A beat has a depth, and six essays held it at one

The beat ladder has six rungs and one parameter it never varied.

Beats are arithmetic sets it out at the start and states it as though it were a definition: “two steady tones are present, each at constant amplitude, and their sum is a wave whose amplitude varies”. Three strings and the note that comes back says it again — “added together, each with its own amplitude, neither affecting the other” — and then, like the four rungs around it, draws two tones of the same size.

Equal amplitude is not one case among many. It is the only ratio at which the trough of a beat goes to zero, and a trough that goes to zero is precisely what makes the tuner’s method work: a tuner counts beats is about listening for the pulsation to stop, and a pulsation that never quite reaches silence is much harder to hear stop.

The arithmetic, which is two lines #

Two sinusoids of amplitudes a and b at nearby frequencies drift in and out of phase, so their sum has an envelope that swings between a + b when they agree and the absolute value of a − b when they oppose. The dip between the two, in decibels, is

Nothing about the beat rate appears in it. The rate is the frequency difference and the depth is the amplitude ratio, and the two are completely independent — which is exactly why six rungs could get the rate right while holding the depth at a value it almost never has.

The rate is what all six of those rungs are about, and they are all correct. What they cannot say is whether the beat they compute is audible, and the answer is that it depends on a quantity none of them contains.

Where the amplitudes come from #

The amplitudes are not free. A tuner listening to an interval is not listening to the two fundamentals — those are far apart and do not beat at all — but to a coincident pair of partials. For an interval of p:q, the beating pair is the lower note’s p-th partial against the upper note’s q-th.

So the amplitude ratio is a property of the timbre, read off at two specific partial numbers, and it is completely determined once the interval is chosen.

For a power-law spectrum there is an exact result and it is worth writing out, because it says the opposite of what this section originally claimed. If the partials fall as 1/nˢ, the beating pair’s ratio is (1/qˢ)/(1/pˢ), which is (p/q)ˢ — the interval’s own ratio raised to the spectral slope. So the slope enters as an exponent on a quantity the interval fixes, and the ordering of the intervals cannot depend on it:

The ordering is identical at every slope, from a nearly flat spectrum to a very steep one, because raising every ratio to the same power preserves their order. What the slope decides is the scale: a flatter spectrum makes every null deeper and a steeper one makes every null shallower, by about eleven decibels across the range of slopes a real instrument might have. So the finding is the reverse of the sentence this paragraph used to carry: which interval is easiest to tune by beats is decided by the interval, and the spectral slope decides how easy any of them is.

That leaves the clarinet result standing and re-explains it. A clarinet’s spectrum is not a power law at all — it alternates — and it is the alternation rather than the slope that reorders the table, by putting some pairs on one strong partial and one absent one. A monotone spectrum, however steep, can never do that.

One small correction falls out of the same expression. The table below puts the major sixth ahead of the minor, at 12.2 against 12.0. For any power-law spectrum it is the other way round, because 8:5 is a ratio of 1.60 and 5:3 is 1.67, and the smaller ratio gives the deeper null. The reversal in the table is an artefact of the rounded amplitudes in this site’s own partial list — 0.17 where a pure 1/n wants 0.167 — and not a property of any spectrum.

For a string spectrum falling as one over n, adjacent partials are close in strength high up and far apart low down. Partial 6 against partial 5 is a ratio of 1.18; partial 2 against partial 1 is 2.00. So the ordering comes out:

minor third   6:5    1.18:1    21.8 dB
major third   5:4    1.25:1    19.1
fourth        4:3    1.32:1    17.2
fifth         3:2    1.52:1    13.8
major sixth   5:3    1.65:1    12.2
minor sixth   8:5    1.67:1    12.0
octave        2:1    2.00:1     9.5

The octave is the worst interval in the list, and the expression above says it is the worst for every monotone spectrum there is: 2:1 is the largest ratio any of these intervals has, so its pair is the most unequal whatever the slope.

