An elaboration that arrives early makes every goal tone the bar to tune first

The bar a gamelan can least afford wrong is the one its skeleton lands on priced a tuning error in a two-instrument gamelan texture under a stated rule for how the parts fit: the skeleton moves in units of four strokes, each ending on a goal tone, and the elaborating part meets the skeleton’s pitch on each unit’s last stroke. A skeleton that ends every unit on one degree made that degree’s bar three and a half to five times as costly to leave wrong as any other. But a mode has several goal tones, and with the goals shared among three degrees the ranking went flat: a low goal could end up no dearer than a high bar the skeleton never ends a unit on, because a beat at the same number of cents is slower and softer lower down.

That essay’s rule was the thinnest version of the convergence. It put the elaborating part on the goal at exactly one stroke. An elaboration in the style does something thicker: it reaches its goal tone early — anticipating the arrival, sometimes by most of the unit — and stays there, or returns to it, into the goal stroke. The question left was how much of a goal bar’s cost that anticipation carries, and whether it makes the ranking steep even for a mode with three goals.

It does both. Arriving seven of its own strokes early, the elaborating part makes every one of three goal tones cost more than three times the dearest bar that is not a goal. A single goal’s cost rises more than threefold between meeting on the stroke and arriving eleven strokes early.

Arriving early, stated #

The rendering is the earlier essay’s: the measured slendro at 0, 231, 474, 717 and 955 cents over middle C, on a bar’s inharmonic spectrum; a ringing elaborating part striking every 0.15 seconds over a damped skeleton four times slower; roughness summed over every pair of notes sounding together, with the beating between the same degree on the two instruments counted, as a scale is not a set of pitches and a wrong bar beats the same on either instrument set it up. Passages run for 128 bars. The skeleton’s four-stroke units end on stated goal degrees, and the elaborating part plays each unit’s goal on the goal stroke.

The one change is a number: how many of the elaborating part’s own strokes before the goal stroke also play the goal. At nought, the earlier rule. At three, the elaborating part lands on the goal three strokes — just under half a second — before the skeleton does, and holds it. At seven, it arrives about a second early, most of the way through the last skeleton stroke before the goal. At eleven, it has been on the goal for three-quarters of the unit. Every other elaborating stroke is drawn at random from the scale, as before.

A tuning error is a bar moved ten cents flat or sharp, on either instrument, and its cost is what that adds to the texture’s roughness per second, averaged over the two instruments and the two directions.

Three goals become three dear bars #

The first figure is the mode the earlier essay found flattest, goals on 6, 2 and 3. Meeting only on the stroke, its goals cost 0.161, 0.097 and 0.082, and the third degree’s cost is no more than it is under a skeleton drawn at random. Three strokes early, they cost 0.221, 0.115 and 0.106. Seven strokes early, 0.295, 0.159 and 0.136, while the fifth degree, which the skeleton never ends a unit on, falls from 0.064 to 0.043.

So the goal tones separate from the rest. At seven strokes early the cheapest goal costs 3.2 times the dearest single bar that is not a goal; meeting on the stroke it was 1.3 times. The tonic, which in this rendering is two bars — its own and its octave — is left out of the comparison; it falls as the goals rise, from 0.171 to 0.100, because the strokes the elaborating part spends on goals are strokes it does not spend at random on the tonic.

The second mode shows the crossing. With goals on 5, 1 and 2, meeting only on the stroke, the goal on 2 costs 0.97 of what degree 6 costs, though 6 is never a goal there: the case the earlier essay flagged, where register beats goal. One stroke early the goal on 2 overtakes it. By seven strokes early every goal in both modes costs more than three times every non-goal single bar, and by eleven the ratios are fifteen and nearly eight.

The steep ranking the earlier essay found for a single dominant goal is what anticipation produces for any number of goals. A tuner working for a mode’s three goal tones would, under an elaboration that arrives early, find all three bars the costliest to leave wrong, whatever their register.

Why the flat ranking was flat, and why anticipation undoes it #

The earlier essay’s three-goal ranking went flat for a reason that can be stated in two numbers. Every doubling of a degree costs something when that degree’s bar is wrong, and a doubling of a high degree costs more than a doubling of a low one, because the same ten cents is more hertz higher up: the earlier pricing put a single coincidence at 0.076 on the tonic and 0.098 on the sixth degree. A goal tone gets more doublings than a non-goal, because the skeleton lands on it more often. With one goal the extra doublings are many and they swamp the register; with three goals each gets a third of them, and the register’s quarter-again difference per doubling can then outweigh a low goal’s modest excess.

