparatus that produced this site’s scale results returns nothing useful about a mode that a hundred million people would recognise. Not because the mode is exotic — it is in twelve, it can be played on a keyboard, it has a name in Western theory too — but because the properties the census measures are properties of generated sets, and this one was not generated. It was assembled.

How much the census cannot see #

The size of the gap can be counted.

Fill the fourth on a grid, with a minimum step, and count. On a genuine quarter-tone grid — 50 cents, which is the resolution the maqam tradition notates in — there are 36 such tetrachords, of which six have all four degrees within an eighth of a tone of a step of twelve. On an eighth-tone grid of 25 cents there are 153, and the same six.

Six of thirty-six is one in six; six of a hundred and fifty-three is one in twenty-six. The numerator is fixed and the denominator is the grid, so the ratio is a statement about how finely the fourth is being sliced and not about the tradition. The six are the six the twelve-note census can see, whatever resolution the question is asked at, and that is the durable half of the count.

One in six at the tradition’s own resolution, and it is the wrong level to ask at. The census is over scales, not over tetrachords, so the number that answers the question is the one at the octave: taking every ordered pair of the thirty-six with a disjunctive tone between them gives 1,296 octaves, and 36 of them — 2.8 per cent — have every degree within an eighth of a tone of a step of twelve.

That is the honest measure of what a subset-of-twelve census is a census of, and it is much smaller than the tetrachord-level figure because both halves of an octave have to land in twelve at once. It is also, satisfyingly, thirty-six again: the visible octaves are exactly the pairs drawn from the six visible tetrachords, six by six, which is what the arithmetic has to give and is worth seeing rather than assuming.

The fraction is a much smaller share of the available scales than the census’s own completeness suggests. The four-property census is exhaustive over all 349 shapes in twelve and exhaustive over every universe from four to thirty, and it is exhaustive over a domain that excludes almost everything this construction makes.

The fourth is doing the work, and it is worth asking why #

Nothing so far explains why the bounding interval is a fourth. It could have been a fifth, or an octave, or a third, and the choice is not arbitrary.

Three reasons converge on it and only the first is usually given.

A fourth is a consonance, and it is the largest interval that is a consonance and still small enough to be filled with two steps of usable size. Fill a fifth with two interior notes and the steps average 234 cents; fill a fourth and they average 166; fill a third and they average 129, which is a scale of quarter-tones with nothing to distinguish the fillings.

Two fourths and a tone make an octave exactly. That is the arithmetic 498 + 204 + 498 = 1200, and it is exact because the fourth is 4:3 and the tone is 9:8 — the ratio arithmetic closes with no comma left over, which is the one place in this whole collection where a tuning construction closes cleanly. Two fifths overshoot the octave by a whole tone and have to be corrected; two fourths undershoot by exactly a whole tone, which is a usable step.

And a fourth is what a hand spans on most instruments. The Greek lyre’s tunings, the fretting patterns of the lutes the maqam tradition was theorised on, and the four-string groupings of a great many instruments are fourths, so the unit was also a physical one.

Which makes the two accounts even less opposed than they looked. The generator account’s interval is the fifth; the tetrachord account’s is the fourth; and they are the same interval inverted. What differs is not the ratio chosen but what is done with it — iterated, or used as a boundary.

Two different kinds of theory #

It is worth setting the two accounts side by side as accounts rather than as results, because they answer different questions and the difference is not about which notes come out.

A generator account asks what one interval makes. Its natural questions are how many notes, what step sizes, when the chain closes. Its natural results are the moment-of-symmetry cardinalities and Myhill’s property.

A tetrachord account asks what fills a fourth. Its natural questions are which fillings a tradition uses, how they are joined, and which one a phrase is currently in. Its natural results are a catalogue — the genera, the ajnas — with a grammar over it.

And the generator account’s own construction is the same interval turned over. Take a pure fifth and iterate it and the scale sizes that come out — two, three, five, seven, twelve, seventeen, twenty-nine — are the sizes at which every generic interval has two specific sizes, which is the property the diatonic set has. The tetrachord account’s interval is the fourth, the generator account’s is the fifth, and a fourth is a fifth inverted. What differs is not the ratio chosen but what is done with it: iterated, or used as a boundary.

The second is a grammar in a sense the first is not, and the sense is precise. A maqam is a sequence of ajnas: a lower jins from the tonic and an upper one from the fourth or the fifth, with the upper one often exchanged during a performance. What that exchange changes is one note or two, at a stated place, with the rest of the scale untouched — a modular operation on a scale, which a set has no structure to support.

The enharmonic genus, and a limit this ladder already measured #

One of the genera is worth a paragraph on its own, because it runs into a number this ladder produced three rungs ago.

Archytas’ enharmonic tetrachord has interior steps of 63 and 49 cents. Three answers to how finely a pitch can be heard found that a melodic interval’s identification category is 25 to 50 cents wide for trained listeners — which puts the enharmonic genus’s second step at the edge of the width of a category.

