Acoustics & Music Theory
why it sounds like that
Concept

Gamelan — where it appears

An ensemble of tuned gongs, metallophones and drums from Java and Bali, playing layered parts around a slow gong cycle. Its paired instruments are tuned slightly apart on purpose, so that every struck degree shimmers with a beat of its own.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

What a chorus costs on one bridge. The sixty-decibel time of the longest-lived mode of two coupled strings at 262 hertz, against how far apart they are tuned. Below the bifurcation at 4.5 cents the pair splits its decays and one mode rings on; above it the pair splits its frequencies instead, both modes carry the bridge's loss, and the sustain sits at 2.8 seconds however much further the tuning is opened. The flat line is what the same two strings would do if they did not share a bridge. A chorused sound and a long one cannot be had from one bridge, and the first four and a half cents cost all of it.

The pair tuned apart on purpose

A piano's unison buys its sustain with a detuning of two and a half cents and loses it past four and a half. Read from the other side that is a design constraint on every instrument that wants the beating instead: past the bifurcation the sustain is gone and no further detuning costs anything more, which is why every chorused voice in the world — the celeste rank, the musette reeds, the paired gamelan — is built out of sources that do not share a bridge.

tuning · Beating Carried across three octaves, a rise as the pitch to the power 2 leaves 4 pairs no longer beating. Every degree's pair across three octaves of slendro, the geometric mean beat rate held at 5.0 a second, when the rate rises as the pitch to the power 2 across the whole compass, and when the middle octave's rates are repeated in every octave. Carried on: low octave 0.7, 0.9, 1.2, 1.7, 2.2; middle octave 2.9, 3.8, 5.0, 6.6, 8.7; high octave 11.6, 15.1, 20.0, 26.5, 34.9, 4 of fifteen pairs above fifteen a second. Repeated: every octave 2.9, 3.8, 5.0, 6.6, 8.7. The fifteen-hertz line is where these essays stop calling a fluctuation a beat.

A gamelan's shimmer has to start again at every octave

Within one octave, a gamelan tuner who lets each pair's beat rate rise faster than its pitch names the goal tones and pays nothing in roughness for it. Carried across a real instrument's three octaves, the same rise breaks twice. The steepest rise that keeps every pair beating falls from a power of four to a power of one, and the roughness saving turns into a cost: a rise as the square of the pitch is seven per cent rougher than one rate everywhere. A tuning that gives each degree its middle-octave rate in every octave keeps both the clock and the saving at any compass. The beat rate then names the degree, not the pitch.

scales · Beyond twelve To a listener who fuses a degree's swells, a rise carried across the compass is worth nothing to the clock. Bits of position left unknown just before each goal in a sixteen-step cycle of 160 seconds, goals on degrees 1, 2, 1 and 4 of slendro, each preceded by its degree's swell of beating struck in three octaves at once, against the power the pair's beat rate rises as. fuses agreeing swells, rise carried on: 0 0.40, 0.25 0.40, 0.5 0.40, 0.75 0.40, 1 0.40, 1.5 0.40, 2 0.40, 2.5 0.40, 3 0.40; fuses agreeing swells, rates repeated: 0 0.40, 0.25 0.40, 0.5 0.40, 0.75 0.23, 1 0.15, 1.5 0.15, 2 0.15, 2.5 0.15, 3 0.15; attends to one octave, either family: 0 0.40, 0.25 0.40, 0.5 0.40, 0.75 0.23, 1 0.15, 1.5 0.15, 2 0.15, 2.5 0.15, 3 0.15. The skeleton alone leaves 0.65 bits; swells that all sound alike leave 0.40. A listener who attends to one octave hears both families identically; a listener who fuses a degree's three swells only when they agree finds, under the rise carried on, that no degree's swells agree.

A degree's three swells are one landmark only if they agree

A gamelan's goal tone is struck in several octaves at once, so the swell of beating that warns of it is three swells. A tuning whose beat rate rises with pitch across the whole compass gives each octave its own rate, and a listener who hears a degree's swells as one event only when their rates agree finds that, at any useful rise, none of them do: every goal becomes a swell of no particular rate and the clock is left exactly where swells that all sound alike leave it. No rise can both name the goals and let their swells agree, since naming needs ten times the power agreeing allows. Repeating the middle octave's rates in every octave does both.

scales · Beyond twelve The tuning that saves roughness rises only downward, and keeps the clock only if the low octave leaves the goals alone. Five ways of setting the beat rates of a gamelan's paired instruments over three octaves, the rate rising as the pitch to the power 2 where it rises: the roughness the detuned pairs add to the passage, against the bits of position a listener who fuses a goal's swells has left before each goal — filled with the goals struck in all three octaves, open with them struck only in the middle and high octaves. middle octave's rates repeated: roughness 3.06, bits 0.15 and 0.15; rise carried on: roughness 3.37, bits 0.40 and 0.40; repeated at the goals, rising elsewhere: roughness 3.42, bits 0.15 and 0.15; rising below the middle, repeated above: roughness 2.45, bits 0.40 and 0.15; the same, goals repeated in every octave: roughness 2.96, bits 0.15 and 0.15.