That is a stronger statement than a table of one timbre can make, and it has a consequence for the whole family of instruments. Any instrument whose partials fall smoothly — a string, a bowed string, a reed with a full series, an organ flue — has the same ordering of tuneable intervals, differing only in how deep all seven nulls are. The instruments that break the ordering are the ones with a hole in their spectrum, and there are exactly two kinds: stopped pipes, which lose their even partials, and struck bars, whose partials are not multiples of anything.

Which is what piano tuners do #

That ordering is not a curiosity. It is the practice.

A piano tuner sets a temperament by counting beats on thirds and sixths — the standard sequences all work through major thirds, minor thirds and sixths, with published beat rates for each — and checks octaves last, by other means. Every manual says so and none of them says why in these terms; the usual justification is that a third’s beat rate is conveniently fast enough to count and an octave’s is too slow.

That justification is about the rate. This figure is about the depth, and it says something stronger: an octave’s beat is not merely slow, it is shallow, by twelve decibels against a minor third’s. A tuner tuning an octave is listening for a fluctuation that never falls below a third of its own peak, and one tuning a third is listening for something that nearly disappears.

The two accounts are not in competition. A fixed detuning in cents gives a beat rate proportional to the coincidence frequency, which the fourth rung draws — so the high coincidences that give the deepest nulls also give the fastest rates, and the same choice optimises both. What has been missing is that there were two things being optimised.

The instrument that cannot be tuned this way at all #

Change the timbre and the whole table moves, and one change moves it catastrophically.

A clarinet’s spectrum is odd-dominant: partials 1, 3, 5 and 7 are strong and 2, 4, 6 and 8 are down by more than twenty decibels. So a coincidence pair that lands on one odd partial and one even partial is a pair with an amplitude ratio of ten or twenty to one, and the beat between them has a dip of about a decibel.

the major sixth   5:3    1.52:1    13.8 dB
the major third   5:4   11.00:1     1.6
the fifth         3:2   12.50:1     1.4
the octave        2:1   25.00:1     0.7

Six of the seven intervals are unusable, and the one that survives is the major sixth, whose pair is 5 against 3 — both odd.

This is a real and slightly startling claim about a real instrument: two clarinettists cannot tune an octave to each other by listening for the beat to stop, because there is no null there to hear. What they can do is tune the unison, where the pair is partial 1 against partial 1 and the amplitudes are equal by construction — which is exactly what a wind section does.

What a shallow beat looks like #

The two envelopes are worth seeing beside each other, because the difference is not subtle and it is not what the word beat suggests.

At equal amplitudes the envelope touches zero once per cycle: the sound goes away and comes back, and what a tuner hears is a series of silences whose spacing can be timed. At two to one it swings between three and one, which is a pulsation — audible, countable, and never silent. At ten to one it swings between eleven and nine, which is a tremor.

Where the equal-amplitude case actually occurs #

It is worth saying that the assumption is not perverse, because there is one very common case in which it is exactly right.

A unison between two nominally identical sources — two strings of a piano’s trichord, two singers on one note, two ranks of one organ stop — pairs partial one against partial one, partial two against partial two, and so on down the spectrum, with the amplitude ratio equal to one at every pair by construction. Every beat in a unison is a true null.

That is the case the fourth and fifth rungs are about, and both of them are about tuning practice — a technician setting a trichord, a voicer setting a celeste rank. So the ladder’s most practical rungs happen to live at the one point where its unstated assumption holds, and the rungs that generalise to intervals are the ones the assumption quietly broke.

Which computation produced the numbers #

The amplitudes are the collection’s own TIMBRES lists, unchanged — the string’s one-over-n series and the clarinet’s measured odd-dominant one, both of which have been in every spectrum figure here from the beginning.

The coincidence pair for an interval p:q is the lower note’s p-th partial against the upper note’s q-th, which is the same identification the sixth rung uses to find where a detuned pair crosses into roughness. It is exact only for just intervals; a tempered interval’s coincidence is a near coincidence, which is the whole of why it beats at all, and the amplitudes are the same either way because a few cents does not move a partial’s strength.

The dip is the formula above, with the infinite case at exactly equal amplitudes reported as infinite rather than clipped. The modulation index — peak minus trough over peak, which is what a demodulator would report — is computed alongside it and is the quantity to compare against published detection thresholds, which are in the region of a few per cent for a slow modulation.

