A gamelan's shimmer should rise only downward

A gamelan’s paired instruments are tuned a few cents apart so that a note played on both shimmers, and the rate of that shimmer is a choice the tuner makes key by key. A gamelan tuner can name the goals for nothing found that within one octave a rate rising faster than the pitch costs nothing in roughness and gives each goal tone its own rate of swell, so that a listener keeping their place in a long cycle can tell which goal is coming. A gamelan’s shimmer has to start again at every octave carried that rise across three octaves and found it breaking: rougher overall, with the high octave’s pairs beating too fast to be beats, while repeating the middle octave’s rates in every octave kept both ledgers. A degree’s three swells are one landmark only if they agree then found that a listener who fuses a goal struck in three octaves at once — hearing its three swells as one event only if their rates agree — gets nothing from the carried rise, because no rise steep enough to name the goals lets a degree’s three swells agree.

That essay ended with a compromise. The dilemma bites only where a goal sounds in several octaves at once. A tuner might repeat the middle octave’s rates on the degrees and registers the ensemble doubles at the goals, keeping the clock, and let the rate follow the pitch everywhere else, where the earlier essays found a rise both names and saves roughness. The question was whether that mixture beats repeating everywhere.

It does not. A different mixture does, and the axis it is chosen along is not the one the lead proposed.

Where the clock was left #

The clock is the earlier essays’: a sixteen-step cycle of 160 seconds, its four goals falling on three of slendro’s five degrees, each announced a step early by its degree’s swell of beating. A listener who fuses a goal’s swells is left 0.15 bits of position before each goal when the swells name the goals, 0.40 when they all sound alike, 0.65 with no swells. The roughness ledger is the passage the goal-tone essays rendered — two instruments, the elaboration meeting its skeleton every four strokes, arriving a stroke early — played in each of three octaves, with each pair detuned by the cents its beat rate requires, and reported as the roughness added over the same passage in tune. The pairs’ rates are held to a fixed geometric mean, so every tuning below has the same average shimmer, and the tunings differ only in how it is shared out.

The rise saves below and costs above #

The first thing to find out is where a rising rate saves roughness, because the lead’s proposal assumed it saved roughness wherever it was allowed.

At the square, the rising family adds 0.44 to the low octave’s passage against 1.05 for the repeated rates, the same 1.09 to the middle octave — the two families are one tuning there — and 1.84 to the high octave against 0.92. The rise saves 0.61 below and costs 0.92 above. Roughness follows the speed of the beating: under the rise the low octave’s pairs beat slowly, a heave under a second to two a second, and the high octave’s race to fifteen a second and past, into the range where a fluctuation stops being a beat and becomes roughness.

That asymmetry was hidden in the earlier ledger, which summed the octaves. It changes what a mixture should do. The lead proposed freeing the degrees the goals do not use, in every octave, to follow the pitch; but freeing a pair in the high octave is exactly the move that costs roughness.

Pair by pair, the rule has no exceptions. Letting any one pair in the low octave follow the pitch saves roughness, by 0.06 to 0.28 depending on the degree; letting any one pair in the high octave follow it costs between 0.14 and 0.29. Whether the degree carries a goal does not change the sign, only the size: the goal degrees happen to sit where the savings and costs are largest, but every degree saves below and costs above. The direction of the rise, not the role of the degree, decides whether following the pitch pays.

The five tunings, side by side #

At the square:

• The middle octave’s rates repeated everywhere: roughness 3.06, and the clock at 0.15 bits.
• The rise carried on everywhere: 3.37, and the clock at 0.40, because no goal’s swells agree.
• The lead’s mixture — repeated at the goals in every octave, rising elsewhere: 3.42. The clock is kept at 0.15, by construction, but the tuning is rougher than repeating everywhere and even a little rougher than rising everywhere. It frees the non-goal degrees in the high octave to race and gives them nothing in the low octave that the goal degrees, held back, could not have given as well; and the racing high pairs against held goal pairs beside them are rougher together than either family alone.
• Rising below the middle octave, repeated above: 2.45, a fifth less roughness than repeating everywhere. Its clock depends on where the goals sound: with the goals struck in all three octaves the low octave’s goal swells disagree with the others and the clock falls to 0.40; with the goals struck only in the middle and high octaves, it stays at 0.15.
• The same, but with the goal degrees repeated in the low octave too: 2.96, a saving of only 0.10, and the clock at 0.15 wherever the goals sound.

