Scales and modes

A scale is committed to how long its instrument rings

A traditional scale's standing on the roughness model moves when the instrument changes, and that movement was read as a commitment to the instrument's spectrum. Weight every pair of notes a melody sounds by how much of the earlier note is still ringing when the later one begins, and the movement grows — the tempered diatonic's by 37 percentile points at a note every 0.6 seconds — but almost none of it is the spectrum. Put four instruments' rings on one spectrum and the scale moves 38.6 points; put four spectra under one ring and it moves 9.4. And the partials that make a fifth smooth are the first to stop sounding.

Assumes: The smoothness is in the skips · The scale least committed to its own instrument

The smoothness is in the skips counted a scale’s roughness over the pairs of notes its plainest melody sounds, and counted them by distance: each note with the next, or with every note up to some number later. Counted by the steps, every seven-note scale was rougher than nine random scales in ten; counted by the skips, and above all by the fifths four notes apart, the same scales were among the smoothest there are. And the prediction the scale least committed to its own instrument had recorded — that weighting the pairs would make each scale’s standing depend more on the instrument — held only once enough of the melody was counted.

Its last section said what the distance was standing in for. Roughness is a property of two sounds at once, and a melody’s notes come one at a time, so a step is rough only as far as the note before is still sounding when the next begins. A distance counts every pair within it fully and every pair beyond it not at all. A real instrument does neither: it rings, and the note before fades by an amount set by the instrument and by the tempo. Replacing the distance with the ring puts the instrument into the measurement a second time — once in the partials, and once in how long they last.

The result of doing that is that the second appearance is nearly the whole of it.

Weighted by the ring of the note before, a note every 0.6 seconds, Raga Deshkar moves least. Each scale placed by how smooth it is against 2000 random scales of its size under 4 instruments, each with its own spectrum and its own ring, with every pair of notes its plainest melody sounds weighted by how much of the earlier note is still sounding when the later one begins, a note every 0.6 seconds. Low is smooth. Raga Bhupali: plucked string, 6 s 53.6, long-ringing string, 12 s 35.8, blown note, 2 s hall 41.3, free bar, 4 s 58.5; spread 22.8; Raga Deshkar: plucked string, 6 s 39.0, long-ringing string, 12 s 25.9, blown note, 2 s hall 26.1, free bar, 4 s 42.0; spread 16.1; one measured slendro: plucked string, 6 s 42.9, long-ringing string, 12 s 27.8, blown note, 2 s hall 32.6, free bar, 4 s 46.4; spread 18.6; Rast, Arabic theory: plucked string, 6 s 92.0, long-ringing string, 12 s 65.2, blown note, 2 s hall 92.3, free bar, 4 s 96.1; spread 30.9; Rast, Turkish theory: plucked string, 6 s 88.0, long-ringing string, 12 s 58.1, blown note, 2 s hall 90.8, free bar, 4 s 93.6; spread 35.5; the diatonic major, tempered: plucked string, 6 s 84.8, long-ringing string, 12 s 54.2, blown note, 2 s hall 89.5, free bar, 4 s 91.3; spread 37.0; the diatonic major, five-limit just: plucked string, 6 s 87.7, long-ringing string, 12 s 56.7, blown note, 2 s hall 90.3, free bar, 4 s 92.9; spread 36.2. — Fig. 1 Seven scales, each placed among two thousand random scales of its size, with every pair of notes its plainest melody sounds weighted by how much of the earlier note is still sounding when the later one begins, a note every 0.6 seconds, under four instruments: a plucked string ringing for 6 seconds, a long-ringing string for 12, a blown note heard in a hall of 2 seconds, and a free bar for 4. The four seven-note scales sit between the 54th and the 96th percentile and move 30.9 to 37.0 points across the instruments; the three pentatonic scales sit between the 26th and the 59th and move 16.1 to 22.8.

A ring instead of a distance

The ring is one law with two numbers in it. The note before loses sixty decibels at its fundamental in a decay time, and loses its upper components faster, in proportion to their frequency ratio raised to a power. A power of one means a component at three times the fundamental dies three times as fast; a power of nought means everything dies together. The laws used here are the ones the partials that do not die together set down for a plucked string, a long-sustaining string and a free bar, which that essay was careful to call ordinal rather than measured, and the blown note is a note that stops when the next one starts and is heard in a hall whose reverberation time is two seconds — a room of the kind a concert hall is built to be.

