Where the chain was never closed

The comma has been treated in this ladder as something the subject ran into: a piece of arithmetic that turned up when people stacked fifths, and that every tradition then had to deal with. Eleven rungs of dealing with it have all been European, which is a suspicious pattern for something described as arithmetic.

The pattern is not suspicious once the premises are written out. A comma appears when three requirements are made at the same time, and the requirements are decisions rather than discoveries.

The three requirements #

That the scale is generated by fifths. Five notes are four fifths and seven are six; the whole European system is one chain and every note in it has a position along that chain. Nothing forces this. A scale can be a set of intervals arrived at by other means entirely, and several are.

That the octave is a pure 2:1. This looks like a law and is a consequence — the octave fuses because every partial of the upper note lands on a partial of the lower. It is nearly universal and it is not quite universal, and where it is relaxed the arithmetic changes completely.

That the chain closes. Twelve fifths have to arrive at seven octaves so that the same set of pitches serves every key — which is only necessary if the music transposes, and transposition is a requirement about repertoire rather than about tuning.

Make all three and 23.46 cents is left over. Make two of the three and there is nothing left over, because the closure that fails is the thing that was not being asked for. The fourth column of the table separates the last requirement into its cause and its consequence — a system transposes because its designers wanted every degree available, and it closes because transposing is impossible unless it does — and keeping the two apart is what makes the Thai row legible.

What each tradition declines, and what that costs #

Turkish makam theory makes all three, and answers with a bigger number. The Arel–Ezgi–Uzdilek system divides the octave into fifty-three Holdrian commas of 22.64 cents each, and thirty-one of those steps is a fifth of 701.887 cents — 0.07 cents from pure, which is thirty times better than twelve-tone equal temperament manages and far below anything audible. The chain closes at fifty-three, the octave is pure, and the fifth is generated. The comma has not been hidden; it has been made smaller than the resolution of the system that contains it.

That is not a different answer from Europe’s. It is the same answer with a different note count, and it is the answer this site has already computed from the other direction.

Hindustani practice declines closure. A performance has a drone, the drone fixes the tonic for its whole duration, and nothing modulates in the European sense. So the chain never has to come back to where it started, the fifth can be exactly 3:2 forever, and the twenty-two shrutis of the theoretical literature are a description of a fixed set rather than a system that must circulate. Nothing in it corresponds to the reassignment of a home degree that produces a European mode, because there is no reassignment: the tonic is given and the material is built above it.

The price is that the whole apparatus is relative to one tonic. Two ragas with the same five notes are different ragas because of which degrees are weighted, which is a kind of information a transposable system cannot carry — and a system that cannot transpose has no use for a temperament and no comma to distribute.

Thai classical tuning declines generation. Its seven steps are near-equal, at about 171 cents each, and they are not built from anything: the tuning divides an octave rather than stacking an interval. It closes, it transposes freely between modes, and its best approximation to a fifth is four steps at 685.7 cents — 16.2 cents narrow, which is eight times equal temperament’s error and well over the limen.

So Thai tuning has paid for its closure and its transposability in the fifth itself, permanently and by design. A comma is a leftover of about twenty cents; Thai practice has a standing error of about sixteen and no leftover, which is the same order of quantity relocated from the arithmetic into the interval.

Javanese slendro declines everything. Its five steps are near-equal at around 240 cents, the octave is often deliberately not a pure 2:1, no interval generates the set, and no two gamelan are tuned alike — which is not a failure of standardisation but the point: a gamelan is a set of instruments tuned to each other, and its tuning belongs to it.

The nearest thing to a fifth in a slendro tuning is three steps, at about 717 cents, which is 15 cents wide. It is the mirror of the Thai case: a standing error of the same size in the other direction, in a system that has no use for the interval at all.

The one that closes on something other than a fifth #

Thai tuning is the case worth dwelling on, because it is the only system here that makes the transposition requirement without making the generation one, and it shows what that combination costs.

