Acoustics & Music Theory
why it sounds like that
Series

Beyond twelve — the series

24 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Maqam Rast, Arabic theory. Maqam Rast, Arabic theory: its degrees in cents above the tonic, drawn against the twelve equal steps of a keyboard. the third and the seventh sit halfway between major and minor — by convention, exactly halfway. Source: the quarter-tone convention fixed at the Cairo congress of 1932.

    A scale is not a set of pitches

    Two ragas can have identical pitch sets and be different ragas. Two national theories of one maqam put its third degree thirty-five cents apart. Both facts are fatal to the idea that a mode is a collection of notes, and both are ordinary in the traditions concerned.

    part 1 · scales
  2. Roughness across an octave. Sensory dissonance between two complex tones as the upper one is swept through 19.02 semitones, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit on 7/5, on 11/7, on 5/3, on 9/5, on 11/5, on 7/3, on 13/5, found by scanning the curve rather than by marking them. The wells are where this spectrum's partials coincide, and for a spectrum with no even partials they are not where an octave-based scale would put them.

    A scale without an octave, and the spectrum that asks for it

    Every scale so far repeats at the 2:1. That looks like a law and it is a consequence — the octave fuses because every partial of the upper note lands on one of the lower. Take a spectrum with no even partials and the 2:1 stops doing that, while the 3:1 starts.

    part 2 · scales
  3. Roughness across an octave. Sensory dissonance between two complex tones as the upper one is swept through an octave, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit on 4/3, on 3/2, on 5/3, found by scanning the curve rather than by marking them. And the tempered minor third sits 198 cents below the nearest well, and the tempered major third sits 98 cents below the nearest well, which is a real feature of this model and not a defect of the drawing.

    The third the model has no opinion about

    A neutral third — halfway between major and minor, and a scale degree in most of the music between Morocco and Iran — is nothing at all on an ordinary roughness curve. It becomes a well at 347 cents, which is exactly 11:9, only for a spectrum whose ninth and eleventh partials are at least three times their natural strength. No instrument has that spectrum.

    part 3 · intervals
  4. Three answers to how finely a pitch can be heard. Three resolutions across five octaves, on a logarithmic scale of cents. Two notes one after the other are told apart at 4.0 cents at A440 and 8.6 cents three octaves down. Whether a melodic interval is in tune is a judgement an order of magnitude coarser, 25 to 50 cents. And two notes held a fifth apart are heard to beat once every 2 seconds at 1.31 cents, which is finer than either. The horizontal lines are the step sizes of the equal divisions that have been built: 12 at 100.0 cents, 24 at 50.0 cents, 53 at 22.6 cents, 72 at 16.7 cents. Every one of them is coarser than discrimination and finer than melodic judgement.

    Three answers to how finely a pitch can be heard

    Two notes one after the other are told apart at about four cents at A440. Whether a melodic interval is in tune is a judgement an order of magnitude coarser. And two notes held together are heard to beat at a third of a cent, because the question is answered by counting rather than by hearing pitch at all. Every equal division ever built sits between the coarsest and the finest.

    part 4 · perception
  5. The roughness curve for ideal bar. The partials are at 1 : 2.756 : 5.404 : 8.933 : 13.340 times the fundamental, which is not a harmonic series, so nothing about where the wells fall can be read off the small whole numbers. Sensory dissonance between two complex tones as the upper one is swept through 13.00 semitones, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit 7 cents below 3/2, 14 cents below 5/3, 13 cents above 7/4, found by scanning the curve rather than by marking them. The wells are where this spectrum's partials coincide, and for a spectrum with no even partials they are not where an octave-based scale would put them.

    The spectrum that was supposed to explain the gamelan

    Run the model that designed the Bohlen–Pierce scale on a bar's partials and it asks for a compressed pseudo-octave at 1166 cents and wells at 694 and 870. A measured slendro's degrees are at 231, 474, 717 and 955, and only one of the four is near anything the model wants. Sweep the partials and a spectrum wanting a 240-cent step can be found — sitting sixty per cent of the way up its own roughness curve, which is what a scale-level test says about the whole comparison.

    part 5 · timbre
  6. Two ways to detune a pair, and they disagree by an octave. A pair of instruments tuned deliberately apart, drawn across 4 octaves. Holding the detuning at 10 cents gives a beat rate that rises from 0.64 to 10.20 beats a second — a factor of 16, one doubling per octave, because a fixed ratio is a growing number of hertz. Holding it at 3 beats a second instead gives a flat rate and an interval that shrinks from 46.6 cents at the bottom to 2.9 at the top. A tuner has to choose, the choice is audible across the range, and a table of cents can only write down the first of the two.