Turn the dial and the ordering holds. At a power of one the tuning that rises only downward adds 2.78 against 3.15 repeated; at the cube 2.24 against 2.95. The lead’s mixture is the roughest of the five at every power above one.

Drawn against the power, the five tunings fan out. Repeating everywhere grows slightly smoother as the power rises, from 3.17 at a half to 2.95 at the cube, because a steeper rise across the middle octave’s own degrees spreads its rates in a way the ledger happens to favour. Rising everywhere grows rougher, from 3.24 to 3.55. The lead’s mixture grows roughest of all, from 3.25 to 3.91, its racing high pairs getting faster with every step of the power. Rising only downward falls fastest, from 2.98 to 2.24 — the steeper the rise, the slower the low octave’s pairs and the more the low octave saves, while the high octave, held at the middle octave’s rates, costs nothing more. The version with the goals held in the low octave falls more gently, from 3.15 to 2.85, keeping the clock in every ensemble and taking only the low octave’s non-goal saving.

At a power of a half the five are within a third of a unit of each other; at the cube they span 1.7. The steeper the tuner wants the rise across the degrees — which the earlier essays found is what lets a coarse ear tell the goals apart — the more the direction of the rise across the compass decides the roughness. A tuner content with a gentle rise can choose almost freely among these; a tuner who wants a steep one, to name the goals for coarse ears, has to choose the downward tuning or pay for it.

So the answer to the question has two parts. Repeating only at the goals is the wrong mixture: it keeps the clock but costs roughness. Rising only downward is the right one, and how much of its saving a tuner can keep depends on a fact about the ensemble rather than the tuning — whether its lowest octave doubles the goal tones.

What the ensemble’s parts decide #

That fact is the lead’s own “goals’ octaves taken from a real ensemble’s parts”, and it enters as a single switch. If the instruments that play the low octave strike the goal tones together with the rest of the ensemble, then a fusing listener hears each goal as three swells including the low octave’s, and the low octave’s goal pairs must beat at the middle octave’s rate for the swells to agree. Then the downward rise is available only on the low octave’s non-goal degrees, and it saves 0.10 at the square, three per cent of the ledger.

If the low octave does not double the goals — if the instruments in that register play something else at the structural strokes, or are silent there — every pair in the low octave can follow its pitch down, and the saving is 0.61, a fifth of the ledger, with the clock untouched. The arithmetic does not say which is true of any particular ensemble or style, and the essays have deliberately not tied the model to one. It says that the question to ask of a gamelan’s parts is not which degrees the goals use but which registers sound at the goal strokes, because that, and only that, limits how far the low octave’s shimmer may slow.

What the downward tuning sounds like is visible in its rates. The middle and high octaves beat at the middle octave’s rates, from 2.9 to 8.7 a second across the five degrees. The low octave slows to 0.7 to 2.2 a second: a heave rather than a shimmer. No pair anywhere leaves the range of beating, which the rise carried on could not manage at the square. In the earlier essay’s terms, a low-octave goal swell beating less than once a second is barely an event a step before the goal — and that is the second reason the downward tuning wants the low octave kept off the goals. A pair that slow could not announce a goal in time even if a listener heard it alone.

Why the direction matters #

The asymmetry has a plain cause. A roughness ledger weights a pair’s beating by how fast it is, rising steeply through the range where beats turn into roughness, and the pairs’ rates are spread round a fixed average. A rise across the compass moves the low octave’s rates down by the same factor that it moves the high octave’s up, but roughness is not symmetric in the factor: slowing an already slow beat saves little in absolute terms per pair, while speeding a fast one toward fifteen a second costs a great deal. Repeating the middle octave above it removes all the cost; following the pitch below it keeps most of the saving. The tuning that does both is simply the cheapest point in the family, and the lead’s partition, which was drawn by the degrees’ roles, cut across it at right angles.

