The tempo moves a scale further than the touch

The essay that weighted every pair by the ring before it found that a scale is committed to how long its instrument rings, not only to what its instrument’s spectrum is: the same pitches score differently when the notes overlap and when they do not.

Its closing paragraph named the case that turns that from acoustics into practice. A gamelan is played two ways on the same bars. A saron’s bars are damped at every stroke — the player pinches the ringing bar with the left hand as the right strikes the next — and a gendèr’s are left to ring over tuned resonator tubes and damped selectively. Both are the same bar with the same spectrum, and the difference is entirely in how many later notes a struck bar is still sounding against.

What the touch can change #

That count is the whole of what the player’s hand controls, and it is arithmetic.

A free bar stays within twenty decibels of its strike for about 1.3 seconds. At a fast elaboration of 0.15 seconds a note it is still sounding under eight later notes; at 0.6 seconds under two; at 2.4 seconds under none at all.

Where that count is zero the damping decides nothing, because the bar is already silent when the player’s hand arrives. Where it is eight, damping removes eight simultaneities that would otherwise have been there. So the touch is a dial whose range is set by the tempo before the player touches it.

What the touch does change #

At six tenths of a second a note — an ordinary middle tempo — a measured slendro sits at the 63rd percentile with the bar damped and the 46th with it ringing. Left to ring it is smoother than most five-note scales; damped it is rougher than most. Seventeen percentile points, and the pitches never moved.

The diatonic sets in the same figure are at the 93rd to 99th percentile under a bar’s spectrum whatever the touch, which is the long-standing point of these essays and not this one’s: a scale built for harmonic tones is a bad scale for a bar, and no way of striking it repairs that.

What is worth noticing is which sets the touch moves most. The slendro moves 17 points and Raga Bhupali 15; the tempered diatonic moves 6. A scale matched to its instrument is a scale whose standing depends on being allowed to sound — which is nearly the definition the scale least committed to its own instrument arrived at from the other end, and a scale that is badly matched is badly matched under any touch.

And what the tempo does #

Left to ring, the slendro sits at the 29th percentile at 0.15 seconds a note and the 83rd at 2.4 seconds. That is a span of fifty-four percentile points, against seventeen for the touch at its most consequential — and the two touches converge at the slow end, to a difference of 1.3 points at 2.4 seconds a note, for the reason the overlap figure gives.

So the ordering of the two effects is not close. A slendro is a smoother-than-average scale when it is played fast and a rougher-than-average one when it is played slowly, on the same bars, struck the same way, with the same five pitches.

That is a strange thing to be able to say about a scale, and it is worth being clear what it is not. It is not that the scale sounds better fast; percentile among random scales is a measure of how much sensory roughness a scale’s own intervals produce, and it says nothing about what anybody prefers. What it says is that the property the eleventh essay measured — a scale’s fit to its instrument — is not a property of the scale and the instrument. It is a property of the scale, the instrument and the tempo, and the third is the largest term.

Why a slow tempo makes every scale rougher #

The direction of the tempo effect is worth a paragraph because it is the opposite of the intuition and the reason is one line.

The intuition is that a slow tempo gives the notes room and a fast one crowds them, so slow should be smoother. The percentile says the reverse, and the reason is that a percentile is a comparison. At a slow tempo nothing overlaps, so every scale’s roughness comes from its melodic intervals alone, and on that measure a random scale and a chosen one are much more alike — the chosen scale’s advantage is in what it does when its notes sound together, and it has no chance to do it.

So the slow end of the range is not a place where scales sound rough. It is a place where scales stop being distinguishable, and a scale that was chosen for its simultaneities drifts back toward the middle of the pack. The 83rd percentile at 2.4 seconds a note does not mean the slendro is grating; it means that at that tempo a random five-note scale would do about as well and slightly better.

