The third the model has no opinion about

The first rung of this ladder put two codifications of Maqam Rast side by side: the Arabic one, whose third degree is at 350 cents, and the Turkish one, whose third degree is a pure 5:4 at 386. The gap is 35 cents, both are written down as theory, and both describe the same maqam.

The Arabic third is a neutral third — neither major nor minor, roughly halfway. It is a scale degree across an enormous geographical range, and this site has a machine for asking whether a given interval is consonant. Running it is the obvious next question and the answer is the interesting kind of negative.

Nothing is there #

The site’s well-finder requires a minimum to have some prominence before it is called a well — the curve has to climb at least a hundredth of its full range to get out of it in both directions — because a bare local minimum at four-hundredths of a per cent would announce a consonance nobody claims.

On a 1/n harmonic spectrum, the neutral third does not have a minimum at all, prominent or otherwise. Nor does it acquire one with more partials: sixteen partials at 1/n give wells at 498, 702 and 884 cents and nothing else. Nor with an odd-only spectrum, which is what a clarinet approximates — that one has wells at 782 and 884 and loses the fifth’s almost entirely.

Whatever a neutral third is, it is not a roughness minimum for any ordinary instrument.

That is worth stating plainly because the roughness account has been the site’s workhorse. It puts the wells on the simple ratios for harmonic spectra, it explains why a clarinet’s fourth has no well, and it was used to design a scale for a spectrum with no even partials. Here it returns nothing, on an interval that a great deal of music treats as a scale degree.

The ratio it would be #

A neutral third does have a small-integer ratio available: 11:9, which is 347.4 cents — two and a half cents from the Arabic theory’s 350.

The model has no opinion about the third because it has no opinion about intervals at all — it has opinions about spectra. Give it partials that are not harmonic and the wells move to match, which is the strongest evidence that what it computes is a property of the sound rather than of the interval named on top of it.

Eleven and nine are small numbers and the ratio is nowhere near consonant, which is a useful reminder of what the small-integer story actually rests on. It is not smallness as such: it is that the partials of the two tones coincide, which is the whole content of the consonance ladder’s second rung and the reason a spectrum can be made to choose a different scale. Two tones a fifth apart share every second partial of the lower with every third of the upper, starting immediately. Two tones at 11:9 share nothing until the ninth partial of the upper meets the eleventh of the lower.

So the coincidence exists and it is too high up the series to matter — unless the spectrum puts energy there.

What it would take #

The question has a numerical answer. Take an ordinary 1/n spectrum, raise the ninth and eleventh partials to some amplitude a, and ask at what a the neutral third acquires a well.

The threshold is about 0.3 — roughly three times the natural amplitude of a ninth partial and three and a half times an eleventh. At 0.3 the well has one per cent prominence, which is the bare minimum this site will call a well. At 0.5 it is 7 per cent. At 0.9 it is 22 per cent, essentially equal to the fifth’s — but the fifth’s has fallen from 36 to 22 on the way, because those two loud partials are creating roughness everywhere else as well.

No instrument has a spectrum like that. A ninth partial at nine-tenths of the fundamental’s amplitude, with the tenth at a tenth, is not a shape any string, pipe, reed or voice produces. The spectrum that would make a neutral third consonant is not merely unusual; it is one nobody has built.

The same question, asked of the other neutral intervals #

A neutral third is not the only degree of its kind. Rast’s seventh at 1050 cents is a neutral seventh, and several maqamat have a neutral second or a neutral sixth. The model’s answer is the same in every case and worth recording as a set rather than one at a time.

At natural amplitudes the wells across an octave are at 498, 702 and 884 cents and nowhere else, so not one of the neutral degrees is at a well. The 11-limit ratios they are nearest to — 11:9 at 347, 11:8 at 551, 11:6 at 1049 — are all ratios whose coincidence is at the eleventh partial, which is where the energy is not.

That uniformity is what makes the result a boundary rather than a curiosity. It is not that one degree happens to fall in a gap; it is that the whole family of intervals whose justification would be an eleventh-partial coincidence is invisible to a model built on spectra that fall as 1/n.

So the degree is not there for that reason #

This is the point where the model’s silence becomes informative rather than embarrassing.