A gamelan's shimmer should rise only downward

A gamelan tuner who lets each pair's beat rate follow its pitch saves roughness in the low octave, where the pairs slow, and pays it in the high octave, where they race. Repeating the middle octave's rates only at the goal tones and letting the rest follow the pitch, as the lead proposed, is rougher than repeating everywhere. The tuning that saves is the one that lets the rate follow the pitch below the middle octave and repeats it above — a fifth less roughness at the square — and it keeps the clock only if the low octave does not double the goals.

scales · Beyond twelve Held to the shimmer it promised, the downward tuning is rougher than repeating the middle octave's rates. A gamelan's paired instruments over three octaves, the rate rising as the pitch to the power 2 where it rises: the roughness the detuned pairs add to the passage for each tuning at the average shimmer it actually has (open) and with every rate rescaled so the average is the one the families promised (filled). middle octave's rates repeated: average 100% of the promised, roughness 3.06, held 3.06; rise carried on: average 100% of the promised, roughness 3.37, held 3.37; repeated at the goals, rising elsewhere: average 100% of the promised, roughness 3.42, held 3.42; rising below the middle, repeated above: average 63% of the promised, roughness 2.45, held 3.26; the same, goals repeated in every octave: average 83% of the promised, roughness 2.96, held 3.32; repeated, slowed to the same average: average 63% of the promised, roughness 2.27, held 3.06.

A gamelan's shimmer has no cheaper shape, only a slower one

The tuning that let each pair's beat rate follow its pitch below the middle octave and repeated it above saved a fifth of the roughness — and slowed the ensemble's average shimmer by more than a third while doing it. Held to the shimmer it promised, it is rougher than repeating the middle octave's rates everywhere, and the middle octave's rates simply slowed to the same average are smoother than either. With the average fixed, the cheapest shape runs the other way, and a knee is never cheaper than a step.

scales · Beyond twelve Slowing every pair by one factor smooths the shimmer steadily and leaves the goal-tone clock alone until a fifth of the promised speed. A gamelan's paired instruments with the middle octave's rates repeated in every octave, the rate rising as the square of the pitch across the degrees, and every rate multiplied by the factor on the axis; a cycle of 160 seconds, sixteen skeleton steps of 10.0 seconds, with a swell of beating in the step before each of four goal strokes. The roughness the pairs add, as a share of the promised shimmer's 3.06 (the falling line), and the share of the swells' value to the clock that survives (the other line): the skeleton alone leaves 0.646 bits of position unknown and the swells at the promised shimmer 0.146. 0.1: clock 49%, roughness 15%; 0.12: clock 84%, roughness 18%; 0.14: clock 84%, roughness 21%; 0.16: clock 84%, roughness 24%; 0.18: clock 84%, roughness 27%; 0.2: clock 84%, roughness 29%; 0.21: clock 100%, roughness 31%; 0.22: clock 100%, roughness 32%; 0.25: clock 100%, roughness 36%; 0.3: clock 100%, roughness 42%; 0.35: clock 100%, roughness 48%; 0.4: clock 100%, roughness 53%; 0.5: clock 100%, roughness 63%; 0.6: clock 100%, roughness 72%; 0.7: clock 100%, roughness 80%; 0.8: clock 100%, roughness 87%; 0.9: clock 100%, roughness 94%; 1: clock 100%, roughness 100%; 1.1: clock 100%, roughness 105%; 1.2: clock 100%, roughness 110%. The clock first loses at 0.20; the floor is 0.21.

A gamelan's shimmer can slow to a fifth before its clock notices

Roughness falls steadily as every pair's beating is slowed by one factor, and the listener who times a goal's swell does not object until the shimmer is a fifth of the promised speed. The limit is not how many beats a swell holds — a ten-second step holds eight even there — but the once-a-second boundary below which a beat is waited out rather than counted, and it bites when the second-slowest goal degree crosses it. At the quick tempi where a swell could hold too few beats, the skeleton already carries the clock alone.

scales · Beyond twelve A goal degree that returns is better marked once than twice, and best told apart. A cycle of 160 seconds, sixteen skeleton steps, a swell of beating in the step before each of four goal strokes; two of the four goals share a slendro degree and so a beat rate, at every pair of positions. Bits of position left unknown with both of that degree's swells, with the first or the second silenced, with neither, and with the two told apart by some other mark. steps 3 and 7: both 0.034, first silenced 0.049, second 0.062, neither 0.143, told apart 0.003; steps 3 and 11: both 0.146, first silenced 0.049, second 0.082, neither 0.128, told apart 0.003; steps 3 and 15: both 0.046, first silenced 0.049, second 0.067, neither 0.126, told apart 0.003; steps 7 and 11: both 0.066, first silenced 0.062, second 0.082, neither 0.202, told apart 0.003; steps 7 and 15: both 0.101, first silenced 0.062, second 0.067, neither 0.127, told apart 0.003; steps 11 and 15: both 0.077, first silenced 0.082, second 0.067, neither 0.188, told apart 0.003.

A goal tone that returns should swell only once

When two of a gamelan cycle's four goal strokes land on the same degree, their swells of beating sound alike, and a listener who hears one cannot tell which goal is coming. Across all 625 ways five degrees can fill four goals, 365 are served better by an elaboration that anticipates fewer goals than four. The repeated pair costs most at slow cycles, can cost more than every swell is worth at moderate ones, and is spared only when its first swell comes straight after the gong.

scales · Beyond twelve

Named alongside it

The objects these essays reach for when they reach for this one.

BeatingSlendroRoughnessOctave equivalenceTempoChorus effectCoupled oscillatorCycleDetuningNormal modeSustainTuning by ear

All concepts