A wrong bar beats the same on either instrument

The scale belongs to the ringing instrument played one slendro on two instruments at once — a gendèr-like part at a note every 0.15 seconds with its bars left to ring, over a saron-like part four times slower with its bars damped at the next stroke — and found that the ensemble’s standing among random five-note scales is the ringing part’s. The damped part alone sits at the ninety-sixth percentile; the ensemble and the ringing part both sit near the sixth.

From that it drew a prediction a tuner could act on. If the ringing instruments carry the scale, an error in a ringing instrument’s pitch should cost more than the same error in a damped one’s, by roughly the ratio of their contributions. And it named what could make the prediction wrong. Its model had no term for two instruments tuned apart. A degree that is the same on both instruments was a unison and scored nothing, and a gamelan’s errors arrive as one instrument tuned a few cents from another, which is beating.

With that term added, the prediction fails, and the reason is simple. A beat belongs to both of the bars that make it. A wrong bar on the ringing instrument beats against the right bar on the damped one exactly as hard as a wrong bar on the damped instrument beats against the right one on the ringing. Beating is almost all of what a tuning error costs, so the two tolerances come out the same.

The term the model lacked #

The rendering is the one the earlier essay built. Each part is an explicit list of onsets drawn from the scale with a fixed seed. A ringing note sounds until it has decayed forty decibels, and a damped one only until its own part’s next stroke. Every pair of notes that overlap in time contributes the spectral roughness of their two pitches under a bar’s inharmonic spectrum — the scoring that has measured a scale by the pairs its music actually sounds since a scale is not a set of pitches. That is Plomp and Levelt’s curve summed over every pair of the bar’s partials, 1, 2.76, 5.40, 8.93 and 13.34 times the fundamental.

One rule in that rendering decides this question. A pair of notes on the same degree was skipped, because on one tuning table a unison is not an interval, and the roughness of a tone against itself is only the roughness among its own partials, which it carries whether or not a second instrument doubles it.

Once the two instruments are tuned to different tables, that pair stops being a unison, and what it becomes is the oldest measurement in tuning: a tuner counts beats between two nearly equal tones because the rate is exactly their difference in hertz, as beats are arithmetic derives. Every partial of one bar sits a little away from the corresponding partial of the other, and every such pair of partials beats. The new term counts it. A cross pair on one degree scores the roughness of the two slightly different tones, less the roughness either tone has against an exact copy of itself. With no detuning that difference is zero, which is how the new rendering reproduces every number of the old one exactly. With detuning it is the beating.

Two kinds of detuning are priced. A single degree can be wrong on one instrument, moved flat or sharp while every other bar of both instruments stays put. Or a whole instrument can be offset, every bar of the ringing instrument moved by the same number of cents against the damped one, which is the way instruments made as a set by different hands, or retuned at different times, actually disagree.

One wrong bar, on either instrument #

The hero figure prices the first kind at ten cents. A ten-cent error on one bar is small by any practical standard — at the tonic of this texture, middle C, it is a beat of about one and a half hertz between the fundamentals.

Degree by degree, the costs on the two instruments are close and not identical. The tonic costs 0.142 flat and 0.202 sharp on the ringing instrument and 0.163 and 0.185 on the damped one. The third degree, at 474 cents, costs 0.106 flat and 0.021 sharp on the ringing instrument, and 0.069 and 0.057 on the damped. The sixth degree costs about two hundredths either way on either.

The totals are the comparison the prediction makes, and they agree to two decimal places: 0.85 for the ringing instrument’s ten errors and 0.85 for the damped one’s. The beating in them is the same on both, to the sixth decimal place, because it is one number computed from one pair of bars.

The degrees differ from one another for a reason that has nothing to do with their pitch. The tonic’s bar is also the octave’s in this rendering, so a tonic error is two wrong bars, and it costs about twice what the second degree does. The rest is how often each degree happens to sound on both instruments at once in the seeded passage: in sixteen bars the second degree lands on both instruments at once about eleven times, the third seven, the fifth eight, and the sixth only twice, so there is least of the sixth to beat. Each coincidence costs a little more the higher the degree, from 0.076 at the tonic to 0.098 at the sixth, since the same cents are more hertz higher up, but that spread is a quarter and the spread in coincidences is a factor of five. A tuning error costs in proportion to how much a degree is doubled, and doubling is a fact about the music, not about the bar.

