The scale belongs to the ringing instrument

The essay that compared the two touches compared a gamelan’s two touches — a saron’s bars damped at every stroke against a gendèr’s left to ring — and found that the tempo moves a scale’s standing further than the touch does. Its closing paragraph named what the comparison had not done. A gamelan does not choose between the two. It plays both at once, at densities that differ by a factor of two or four, so the damped instrument and the ringing one are sounding the same five pitches at the same moment.

That puts a third kind of pair into the arithmetic. Every figure so far computes pairs within one melodic line — which has been the shape of the measure since a scale stopped being a set of pitches; an ensemble has pairs that cross between the lines, and they fall where neither line’s own intervals do.

A quarter of it falls between the two #

At one slow note to four fast ones — an ordinary relation between a saron and a gendèr — the fast part’s own pairs contribute 5.50 units of roughness a second, the cross pairs 1.91, and the slow part’s own 0.05.

The slow part’s own contribution is a rounding error and the reason is the previous essay’s: a bar damped as its successor is struck is sounding against nothing at all, so its intervals never meet. Whatever roughness the slow part is responsible for arrives entirely through its pairs with the fast one.

Those cross pairs are 26 per cent of the whole. At a denser skeleton, one slow note to two fast, they are 36 per cent; at a sparser one, one to eight, 16. A quarter of what a listener receives has never appeared in any figure the account here has drawn.

And they behave #

The reason to compute them is that they could have been bad. A scale chosen so that its intervals are smooth in a line need not be smooth across two lines: the cross pairs sample the same interval set at different lags and different amplitudes, and a scale whose smoothness depended on a particular melodic ordering would come apart.

It does not come apart. The ensemble sits at the sixth percentile — smoother than nineteen random five-note scales in twenty — at every ratio drawn, and never worse than either of its parts.

So the slendro’s smoothness is a property of its interval set rather than of any arrangement of it. That is a real check and not a triviality: had the cross terms been worse, the finding would have been that the scale is tuned for a melodic line and the ensemble pays for it, and there is nothing in the arithmetic that had to come out this way. It is the same kind of robustness the smoothness is in the skips found when it asked which intervals a scale’s smoothness actually rests on.

The control: a scale that has nothing to gain #

The ensemble arithmetic can be run on a scale that was not designed for bars, and doing so says whether the sixth percentile is about the slendro or about the rendering.

The diatonic set under a bar’s spectrum sits at the ninety-fifth percentile at six tenths of a second a note and does not come down until the tempo is fast enough that the distinctions blur. Its simultaneities do not help it, because its intervals were chosen against a harmonic series and a bar has none.

So the rendering does not flatter everything. A scale with something to gain from being sounded together gains; a scale with nothing to gain does not. That is what makes the slendro’s sixth percentile a statement about the slendro rather than about the two-stream model, and it is the check worth having before any of the tuning claims above are taken seriously.

It also puts a floor under how much the correlation between the real streams could be doing. Correlated streams would add unisons and near-unisons to the cross pairs, which would help the diatonic set exactly as much as the slendro — so whatever the correlation is worth, it does not close a ninety-point gap.

Which instrument the tuning is for #

The striking figure is the third line. The slow damped part on its own sits at the ninety-sixth percentile — rougher than nearly every random scale of its size — and the ensemble sits at the sixth.

That is the previous essay’s tempo finding at its extreme. A damped bar at a slow rate has essentially no simultaneity, so its roughness comes from its melodic intervals alone, and on that measure the slendro has no edge whatever. It is not that the saron sounds rough; it is that a saron gives a scale nothing to be smooth with.

And the ensemble takes the fast part’s figure and not an average of the two. At one to four the ensemble is at 5.6 and the ringing part alone at 7.6; at one to eight both are at 5.6. The instrument with the simultaneities decides, and the instrument without them contributes a quarter of the roughness and none of the standing.

The conclusion is a claim about what a gamelan’s tuning is a tuning for, and it is sharper than the account here has managed before. A slendro is tuned for the instruments that ring. A saron plays the same five pitches and would be no worse off, by this measure, on almost any five pitches at all.

Why the cross share falls as the skeleton thins #

The cross share runs 36, 26 and 16 per cent as the slow part goes from one note in two to one in eight, and the shape of that is worth a line because it is not quite the obvious arithmetic.

The obvious expectation is that halving the slow part halves its cross pairs, so the share should halve. It does not quite: 36 to 26 is a fall of a quarter, not a half, and 26 to 16 is a fall of two fifths. The reason is that the fast part’s own contribution is not constant either — at a wider ratio there are more fast notes in the same span, so the denominator grows while the numerator shrinks, and the two effects partly cancel.

Which means the cross share is not a free parameter of the ensemble. It is fixed by the two rates together, and a gamelan playing at a stated irama has a stated share whether anybody intends one. At the ratios gamelan actually uses — the elaborating parts at two, four or eight times the skeleton — the share sits between a sixth and a third, which is the range this figure covers.

That also bounds how much the finding could have changed. If the cross pairs had been much rougher than the within-stream ones, a quarter of the roughness arriving from them would have moved the ensemble’s percentile a long way. They are not, so it does not, and the bound on the effect is the share itself.

What that predicts about the tuning process #

If the tuning serves the ringing instruments, a tuner has an ordering: get the gendèr and the gambang right and let the sarons follow.

That is testable outside this collection and it is the sort of thing ethnographic accounts of gamelan tuning record. What the arithmetic contributes is the reason it would be true, and a number: the ringing instruments carry the whole of the scale’s standing and the damped ones carry a quarter of the roughness and none of the information about whether the tuning is any good.