Which makes the finding a statement about what a scale is for. A set of five pitches is a choice that pays off only when the pitches meet, and how often they meet is a performance decision. A tradition that plays its scale slowly and singly has chosen pitches whose choosing does very little work.

Where the touch decides anything #

The tempo and the touch are not independent, and the interaction is the useful part.

The touches differ by 12.8 points at the fastest tempo, 17.0 in the middle, 5.2 at 1.2 seconds and 1.3 at the slowest. The player’s hand matters most in the middle of the range and hardly at all at either end.

At the slow end the reason is the overlap figure: a bar that has died before its successor is sounding against nothing, and damping silence changes nothing. At the fast end the reason is the opposite — every note overlaps several others whatever the player does, and one stroke of ring is already most of what the bar has to give.

So the touch is a real decision in a band of tempi and a formality outside it, and gamelan practice has both kinds of instrument playing at once. A saron plays the skeleton at the slow end of the range and a gendèr elaborates at the fast end, which puts the damped instrument where damping matters least and the ringing one where ringing matters most.

That is either a coincidence or the point, and this arithmetic cannot tell which. What it can say is that the arrangement is efficient: the instrument whose hand is busiest with damping is the one whose damping buys the most.

The control, which repeats an earlier finding #

Scored under a string’s spectrum and a string’s decay rather than a bar’s, the slendro sits at the 43rd percentile when ringing against the 46th under a bar. Three points. The spectrum that was supposed to explain the gamelan found that a bar’s partials do not recommend a slendro, and this is the same result arriving from the ring: the slendro is barely more at home on a bar than on a string.

Which sharpens the finding above rather than softening it. If the slendro’s standing were a tight property of the bar’s spectrum, a fifty-four-point swing with tempo would be surprising. It is not a tight property of anything about the instrument — so what moves it is what moves any scale’s roughness, which is how much of it is sounding at once.

What this does to the eleventh essay’s claim #

The eleventh essay’s title is that a scale is committed to how long its instrument rings, and the finding here does not contradict it. It supplies the missing variable and reverses which of the two is the headline.

A scale is committed to how long its instrument rings relative to how fast it is played. The ring is a property of the bar and is fixed by its maker; the tempo is a property of the performance and changes several times inside one piece. What matters to the arithmetic is the ratio, and only one of its two terms is an instrument.

That is why the tempo term is the larger one. A gamelan’s tempo levels span a factor of sixteen in note rate, and no instrument’s decay time varies by anything like that between one bar and another of the same set. A performance moves the ratio further than a workshop can, and the scale’s standing moves with the ratio.

It also says which of these essays are about instruments and which are about performances, which has not been distinguished before. Everything about spectra — what a bar’s partials recommend, which scale a curve’s minima name — is about instruments and does not move. Everything about overlap is about performances and moves a great deal. The account has been mixing the two since its second essay, and this is the first number that separates them.

Which computation produced the numbers #

A scale’s roughness is the sensory dissonance summed over every pair of its degrees, with each degree given the stated spectrum, and each pair weighted by how much of the earlier note is still sounding when the later one begins. The weighting is the eleventh essay’s: a component at $r$ times the fundamental has fallen to $10^{-3t\,r^{p}/T_{60}}$ after $t$ seconds, with $T_{60}$ and $p$ the instrument’s own decay time and exponent.

The touch enters as which lags are summed over. A bar left to ring sounds against every later note its decay reaches at the tempo played; a bar damped at its successor sounds against one; the damped limit sounds against none, which is a scale scored on its melodic intervals alone and is drawn here only as the limiting case it is.

The percentile is against two thousand random scales of the same size drawn once per spectrum from a fixed seed, which is the eleventh essay’s own pool and is why the numbers here can be set beside its numbers.

The melodic cycle each scale is scored over is its own: a raga’s ascent and descent where the tradition supplies them, and a plain cycle otherwise. The slendro has no grammar in this collection and is scored on a plain cycle, which is a simplification and is stated as one.