The depth as a slider #

The dip is a smooth function of one number, and it is worth being able to move it: the interesting thing about the curve is not any point on it but how fast it falls away from the left-hand edge.

Half of the total collapse happens between a ratio of one and a ratio of about 1.4 — a difference of three decibels between the two tones, which is a smaller imbalance than any two instruments will ever have by accident. By two to one, which is the octave’s own ratio on a string spectrum, two thirds of the available depth is gone. After five to one there is nothing left to lose.

That shape is why the assumption survived six rungs without being noticed. It is not that the equal-amplitude case is a good approximation to the unequal one; it is that the equal-amplitude case is a singularity, and every figure drawn at it is drawn at the one point where the quantity is not finite. A model evaluated at a singularity looks clean and tells nobody how steep the ground is beside it.

Where the model stops #

Two sinusoids is the whole model. Real notes have many partials and several pairs of them are near coincidence at once, which the sixth rung is entirely about — so the envelope a listener actually hears is a sum of several fluctuations at different rates and different depths, and the deepest is not necessarily the most audible.

The threshold is not here. A dip of nine and a half decibels is a fact about a waveform. Whether it is audible as a beat, and at what rate, is a psychoacoustic question with a published answer that this figure does not use — modulation detection thresholds depend on rate, level and carrier frequency, and all three are outside the arithmetic.

And the partials are a table. A real string’s partial amplitudes depend on where it was struck, how hard, and on the soundboard, all of which this collection has ladders for and none of which is in these two lists. A hammer at one eighth silences partial eight, which is one of the amplitudes in the minor-sixth row.

What the picture cannot show #

It cannot show what a tuner listens to instead. A tuner who cannot hear a null counts a rate, and counting a shallow fluctuation is harder but not impossible. The claim here is about how much help the waveform gives, not about what can be achieved.

Nor can it show the unison, which is the case that matters most. A unison’s pair is partial 1 against partial 1 on two nominally identical instruments, so the amplitude ratio is one by construction and the dip is infinite. That is why every tuning practice in every tradition starts from unisons and octaves-by-instrument rather than from intervals — and the octave row in these tables is an octave between two notes, not the octave between two ranks of a stop.

And it cannot show the coupling. Two piano strings on one bridge are not two independent sources, and the fourth rung is about what that does. A coupled pair’s envelope is not the sum of two constant-amplitude tones at all, so the arithmetic above does not apply to the case the ladder’s most practical rung is about.

Whose tuning, and when #

The interval sequences a piano tuner uses are nineteenth- and twentieth-century, and the published beat rates for each interval at each pitch are a twentieth-century apparatus — a table of numbers to be counted against a watch. What is older is the practice of setting a temperament by ear on thirds and fifths, which goes back as far as the instructions do: Werckmeister’s and Neidhardt’s directions are in terms of how much each fifth is narrowed and how the thirds should sound.

The depth table says something about why the tradition works the way it does. A tradition built on fifths — and the Pythagorean and meantone traditions are — is working with a thirteen-decibel dip; one built on thirds is working with nineteen. A fraction of a comma is the essay about the historical move from tuning fifths to tuning thirds, and it treats the move as a question about where the comma goes. This adds a second reason for it: the thirds were easier to hear.

Where this ladder goes next #

Seven rungs. Beats are arithmetic anybody can hear; a tuner counts them; a cellist’s wolf is the same arithmetic coupled; a piano’s unison is a detuning below a bifurcation; a chorus is that pair past it; every partial beats at its own rate; and now every beat has a depth, which is the amplitude ratio the first six held at one.

The rung after it is the one the threshold names. This figure produces a modulation index for every interval on every timbre, and whether a fluctuation of that index at that rate is detectable is a published function of exactly those two quantities plus the level. Running one against the other would turn a table of decibels into a table of yes and no, and would say — for a stated instrument, at a stated pitch — which intervals a tuner can actually work with. That is the same shape as the echo threshold laid over an echogram, and it needs nothing this collection has not got.