The third sound magnifies cents, not hertz

Play a major third on a violin, loudly, and a low note appears that neither string is playing. Tartini taught his students to listen for it and to move the upper finger until it sat where it belonged, and the method is still taught. The ear makes its own sound worked out where that note comes from — the quadratic distortion of the cochlea, which puts a tone at the difference of the two frequencies — and said of the method that on a tempered interval the product is “a few cents off and much harder to identify”.

The arithmetic says something else about the size. A tempered major third on D4 has its upper note at 369.99 hertz where a just 5:4 would put it at 367.08. The difference tone of the just third is at 73.42 hertz; the tempered one’s is at 76.33. Those two numbers are 67 cents apart, which is two thirds of a semitone, from an interval that is 13.7 cents wide of just.

That is not a small discrepancy dressed up by a choice of unit, and it is not specific to the major third. Every interval has a factor like it, the factor can be computed from the ratio alone, and what it turns out to be worth to a listener is a separate question with a much less flattering answer.

A difference tone is geared to its interval #

Put the just interval p:q on a fundamental of one unit, so that the lower note is harmonic q and the upper note harmonic p. The difference tone is harmonic p − q. Move the upper note sharp by a small fraction ε and it rises by pε units; the difference tone, which is the upper note minus a lower note that has not moved, rises by the same pε units — but it started at p − q units rather than p, so in proportion it has moved p/(p − q) times as far.

Proportion is what cents measure. So a difference tone moves p/(p − q) times as many cents as the upper note did: five times for a major third, six for a minor third, three for a fifth, sixteen for a minor second.

Read across the octave, the tempered intervals fall into three groups by what they do to their product. The fourth and the fifth barely move it: two cents of temperament become eight and six, well inside a limen. The thirds and sixths move it by a third to a whole semitone: the major third’s product is 67 cents sharp, the minor third’s 96 flat, the major sixth’s 39 sharp and the minor sixth’s 37 flat. And the seconds are geared so steeply that the product is somewhere else entirely: a tempered minor second’s difference tone is 198 cents flat, a whole tone below where a 16:15 would put it.

The ordering is the ordering of the damage equal temperament does to the intervals themselves, multiplied. Where to hide the comma is the account of why the fifths are nearly pure and the thirds are not in that tuning, and the products inherit the decision: the fifth, which the tuning barely touches, has a product still barely touched, and the thirds, which pay most of the tuning’s cost, have products that pay five and six times as much.

The products also land somewhere specific in the ensemble’s register, and it is lower than intuition suggests. A major third on a violin’s two lowest strings, G3 and B3, has its difference tone at 49 hertz — below the open C of the cello, which is 65. On the A string the product is at 110 hertz, the cello’s open A. The third sound under a violin’s lowest double stops is the lowest note in the quartet, played by nobody, and a tempered third puts that note two thirds of a semitone from where a just one would.

Each interval has its own gearing #

A table of tempered errors is a single point on each interval’s curve. The curve itself is the product’s error against the upper note’s for every mistuning a player might produce, and it is nearly a straight line through the origin with the gearing as its slope.

Two things about the fan are worth reading off it. The first is that the lines are very nearly straight across forty cents, bending only slightly as the product’s own frequency changes, so the gearing is a good description of any mistuning a player could make by accident. The second is that the minor third is geared more steeply than the major third although it is the smaller interval, and the reason is the formula: 6/(6 − 5) is larger than 5/(5 − 4). The closer the two harmonics, the lower the product sits and the more cents a fixed movement is.

That is the whole of the mechanism, and it is worth stating it before looking at what a listener gets from it, because the mechanism has a consequence that the word “magnifier” hides.

In hertz there is no gearing at all #

Go back to the derivation and read it in hertz instead of in proportion. The upper note moves by some number of hertz; the difference tone is that note minus a stationary one; so the difference tone moves by exactly the same number of hertz. Nothing was multiplied. The factor of five appeared only because the same number of hertz is a larger fraction of 73 hertz than of 367.

The magnifier magnifies the unit, not the mistuning. It relocates a frequency error of fixed size to a place on the spectrum where that size is a larger interval, and whether that helps depends entirely on how well the ear resolves hertz at the new place compared with the old one.