The number that would falsify this #

A finding of this shape invites the reply that the measure is doing the work rather than the music, and there is one quantity that would settle it.

The percentile is a comparison against random scales of the same size, and it moves with tempo partly because the pool moves with tempo. If random five-note scales get smoother as the tempo slows while the slendro stays put, the slendro’s percentile rises without the slendro changing at all — and the essay above would be describing an artefact of the comparison.

The raw roughness says which it is. Under a bar’s spectrum the slendro’s own roughness falls by a factor of a hundred as the tempo slows — 0.215 at a sixth of a second a note, 0.054 at six tenths, 0.002 at two and two fifths — because fewer pairs overlap and there is less to be rough about. So does the pool’s, and the pool’s falls less. The slendro loses ground because its advantage was in the overlaps and the overlaps are what the slow tempo removes.

So the percentile and the raw number tell the same story in opposite directions, which is the check: a scale that is genuinely smoother at a slow tempo and merely less exceptional would be a different finding, and it is not this one. What the arithmetic says is that a slendro’s edge is entirely in what its notes do together, and a performance that does not let them sound together has thrown the edge away.

Where the model stops #

The resonator is not in the model. A gendèr’s bars sound over tuned bamboo tubes, and a tube tuned to the bar’s fundamental lengthens that fundamental’s decay and not its upper partials’. That would make a ringing bar’s late sound more nearly a pure tone than the model’s, which would lower the roughness of every overlap and move the ringing curve down — in the direction that widens the gap this essay is about, and by an amount nothing here computes.

And the damping is treated as instantaneous and total. A pinched bar does not stop; it decays fast, over some tens of milliseconds, and the pinch is on one bar while the hand is travelling. A real saron part has a little ring in it.

The random pool is drawn uniformly over the octave, which is what makes a percentile interpretable and is not what any tradition’s candidate scales look like. A pool drawn from scales anybody might actually have chosen would be smoother throughout and every percentile here would rise; the ordering across tempi would not change, because it comes from the slendro’s own roughness falling faster than the pool’s.

The tempo range is a range of note rates and not of irama. A gamelan’s tempo levels change the density of the elaborating parts as well as the rate of the skeleton, so the saron and the gendèr are not moving along one axis together.

What the picture cannot show #

It cannot show a scale’s intonation drifting. A gamelan’s tuning is set per instrument by the maker and the pairs are deliberately mistuned to beat, which a tuning is not a table of cents is the essay about, which is a chorus effect rather than a scale and adds roughness this model has no term for.

Nor two instruments at once. Every figure here is one instrument playing one melodic line. A gamelan is a dozen instruments playing at several densities against one another, and the simultaneities that matter are between parts rather than within one.

And it cannot show what anybody prefers. A percentile among random scales is a statement about sensory roughness. The smoothness is in the skips is the essay that found where a scale’s smoothness actually lives, and even that is a statement about a model rather than about a tradition’s ears.

Still open: whether the two instruments are scored on the same scale #

The finding above compares one scale under two touches and one tempo range. A gamelan does not do that: it plays the saron and the gendèr simultaneously, at densities that differ by a factor of two or four, so the damped instrument and the ringing one are sounding the same pitches at different rates at the same moment.

What that means for the arithmetic is that the roughness a listener receives is not either curve. It is a sum over pairs drawn from two streams with different overlap structures — a saron’s notes against a gendèr’s, as well as each against itself — and the cross terms are the ones nothing here computes.

They are computable and the pieces are nearly all present: two cycles at a stated density ratio, each with its own ring, and every pair weighted by what is still sounding. What it would settle is whether the slendro’s standing in the ensemble is the average of the two curves, or better than both because the fast part supplies the smoothness while the slow part supplies the frame, or worse than both because the cross terms fall where neither stream’s own intervals do. The third would be the interesting answer, and it is the one an arrangement that puts a fast ringing instrument over a slow damped one would be designed to avoid.