A ten-cent error adds about 1.1 per cent to the texture’s roughness. That is the size of the thing a tolerance is about, and it is small beside what the tempo does to the same scale: the tempo moves a scale further than the touch found its standing moving by fifty-four percentile points across the tempo levels of one piece.

Why the ringing part does not take the blame #

The earlier essay’s argument was not wrong about where the scale’s standing lives. It was wrong about what an error changes.

Take one degree apart. When the third degree moves twenty cents flat on the ringing instrument, the added roughness is 0.197: beating 0.109, a change of 0.077 in the ringing part’s own intervals, and 0.011 in the other pairs that cross between the parts. The same error on the damped instrument adds 0.122: the same beating 0.109, nothing at all in the ringing part’s own intervals, which a damped bar cannot touch, and 0.011 across.

Sharpen it instead, and the ringing instrument’s own term turns negative. At twenty cents sharp the ringing error costs 0.022 — beating 0.110 less 0.080 saved in its own intervals — while the damped error costs 0.095.

That is the whole of the ringing instrument’s special status in a tuning error. A bar on the ringing instrument also forms intervals with that instrument’s own other notes, which ring over it, and those intervals are what gives the scale its standing. Moving the bar moves them, and those are the pairs that the smoothness is in the skips found carry a scale’s advantage — and for any one degree, moving it one way makes them rougher and the other way smoother, since a measured slendro is not at a minimum of this model in every direction. The term is real and it is large for a single degree in a single direction. It is not a cost of error. It is a change of scale, and over both directions and all five degrees it nearly cancels.

The beating does not cancel, because it has no good direction. Any departure from the other instrument’s bar is a beat, flat or sharp.

The tolerance is one curve #

Averaging over the five degrees removes the accident of which way each degree’s intervals lean, and leaves the curve a tuner would use.

The damped instrument’s curve lies on the beating curve at every size, because beating is all a damped error can cost. The ringing instrument’s curve runs a little above it for flat errors and a little below for sharp ones: at thirty cents flat 0.217 against the damped 0.190, at thirty cents sharp 0.169 against 0.192. At every size tried, the ringing instrument’s cost is between 0.86 and 1.14 times the damped one’s.

Both curves bend. Two cents costs about two hundredths, ten cents under a tenth, thirty cents about a fifth — each doubling of the error buys less than a doubling of the cost. That is a property of Plomp and Levelt’s curve near zero: roughness rises steeply from a unison and then less steeply, and the highest partials of a bar, which are nine and thirteen times the fundamental and beat nine and thirteen times as fast, reach their own peaks first.

So the prediction fails on its own terms. A gamelan’s tuning tolerance is not tighter on its ringing instruments. In this model it is the same on both, set by beating, and the only asymmetry is that a ringing bar’s error can be partly paid back by the scale leaning its way.

A whole instrument out #

The second kind of detuning is the one the earlier essay said a gamelan actually has: every bar of the ringing instrument moved by one amount against the damped one.

Ten cents on the whole instrument costs 0.38, which is 5.1 per cent of the texture’s roughness in tune. Ten cents on each of the five degrees separately costs 0.38 as well. The errors add. Nothing about moving the whole instrument at once is worse or better than the sum of its parts, because each degree beats only against its own counterpart and the intervals within the ringing part do not change when all its bars move together.

That last point shows in the split. The beating at ten cents is 0.43, more than the whole cost, because an offset of the ringing instrument moves every cross interval that is not a unison as well, and on this slendro those move very slightly toward smoothness. At two cents the offset costs 0.09, at twenty 0.65, at fifty 1.05. It peaks near eighty cents, at 1.08 above the in-tune figure, and has begun to fall by 120.

Set against the scale’s own margin the sizes become readable. In tune, the texture scores 7.47 and the median random five-note scale 9.79. Ten cents of offset spends 17 per cent of that margin, and no offset in the range tried spends all of it: at its roughest, eighty cents out, the slendro is still smoother than the median random scale in tune.

In hertz the range is a familiar one. At middle C, 30 cents is a beat of 4.6 hertz between fundamentals and 50 cents is 7.7. That is the range the pair tuned apart on purpose describes for instruments built to beat, and it is where this curve is flattening. Past the point at which a pair’s sustain has gone, a further offset buys the beating that is wanted and costs little more roughness.

The standing does not move #

The percentile is the measure the earlier essays used for a scale’s standing, and an offset raises a question about it that has a trap in it.