The traditions that use a neutral third — Arabic maqam, Turkish makam, Persian dastgah, and by extension a great deal of music across North Africa, the Balkans and Central Asia — are overwhelmingly monophonic or heterophonic. There is a melody, there are drones, there is ornamented unison. What there is very little of is simultaneous intervals of independent voices, which is the situation roughness is a theory of.

The wells barely move and their depths change a good deal, which is the honest scope of the model: it is robust about where a minimum is and weak about how deep. Any argument that ranks two intervals by how much smoother one is than the other is resting on the half of the output that the body reshapes.

Roughness is a claim about two tones at once. It computes the beating between the partials of two simultaneous complex tones, and every number it produces is a number about simultaneity. Asked about a melody, it has nothing to say, because a melody presents its intervals one at a time and the ear’s response to a succession is a different mechanism entirely.

A scale degree in a monophonic tradition has no reason to be at a roughness minimum. The obvious next sentence is that European scale degrees mostly are, as a consequence of three centuries of writing them in chords, and the section below runs that as a control and finds it false.

A drone changes the question #

There is one situation in this repertoire where simultaneity is unavoidable, and it deserves its own arithmetic because it is the case where the model is entitled to an opinion.

A drone is a held tone, usually the tonic, sometimes the tonic and its fifth. Every scale degree is therefore heard against it at some point, so every degree has a roughness value that can be computed — not as a well or the absence of one, but as a number.

The reading that follows is not “the neutral third is dissonant”. It is that a degree can be a place of tension in a line without being a place a line stops, and a tradition whose grammar is about motion through a tetrachord has every reason to include such a degree. European tonality, which stops on its degrees and stacks them, does not.

That is a real difference between the two kinds of music and it is the sort of difference this model can speak to, because it is a claim about what happens when two notes sound together and one of them is held.

The control case, which fails #

The argument above has a control built into it: if roughness selects the degrees of a harmonising tradition, the degrees of the most thoroughly harmonising tradition there is should be at the wells. Running the same curve over the just major scale says they are not.

For a string spectrum the wells are at 498, 702, 885 and the octave, and that is the whole list at any number of partials. A seven-note scale cannot have its degrees at four wells, so at most four of them can be, whatever anyone writes in chords — and in fact three are. The major third is not one of them.

The interval European harmony is built on has no roughness well. Neither has the minor third, the minor sixth or either seventh. The region from about 300 to 400 cents is a smooth descending flank with no feature anywhere in it, and the major third sits on that flank exactly as the neutral third does, eight percentage points further down.

That changes what the negative result of this rung is. It is not that a neutral third has no roughness justification while its neighbours do; it is that no third of any size has one, so the model was never going to distinguish 347 from 386 and its silence at the first is the same silence it keeps at the second. The essay’s finding survives and its scope was wrong: the boundary is not between monophonic and harmonising traditions, it is between the four intervals this curve has an opinion about and the eight it does not.

What the continuous value does say is worth keeping, because it is the one thing here that is not a null. The neutral third is not anomalously rough: at 53.4 per cent it sits between the minor third’s 57.7 and the major third’s 44.1, which is where a third halfway between them belongs and is a good deal smoother than the major second above it. A degree in that region is paying an ordinary price for its position, not an unusual one, and nothing about the arithmetic makes it a strange place to put a scale degree. The model’s opinion about the neutral third is not “no” and never was; it is that it has none about any third, and the first version of this rung read the absence of a feature as a verdict.

What does fix a neutral third, then #

This site cannot answer that and can say what the candidates are, which is more useful than silence.

Melodic position. In maqam practice a degree is defined by its position in a tetrachord — the jins — and by which degrees it moves to. The third of Rast is what it is partly because the tetrachord it belongs to is a recognisable shape, and a shape can be recognised at any absolute size.

Equidistance as a design. The Cairo congress of 1932 fixed a 24-step equal division precisely because a quarter-tone convention makes an existing practice writable. That is a notation decision, and the 350 in the figures above is a theory’s number rather than a measurement: measured performances put Rast’s third anywhere between about 340 and 365 cents depending on the performer, the region and the phrase.