The same bookkeeping explains the beat, which is the method a tuner actually counts. A p:q interval’s lowest coincidence is between the lower note’s partial p and the upper note’s partial q, and when the upper note moves by some hertz its q-th partial moves by q times as many, so the beat moves by q times as many hertz. The unison is the coarsest thing in the room found that multiplier from the other side, as a beat that betrays a mistuning faster on a fifth than on a unison. In hertz, the beat multiplies a mistuning by q and the third sound multiplies it by one. Before any perception enters, the beat is the better-geared instrument for every interval whose q is larger than one.

What the gearing is worth: the ear’s hertz at a low frequency #

The third sound can still win if the ear hears a hertz more clearly down where the product lands than up where the upper note sits. It does, a little. The smallest frequency difference a listener detects in a steady tone, the quantity how small a difference is audible is about, is roughly constant in hertz at low frequencies and grows above a few hundred: about half a hertz near 70 and nearly a whole hertz near 370.

So on a one-second major third on D4, a listener attending to the difference tone can find an error half the size in hertz that the same listener could find by attending to the upper note alone — a gain of 1.85. On G3 the gain is 1.5; on E5 it is 2.6, because the upper note there has climbed into the region where the limen grows fastest. That is the real magnification, and it is a factor of about two rather than five.

The three curves sit in the same order at every register. Comparing the two notes’ pitches is the coarsest, at 6.5 cents on D4, because it adds two limens rather than using one. The difference tone comes next, at 2.35 cents — a real improvement over the pitch comparison, and the one Tartini’s students were being taught. Counting the beat is finest, at 1.18 cents, because it multiplies the mistuning by four in hertz where the difference tone multiplies it by one, and a beat is counted rather than estimated.

So the three answers to how finely a pitch can be heard have a fourth, and it sits between the second and the third. The difference tone is better than listening to an interval as an interval and worse than listening to it as a rate.

The ordering matches one already measured from a different direction. An orchestra is given a note put matching a unison by comparing two pitches at 5.7 cents on A, and nulling the beat between them at 2.0. The difference tone of a double stop on that A comes in at 1.7 cents for a second-long note — between the two, nearer the beat, and reached by listening to a pitch rather than to a rate.

A short note leaves the third sound nothing #

Every number above is for a one-second note, and the gain the difference tone had came from the ear’s steady-state limen being finer at low frequencies. A short note removes it.

A tone lasting T seconds cannot have its frequency known to better than about one over 2T hertz, however good the ear, and at three tenths of a second that bound is 1.67 hertz at every frequency on the drawing. The difference tone’s low frequency no longer buys anything, because the limen there and the limen at the upper note are the same number. The product is geared by one, the note is geared by one, and the gain over hearing the upper note directly is exactly nothing.

On D4 the three methods now give 7.8 cents through the difference tone, 3.9 by the beat and 12.6 by comparing pitches. The beat keeps its factor of two because its advantage was in the hertz, and the difference tone has lost the only advantage it had that was not in the hertz.

The practical reading is narrow and definite. The third sound is a sustained-note method. In a passage of quavers at any ordinary tempo a player has nothing to gain from it that the notes themselves do not already offer.

The intervals whose beats are not there #

The comparison so far has assumed the beat exists, and it exists only if the lower note has a partial p and the upper note a partial q strong enough to beat. For thirds and sixths those are partials four to six, which any bowed or blown tone has. For seconds and sevenths they are partials eight to sixteen.

The difference tone’s threshold is almost flat across the octave, between 2.3 and 2.7 cents, and that is not a coincidence. The product moves by the upper note’s hertz, its limen is roughly constant in hertz at the low frequencies all these products occupy, and so the threshold in cents on the upper note is roughly the limen in hertz divided by the upper note’s frequency — the same for every interval with the same lower note, give or take how high the upper note is.

So the ranking reverses on the intervals that matter least to a violinist and most to anybody tuning a tone with few partials. Where partials are available the beat wins, by between a fifth for the fifth and a factor of three for the minor sixth; where they are not, the difference tone is the only measurement there is. A recorder in its upper register, a flute stop on an organ, or a pair of whistles have fundamentals and little above them, and for those the third sound is not a folk method but the only method the physics leaves.

Just or tempered is a large difference on the product #

The thresholds above are about finding the last few cents. The question a player more often has is coarser — whether a third is just or tempered — and for that question the three methods are not in the same order.

The difference between a just and a tempered major third is 13.7 cents of the interval. Comparing pitches, that is about two thresholds on D4, which is detectable and not comfortable. Through the product it is 67 cents, and a one-second note’s threshold at the product is about twelve cents, so the choice between the two tunings is nearly six thresholds wide: on the third sound, just and tempered are different notes rather than slightly different tunings of one.

