A tuning is not a table of cents

Every scale on this site has been written as a list of numbers. Seven positions out of twelve; five degrees at 0, 231, 474, 717 and 955 cents; the Turkish Rast as nine, eight, five, nine, nine, eight and five commas. A list of cents is how a tuning gets into a table, into a paper, and into this site’s own data files.

The first rung of this ladder showed that a list of pitches is not a mode — two ragas can share one, and do. This rung is the same argument about the pitches themselves: a list of cents is not a tuning either, and three things it cannot record are each large enough to change what an ensemble sounds like.

Two instruments, one note, and a choice #

A great many traditions tune instruments in pairs slightly apart on purpose. Balinese gamelan does it — the two halves of a gangsa pair are the pengumbang and the pengisep, and the shimmer their difference produces is called ombak, the wave. So do the multiple ranks of a celesta, the courses of a mandolin, the voix céleste stop on an organ, and the honky-tonk piano.

In every case the interval between the two is not an error and is not a target the tuner is failing to hit. It is the point, and it is heard as a rate — beats are arithmetic, and anybody can count them.

That immediately poses a question a table cannot answer. Is the pair a fixed number of cents apart, or a fixed number of beats apart? The two are the same thing at one pitch and nowhere else.

A detuning of c cents at frequency f produces a beat rate of f(2^(c/1200) − 1) — proportional to f. So a fixed detuning in cents doubles its beat rate every octave: ten cents beats 0.64 times a second at A110 and 10.2 times a second at A1760, a factor of sixteen across four octaves. Ten cents is one number and it describes sixteen different sounds.

A fixed beat rate is the other extreme. Three beats a second everywhere means an interval of 46.6 cents at A110 — nearly a quarter-tone — falling to 2.9 cents at A1760, which is below the discrimination threshold — inaudible as a pitch difference and perfectly audible as three beats a second, which is the whole point of that essay arriving one rung earlier.

This is the same arithmetic a tuner setting a temperament works with, used for the opposite purpose. A tuner counting beats is removing them; a gamelan tuner counting beats is placing them. The method is identical and the sign of the target is reversed.

A tuner has to choose and neither choice is writable as a scale. Writing the pair as two entries in a cents table records the first policy and silently asserts that the beat rate was not the quantity being controlled. Writing one entry records neither.

What the middle ground looks like #

In practice neither extreme is what anybody does, and the practical answer is itself informative.

The reported behaviour in Balinese tuning is that the beat rate rises with register but by much less than a factor of two per octave — perhaps four to eight beats a second across the instrument’s range rather than four to sixty-four. That is neither policy: it is a third curve, between the two, and it is the shape one would expect if what is being controlled is the sound of the shimmer rather than either of the two quantities that are easy to write down. A shimmer that is pleasant at four beats a second in the bass is a rattle at sixteen in the treble, so a tuner working by ear will compress the range whether or not they can say why — which is the same kind of correction a piano tuner makes to the octave and for the same reason, that the ear is the instrument and the table is a record.

This site cannot check those numbers and says so. What it can say is that the two writable policies bracket the practice, that they differ by a factor of sixteen at the edges, and that the difference is not a subtlety: at the top of the range one policy gives a slow pulse and the other gives a rattle.

The quantity being controlled is one a cents table has no column for.

It can also say what the third curve is, which costs one line of arithmetic and gives the observed practice a shape rather than a description. Write the beat rate as proportional to frequency raised to some power. The cents policy is that power at one; the beat policy is that power at zero. Four to eight beats a second across four octaves is a factor of two against a factor of sixteen, so the reported practice is the exponent one quarter, and the detuning it implies runs from 61.8 cents at A110 down to 7.9 at A1760. Both writable policies are corners of a one-parameter family and the practice is an interior point of it — which is a better statement of the problem than “neither”, because it says exactly which single number a table would have to carry in order to record what a tuner does.

What complex tones do to all three #

The closing caveats of this essay note that the beat-rate figures assume pure tones and guess at the consequence: that the direction of the argument survives and its exact shape does not. Both halves are checkable with this site’s own roughness weighting, since a detuned unison’s partials beat pairwise — partial n at n times the fundamental’s rate — and what a listener is exposed to is the rate those contributions put the energy at.