    A tuning is not a table of cents

    Two instruments of a pair are tuned deliberately apart, and the tuner has to choose between a fixed number of cents and a fixed number of beats — which differ by a factor of sixteen across four octaves and cannot both be held. A list of pitches records the first, cannot record the second, and cannot record at all that there are two instruments.

    part 6 · tuning
  7. 3 ways to fill the same fourth. Each row is a tetrachord: a span of 498 cents — a pure fourth — with two notes inside it, drawn in cents from the lower bound. The bounding notes never move, which is what makes the tetrachord a unit; everything that varies is interior. diatonic (ditonic) puts them at 90 and 294 cents, giving steps of 90, 204, 204; intense chromatic puts them at 81 and 231 cents, giving steps of 81, 151, 267; enharmonic puts them at 63 and 112 cents, giving steps of 63, 49, 386. Against the twelve equal steps below, 1 of 3 land within twenty-five cents of a semitone at every degree; the worst miss is 37 cents, which is a quarter of a semitone and has no name on a keyboard. Nothing in this construction is a subset of an equal division, so nothing the census here does applies to any of it.

    A scale built downward from a fourth

    A tetrachord is a bounded span with two free notes inside it, which is a different generator from a chain of fifths and a different object from a subset of an equal division. At the maqam tradition's own quarter-tone resolution the construction admits thirty-six fillings of the fourth and 1,296 octaves built from them; the census here can see six of the first and 2.8 per cent of the second, and one of the octaves it cannot see is an ordinary mode of a living tradition.

    part 7 · scales
  8. Two modes, one set, and every measure of a set that cannot tell them apart. Raga Bhupali and Raga Deshkar drawn as the moves each allows: an arrow from one degree to another means the tradition's ascent or descent goes that way. Raga Deshkar's ascent omits Re, so the two graphs differ by an edge. Below, every standard measure of a scale, evaluated on both — and they are identical in every row, which the drawing checks before it is made. An ascent and a descent that between them use all 5 degrees can be chosen in 63 ways, 62 of them asymmetric. That is how many modes collapse onto one pitch set under the simplest order model there is, and a census over subsets counts the set once.

    A degree is where it goes next

    The first essay on scales beyond twelve said a mode is not a set and listed four things a set cannot record. This one makes the sharpest of them arithmetic. Specify a mode as an ascent and a descent — which is how a living tradition specifies one — and it becomes a directed graph on its degrees; sixty-three such graphs collapse onto a single pentatonic set, sixty-two of them with an ascent that is not the descent reversed, and every standard measure of a scale returns the same value for all of them.

    part 8 · scales
  9. one measured slendro is the scale least committed to a spectrum. Each scale drawn, placed by how smooth it is against 2000 random scales of the same size, under 4 spectra. Low is smooth. Every one of them lands in the smoothest third under every spectrum, so the account that a spectrum chooses a scale survives as a statement about levels. What separates them is the length of each row's line, which is how much the answer moves when the instrument changes: one measured slendro spans 2.9 percentile points and Rast, Turkish theory spans 17.6, a factor of 6.0. The scale usually offered as the case for spectral matching is the one whose standing barely depends on the spectrum, and Rast, Turkish theory is the one that depends on it most.