Anticipation changes the first number, not the second. Each stroke of anticipation adds a doubling of the goal on every unit, so the goal’s excess of doublings over a non-goal’s grows linearly with how early the elaborating part arrives, while the register’s advantage per doubling stays fixed. Past a stroke or two for the flattest mode, the goal’s excess wins everywhere. That is why the crossing in the second figure happens early and the ratios then grow without limit: a mode whose elaboration arrives on its goals most of the way through each unit makes the goals almost the only degrees the two instruments share.

Strokes, and seconds #

The anticipation is counted in the elaborating part’s strokes, and at 0.15 seconds a stroke seven strokes is about a second. In a slower tempo the skeleton’s strokes are longer and an elaborating part typically plays more notes to each of them, so the same musical gesture — arriving on the goal halfway through the last skeleton stroke — is more elaborating strokes. On this arithmetic that makes the ranking steeper at slow tempi, where each skeleton stroke is long enough for the elaborating part to settle on the goal well before it arrives, and flatter at fast ones, where there is little room to anticipate at all. The tempo moves a scale further than the touch found tempo to be the setting that changes a gamelan scale’s standing most; on this arithmetic it would also change which bars a tuner can least afford wrong, through how much time the elaboration has to reach its goals early. That has not been computed here; it follows from the stroke count if the gesture is fixed in musical rather than clock time.

One goal, and where its cost comes from #

With a single goal the effect is large and monotonic. Each stroke the elaborating part spends on the goal early raises the goal’s cost: 0.305 meeting on the stroke, 0.393 one early, 0.454 two early, 0.509 three, 0.765 seven, 1.011 eleven. Every other bar’s cost falls in step, because each of those strokes is one the elaborating part would otherwise have played on some other degree. The fifth degree goes from 0.064 to 0.012, the tonic from 0.171 to 0.052.

The reason is the same one the earlier essays found for everything a tuning error costs: a beat belongs to both of the bars that make it, and it happens whenever the two instruments sound the same degree at once. An elaborating part that holds the goal for several strokes before the skeleton strikes it is sounding the goal while the skeleton’s previous note still sounds, and then again as the skeleton’s goal arrives. Every one of its goal strokes is a moment of doubling the goal’s bar will pay for if it is out of tune — and in a ringing part, each such stroke keeps sounding after it is struck.

The touch #

That last point is why the touch matters. A ringing elaborating part’s early strokes on the goal go on sounding into the goal stroke and past it, so each adds a long doubling: between meeting on the stroke and arriving seven early the goal’s cost rises from 0.305 to 0.765, two and a half times. A part given one stroke of ring rises from 0.092 to 0.180, and a damped part from 0.084 to 0.141. Anticipation raises the goal’s cost under every touch, and most under the one the gendèr has — the instrument the scale belongs to in these essays.

The comparison with the earlier essay’s meeting rule is instructive. There, meeting the skeleton on the goal stroke added only three per cent to a ringing part’s errors, because a ringing part doubles degrees constantly and one extra doubling per unit is lost in the rest. Anticipation is different in kind: it is not one extra doubling per unit but a run of them on one degree, and a ringing part turns each into a long one.

A tuner’s order of work #

Put as an order of work, the result is short. For an ensemble whose elaboration arrives about a second early on the goals of a mode ending on 6, 2 and 3, the bars to match first across the two instruments are the sixth degree’s, at 0.295 for a ten-cent error, then the second’s at 0.159 and the third’s at 0.136; then the tonic’s pair of bars at 0.100; and last the fifth degree, at 0.043, which a ten-cent error barely touches. Meeting on the stroke, the same mode’s order would have been the tonic, the sixth, the second, the third and the fifth, with little between the last three. Anticipation turns an order the register mostly decided into one the mode decides.

