Intervals and chords

The thirds and the fourths check different octaves

Every interval counted to its printed beat rate on a stiff string is left wide, by an amount set by the squares of the partial numbers that meet in it. Add the links of a chain and the sum lands on an octave type: a fourth and a fifth make the 4:2, a minor third and a major sixth make the 6:3, and three major thirds come just short of the 6:3. So the two most familiar checks in a bearing are exact tests of two different octaves, and on a stiff piano no octave passes both. Each fails on the other's octave by a beat a second at A3.

Assumes: The fourth–fifth check cannot hear the note it checks · A stiff string slows the fourths and quickens the fifths

The fourth–fifth check cannot hear the note it checks found that the tuner’s most familiar test — a fourth below a note and a fifth above it, inside one octave, compared for their beats — is blind to the middle note and reads only the octave, and that the octave it passes is the 4:2: the one whose lower note’s fourth partial agrees with its upper note’s second. On a stiff piano that octave is about 1.6 cents wider than 2:1 at A3.

The fourth–fifth check is not the only test in a bearing, and a tuner does not lay a temperament from fourths and fifths alone. Many aural sequences begin with major thirds: the three thirds F3–A3, A3–C♯4 and C♯4–F4, which divide the octave into equal parts, are set to their printed rates first, and everything else is fitted between them. And there is a second contiguous test with the same shape as the first: a minor third up from the bottom of an octave and a major sixth from there to the top, compared for their beats. The question the earlier essay left was whether these build the same octave as the fourths and fifths. They do not, and the reason is a piece of arithmetic short enough to state in a line.

A chain leaves its octave wide by a sum of squares

On a stiff string the k-th partial of a note is raised, to first order, by a fraction $Bk^2/2$ of its frequency, where B is the string’s inharmonicity coefficient. An interval whose coincident partials are the m-th of the lower note and the n-th of the upper — 3 and 2 for a fifth, 5 and 4 for a major third — has its lower partial raised by $Bm^2/2$ and its upper partial by $Bn^2/2$ . A tuner who adjusts the upper note until the pair beats at the rate printed for whole-number partials undoes the difference: the interval is left wide by $(m^2 - n^2)\,B/2$ , as a stiff string slows the fourths and quickens the fifths found every interval is, narrow or wide.

Stack a chain of such intervals to fill an octave and, if every string had the same stiffness, the widenings would simply add. The octave would be wide by $B/2$ times the sum of $m^2 - n^2$ over the links. An octave type is defined the same way: the 2k:k type silences the lower note’s 2k-th partial against the upper’s k-th, so it is wide by $B/2$ times $(2k)^2 - k^2$ , which is $3k^2$ . The 2:1 type is 3, the 4:2 is 12, the 6:3 is 27, the 8:4 is 48.

With one stiffness everywhere, a chain's octave is the sum of its squared partial numbers. At A3, with every string given A3's stiffness: the octave each chain of intervals builds when every link is counted to its printed rate, in units of the stiffness coefficient (half of it, in cents), against the sum over its links of the squared partial numbers that meet. fourth, then fifth: 4²−3² + 3²−2² = 12, computed 11.9; fifth, then fourth: 3²−2² + 4²−3² = 12, computed 11.9; three major thirds: 5²−4² + 5²−4² + 5²−4² = 27, computed 26.2; minor third, then major sixth: 6²−5² + 5²−3² = 27, computed 26.9; four minor thirds: 6²−5² + 6²−5² + 6²−5² + 6²−5² = 44, computed 44.7. The octave types 2:1, 4:2, 6:3 and 8:4 are 3, 12, 27 and 48 by the same rule (3.0, 12.0, 26.8, 47.2 computed). So both orders of fourth and fifth are the 4:2, the thirds and the minor-third-and-sixth are the 6:3, and four minor thirds fall between the 6:3 and the 8:4. — Fig. 1 With every string given A3’s stiffness, the octave each chain builds when every link is counted to its printed beat rate, in units of half the stiffness coefficient, against the sum of the squared partial numbers that meet in its links. The vertical lines are the octave types at 3, 12, 27 and 48. A fourth and a fifth sum to 12 in either order; three major thirds and a minor third with a major sixth sum to 27.