How much of the note before is still sounding, a note every 0.3 seconds. For each instrument, the level of the earlier note's fundamental and of a component at three times its frequency, in decibels below the strike, when a later note begins some number of notes afterwards at 0.3 seconds a note. plucked string, 6 s, losses rising as the component's ratio to the power 1: fundamental -3.0 dB 1 note later, -6.0 dB 2 notes later, -12.0 dB 4 notes later, -24.0 dB 8 notes later; the component at three times -9.0 dB at 1, -18.0 dB at 2, -36.0 dB at 4, -72.0 dB at 8; long-ringing string, 12 s, losses rising as the component's ratio to the power 0.7: fundamental -1.5 dB 1 note later, -3.0 dB 2 notes later, -6.0 dB 4 notes later, -12.0 dB 8 notes later; the component at three times -3.2 dB at 1, -6.5 dB at 2, -12.9 dB at 4, -25.9 dB at 8; blown note, 2 s hall, losses rising as the component's ratio to the power 0: fundamental -9.0 dB 1 note later, -18.0 dB 2 notes later, -36.0 dB 4 notes later, -72.0 dB 8 notes later; the component at three times -9.0 dB at 1, -18.0 dB at 2, -36.0 dB at 4, -72.0 dB at 8; free bar, 4 s, losses rising as the component's ratio to the power 0.5: fundamental -4.5 dB 1 note later, -9.0 dB 2 notes later, -18.0 dB 4 notes later, -36.0 dB 8 notes later; the component at three times -7.8 dB at 1, -15.6 dB at 2, -31.2 dB at 4, -62.4 dB at 8. — Fig. 2 How far below its strike the note before has fallen when a later note begins, at 0.3 seconds a note: its fundamental solid and a component at three times its frequency dashed. The plucked string’s fundamental is 3 decibels down one note later and 12 four notes later, and its component at three times is 9 and 36 down. The long-ringing string’s fundamental is 1.5 and 6 down. The blown note in the hall loses 9 decibels a note at every component, and the free bar’s fundamental loses 4.5.

A pair of notes is then scored by the same Plomp–Levelt model every roughness figure in these essays has used, but between two different spectra: the later note’s, as struck, and the earlier note’s, as it stands at that moment. Every pair the melody sounds is counted, at every distance, until the earlier note’s fundamental is sixty decibels down. A score per note of melody is compared with the same score for two thousand random scales walked by the same melody, as before, so a percentile still says where a tradition’s scale sits among the scales it could have been.

Two properties of the construction matter for everything that follows. With every component decaying at the fundamental’s rate, the ring at one distance is a single number for every pair, so a ring that stops after one note is exactly the count of steps, and each column of pairs at one distance is exactly the earlier count at that distance — both are checked percentile for percentile. And the score reads a ring only through the time between notes divided by the decay time, so halving every decay time is exactly the same as doubling the time between notes. A decay time that is wrong by a factor is a tempo that is wrong by the same factor, and that is the most a borrowed decay time can distort here.

With nothing decaying faster, the skips still carry it

The plucked string, 6 s with every component decaying together: where the smoothness is, a note every 0.3 seconds. For each scale, its percentile among 2000 random scales under a plucked string, eight harmonics, counting only the pairs of notes its plainest melody sounds at each distance, each weighted by how much of the earlier note the plucked string, 6 s leaves sounding when the later begins, a note every 0.3 seconds, with every component decaying at the fundamental's rate. Low is smooth. Raga Bhupali: 1 apart 56.8, 2 apart 3.0, 3 apart 38.0, 4 apart 5.3, 5 apart 20.4, 6 apart 5.3, 7 apart 38.0; Raga Deshkar: 1 apart 37.5, 2 apart 7.8, 3 apart 26.1, 4 apart 17.8, 5 apart 17.8, 6 apart 26.1, 7 apart 7.8; one measured slendro: 1 apart 44.5, 2 apart 3.6, 3 apart 24.9, 4 apart 8.0, 5 apart 39.7, 6 apart 8.0, 7 apart 24.9; Rast, Arabic theory: 1 apart 98.6, 2 apart 7.0, 3 apart 38.9, 4 apart 0.1, 5 apart 35.5, 6 apart 9.3, 7 apart 39.0; Rast, Turkish theory: 1 apart 97.8, 2 apart 3.6, 3 apart 20.9, 4 apart 0.0, 5 apart 35.6, 6 apart 11.6, 7 apart 33.1; the diatonic major, tempered: 1 apart 97.0, 2 apart 4.3, 3 apart 13.4, 4 apart 0.0, 5 apart 41.5, 6 apart 17.4, 7 apart 33.0; the diatonic major, five-limit just: 1 apart 97.8, 2 apart 2.1, 3 apart 13.9, 4 apart 0.1, 5 apart 38.5, 6 apart 18.1, 7 apart 29.1. — Fig. 3 Each scale’s percentile under the plucked string, counting only the pairs of notes its melody sounds at each distance, with every component of the note before decaying at its fundamental’s rate, a note every 0.3 seconds. The seven-note scales are at 97.0 to 98.6 one note apart, at 2.1 to 7.0 two notes apart, and at 0.0 or 0.1 four notes apart, which is where the fifths are.