    The scale least committed to its own instrument

    The essays on scales beyond twelve draw scales without spectra and spectra without scales, and none of them had put a tradition's own degrees through the roughness model. Doing it for six scales and five spectra says the account survives — every tradition lands between the eighth and the twenty-sixth percentile of random scales of its size, and under a pure tone not one of them does. But the scale whose standing moves least when the instrument changes is the gamelan's, at 2.9 percentile points, and the one that moves most is the five-limit diatonic at 17.4. The scale the argument is usually made about is the one it explains least.

    part 9 · scales
  10. Weighted by the pairs a melody sounds next to each other, every seven-note scale is rougher than most. Each scale placed by how smooth it is against 2000 random scales of the same size, under 4 spectra, with each pair of its degrees weighted by how often the plainest melody its ascent and descent permit sounds the two — here the pairs a melody sounds next to each other. Low is smooth. Raga Bhupali: 40.3 to 56.8, spread 16.5; Raga Deshkar: 25.1 to 37.5, spread 12.4; one measured slendro: 31.8 to 44.5, spread 12.7; Rast, Arabic theory: 92.0 to 98.6, spread 6.6; Rast, Turkish theory: 91.8 to 97.8, spread 6.0; the diatonic major, tempered: 90.7 to 97.0, spread 6.3; the diatonic major, five-limit just: 91.8 to 97.8, spread 6.0.

    The smoothness is in the skips

    Traditional scales come out smoother than random scales of their size because every pair of their degrees is counted once. Count only the pairs a melody on the scale actually sounds next to each other, and every seven-note scale here is rougher than ninety per cent of random ones, under every spectrum, and under a pure tone as well. The smoothness is carried by the pairs a melody reaches by skipping — its thirds and above all its fifths, which are smoother than all but two random scales in two thousand. Counted by their own ascents and descents, the two ragas that share one set of notes stand in different places at last: Deshkar's skipped Re buys a smoother ascent and costs it the fifths.

    part 10 · scales
  11. 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.

    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.

    part 11 · scales
  12. The tempo moves it further than the touch does. One measured slendro, scored among random scales of its size under a free bar, 4 s, at five tempi and under each touch. Left to ring it runs from 29 at 0.15 seconds a note to 83 at 2.4 — a span of 54 percentile points, where the two touches differ by at most 17. So the scale is smoother than most of its size when the music is fast and rougher than most when it is slow, and how the bar is damped is the smaller decision. The two touches converge at the slow end because a bar that has died before its successor is sounding against nothing whatever the player does.

    The tempo moves a scale further than the touch

    A gamelan is played two ways on the same bars: a saron's are damped as the next is struck and a gendèr's ring over their resonators. That decision moves a slendro's standing among random scales of its size by up to seventeen percentile points, which is real. Over the tempo levels a piece actually moves through it moves by fifty-four — from the twenty-ninth percentile at a fast elaboration to the eighty-third at a slow one. The same five pitches on the same bars are a smoother-than-average scale and a rougher-than-average one, and which depends on how fast they are played.

    part 12 · scales
  13. The scale's standing belongs to the ringing instrument. Where a measured slendro sits among random five-note scales, at three density ratios, scored three ways: the two instruments together, the fast ringing part on its own, and the slow damped part on its own. At one slow note to 2 fast ones the ensemble is at the 7th percentile, the ringing part alone at the 6th, and the damped part alone at the 95th; At one slow note to 4 fast ones the ensemble is at the 5th percentile, the ringing part alone at the 7th, and the damped part alone at the 95th; At one slow note to 8 fast ones the ensemble is at the 5th percentile, the ringing part alone at the 5th, and the damped part alone at the 95th. A low percentile is a scale smoother than most of its size. The damped part on its own is rougher than nineteen random scales in twenty, because it has almost no simultaneity for its intervals to be smooth in; the ensemble is at the ringing part's figure throughout. And the cross pairs, which are a quarter of the roughness, do not make the ensemble worse than either part alone.

    The scale belongs to the ringing instrument

    A gamelan plays two instruments on one scale at once — a saron damped at every stroke and a gendèr several times faster with its bars left to ring — and a quarter of the roughness a listener receives falls on pairs that cross between them, which no figure had computed. The cross pairs turn out to be no worse than either stream's own. What decides the scale's standing is the fast part: the ensemble sits at the sixth percentile among random five-note scales and so does the ringing instrument alone, while the damped one alone sits at the ninety-sixth.