So the genus that fell out of use is the one whose steps are the hardest to identify, and it is reported by the Greek writers themselves as difficult, requiring training, and eventually as obsolete. That is not a proof — a great deal falls out of use — but it is a case where a psychophysical bound and a documentary record point the same way, and the bound was measured independently for another rung.

What a unit buys that a set does not #

There is one further consequence of building from a bounded unit, and it is the reason the account survived in the traditions that kept it.

A unit can be transposed on its own. Moving a jins from the tonic to the fourth degree is one operation, and the notes it produces are whatever that jins produces there. In a set-based description the same move is a wholesale substitution of pitch classes with no structure connecting the two collections, which is why the parallel and relative maps had to be computed separately rather than falling out of anything.

A unit can be swapped for another of the same span. The bounding fourth is fixed, so exchanging the upper jins of a maqam for a different one leaves the tonic, the fourth and the fifth untouched and changes two notes. That is a modulation with a defined cost and a defined invariant, and it is much more like a pivot chord than like a change of key — the shared frame is the bounding interval rather than a shared chord.

And a unit carries an identity a set has no room for. Two ajnas can contain the same pitch classes and be different ajnas, because which note is the lower bound is part of what they are. That is exactly the property the diatonic census proved it could never have: all four of its properties are invariant under rotation, so nothing in it can distinguish a mode from its own rotation. A tetrachord is oriented by construction.

That last point is the one worth carrying away. The census’s closing result was that it could not name a tonic and that every question about which note is home belonged elsewhere. Here is one of the elsewheres, and it does not answer the question by adding a tonic to a set — it answers it by never having had a set in the first place.

Whose theory, and when #

The tetrachord account is Greek and it is the oldest complete scale theory that survives. Archytas’ tunings are fourth-century BCE and survive only because Ptolemy recorded them; Ptolemy’s Harmonics, second century CE, tabulates several tunings of each genus as ratios and argues about them. The system is descending because Greek notation and Greek instruments were described from the top down.

The chromatic and enharmonic genera were already archaic in Ptolemy’s own time, and the diatonic is what survived into the medieval European tradition — where the tetrachord persisted as a unit of description for centuries, in the hexachord system and in the naming of the church modes as authentic or plagal according to where the fourth sits relative to the fifth.

The living version is not European. Arabic, Turkish and Persian theory has used the same construction continuously: al-Farabi in the tenth century tabulates tetrachords in the manner of Ptolemy, and the modern ajnas are the same objects with the same role. The Cairo congress of 1932 fixed a twenty-four-step notation for it, which is a writing convention rather than a claim that the degrees are at quarter-tones — and measured performances put a neutral third anywhere from 340 to 365 cents, which is the first rung of this ladder’s own finding.

What the picture cannot show #

These are theorists’ tunings, not measurements. Every ratio drawn here is from a treatise. What performers do is a separate question with a separate literature, and the spread in it is larger than the differences between several of the tunings above.

The Greek system is a reconstruction. It is assembled from treatises that disagree, that describe practices already old when they were written, and that were themselves working from earlier sources. The cent values are exact because the ratios are exact; what they were exact about is a question for somebody with different evidence.

The count is a count on a grid, and the ratio is more sensitive to it than this essay first claimed. Sweeping the grid at a fixed tolerance: 6 of 6 at whole semitones, 6 of 36 at quarter tones, 6 of 153 at eighth tones, 6 of 253 at 20 cents — and then 78 of 1,128 at 10 cents, because at that spacing three grid points rather than one fall within an eighth of a tone of each multiple of a hundred. So the fraction runs from 100 per cent to 2.4 and back up to 6.9, and it is not monotone: it is partly a coincidence between the grid and the tolerance. Only the numerator is stable. Six tetrachords are visible to the census at every resolution from a semitone to an eighth of a tone, and the honest statement is about that six rather than about any ratio.

And the tolerance is inert at the resolution that matters. On a 50-cent grid every degree is either exactly on a multiple of 100 or exactly 50 away from one, so moving the “eighth of a tone” from 5 cents to 30 changes nothing at all — the count stays at six throughout. A stated parameter that cannot affect the answer is worth naming as such: it is doing work only on the finer grids, where it is deciding an arithmetic coincidence rather than a musical question.

And nothing here is a grammar. The essay says a maqam has one and does not compute one. Which jins follows which, where a modulation can occur and what a phrase does on the way down are the content of the tradition, and this site has drawn four-note spans and their arithmetic.

The ladder from here #

The other rung the previous one named as owed is still owed: what it means for a degree to be defined by where it goes next rather than by where it sits. That is the question the tetrachord account raises and does not answer — a jins is recognised as a shape and used as a path — and it is the point at which a theory of scales has to become a theory of melody.