Seven equal steps of 171.43 cents divides the octave exactly, so any degree can be a tonic and every mode is available from every note — the strongest form of the transposability European practice went to such trouble for. The whole comma problem is absent, not because the tradition avoided a hard question but because it never built the chain that raises it.

What it gives up is the fifth. Four steps is 685.71 cents, and 16.24 cents narrow is four times the difference limen — an interval a listener can hear is not pure, sounding continuously, on every instrument in the ensemble. That is a much larger standing error than any European temperament tolerates in a fifth: the worst fifth in a well temperament is under six cents, and the worst in equal temperament is under two.

Whether that matters depends on the timbre, and the timbre is not incidental. A spectrum’s roughness minima are wherever its partials coincide, and a metallophone-heavy ensemble has partials that are not whole-number multiples of anything — so the intervals that would be rough for a violin are not necessarily rough for a gamelan, and the question of how far a fifth is from 3:2 is a question about a spectrum that these ensembles do not have.

What all of this is evidence about #

The natural reading of the comparison is relativist — that European tuning is one option among several and its problems are its own. That reading is available and it is not quite what the numbers say.

What the numbers say is narrower, and it is not the tidy version this section used to give. Setting the five payments side by side in the one unit they can share:

Three of the five pay between fifteen and twenty-three cents and two pay nothing measurable at all, which is not a conserved quantity. Turkey’s cost is fifty-three frets rather than twelve and Hindustani’s is a music that cannot modulate; both are real and neither is an interval error, so no arithmetic converts them into the column above. The currencies are genuinely different and they are not commensurable, which is the honest form of the comparison.

And the three that do agree agree for a reason that has nothing to do with the comma. Sweeping every equal division from five to sixty and reading off its best fifth: the value is +18.0 cents for every multiple of five and −16.2 for every multiple of seven, because a five-fold division’s best fifth is three of its steps whatever the multiple, and a seven-fold division’s is four. Slendro is a five-note division and Thai a seven-note one, so their errors are +15 and −16.2 for the same reason a clock has twelve hours on it. The Pythagorean comma is 23.46 because twelve fifths overshoot seven octaves. These are three different pieces of arithmetic that happen to produce numbers of one size.

Which turns the finding into a sharper one about the gamut rather than about the fifth. At five near-equal steps the best available fifth is eighteen cents wide and at seven it is sixteen narrow, and no tuning of five or seven equal steps can do better — so neither tradition declined a good fifth. Each chose how many notes to have, and the fifth it could then contain was decided for it. The first equal division that beats twelve’s two cents is fifty-three’s seven hundredths, and Turkish theory is at exactly that number.

And the four gamut sizes in this essay are not four arbitrary choices. Listing the divisions that beat every smaller division at the fifth — the record-holders, in order — gives 2, 3, 5, 7, 12, 29, 41 and 53, and the traditions here occupy 5, 7, 12 and 53 of that list. Every system in the comparison that closes its octave sits on it; not one of them uses a division that some smaller division already beats. Java is at the first size where a fifth is recognisable at all, Thailand at the next improvement, Europe at the one after that, and Turkish theory at the one where the fifth stops being an approximation. Four traditions with no contact between them, choosing four numbers off one sequence that nobody wrote down.

That is a stronger claim than the twenty cents ever was, and it is the one this section should have been making. It also says what “declining to build on the fifth” really means: a tradition at five or seven steps is not indifferent to the fifth — it has taken the best one available at a gamut it wanted for other reasons, and the best one available there is sixteen or eighteen cents out.

So the comparison’s real content is a trade between gamut size and interval purity, with the comma as one special case of it: a system with few notes has bad fifths and no comma, a system with fifty-three has excellent fifths and no comma, and twelve is the only place on the curve where the fifth is good enough to build a chain on and bad enough that the chain does not close. Only that combination, the leftover to be distributed among keys, produces a literature about where it should go.

Four quantities of the same order, one of which is not a comma: the Pythagorean at 23.5 cents, the syntonic at 21.5, the Holdrian step at 22.6 and the diaschisma at 19.6. A Holdrian comma is a unit of measurement and the others are errors — the Turkish system’s step is a thing to count in, and the European commas are things to dispose of. That two of them are the same size to within a cent is arithmetic and not evidence of anything.