    part 13 · scales
  14. A wrong bar costs the same whichever instrument it is on. What moving one degree of a measured slendro by 10 cents, flat or sharp, adds to the roughness per second of a two-instrument texture — a ringing part at 0.15 s a note over a damped one four times slower — on the ringing instrument and on the damped one. The ensemble in tune scores 7.47. Degree 1 (0¢): ringing 0.142 flat and 0.202 sharp, damped 0.163 and 0.185, of which beating 0.178; Degree 2 (231¢): ringing 0.119 flat and 0.068 sharp, damped 0.096 and 0.090, of which beating 0.093; Degree 3 (474¢): ringing 0.106 flat and 0.021 sharp, damped 0.069 and 0.057, of which beating 0.063; Degree 5 (717¢): ringing 0.079 flat and 0.072 sharp, damped 0.075 and 0.078, of which beating 0.077; Degree 6 (955¢): ringing 0.023 flat and 0.018 sharp, damped 0.019 and 0.022, of which beating 0.020. Over all ten errors the ringing instrument's cost 0.85 and the damped one's 0.85, and the wrong bar beating against the other instrument's right one comes to 0.86 on either — as much as the whole, because the intervals the error changes add as often as they save. One error costs about 1.1% of the texture's roughness.

    A wrong bar beats the same on either instrument

    The ringing instrument carries a gamelan scale's standing, so a tuning error on it was predicted to cost more than the same error on the damped instrument. Put in the beating the model lacked, and the prediction fails: moved ten cents, a degree costs the same 0.85 summed over the scale on either instrument, because almost all of the cost is the wrong bar beating against the right one on the other instrument, and a beat belongs to both of the bars that make it. Moving a whole instrument costs exactly what its five degrees cost separately, and it takes nothing from the scale's standing.

    part 14 · scales
  15. The bar a tuner can least afford to leave wrong is the one the skeleton lands on. What a 10-cent error on each degree of a measured slendro adds to the roughness per second of a two-instrument texture over 128 bars, averaged over both instruments and both directions, with the elaborating part meeting the skeleton on the last stroke of every four-stroke unit. Top row: the skeleton's degrees drawn at random. Each row below: the skeleton ending every unit on one degree. Random: 1 0.219, 2 0.054, 3 0.083, 5 0.095, 6 0.104. Ending on 1: 1 0.348, 2 0.028, 3 0.049, 5 0.062, 6 0.068. Ending on 2: 1 0.171, 2 0.228, 3 0.049, 5 0.064, 6 0.067. Ending on 3: 1 0.170, 2 0.028, 3 0.247, 5 0.064, 6 0.068. Ending on 5: 1 0.170, 2 0.028, 3 0.049, 5 0.285, 6 0.069. Ending on 6: 1 0.171, 2 0.027, 3 0.049, 5 0.064, 6 0.305. The tonic's error moves its octave too, so it is two bars. In every row the degree the units end on costs the most, three and a half to five times what the dearest other single bar costs.

    The bar a gamelan can least afford wrong is the one its skeleton lands on

    A wrong bar in a gamelan costs what its beating against the other instrument costs, and that depends on how often the music doubles the degree. The obvious place for doubling is where an elaborating part meets the skeleton at the end of each four-stroke unit, so the prediction was that the meeting rule would rank the degrees. It barely does: a part left to ring doubles a degree whenever it sounds it while the other instrument's bar still sounds, and the meetings add three per cent. What ranks the degrees is where the skeleton lands. A skeleton that ends its units on one degree makes that bar three and a half to five times as costly to leave wrong as any other single bar.

    part 15 · scales
  16. An elaboration that arrives early makes all three goal tones the dearest bars. What a 10-cent error on each degree costs, for a skeleton whose four-stroke units end on 6, 2 or 3, with the elaborating part meeting the goal on the unit's last stroke and arriving on it 0, 3 or 7 of its own strokes earlier. 0 early: 1 0.171, 2 0.097, 3 0.082, 5 0.064, 6 0.161. 3 early: 1 0.139, 2 0.115, 3 0.106, 5 0.056, 6 0.221. 7 early: 1 0.100, 2 0.159, 3 0.136, 5 0.043, 6 0.295. Meeting on the stroke, the goal on 3 costs no more than degree 3 does with the skeleton at random; arriving seven strokes early, the cheapest goal costs 3.2 times the dearest single bar that is not a goal.