With every component decaying together, the ring reproduces the earlier finding under the plucked string’s spectrum as well as the tanpura’s. The steps are the roughest pairs a seven-note scale has and the fifths the smoothest: 97.0 to 98.6 one note apart, and 0.0 or 0.1 four notes apart. The pentatonic scales keep their own pattern, with Bhupali’s steps at 56.8 and its thirds and fourths two notes apart at 3.0, and the slendro’s at 44.5 and 3.6.

That is the control. It says that the change of instrument, as far as the spectrum goes, does not alter where a scale’s smoothness is located.

The partials that make a fifth smooth are the first to go

A fifth is smooth in this model for one reason, which a spectrum chooses its own scale put at the centre of the account: the third partial of the lower note lands on the second partial of the upper, and a coincidence contributes no roughness. A plucked string’s third partial is the component that, on its law, dies three times as fast as its fundamental. By the time a note four notes later begins, at a note every 0.3 seconds, it is 36 decibels down.

The plucked string, 6 s, upper components decaying first: the seven-note scales' fifths at the 13.1 to 18.8 percentile. For each scale, its percentile among 2000 random scales under a plucked string, eight harmonics, counting only the pairs of notes its plainest melody sounds at each distance, each weighted by how much of the earlier note the plucked string, 6 s leaves sounding when the later begins, a note every 0.3 seconds, with the upper components decaying faster than the fundamental. Low is smooth. Raga Bhupali: 1 apart 68.3, 2 apart 11.8, 3 apart 57.2, 4 apart 16.4, 5 apart 34.1, 6 apart 19.3, 7 apart 62.0; Raga Deshkar: 1 apart 46.3, 2 apart 17.8, 3 apart 37.9, 4 apart 31.8, 5 apart 31.5, 6 apart 42.0, 7 apart 21.4; one measured slendro: 1 apart 59.1, 2 apart 6.7, 3 apart 38.8, 4 apart 18.6, 5 apart 43.7, 6 apart 21.9, 7 apart 41.5; Rast, Arabic theory: 1 apart 99.8, 2 apart 9.2, 3 apart 48.8, 4 apart 18.8, 5 apart 48.2, 6 apart 27.1, 7 apart 40.3; Rast, Turkish theory: 1 apart 99.5, 2 apart 7.3, 3 apart 35.5, 4 apart 14.8, 5 apart 48.3, 6 apart 32.5, 7 apart 41.7; the diatonic major, tempered: 1 apart 98.9, 2 apart 6.2, 3 apart 27.4, 4 apart 13.1, 5 apart 48.3, 6 apart 36.4, 7 apart 45.1; the diatonic major, five-limit just: 1 apart 99.4, 2 apart 6.3, 3 apart 30.7, 4 apart 15.9, 5 apart 53.8, 6 apart 35.4, 7 apart 43.9. — Fig. 4 The same table with the plucked string’s upper components decaying faster than its fundamental, as its law says they do. The seven-note scales’ steps are at 98.9 to 99.8; their fifths, four notes apart, rise to between 13.1 and 18.8; and the smoothest distance is now two notes apart, at 6.2 to 9.2.

Let the upper components decay at their own rate and the fifths lose their place. Four notes apart, the seven-note scales go from 0.0 or 0.1 to between 13.1 and 18.8 — still smoother than most random scales, no longer smoother than all of them. The smoothest distance becomes two notes apart, where the pairs are mostly thirds and the note before has had half as long to lose its upper partials, at 6.2 to 9.2. The steps are rougher still, at 98.9 to 99.8, because the earlier note of a step is now closer to a pure tone, and a pure tone keeps the rough steps: they belong to the critical band and not to the partials.