It also matters that the middle octave is the reference. The earlier essays held the average shimmer to the middle octave’s, so the repeated rates are its rates. A tuner who repeated a slower octave’s rates upward — the low octave’s, say — would shift every rate in the ensemble down and change the shimmer’s character, which none of these tunings does. Within the constraint that the ensemble’s shimmer keeps its middle-octave character, the rule is: never let it rise above the middle, and let it fall below as far as the goals allow.

Read against the earlier essays #

A gamelan tuner can name the goals for nothing found that within one octave a rising rate names the goals at no cost in roughness. That finding is intact: within the middle octave every tuning here is the same. What it does not survive is extension in both directions. The pair tuned apart on purpose described the shimmer as the reason the instruments are paired, and a shimmer that slows to a heave in the lowest register is still a shimmer; one that races to roughness in the highest has stopped being one. A degree is where it goes next bound a gamelan listener’s sense of a degree to its goals; the downward tuning keeps every goal’s swell the same in every register that sounds it, which is the property two landmarks in a cycle help only if they can be told apart asked of a landmark.

What octave equivalence means for a shimmer #

A scale without an octave asked what a tuning loses when its degrees do not repeat at the octave, and the gamelan essays have been asking a narrower version of the same question about the beating. The earlier essay found that repeating a degree’s rate in every octave makes the degree, rather than the pitch, the unit a fusing listener’s clock runs on — octave equivalence carried into the shimmer. The downward tuning shows that the equivalence does not have to be complete to be useful. It needs to hold where the goals are heard together and above the middle, where it also saves roughness; below the middle, where it is not needed for the clock, breaking it saves roughness instead.

That is an unusual shape for an octave equivalence: one-sided, holding upward and failing downward. Nothing about it is arbitrary, though. It follows from two facts already in the collection — that roughness rises with the speed of beating, as beats are arithmetic and the roughness essays have measured, and that a goal’s swells must agree only in the registers where the goal is struck — and from one fact about a particular ensemble that the arithmetic leaves open.

The model under the tunings #

The rates are a family with one free choice per cell. Each degree in each octave either repeats the middle octave’s rate for that degree or follows its own pitch at the stated power, both normalised to the same average. A real tuner sets each key by ear, and nothing obliges a tuning to lie in this family.

The roughness ledger is the earlier essays’ passage, played in three octaves, the elaboration arriving a stroke early on goal degrees, with the roughness of every pair of partials summed. It is the same passage in every octave, so the ledger compares registers on equal terms; a real piece uses its registers unequally, and a piece that lived mostly in the high octave would weigh the downward tuning’s saving less.

The clock’s fusing listener compares a goal’s swells only in the octaves where the goal is struck, and fuses them when their rates agree within fifteen per cent.

What the ledgers cannot say #

What a tuner hears as right. Gamelan tuning is a craft with its own judgements of what an ensemble’s shimmer should sound like across its range, and these ledgers price two things a listener could get from it, not what the tradition values. A slow heave in the low register may be prized or avoided; the arithmetic says only what it costs and saves.

Which instruments double the goals. The switch the result turns on — whether the low octave sounds at the goal strokes — is a fact about a repertory and a style, and the figures leave it as a switch rather than claim either value for any particular ensemble.

Still open: a rise that stops at the beating limit #

The downward tuning sets the high octave to the middle octave’s rates exactly. Between repeating and rising lies every rate that rises a little above the middle and stops before the high octave’s pairs reach the edge of beating — a rise that is steep below the middle and flat above it, with a knee. The pair-by-pair figure says every high-octave pair allowed to rise costs roughness at the square, but a gentler power above the middle than below it, chosen so that the high octave’s goals still agree within the fusing listener’s fifteen per cent, might name the goals more strongly in the upper registers at little cost. The computation is this one with separate powers above and below the middle octave, and it would say whether the cheapest tuning is a knee or a step — whether the rate should merely stop rising at the middle octave, or bend.