The sums can be read off the figure. A fourth is $4^2 - 3^2 = 7$ and a fifth $3^2 - 2^2 = 5$ : together 12, the 4:2 type, in either order. A major third is $5^2 - 4^2 = 9$ , and three of them make 27, the 6:3 type. A minor third is $6^2 - 5^2 = 11$ and a major sixth $5^2 - 3^2 = 16$ : together 27, the 6:3 again. Four minor thirds make 44, between the 6:3 and the 8:4. Computed rather than counted, the chains land within a few per cent of these numbers, the remainder being the second-order terms the rule drops.

So on strings of one stiffness, a bearing built from thirds builds a 6:3 octave and a bearing built from fourths and fifths builds a 4:2, and no one has to choose either. The numbers are fixed by which partials meet in the intervals, as small whole numbers have fixed everything else about a beat, and as twelve fifths and seven octaves fixed the temperament the table prints in the first place.

On real strings

A piano’s strings are not one stiffness: on the grand modelled here the coefficient is 2.87 times larger an octave up. Two of the chains do not care. The fourth then the fifth share the middle note’s third partial, and the minor third then the major sixth share the middle note’s fifth partial — in each case the shared partial’s displacement cancels, exactly as it did in the fourth–fifth check, and the chain lands on its octave type whatever the stiffness does from note to note. The thirds share nothing. F3’s fifth partial meets A3’s fourth, and A3’s fifth partial meets C♯4’s fourth, so the chain’s displacements do not telescope and the rising stiffness pulls it off its type.

Fig. 2 On each lower note from F3 to F4 on the grand, each note with its own stiffness: the octave reached by a fourth then a fifth, three major thirds, a minor third then a major sixth, and four minor thirds, every link counted to its printed rate, against the 4:2, 6:3 and 8:4 types drawn dotted in the same colours. The fourth-and-fifth line hides its 4:2 type and the third-and-sixth line its 6:3. The dial changes the strings’ stiffness.

At A3 the fourth and fifth build an octave 1.60 cents wide, which is the 4:2 type; the minor third and major sixth build 3.57, the 6:3 type at 3.56; three major thirds build 3.20, just short of it; four minor thirds build 8.09, past the 8:4’s 6.22. Through the bearing octave the thirds’ octave runs from 2.24 cents at F3 to 6.45 at F4, and at every note it is twice the fourths and fifths’ — 2.00 at A3, between 1.98 and 2.01 everywhere on the grand. The dial scales every number with the stiffness and keeps the ratio: a piano four times as stiff has the fourths and fifths at 6.22 cents and the thirds at 11.89.

The doubling is close to the sum-of-squares ratio of 27 to 12, which would make it 2.25; the rising stiffness takes about a tenth of that off, because the thirds’ upper partials sit on stiffer strings than their lower ones.

The checks disagree most on strings of one stiffness, and less the faster stiffness rises. At A3, the octave each chain builds, counted to the printed rates, against how many times stiffer a string is an octave higher, with the 4:2 and 6:3 types. 1 times: fourth, then fifth 4.25, three major thirds 9.33, minor third, then major sixth 9.56, four minor thirds 15.94; 2 times: fourth, then fifth 2.83, three major thirds 6.26, minor third, then major sixth 6.34, four minor thirds 12.12; 3 times: fourth, then fifth 1.41, three major thirds 2.71, minor third, then major sixth 3.15, four minor thirds 7.42; 4 times: fourth, then fifth 0.00, three major thirds -1.04, minor third, then major sixth 0.00, four minor thirds 2.37. The modelled grand's strings are 2.87 times stiffer an octave up. On uniform strings the thirds' octave is more than twice the fourths and fifths'. At four times, where a partial's rise depends only on its own frequency, the slow-beating fourth and fifth build a 2:1 octave exactly, and the thirds, whose partials are eight hertz apart and so raised by different amounts, miss it by about a cent the other way. — Fig. 3 At A3, the octave each chain builds against how many times stiffer a string is an octave higher, with the 4:2 and 6:3 types dotted. On strings of one stiffness the thirds and the third-and-sixth both build the 6:3 and the fourth and fifth the 4:2; the faster stiffness rises, the closer the chains come together, and at four times the fourth and fifth build a 2:1 octave.