So the division of labour the earlier essay found is sharpened rather than overturned. The critical band makes a scale’s steps rough whatever is sounding, and the partials make its fifths smooth — but only while the partials last, and on a plucked string they last a third as long as the note does.

Rings, not spectra

Back at a note every 0.6 seconds and with each instrument’s own spectrum and ring, the four seven-note scales move 30.9 to 37.0 percentile points across the four instruments, which is more than any distance weighting moved them: counting only steps they moved 6.0 to 6.6, and counting everything within eight notes 16.6 to 22.1. The instrument matters more once it is allowed to ring. What the first figure cannot say is which half of the instrument is doing it.

A note every 0.6 seconds: for 7 of 7 scales the rings alone move them as far as the instruments do, and the spectra alone half as far or less. For each scale, how far its percentile among 2000 random scales moves across 4 instruments when every pair its plainest melody sounds is weighted by the ring of the earlier note, a note every 0.6 seconds: with each instrument's own spectrum and ring; with every ring put on a tanpura, sixteen harmonics; with every spectrum under the ring of the plucked string, 6 s, all its components decaying together; and the same with its upper components decaying faster. Raga Bhupali: each instrument its own 22.8, every ring on the tanpura 19.2, every spectrum, ring of the plucked string 6.0, the same, upper parts decaying faster 9.9; Raga Deshkar: each instrument its own 16.1, every ring on the tanpura 13.1, every spectrum, ring of the plucked string 5.0, the same, upper parts decaying faster 11.4; one measured slendro: each instrument its own 18.6, every ring on the tanpura 15.0, every spectrum, ring of the plucked string 4.7, the same, upper parts decaying faster 10.0; Rast, Arabic theory: each instrument its own 30.9, every ring on the tanpura 30.7, every spectrum, ring of the plucked string 13.8, the same, upper parts decaying faster 3.1; Rast, Turkish theory: each instrument its own 35.5, every ring on the tanpura 35.4, every spectrum, ring of the plucked string 9.5, the same, upper parts decaying faster 2.5; the diatonic major, tempered: each instrument its own 37.0, every ring on the tanpura 38.6, every spectrum, ring of the plucked string 9.4, the same, upper parts decaying faster 2.8; the diatonic major, five-limit just: each instrument its own 36.2, every ring on the tanpura 36.1, every spectrum, ring of the plucked string 8.3, the same, upper parts decaying faster 2.9. — Fig. 5 For each scale, how far its percentile moves across the four instruments at a note every 0.6 seconds, four ways: each instrument with its own spectrum and ring; all four rings on the tanpura’s spectrum; all four spectra under the plucked string’s ring with every component decaying together; and the same with the upper components decaying faster. The tempered diatonic moves 37.0, 38.6, 9.4 and 2.8. For all seven scales the rings on one spectrum move it at least three quarters as far as the instruments do, and the spectra under one ring half as far or less.

The four columns separate the two halves. Put all four rings on one spectrum — the tanpura’s sixteen harmonics — and every scale moves almost exactly as far as it does with the real instruments: the tempered diatonic 38.6 against 37.0, the Turkish Rast 35.4 against 35.5, the slendro 15.0 against 18.6. Put all four spectra under one ring, the plucked string’s with its components decaying together, and the tempered diatonic moves 9.4; with its upper components decaying faster, 2.8. For every one of the seven scales, the rings alone move it at least three quarters as far as the instruments do, and the spectra alone half as far or less.

The reason the tilted column is the smallest of all for the seven-note scales is the section above. A ring that strips the upper partials from the note before strips out exactly the thing that distinguished the spectra, and four spectra with their upper partials gone are four spectra much more alike. The long ring is what separates the instruments: the long-ringing string is the smoothest of the four for every scale, because it keeps the note before sounding long enough for the skips to be counted, and the skips are where a scale’s smoothness is.

The order of the four instruments is as regular as that. At a note every 0.6 seconds the long-ringing string, losing three decibels a note, is the smoothest instrument for every one of the seven scales, and the free bar is the roughest for every one of them: its partials do not coincide at the fifth, and its ring of four seconds is long enough to count its rough skips as well as its steps. The two in between trade places by scale size. The blown note, which the hall takes down eighteen decibels a note at this tempo, is the second smoothest instrument for all three pentatonic scales and the second roughest for all four seven-note ones. A ring that short counts little beyond the steps, and the steps of a pentatonic scale are wide enough not to be rough while the steps of a seven-note scale are the roughest pairs it has. The instrument is not ranked by its spectrum or its ring alone but by what its ring lets a scale’s own steps and skips do.