What the arithmetic does say universally #

Two things survive the comparison and it is worth separating them from the parts that do not.

The octave is nearly universal and the fifth is close behind it. Almost every tuning tradition that has been measured has something within a few tens of cents of a 3:2, and the reason is not cultural: it is the second interval at which a harmonic spectrum’s partials coincide, and it is the strongest coincidence after the octave. Even the systems that decline to build on the fifth contain something near it.

And a system that transposes has to close. That is a theorem rather than a preference. A finite set of pitches in which every degree can serve as tonic is a set closed under transposition by its own steps, which forces the steps to divide the octave a whole number of times — so the choice is between an equal division and a system that does not transpose, and every tradition here is at one end of it or the other.

Read as a necklace the European answer is a selection of positions from twelve, and the pentatonic is a smaller selection from the same twelve — but every tradition above has a different universe to select from, and three of them have no universe of equal positions at all. The necklace is a picture of one system’s own decisions, and it does not survive being pointed at another’s.

Whose music, and when #

Each row of the table is a claim about a documented practice, and each has a source that is worth stating because the traditions are not equally well described in these terms.

The Turkish system is a theory, fixed by Arel, Ezgi and Uzdilek in the early twentieth century, and Turkish performance practice does not always follow it — a fact Turkish theorists themselves discuss at length. The Thai near-equal division has been measured since Ellis’s work on the tunings of piphat ensembles in 1885 and confirmed repeatedly since, and it is a description of instruments rather than a doctrine. The Hindustani shruti system is a theoretical tradition of long standing whose relation to what is played is a live question in its own literature. Javanese slendro is a measurement of particular gamelan, and the variation between them is larger than the differences between any two of the systems in this essay.

None of the four is a temperament in the European sense, because none of them is a scheme for distributing a known error over a known set of intervals. That is the finding: the object this ladder has spent eleven rungs on is specific to a tradition that made three requirements at once, and it turns out to be the only one that did.

Where the model stops #

A tuning system is not a performance. Every number here is a specification, and the site’s own essay on this makes the point at length: the pitch set is the least of what a modal tradition carries. Measured performances depart from every one of these tables, in ways that are part of the tradition rather than errors in it.

The four requirements are a model of the traditions and not a description of them. Real systems answer “does it transpose?” with qualifications, and the yes-or-no grid flattens them. The grid is worth having because it isolates the one combination that produces a comma; it is not worth mistaking for an account of any of the practices in it.

The best-fifth column flattens a real difference. For the European and Turkish rows it is the interval the system is built out of; for the Thai and Javanese rows it is the closest thing the system happens to contain to an interval its own theory never names. Those are not the same measurement and the table prints them in one column.

And the cent is a European unit. Expressing a slendro interval in cents is a translation, not a measurement in the tradition’s own terms, and it imports the assumption that pitch distance is logarithmic and octave-relative — which is exactly one of the assumptions two of these systems decline.

What the picture cannot show #

It cannot show the instruments. A fifty-three-step system needs frets or an instrument that can produce the pitches, and Turkish practice uses fretted long-necked lutes whose fret positions are the theory made physical. The cost of a fine division is paid in instrument-building, and the grid has no column for it.

And it cannot show the drone. The single feature that most decisively removes the comma from Hindustani practice is a continuously sounding tonic, which is not a property of a tuning at all. It appears in the table as “does not transpose”, which is true and is a thin description of the most consequential fact about the music.

Where this ladder ends #

Twelve rungs, and the table its own plan named twelve years’ worth of arguments for is now complete. The through-line the last five rungs found was not in that table: the comma is conserved. A regular temperament locks the fifth’s error to the third’s; the twelve thirds of any closing chain total four octaves whatever is done to them; an ensemble’s drift and its vertical impurity sum to exactly one comma; the only gap that escapes is the one below the threshold of the ear; and the debt itself exists only where three requirements were made together. Every rung of this ladder has been about where a fixed quantity is paid, and none of them has ever been about reducing it.