The composite runs about twice the nominal rate under both policies, because the upper partials beat faster and still carry weight, and a tuner setting three beats a second on the fundamental of a complex tone is placing a fluctuation nearer seven. That is a level correction and it applies to both, so it does not change which policy is which.

The shape barely moves. The fixed-cents policy comes out at an exponent of 0.90 rather than 1.00 and the fixed-beat policy at −0.06 rather than 0. So the guess in the caveats was right about the direction and wrong about the size: the exact shape survives nearly intact, and what changes is the constant.

Which leaves the compression the tuners actually perform unexplained, and that is the useful negative result. The partials do not produce the middle curve — the fundamental is the loudest-beating pair at every register under both policies, so timbre supplies no mechanism that would flatten the range on its own. The quarter-power the practice sits at is a choice somebody is making by ear, and not something the physics of a complex tone was doing anyway.

Why the choice cannot be split #

There is an obvious compromise — detune by a fixed number of cents in the bass and a fixed number of beats in the treble, or something smoothly in between — and it is worth showing that this does not rescue the table.

Any such policy is a function of register, and a table of cents has one row per degree and no register axis. Writing it down would take a curve: a beat rate, or equivalently a detuning, specified for every note of the instrument. That is exactly the shape of the Railsback curve below, and exactly the shape a table cannot have.

The two policies cross once, always, whatever constants are chosen, because one is proportional to frequency and the other is flat, and a line through the origin meets a horizontal line exactly once. That is why the choice is forced rather than negotiable: there is no setting at which a tuner is doing both.

The octave is not 2:1 either #

The second omission is one this site has already measured twice from other directions, and it belongs here as a case of the same failure.

A piano’s strings are stiff, so their partials run progressively sharp of whole multiples. A tuner setting an octave by ear removes the beat between the lower note’s second partial and the upper note’s first — and that partial is sharp, so the octave comes out wide. The amount depends on the string, so it varies down the keyboard, and the resulting curve is the Railsback stretch.

The stretch is not small by the standards the previous rung established. Tens of cents in the top octave is several discrimination limens and a substantial fraction of a melodic category, and it is why a piano tuned to a mathematically exact 2:1 sounds flat in the treble to everybody who plays one.

A tuning that varies with register is not a set of pitch classes. It is a function of the note, and every table of twelve is a claim that no such variation exists.

And the tuning belongs to the instruments #

The third omission is the largest and the least tractable.

The measured slendro this site carries is one gamelan’s. The tradition’s own statement is that no two sets are alike and that this is deliberate: a gamelan is tuned as a unit, its tuning is part of its identity, and instruments from two sets are not interchangeable.

A table of cents says the same number at both pitches and the ear does not. A cent is a ratio and a beat is a difference, so a tuning stated in cents has already thrown away the quantity a tuner and a listener both work in — which is the first of this essay’s reasons that the table is not the tuning.

So the row in this site’s own table labelled “Javanese slendro” is a fiction of exactly the kind this essay is about — and the table says so, in its source field: one gamelan’s measured tuning; no two sets are alike. That caveat is doing more work than a caveat usually does, because the variation between sets is larger than the entire distance between this set and a five-fold equal division.

Reporting a tuning as a table of cents is therefore not merely lossy; it can be wrong in the direction that matters most, by making a distributed property of a tradition look like a single object with an error bar.

What this site does about it #

An essay collection that computes from tables has to say what it does when the tables are inadequate, and the answer here is three things, none of them a fix.

The tables carry a source field and it is read aloud. Every tradition in this site’s data has a source line, and the slendro’s says one gamelan’s measured tuning; no two sets are alike. Figures that draw it print that line. It is the smallest possible remedy and it is better than nothing, because it stops the number being quoted as though it were the tradition’s.

Figures that depend on register take a register. The stretch figures above are functions of frequency rather than of pitch class, which is the right shape; nothing forced that except the physics refusing to fit in twelve rows.