That answers the question the debt put. A scale’s commitment to its instrument is a commitment to how long the instrument sounds, at least at this tempo and on these four laws. The account that a spectrum chooses a scale explains where a scale’s smoothness is; it does not explain why the same scale stands in a different place on a different instrument.

Two ragas on one set, and two Rasts

A degree is where it goes next made Bhupali and Deshkar the test case for any measure of a scale’s sound, because they have the same five pitches and different ascents. Counted by distance, they came apart in two directions: one note apart Deshkar was the smoother, because its ascent leaps a third where Bhupali takes two whole tones, and two and four notes apart Bhupali was, because the leap puts Deshkar’s later pairs out of step.

The ring decides between those two answers, and it decides the same way on every instrument. At a note every 0.6 seconds Deshkar is the smoother raga under all four, by between 9.9 and 16.5 percentile points: 39.0 against 53.6 on the plucked string, 25.9 against 35.8 on the long-ringing one, 26.1 against 41.3 for the blown note, and 42.0 against 58.5 on the bar. The distance tables under the plucked string’s own ring at 0.3 seconds keep the split the earlier essay found — Deshkar smoother one note apart, 46.3 against 68.3, and Bhupali smoother two and four notes apart — but a ring does not weight the distances alike. It counts the nearest pairs most, because the note before has faded least by the time they sound, and the nearest pairs are where Deshkar’s leap buys it smoothness. Weighted by what is actually still sounding, the leap in Deshkar’s ascent is worth more than it costs.

The two conventions for Rast differ in one respect, the size of their neutral steps, and they come out in one order too. The Arabic convention, four of whose seven steps are 150 cents and sit on the peak of a pure tone’s roughness, is rougher than the Turkish on every instrument: by 4.0 points on the plucked string, 7.1 on the long-ringing string, 1.5 for the blown note and 2.5 on the bar. The margin is widest on the instrument with the longest ring, which counts the most pairs and so gives a difference in step size the most pairs to show up in.

At every tempo but the fastest

Across tempi, the instruments move a scale further than their spectra do. For each scale, how far its percentile moves across 4 instruments with each instrument's own spectrum and ring (solid), and across the same spectra all under the ring of the plucked string, 6 s with every component decaying together (dashed), against the seconds between notes. one measured slendro: 0.15 s 12.9 against 6.9, 0.2 s 10.7 against 6.3, 0.3 s 9.2 against 4.8, 0.45 s 13.4 against 4.1, 0.6 s 18.6 against 4.7, 0.9 s 27.0 against 7.7, 1.2 s 32.2 against 9.4; Raga Bhupali: 0.15 s 20.4 against 14.6, 0.2 s 17.4 against 13.5, 0.3 s 14.1 against 12.4, 0.45 s 17.9 against 8.3, 0.6 s 22.8 against 6.0, 0.9 s 28.7 against 6.4, 1.2 s 30.7 against 9.3; the diatonic major, tempered: 0.15 s 15.5 against 14.8, 0.2 s 21.8 against 12.9, 0.3 s 35.4 against 9.5, 0.45 s 38.7 against 8.6, 0.6 s 37.0 against 9.4, 0.9 s 27.0 against 11.5, 1.2 s 16.8 against 11.2. — Fig. 6 For the slendro, Bhupali and the tempered diatonic, how far each moves across the four instruments with their own rings (solid) and across the four spectra under the plucked string’s ring with every component decaying together (dashed), from a note every 0.15 seconds to one every 1.2. The tempered diatonic moves 15.5 against 14.8 at 0.15 seconds, 35.4 to 38.7 against 8.6 to 9.5 from 0.3 to 0.6, and 16.8 against 11.2 at 1.2.

The split depends on the tempo, and it holds everywhere except at the fastest. At a note every 0.15 seconds — a quick ornamental run — the tempered diatonic moves 15.5 points across the instruments and 14.8 across the spectra under one ring, so there the spectra account for nearly all of it: every ring is long compared with the notes, all four instruments count the skips, and what is left to differ is the partials. From a note every 0.3 seconds to one every 0.6 the instruments move it 35.4 to 38.7 points and the spectra 8.6 to 9.5. By 1.2 seconds a note both are smaller again, 16.8 against 11.2: on every instrument the note before has faded further by the time each later note begins, the steps count for more on all four, and the four differ less.