The beat does something stranger. A tempered third’s beat is not slow: on D4 the lower note’s fifth partial and the upper note’s fourth are 11.7 hertz apart, on the A string 17.5, and on the E string 26. The beat a tuner can actually use is a slow one, and above ten or fifteen a second a beat stops being counted and becomes a roughness. So on the violin’s two upper strings the tempered third does not offer a beat to count at all; it offers a harshness, and the question of how far to move a finger to remove it has to be answered by moving the finger and listening for the harshness to slow into a beat.

The difference tone has no such regime. At every register the tempered product is a pitch two thirds of a semitone from the just one, and the direction it is displaced in is the direction the finger has to move. For telling just from tempered, the third sound is the one method that works the same way across the whole instrument.

Who the method was for #

The repertoire Tartini taught it for is the violin’s double stops — thirds and sixths, held, at a firm dynamic, in the middle of the instrument — and the arithmetic sorts that practice into two parts.

On precision it is second best. A violin’s tone has every partial the beat needs, and a player who could attend to the beat would find a mistuning two to two and a half times smaller. Players who report tuning double stops by the third sound are either using it on sustained notes where it gives two to three cents, which is good enough for nearly anything, or are hearing the beat and calling it the third sound, which the two phenomena make easy since a mistuned interval produces both at once.

On identification it is unique, and that may be what it was really for. A beat says that an interval is out and not which way. A difference tone is a pitch, it sits at a note, and whether that note is sharp or flat of the note it should be is a judgement of direction. It is also a note with a function — the product of a 5:4 is two octaves below the lower note, which is the root of the chord the third implies and the place a missing fundamental would be heard — and a note that is sharp because of where it goes reminds that direction is the decision a performer actually makes. A violinist who wants the third just rather than tempered can hear, through the product, whether the bass it implies agrees with the bass that is sounding. No count of beats offers that.

The measurements this rests on #

The positions of the products are arithmetic on two frequencies and depend on nothing measured. The magnification is a derivative of that arithmetic and depends on nothing measured either.

The resolutions depend on three measured quantities and one convention. The limen for a steady tone is Wier, Jesteadt and Green’s fit, which rises with the square root of frequency; the bound for a short tone is the length of the note, one over 2T hertz. The limen for an interval heard as two pitches adds the two notes’ limens as independent errors. And the beat’s threshold is the mistuning whose beat completes one cycle within the note, which is a convention — a listener may need half a cycle to notice a change of level or two cycles to be sure of one, and either moves the beat’s curve by a factor of two in the direction that shows.

What the lens leaves out #

The limen used for the difference tone was never measured on a difference tone. Wier’s fit covers 200 to 8,000 hertz and every product of a double stop below the top of the violin’s range sits below 200, so the difference-tone curves are drawn with the fit extrapolated, and the drawing dashes them where they are. More seriously, the fit is for a real tone at a comfortable level, and the difference tone is a tone the cochlea generates from the primaries at a level well below theirs. A frequency limen rises steeply at low sensation levels, so the difference tone’s true threshold is probably coarser than drawn, and the gap to the beat is probably wider.

Level decides whether there is a product to use at all. The quadratic difference tone grows steeply with the level of the pair and vanishes when they are soft, and nothing here draws a level. Every threshold on this page is conditional on the product being audible, and at piano it is not.

The upper note is the one that moves. A player can move either finger, and when the lower note moves the product moves by the lower note’s hertz with the opposite sign — so the gearing in hertz is still one and nothing above changes, except that the direction of the correction is reversed, which is exactly the judgement the method was supposed to make easy.

And vibrato defeats all of it. A violin’s vibrato is several cents wide on each note, two uncorrelated vibratos put a difference tone through excursions p/(p − q) times wider in cents, and a sustained note with vibrato is a note on which the product has no settled pitch to judge. The method belongs to held notes played straight, which is how it is taught, and the drawing cannot say how common that is in performance.

Still open: which note the third sound is #

The difference tone of every just interval sits on a harmonic of a fundamental that neither note contains — the major third’s on the fundamental itself, the minor sixth’s on its third harmonic — and the cubic product, which is the one audible at a moderate dynamic, sits on another. Which harmonic each product lands on is fixed by the ratio, it changes from interval to interval, and it decides whether what a player hears underneath a double stop is the root of the chord the interval implies, some other note of it, or a note no keyboard has. That is arithmetic on the same two numbers p and q, and it has not been asked of the octave as a whole.