The minor thirds are worth a sentence of their own, because they behave worst. Four of them fill an octave as the notes of a diminished seventh chord, and a minor third beats fast — near twelve a second at A3 — with its coincident partials, the lower note’s sixth and the upper’s fifth, the highest-numbered of any interval a tuner uses. Counted to the table they build an octave 8.09 cents wide at A3, five times the fourths and fifths’ and past the 8:4 type. A bearing that leaned on minor thirds would be stretched further than any octave type a tuner names, which is one reason the minor third is used as a check against the major sixth rather than chained.

The same arithmetic reaches the checks that span more than an octave. A major third and a major tenth from one lower note share that note’s fifth partial and compare their upper notes’ fourth and second: an exact test of the 4:2 octave between the third’s upper note and the tenth’s. A tenth and a seventeenth compare the second and first: the 2:1. Every contiguous check a tuner uses is an octave-type test of this kind, built on whichever pair of partials its two intervals happen to share, and every partial beats at its own rate is the reason there is always such a pair to find.

Drawn against the rate at which stiffness rises up the keyboard, the chains converge. On uniform strings, at A3, the fourth and fifth make 4.25 cents and the thirds 9.33 — a gap of five cents. At the grand’s rate the gap is 1.60. At four times stiffer an octave up — the rate at which a partial’s displacement depends only on its own frequency, so that two partials meeting near one frequency are raised alike — the fourths and fifths build a 2:1 octave exactly, the third and sixth do too, and the thirds miss it by about a cent the other way, because a major third’s coincident partials are eight hertz apart rather than one, and that is enough for them to be raised by measurably different amounts. Real pianos sit between, which is where the checks disagree.

Neither octave satisfies the other check

Two exact tests of two different octaves cannot both pass. What a tuner hears is the size of the failure, and it is not small.

Neither octave satisfies the other exact check, by a beat a second or more. On each lower note from F3 to F4: on the 4:2 octave, which satisfies the fourth–fifth check, how far apart the minor third and major sixth of the minor third–major sixth check beat; and on the 6:3 octave, which satisfies that check, how far apart the fourth and fifth beat. On the grand modelled here and on a piano four times as stiff, each note with its own stiffness. Grand, the sixth check on the 4:2: F3 0.84, A3 1.51, C♯4 2.72, F4 4.87 a second; the fourth check on the 6:3: F3 0.56, A3 1.00, C♯4 1.79, F4 3.20. Four times as stiff: 5.80 and 3.76 at A3. No octave satisfies both. The axis is logarithmic. — Fig. 4 On each lower note from F3 to F4: on the 4:2 octave, which satisfies the fourth–fifth check, how far apart the minor third and the major sixth beat; and on the 6:3 octave, which satisfies the third–sixth check, how far apart the fourth and the fifth beat. Solid on the grand, dashed on a piano four times as stiff. The axis is logarithmic.

Set the octave at A3 by the fourth–fifth check, so that the fourth and fifth beat alike, and the minor third A3–C4 and the major sixth C4–A4 beat 1.51 a second apart. Set it instead by the third–sixth check, and the fourth and the fifth beat 1.00 a second apart. Through the bearing octave the third–sixth check’s failure runs from 0.84 a second at F3 to 4.87 at F4, and the fourth–fifth check’s from 0.56 to 3.20; on a piano four times as stiff they are 5.80 and 3.76 at A3.