    An elaboration that arrives early makes every goal tone the bar to tune first

    A gamelan's bars are costliest to leave out of tune where the skeleton lands, but a mode with three goal tones spread the cost so thinly that a low goal could end up cheaper than a high bar that is never a goal. The rule tested there put the elaborating part on the goal only at the goal stroke. Elaborations in the style reach their goal early and stay on it. Let them arrive seven strokes early and every goal tone costs more than three times the dearest bar that is not a goal; the goal's own cost rises by about half for every few strokes of anticipation, and a whole instrument out of tune becomes dearer too.

    part 16 · scales
  17. A gamelan's goal-tone beating is a clock hand only if each goal beats at its own rate. Bits of position unknown over the first cycle of a sixteen-step cycle of gong, kempul and kenong, against the cycle's duration, when the elaborating parts arrive early on each goal tone — kenong and gong on degrees 2 3 2 6 — and a pair of instruments tuned apart beats on it for a step before the stroke: with the pairs tuned together (no swell), tuned apart at one beat rate for every degree, tuned apart by 25 cents so that each degree beats at its own rate (told apart when the rates differ by 15 per cent), and with every swell told apart. tuned together: no swell: 40 s 0.112, 60 s 0.203, 80 s 0.329, 120 s 0.417, 160 s 0.646, 240 s 0.870. tuned apart at one beat rate: 40 s 0.104, 60 s 0.192, 80 s 0.301, 120 s 0.376, 160 s 0.404, 240 s 0.576. tuned apart by 25 cents: 40 s 0.076, 60 s 0.114, 80 s 0.125, 120 s 0.137, 160 s 0.146, 240 s 0.285. every swell told apart: 40 s 0.000, 60 s 0.000, 80 s 0.000, 120 s 0.000, 160 s 0.003, 240 s 0.132.

    A gamelan's beating names its goal only if each goal beats at its own rate

    When a gamelan's elaborating parts reach a goal tone early, a pair of instruments tuned slightly apart beats on that degree for the strokes before the goal — a swell of shimmer just before each structural stroke, which is exactly where a recognisable event does a slow cycle's clock most good. Whether it does that good depends on whether one goal's swell can be told from another's. Tune every pair to the same beat rate, the practice that keeps a gamelan's shimmer even, and every swell is the same event: it buys a third of what it could. Tune every pair the same number of cents apart and each degree beats at its own rate, fifteen per cent faster than the degree below, and the swells start naming which goal is coming.

    part 17 · scales
  18. Naming the goals costs no roughness: it saves a little. Pairs tuned apart so that each slendro degree's pair beats at a rate proportional to its pitch raised to a power — nought for one rate on every degree, one for a constant interval, more than one for a rate rising faster than the pitch — with the average rate held at 5.01 beats a second. Line: the roughness the detuned pairs add to the goal-tone passage, as a share of what one rate adds — 0 100.0%, 0.5 99.7%, 0.75 99.4%, 1 99.0%, 1.25 98.5%, 1.5 98.1%, 2 97.3%, 2.5 96.4%, 3 95.6%. Steps: the bits of position the swells leave unknown at 160 seconds a cycle, rates told apart at 15 per cent — 0 0.404 (AAAA), 0.5 0.404 (AAAA), 0.75 0.226 (AAAB), 1 0.146 (ABAC), 1.25 0.146 (ABAC), 1.5 0.146 (ABAC), 2 0.146 (ABAC), 2.5 0.146 (ABAC), 3 0.146 (ABAC). Every steeper rise costs less roughness, and from a constant interval upward the swells name three different goals.

    A gamelan tuner can name the goals for nothing

    A pair tuned so that each goal beats at its own rate lets the swells before a gamelan's structural strokes say which goal is coming, and the question left was what that costs in roughness. Held at the same average shimmer, nothing: a constant interval adds one per cent less roughness than an even beat rate, and a rate rising as the square of the pitch adds three per cent less while naming the goals for a far coarser ear. The even shimmer puts its widest pairs on the low degrees, where a cent of detuning costs most.

    part 18 · scales
  19. 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.

    part 19 · scales
  20. 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.

    part 20 · scales
  21. 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.

    part 21 · scales
  22. 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.

    part 22 · scales
  23. 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.

    part 23 · scales
  24. 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.

    part 24 · scales

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