And a drone. Where there is a drone, there is simultaneity, and roughness has something to say again — but it says it about the interval between the degree and the drone, which is usually the tonic, and a neutral third above a drone is genuinely rough by this model. That is not obviously a problem: a degree can be a place of tension in a line without being a place anybody stops.

What would settle it is a measurement this site cannot make: whether performers’ neutral thirds cluster near 347 cents, near 350, or somewhere with no simple ratio near it at all. The published measurements that exist point at the last, and at variability much larger than the difference between the candidates.

What the negative result is worth #

A model that returns nothing is easy to dismiss and this one should not be, because it makes a prediction that could fail.

The prediction is about repertoire, not about the interval. If roughness explains where scale degrees sit, then traditions that harmonise should put their degrees at wells and traditions that do not should be free not to. That is checkable in principle across a large number of traditions and it is not checked here; what is checked is the one case, and the one case comes out as predicted — a monophonic tradition with a degree at no well at all.

And it fails if a polyphonic tradition is found using a neutral third as a chord tone. That would be a degree at no roughness minimum being stacked into simultaneities, and it would mean the roughness account is not doing the work claimed for it even where simultaneity exists. Georgian polyphony is the obvious place to look, and it is exactly the case this site cannot resolve: its intervals are reported by different investigators as near-equal divisions, as just ratios, and as something in between, and the disagreement is larger than the effect.

So the honest position is that the negative result is real, its boundary is stated, and the test that would sharpen it is one somebody else has the data for.

What the picture cannot show #

The roughness model is one model and a crude one. It is Plomp and Levelt’s curve, fitted to judgements of pure-tone pairs, summed over partials with no account of masking, of phase, or of the fact that a dissonance has to last. Every negative result here is a negative result of that model, and a better model could put a feature at 347 cents.

Amplitude is not the only way to make a partial matter. The sweep above raises two partials’ amplitudes. A real instrument that emphasised the eleventh partial would do so by resonance, which also changes the phase and the decay of that partial — neither of which is in this model.

And “no instrument has that spectrum” is a claim about instruments this site has data for. Strings, pipes, reeds, bars and voices. A synthesiser has whatever spectrum is asked of it, and the interesting experiment — build the spectrum, then ask listeners whether the neutral third sounds consonant — is exactly the one the Bohlen–Pierce scale was designed by and has, as far as this site can tell, never been run for 11:9.

Whose music this is a claim about #

The neutral third is not one interval and not one tradition. Egyptian, Syrian, Turkish, Iranian, Tunisian and Iraqi practices differ from each other on where it sits, and every one of them differs from its own written theory. Treating “the neutral third” as a single object at 350 cents is a convenience of this essay and of the 1932 congress, and it should be read as such.

What is safe to say is that a degree between the major and minor thirds is a normal part of a very large amount of music, that no consonance argument selects it, and that this is not a defect in the music or in the argument. It is a boundary of the argument, and the boundary runs exactly where simultaneity stops.

The European case is not the special one, and the control above is why. A tradition that stacks its scale degrees into chords ends up with three of its seven degrees at roughness minima, because there are only four minima to go round and one of them is the octave. Whatever else stacking does, it cannot have selected the third — which is the degree the whole system turns on and the one the curve is silent about.

One more asymmetry is worth noticing before leaving it. European theory has a word for an interval that is neither major nor minor and it is neutral — a word that defines the thing by what it is not, in a vocabulary built around a pair of categories it does not belong to. The traditions that use it call its degrees by name — sīkāh, segâh — and those names are positions in a scale rather than qualities of an interval.

That is the same coordinate confusion the modes ladder found between parallel and relative descriptions, arriving in a different field: a vocabulary designed for one system, applied to another, and quietly importing the first system’s questions along with its words. Asking whether a neutral third is consonant is such a question. It is a perfectly good question and the answer is no; what this rung establishes is that the answer costs nothing, because nothing in the practice depended on it.

The ladder from here #

If a scale degree is not fixed by consonance, what fixes how finely it has to be specified? The next rung asks how small a pitch difference matters, and finds three different answers an order of magnitude apart — with every equal division anybody has built sitting between the coarsest and the finest.