These are not marginal readings. The earlier essay found the table’s own overshoot of the 4:2 octave leaving the fourth–fifth check a gap of 0.15 a second at A3, a sixth of a beat, and argued that it sat at the edge of what a short comparison resolves. The gaps here are seven to ten times larger. The third–sixth check compares two beats near twelve a second, and a difference of a beat and a half between them is a twelfth of either, against the fourth–fifth check’s sixth. A tuner who has satisfied either check and turns to the other will hear the failure on any piano with steel strings.

The third–sixth comparison is the larger failure because its partials are higher: it compares the octave’s sixth and third partials near 1320 hertz at A3, the fourth–fifth check its fourth and second near 880, and the same width of octave in cents is half as many hertz again at the higher frequency.

The thirds inside the octave the fourths choose

The conflict can be read the other way, as what the fourth–fifth octave does to the thirds that have to fit inside it.

Inside the octave the fourths and fifths choose, every third beats slower than printed. The three major thirds that divide the octave on A3 equally in their fundamentals, on the grand modelled here, as their true beat rate over the printed one, when the octave is 2:1, the 4:2 the fourth–fifth check selects, the octave three thirds counted to the table build, and the 6:3. octave 2:1 (0.00 cents): 0.947, 0.924, 0.890; the 4:2, by fourth and fifth (1.60 cents): 0.986, 0.963, 0.930; by three thirds (3.20 cents): 1.026, 1.003, 0.970; the 6:3, by third and sixth (3.56 cents): 1.035, 1.012, 0.979. No equal division makes all three beat at the printed rate, because the higher third's partials are raised more; the octave the thirds build brings the middle one to it. — Fig. 5 The three major thirds that divide A3’s octave equally in their fundamentals, each beat rate as a share of its printed rate, with the octave set four ways: 2:1, the 4:2 that satisfies the fourth–fifth check, the octave three thirds counted to the table build, and the 6:3 that satisfies the third–sixth check. Inside the 4:2, every third beats slow.

Inside a 2:1 octave the three thirds of A3 beat at 0.947, 0.924 and 0.890 of their printed rates — the slowing the earlier essay found for every wide interval on a stiff string. Widening the octave to the 4:2 recovers some of it: 0.986, 0.963, 0.930. It is not enough. Inside the octave the fourth–fifth check selects, every major third beats slower than printed, by 1.4, 3.7 and 7.0 per cent from the bottom third to the top. Averaged over the three, the shortfall is 2.8 per cent at F3 and 8.1 per cent at F4.

Widening further to the octave the thirds themselves build brings the middle third to its printed rate, at 1.003, with the lowest a little fast and the highest still slow; the 6:3 octave puts the bottom two slightly fast. No equal division of any octave makes all three beat as printed, because each third is a semitone-and-more higher than the last and its partials are raised more. A tuner who wants three thirds at their printed rates has to give them unequal sizes — which is what counting each to its rate does, and why the thirds’ chain lands where it does rather than on a type.

The progression of thirds

The other test tuners make with thirds is the progression. Sound the major thirds on F3, F♯3, G3 and so on up the bearing octave in turn, and in equal temperament on whole-number partials each beats 5.95 per cent faster than the one below, rising from 6.93 a second to 13.86 over the octave. A tuner listens for that steady quickening, and a third that beats slower than its lower neighbour, or much faster than its upper one, marks a note out of place.

On the grand the progression survives, bent. Exactly tempered, the thirds rise from 6.67 a second to 12.34, and the step from one to the next falls from 5.6 per cent at the bottom of the octave to 4.8 at the top, because each third up sits on stiffer strings and is slowed a little more than the last. Every step is still a quickening, so the check is satisfied — but with steps a fifth smaller than printed at the top, and a tuner who expected the printed ratio would read the upper thirds as slow and widen them.