The pentatonic scales are less dramatic and go the same way. The slendro moves 9.2 at 0.3 seconds and 32.2 at 1.2, against 4.8 and 9.4 across the spectra; Bhupali 14.1 against 12.4 at 0.3, where its two numbers come closest, and 30.7 against 9.3 at 1.2. The ordering of the scales changes with tempo too. At a note every 0.6 seconds the scale that moves least is Deshkar, at 16.1, and not the slendro, which moves 18.6; at 1.2 seconds the slendro moves more than Bhupali or the diatonic. The earlier finding that the slendro is the least committed scale was a finding about a measurement with no tempo in it.

The arithmetic

Each melody is the cycle its ascent and descent describe, as in the essay before this one, and the notes are evenly spaced at the stated time between notes. For a pair in which the earlier note sounded t notes before the later, each of the earlier note’s components is multiplied by ten to the power of minus three times t times the time between notes times its ratio to the law’s exponent, divided by the decay time, which is sixty decibels in the decay time at the fundamental. The pair’s roughness is the Plomp–Levelt sum over every component of the earlier note against every component of the later, the tonic at middle C, and the scale’s score is the sum over every pair the cycle sounds, per note of the cycle, until the earlier note’s fundamental has fallen sixty decibels.

Because that score is linear in the earlier note’s component amplitudes, it is computed as a table of component roughnesses for each scale multiplied by a table of ring weights for each melody, instrument and tempo, and the two thousand random scales are the same scales, drawn from the same seed, that every percentile in these essays has used. The laws are the plucked string of six seconds with an exponent of one, the long-sustaining string of twelve with 0.7, a free bar of four with 0.5, and the blown note in a hall of two seconds with none.

What the ring assumes

That the laws are right in order if not in size. They are ordinal laws carried over from the essay on partial decay, and a uniform error in their decay times is a change of tempo, which the tempo figure covers. An error that changed their order — a bar that rang longer than a plucked string at middle C, say — would change which instrument is smoothest and not whether the rings dominate.

That the long-ringing string stands for a tanpura-like instrument. The tanpura’s spectrum is sixteen harmonics; its ring here is the long-sustaining string’s law, borrowed rather than measured. The column that puts every ring on the tanpura’s spectrum is not affected, and it is the one the conclusion rests on.

That the pair is scored at the moment the later note begins. The earlier note goes on fading while the later one sounds, and the later one fades too. Scoring the overlap over the whole of the later note would shrink every weight, the far ones more, which is a small shift towards the steps.

And that notes are evenly spaced and nothing is damped. A saron player damps each bar with the free hand as the next is struck, which cuts the ring of the note before exactly when the pair would begin: under this measure a damped bar sounds no pair of notes at all, and whatever roughness its melody has in succession is the hall’s. A room does not decay evenly either, and a hall’s own colour on the note before is not modelled.

What a ring cannot show

That listeners hear a scale’s roughness in succession. The measure scores what sounds together, and a melody heard one note at a time is also heard in memory, where nothing decays at sixty decibels in a fixed time. The ring is the physical half of the question.

That traditions chose instruments to suit their scales, or scales to suit their instruments. The arithmetic says that a sustaining instrument makes the skips count and a quickly fading one makes the steps count, and that the commitment to an instrument lives there. Roughness cannot choose a scale is still the limit on what that can explain, and consonance is half learned is still the other half.

Still open: the gamelan’s bar, damped and undamped

The free bar here is an ideal one, and the spectrum that was supposed to explain the gamelan found that a bar’s partials do not recommend a slendro. A gamelan is played two ways on the same bars: the saron’s bars damped at every stroke, and the gendèr’s left to ring over their resonators and damped more selectively. With the ring in the measurement those are two different instruments with one spectrum, and the next computation is the slendro’s commitment between them — a ring cut at the next stroke, a ring left to run, and the resonator tubes that lengthen it — at the tempo levels a gamelan piece moves through. It would say whether a slendro’s standing depends on how it is struck more than on what it is struck on, and whether the damping practice is the one that makes the scale’s steps rough or the one that keeps them from sounding at all.