A whole instrument out #

The earlier essay found that a whole instrument out of tune costs nearly the same whatever the skeleton favours, because an offset moves every degree at once. Anticipation changes that, because it changes the total amount of doubling rather than moving it between degrees. Ten cents on the whole ringing instrument costs 0.569 when the elaborating part meets a single goal on its stroke, and 0.912 when it arrives seven strokes early — sixty per cent more. With the skeleton at random the same offset goes from 0.510 to 0.713.

So an elaboration that anticipates its goals makes a gamelan more sensitive to both kinds of error. It concentrates the cost of single wrong bars on the goal tones, and it raises the cost of the whole instruments disagreeing. The pair tuned apart on purpose is the other side of that: where two instruments are offset deliberately for their beating, an elaboration that sits on its goals thickens the beating a listener hears exactly at the structurally strongest moments.

With no favoured goal #

The control case is a skeleton with no favoured goals at all, its units ending wherever the random draw puts them. Anticipation still concentrates doubling on each unit’s last skeleton note, but that note is a different degree each time, so the extra cost is spread. Every degree’s cost rises, the summed cost of the five errors from 0.555 to 0.854 at eleven strokes early, and the ranking stays the register’s: higher degrees a little dearer, because a cent is more hertz higher up and every partial beats at its own rate.

That separates the two effects cleanly. Anticipation raises the total cost of tuning errors whatever the skeleton does. What decides which bars carry it is where the skeleton lands, and a mode with favoured goals turns the extra cost into a ranking.

What ten cents is on a goal #

At the rates involved the errors are small beats. Ten cents on the sixth degree, near 454 hertz, is a beat of 2.6 hertz between the fundamentals; on the second degree, near 299, 1.7 hertz. A tuner counts beats of about that speed, and beats are arithmetic: the rate is exactly the difference in hertz, so the higher goal’s error is the faster beat. What anticipation adds is not a faster beat but more time spent beating, concentrated where the music is heading. A listener’s attention in a gamelan piece is drawn to the goals — the approach to each goal is the point of the unit — so the extra doubling falls on the moments most attended to, which the roughness sum weights no more heavily than any other moment and a listener might weight a great deal more.

Where the model stops #

How early. The number of strokes the elaborating part spends on the goal is a stated parameter. Real elaborations anticipate by different amounts in different instruments, pieces and tempi, and may leave the goal and return to it rather than holding it. The figures cover the range from meeting on the stroke to three-quarters of a unit, and every value in between gives an intermediate answer.

One elaborating part. A gamelan has several elaborating instruments, each with its own conventions for approaching a goal. If they all anticipate, the effect multiplies; if some anticipate and others arrive on the stroke, it is intermediate.

One octave. Both instruments sound each degree in the same octave here. An elaborating instrument often plays its goal an octave above the skeleton, and a bar’s partials, at 1, 2.76 and 5.40 times its fundamental, have nothing at twice the fundamental to beat against the octave above. On bars, an octave doubling scarcely beats at all, so the costs drawn are those of doublings at the unison, and an elaboration that anticipates its goals an octave up would add much less.

Beating counted as roughness. As in the earlier essays on these instruments, the Plomp–Levelt curve prices beats of a hertz or two as small roughness, which overstates the absolute sizes at small errors and leaves the comparisons intact.

What the picture cannot show #

Which goals a mode favours, and how strongly. The goal sets 6, 2, 3 and 5, 1, 2 are parameters chosen to show two modes with different registers, not transcriptions of any pathet’s practice.

Whether gamelan tuning follows the ranking. The prediction is sharper than the earlier essay’s: under anticipating elaboration, every goal tone of the modes an ensemble plays most is a bar to match closely across instruments. Whether makers and tuners do match those bars more closely than others is a question for measurements of real instruments.

Still open: whether the goal tones’ beating is heard as arrival #

The extra beating anticipation puts on a goal tone falls at the approach to each structural point, and an elaboration is a clock hand where the skeleton has none found that a recognisable event shortly before a structural stroke is exactly what helps a listener know where in a cycle they are. If the ensemble’s paired instruments are tuned slightly apart, as the pair tuned apart describes, anticipation makes the goal audible as a swell of beating before the skeleton strikes it — a second marker inside the punctuated cycle that a cycle that says where it is drew. The computation joins the two models: the beating contributed by each stroke of anticipation, read as a recognisable boundary with a probability that rises with its depth, and asked whether a gamelan whose instruments are tuned apart gives a listener a better clock than one tuned together.