On a piano four times as stiff the progression turns over. The thirds rise from 5.91 a second to a peak of 7.98 on E4, and the third on F4 beats at 7.96 — slower than the one below it. On that instrument an exactly tempered bearing fails the progression check at its top, and a tuner who corrects the failure, by widening the F4 third until it beats faster than the E4, is moving away from equal temperament to satisfy a check that equal temperament no longer passes.

The progression is a better-behaved check than the contiguous ones on a grand, since it only asks for a direction and stiffness does not reverse the direction there. On a small upright it is the opposite: it asks for something the instrument cannot give in tune.

What a tuner’s octave is

Put together, the picture is of three families of check that each have an answer of their own. The fourth–fifth check selects the 4:2 octave exactly. The third–sixth check selects the 6:3 exactly. Thirds counted to the table select an octave just short of the 6:3, and the progression of thirds asks for a quickening that on a stiff enough piano cannot be had at all. On a flexible string all of these are the same octave, the 2:1; on a stiff one they spread across two cents at A3 and four at F4.

A beat is never one beat and the octave that is not two to one treated the octave type as a decision a tuner makes when tuning the octave, by choosing which member of its beat family to silence. These figures say the decision is made earlier and less freely. A tuner who trusts the fourth–fifth check has chosen the 4:2; one who trusts the third–sixth check has chosen the 6:3; one who uses both cannot satisfy both, and the octave that results is a compromise between two failures each about a beat a second wide. The order in which the tests are learnt and the weight given to each decide the stretch as surely as listening to the octave does.

This also says something about why the piano is tuned wrong on purpose: stretched octaves are usually explained by the stiffness of the octave’s own strings and the tuner’s preference for one type over another. The checks are a second route to the same stretch, and a more insistent one, since a tuner can ignore an octave’s beat family but cannot easily ignore a check that fails by a beat a second.

What the arithmetic assumes

The stiff-string law, note by note. Each partial is raised by the square root of one plus $Bk^2$ , with B from the modelled grand’s string lengths and gauges. A real piano’s coefficients scatter from note to note, and would make each chain’s octave ragged rather than smooth. The two exact checks are immune to that: the 4:2 and 6:3 octaves they select are defined by the outer notes’ own partials, whatever the middle note’s string is like.

The printed rates are the whole-number rates at equal temperament. That is what bearing plans print. Where a chain’s second link starts from a note the first moved, its printed rate is read at the moved note, which is why the third-and-sixth chain lands within a few hundredths of a cent of the 6:3 rather than exactly on it.

Equal division in the fundamentals. The thirds inside an octave are drawn as three equal steps of whatever octave is set, which is the temperament a tuner aims at once the octave is fixed. A real tuning distributes the conflict among notes rather than holding the thirds equal.

What these figures do not show

Which compromise tuners actually reach. The figures give each check’s answer and the size of its failure on the other’s octave. They do not say how a tuner weighs a fourth–fifth gap of a beat a second against a third–sixth gap of a beat and a half, and real tuners differ: some favour the 4:2 in the middle of the keyboard and wider types above it, some the 6:3 throughout. Those preferences are reported by tuners and are not derived here.

How fast a rate difference between two fast beats is heard. The third–sixth check compares beats near twelve a second, close to where three beats at most found a beat ceasing to be countable, and the threshold for telling two such rates apart is not one the figures supply.

Still open: the octave that fails both checks least

If neither check can pass, a tuner’s ear settles somewhere, and where it settles is a quantity the same arithmetic can price once a rule for weighing the failures is stated. The simplest rule weighs each check’s gap by the beat rates it compares — a difference of a beat a second is a larger fraction of a fourth’s beat near one than of a sixth’s near twelve — and asks for the octave that makes the two fractional failures equal; the beat a tuner can actually use supplies a threshold for each rate that would serve as the denominator. Computed across the bearing octave and up the keyboard, that octave would be a stretch curve derived from the checks rather than from the octaves, and it could be set beside the curve the octave types give to see whether the two agree — and so whether the stretch a tuner arrives at is a choice of octave at all, or the residue of tests that cannot all be passed.