Where the beat comes from
Two tones close in frequency add up to a single tone whose loudness rises and falls, and the rise and fall happens once per hertz of difference between them. Two partials at 660 and 659.255 Hz produce a wobble at 0.745 a second — about three beats in four seconds, slow enough to count on a watch.
Which partials matter depends on the interval. A fifth is nominally 3 to 2, so the third partial of the lower note and the second partial of the upper note are aiming at the same frequency. A major third is 5 to 4, so it is the fifth partial against the fourth. Those are the pairs the page uses, and they are the pairs a tuner is listening to whether or not anyone has ever named them out loud.
Run the defaults — A3 to E4, equal temperament, A4 at 440 — and you get 220 Hz and 329.628 Hz. Three times 220 is 660. Twice 329.628 is 659.255. The interval beats at 0.745 a second, and it is narrow, because the lower note is the one whose partial is higher.
Why a cent is not a fixed amount of anything
A cent is a ratio: the twelve hundredth root of two, about 1.000578. It multiplies. That is why the same one cent buys you a different number of hertz at every pitch, and why the beat rate of an interval doubles when you move the whole interval up an octave while its width in cents does not change at all.
The table on the page makes that concrete for whatever interval you are on. On the default fifth, one cent on the upper note takes the beat from 0.745 to 0.364 — it more than halves it. That is a strong test: half a beat of change for one cent means you can set that fifth to well under a cent by ear if you can count the beat. Move to a slow interval where a cent barely shifts the rate and the same ear gives you far less.
Just, equal, and the gap between them
The page prints three widths for every interval. The just width is the pure small-integer ratio in cents. The equal width is a hundred cents times the semitone count. The actual width is what your offsets produce.
An equal tempered fifth is 700 cents and a just fifth is 701.955, so equal fifths are just under two cents narrow. Equal major thirds are worse: 400 cents against a just 386.31, so they are 13.69 cents wide, and the beat rate on a tempered third in the middle of the instrument is fast enough that it is used as a check rather than as a thing you set directly. That contrast between fifths and thirds is the reason temperament sequences are built the way they are, and it is visible in the numbers here.
What this cannot tell you
It cannot tell you which temperament to use. Historical and modern temperaments differ, they are chosen for musical reasons, and the offsets are a field on this form for exactly that reason.
It cannot hear false beats. A single string with an irregularity in the wire or a poor termination beats against itself, and no amount of moving the other note will stop it. If a beat refuses to change as you move the upper note, the calculation is not describing what you are hearing.
And it treats each note as one string. In the middle and top of a piano each note is two or three strings that must first be brought into unison, and the quality of that unison sets the floor for everything above it. An interval between two ragged unisons has no clean beat to count.
Questions people ask
How many beats a second should a fifth have?
That depends on the temperament you are laying and on where in the instrument you are, which is why the page asks for the offsets rather than supplying them. What is fixed is the arithmetic: pick the lower note, pick the offsets from whatever chart you are working to, and the page tells you what the partials do. In equal temperament with A4 at 440, the A3 to E4 fifth beats at 0.745 a second, and every fifth an octave higher beats twice as fast as the one below it.
Why does the same interval beat faster higher up the keyboard?
Because a beat is a difference in hertz and an interval is a ratio. Moving an interval up an octave doubles both frequencies, so it doubles the gap between them in hertz while leaving the width in cents untouched. Beats a second double. This is also why temperament is set in one octave near the middle and carried out by octaves — continuing upward by fifths would leave you counting beats far too fast to judge.
What do the inharmonicity fields do?
They stretch the partials of each string upward, which changes which frequencies actually coincide. With both at zero the calculation treats the strings as ideal and the partial frequencies are whole multiples. Put in a real value and the partials of the two notes move by different amounts, because the partial numbers involved are different — a fifth uses the third partial below and the second above, and the third partial is displaced more than twice as far as the second. That difference is precisely why a piano cannot be tuned to pure ratios anywhere.
Is a positive beat number different from a negative one?
The page reports the speed as a positive number and says separately which partial is higher, because that is the part that tells you which way to move. If the lower note has the higher partial the interval is narrow and the upper note wants to come up; if the upper note has the higher partial the interval is wide. The speed alone cannot tell you which, and it is the commonest way to spend five minutes going in the wrong direction.
Can I use this on a harpsichord or a guitar?
The interval arithmetic is the same for any two sustained tones, so the beat rates hold. What does not carry over is the inharmonicity, which for a harpsichord or a guitar is small enough that leaving both B fields at zero is usually reasonable, and the unison problem, which a harpsichord has in a different form and a guitar does not have at all. The temperament offsets are as relevant on a harpsichord as on a piano and rather more likely to be something other than equal.