It also predicts which errors would be tolerated. A pitch set slightly wrong on a saron costs a listener a small increase in the melodic terms, which are the terms the percentile says carry nothing. The same error on a gendèr costs it in the simultaneities, which are where all of the scale’s advantage lives. That is the reverse of what the scale least committed to its own instrument describes, where a scale’s indifference to its spectrum is what makes it portable. A gamelan’s tuning tolerance should be tighter on its ringing instruments than on its damped ones, by roughly the ratio of their contributions — which is a prediction with a number attached and one the essays here cannot check.

What the two streams do not do #

There is one thing the cross pairs might have been expected to do and do not, and it is worth recording as a negative.

A gamelan’s parts are not independent melodies. The elaborating part is derived from the skeleton by rules, so the two streams’ pitches are correlated — the gendèr is playing around the saron’s note rather than drawing from the same urn. Every figure here draws both streams at random from the scale, which breaks that correlation deliberately.

A correlated pair of streams would put more unisons and near-unisons into the cross pairs, which are the smoothest possible intervals, so the real ensemble should be smoother than the one computed here rather than rougher. The direction is knowable and the size is not, and it means the sixth percentile is a conservative figure.

The saron’s job is not the one being measured #

A reader who plays in a gamelan will object to the previous section and the objection is correct, so it is worth putting in the essay rather than leaving to a footnote.

The saron is not there to make the scale smooth. It is there to state the melodic skeleton — the balungan — which every other part is derived from, and a listener’s grip on where the piece is comes from it. A cycle that says where it is is the essay on a different account that prices exactly that job, and by that measure the damped instruments are doing most of the work.

So the finding is not that the saron matters less. It is that the saron’s job is not the one this measure measures, and the two jobs make opposite demands on a tuning. A part that states a skeleton wants its pitches distinct and memorable, which is a melodic requirement; a part that fills in wants its pitches smooth together, which is a simultaneity requirement. This account has only ever computed the second.

That is a useful division and it explains something the percentile alone makes look like a puzzle. A scale at the ninety-sixth percentile played on a saron is not a badly chosen scale being played badly; it is a set of pitches being used for a purpose that has nothing to do with roughness, by an instrument whose design removes roughness from the question by damping it away.

Which raises the possibility that the two demands conflict, and that a gamelan’s tuning is a compromise between them rather than an optimum for either. Nothing here can say — the melodic requirement has no measure in this collection — but the shape of the compromise would be visible if it existed: a scale that is neither as smooth as a ringing instrument alone would want nor as evenly spaced as a skeleton alone would want.

Which computation produced the numbers #

Both streams are rendered as an explicit list of onsets rather than through the factorised weighting the earlier essay use, because the factorisation assumes one ring law and here there are two. A fast note occurs every stated interval and a slow one every stated multiple of it, each drawing a degree of the scale at random from a fixed seed.

How long a note is allowed to sound is its touch: a damped note until its own stream’s next onset, a ringing one until it is forty decibels below its strike. Two notes contribute roughness when the earlier is still sounding as the later begins, and the amount is the spectral dissonance between the two degrees, given the stated spectrum.

The total is divided by the span so that the figures are roughness per second of music and can be compared across density ratios, which change how many notes there are.

The percentile is against random scales of the same size, drawn from a fixed seed and scored through the same two-stream rendering — which is what makes the three lines comparable: each is a percentile in its own pool, and the pools differ because a stream structure changes what a random scale scores as well as what the slendro does.

Where the model stops #

The two streams are independent and real ones are not. Stated above, and the direction of the error is toward a rougher figure than the truth.

There are two instruments and a gamelan has a dozen. The bonang, the gambang, the gongs and the voice all sound at once, at several densities and with several ring laws, and a proper computation is a sum over every pair of parts.

And the amplitudes are one or nothing. A note is either sounding or it is not; a real note decays, and a pair of notes twenty decibels apart contributes far less roughness than a pair at equal level. Putting the decay back into the pair weight is what the factorised roughness model does and it is the thing this rendering gave up in order to carry two ring laws. The effect is to overstate the roughness of the late overlaps, which falls mostly on the ringing part — so the fast part’s own contribution here is an overestimate.

What the picture cannot show #

It cannot show the tuning’s own mistuning. A gamelan’s paired instruments are deliberately set apart to beat, which adds roughness the model has no term for and which is a chorus effect rather than a scale.

Nor a scale without an octave. Every set here closes at the equave, and a scale without an octave is the case where a bar’s own partials would recommend a different closing interval — which would change the pool as well as the scale.

Nor the resonators. A gendèr’s tubes lengthen its fundamentals and not its upper partials, which would make its late overlaps more nearly pure tones and less rough than modelled.

And it cannot show anybody’s preference. A percentile among random scales is a statement about sensory roughness and nothing else — the same limit the spectrum that was supposed to explain the gamelan recorded when it found that a bar’s partials do not recommend a slendro.

Still open: whether the tolerance prediction survives a real ensemble #

The claim worth attacking is the one about tuning tolerance, because it is the only thing here a person could act on and because it rests on the most fragile part of the model.

It says the ringing instruments carry the scale’s standing and the damped ones do not, so an error in a ringing instrument’s pitch should cost more than the same error in a damped one’s. The way to price it is direct: displace one degree by a stated number of cents in one instrument only, rescore the ensemble, and read the slope.

Two things would make the answer different from the naive one. The parts are correlated, so an error on the saron’s note is an error the gendèr’s elaboration is built around, and it may propagate rather than stay put. And a gamelan’s instruments are tuned as a set by one maker, so a single degree wrong in one instrument is not how the errors actually arrive — they arrive as a whole instrument slightly differently tuned from another, which is the beating the model has no term for.

So the measurement worth making is the second one: two whole instruments a few cents apart, playing the two streams, with the beating included. That is a different object from anything here and it is the one a gamelan actually is.