Score the offset slendro against random scales played in tune and it appears to lose standing fast: from the 5.5th percentile to the 10th at ten cents and the 23.5th at eighty. That comparison is unfair. It pits a detuned ensemble against ensembles that were never detuned.

Score it against random scales played with the same offset, which is the comparison the percentile exists to make, and the standing does not move: between the 4.5th and 6th percentile at every offset from zero to 120 cents. Beating depends on how often degrees are doubled and on the spectrum, and scarcely on which five pitches the degrees are, so it adds nearly the same roughness to every scale in the pool.

That separates the two things a tuning is. The scale’s shape is a property of the table, and an offset instrument leaves it where it was. What an offset changes is the ensemble’s intonation — how well the instruments agree — and that is a cost every scale would pay alike.

A bar pays the most #

The bar’s spectrum was a choice, and whether beating is a peculiarity of it is worth checking before anything is made of the size.

Every spectrum pays, including a pure tone with no partials at all, which pays 2.6 per cent at ten cents against the bar’s 5.1. A plucked string and a reed pay 3.8, and odd harmonics alone 4.0. The ordering follows the height of the partials. Every partial beats at its own rate, and the rate is the partial’s number times the offset in hertz, so a spectrum whose energy reaches the thirteenth partial has more of its roughness in fast beats. The bar’s inharmonic partials are not what makes it pay the most. Their height is — which is the same conclusion, from the other side, that the spectrum that was supposed to explain the gamelan reached when a bar’s partials failed to recommend the slendro.

By eighty cents the spectra have converged a little, from 14.4 per cent for the bar to 9.9 for the pure tone, because the upper partials have passed their peaks and only the fundamentals are still climbing.

Which computation produced the numbers #

The scale is the measured slendro of the earlier essays, 0, 231, 474, 717 and 955 cents over middle C. The ringing part plays a note every 0.15 seconds and the damped part one note to its four, both drawing degrees at random from one fixed seed, over sixteen bars. A single-degree error moves one table entry of one instrument; the tonic’s error moves its octave with it, since both are the same bar’s pitch class in the rendering. An offset moves every entry of the ringing instrument’s table.

A cross pair on one degree is scored as the spectral roughness of the two tuned frequencies less that of the first frequency against itself, which is zero when the instruments agree. The percentiles are against two hundred random five-note scales drawn from a fixed seed, each rendered through the same two parts with the same offset or none.

Where the model stops #

Beating is scored as roughness. Plomp and Levelt’s curve is a curve of sensory dissonance, and at a beat of a hertz or two a listener hears a slow waxing and waning rather than roughness. The model counts that beat as a small roughness. The comparisons between instruments and between offsets are unaffected, since the same curve scores them all; the absolute size at the smallest offsets is overstated.

The amplitudes are one or nothing. A ringing bar near the end of its decay beats as hard as a fresh one in this rendering. Real beating between a loud new stroke and a quiet old one is shallow, and a beat has a depth that this rendering holds at full.

The errors are one degree or one offset. A real instrument’s bars are each wrong by their own amount, and a real pair of instruments differs by a pattern, not a constant. Since the errors add, any pattern’s cost is the sum of its degrees’ costs from the curves above, weighted by how often the music doubles each degree.

What the numbers cannot say about a tuner #

Whether tuners do tighten the ringing instruments. The earlier essay’s prediction was that they should; this one says the roughness arithmetic gives them no reason to. If gamelan makers do tune the gendèr more carefully, the reason is something else — its register, its role in the texture, or the fact that its errors ring on — and that is a question for accounts of tuning practice rather than for arithmetic.

Whether a listener hears an offset as error or as colour. The model prices beating as a cost. A tradition that tunes its paired instruments apart on purpose prices it as the sound of the ensemble, and nothing here can choose between those.

Still open: whether a doubling pattern sets the tolerance #

The cost of a wrong degree turned out to be set by how often the music doubles it across the two instruments, which in this rendering is an accident of a random seed. In a real piece it is a fact of the style: the elaborating part is built around the skeleton’s pitches and returns to them at structurally strong points, so some degrees are doubled far more than others and at the moments a listener attends to most. Replacing the random passage with parts that coincide on the skeleton’s notes at the ends of phrases, as an elaboration does, would turn the degree-by-degree costs above into a ranking of which bars a tuner can least afford to leave wrong. That ranking would be a property of the repertoire, and computing it needs no corpus — only a stated rule for where an elaboration meets its skeleton.