And the essays say which numbers are theory and which are measurement. The Arabic Rast’s 350 cents is a convention fixed at a congress in 1932. The Turkish Rast’s degrees are a theory in Holdrian commas. One measured slendro is a measurement of one object. Those are three different epistemic statuses in one table and nothing about the table’s shape distinguishes them.

Twenty-four equal steps were adopted in 1932 as a notation for maqam practice, and this is what the notation costs when the practice is measured: a residue large enough to hear as a beat. A grid fine enough to write a tradition down is not the same as a grid it uses, which is the second reason.

What a table is still for #

None of this is an argument against tables. Everything computed on this site needs one, and the essays that use them are not wrong to.

A table of cents is exactly right for a fixed-pitch keyboard instrument in a European tradition after about 1700, because that is the case it was invented for: one instrument, one tuning, no pairs, an octave assumed to be 2:1, and no register dependence anybody was writing down. Under those conditions a tuning really is twelve numbers.

Three tuning systems written as twelve numbers each is the case where the representation is adequate — a fixed-pitch keyboard really is twelve numbers, and nothing about it varies with what is being played. Everything else in this essay is an argument that the adequacy is a special case rather than the general one.

The failure is in the transfer. A representation built for one situation is used to record all the others, and the things it cannot express become invisible rather than becoming disagreements. Nobody argues about ombak in the literature on gamelan scales because it is not in the format the argument is conducted in.

The general shape of the omission #

Three failures, and they have one form.

A table of cents is a function from twelve labels to twelve numbers. Each thing it cannot record is a dependence on something that is not one of those labels:

• Paired detuning depends on which instrument, and there is no instrument axis.
• Stretch depends on which octave, and pitch classes have no octave.
• A tradition’s variation depends on which set of instruments, and there is no set axis.

Written that way the omissions stop looking like three curiosities and start looking like a single statement about the representation: the pitch class is the wrong domain, and it is the wrong domain for every practice in which the sound is a property of an ensemble rather than of a note.

That is a general enough shape to be worth carrying. A representation is not neutral about what it cannot index, and the things it cannot index tend to become things nobody argues about.

What the picture cannot show #

The figures draw pure tones, and the section above prices what that costs. The correction is about a factor of two in the rate and almost nothing in the shape, so the figures are readable as drawn provided the number under them is read as a lower bound. What the figures still cannot show is the spread: a complex pair presents several rates at once rather than one, and a single line has nowhere to put that.

The Balinese numbers are quoted, not measured. The claim that practice sits between the two policies comes from the literature and this site has no recordings to check it against. Everything computed here is the arithmetic of the two policies, which is exact and would be exact if the practice turned out to be either extreme.

And a stretched octave is an example, not a category. The piano’s stretch has a known cause and a computable size. Other register-dependent tunings — a set of gamelan bars, a rank of organ pipes voiced by ear — may have quite different shapes for quite different reasons, and nothing here predicts them.

Whose music this is a claim about #

Every tradition that tunes instruments as an ensemble rather than to a standard is affected: gamelan above all, but also the paired ranks of a European organ, the courses of lutes and mandolins, and any fiddle tradition where the instrument is tuned to the piece rather than to a pitch.

The tradition where it does not apply is the one this site’s machinery came from, and that is worth naming as the special case rather than the default. A twentieth-century European ensemble tunes every instrument to one pitch standard, treats any deviation as an error, and would regard deliberate paired detuning as a fault — except in the one place it survives, which is the organ, where a whole stop exists to provide it and the wolf it produces is a feature. In that world a tuning genuinely is a table of cents, and the representation is adequate because the practice was reorganised to make it so.

The reorganisation is documented on this site as a separate story: four centuries of local pitch standards ending in an international agreement, which is exactly the process of making a distributed property into a single number.

The ladder from here #

This ladder has now taken four things away from the idea of a scale: it is not a set of pitches, it is not fixed by consonance where a tradition does not harmonise, it is not fixed by any threshold of hearing, and it is not a table. What is left is a set of instruments, a practice, and a grammar for moving between degrees — which is a great deal harder to compute with and is what the traditions concerned actually have.

Two rungs of this ladder remain unwritten and both are about that grammar rather than about the pitches: what a tetrachord is as a unit of construction, and what it means for a degree to be defined by where it goes next.