One Scale or One Shimmer

A Balinese gamelan instrument is built twice. Two sets of bronze, tuned a few hertz apart, struck as one voice, so every note arrives with a slow wave in it. The tuner sets that wave as a rate, so many beats per second, and holds it across the whole instrument. That single choice has a consequence nobody chose: the two halves can no longer carry the same scale. Not approximately. Exactly, and by an amount you can compute.

every number recomputed in your browser · one shared engine · research/ombak/verify-ombak.mjs

The word is ombak, wave. Strike a single bronze key and you get a tone. Strike the pair, one bar tuned slightly above its twin, and the two tones add and cancel and add again, so the loudness of the note pulses. The pulse rate is simply the difference between the two frequencies: two bars 8 Hz apart beat eight times a second, whatever pitch they are at. That is the physics, and it is the whole of the physics.

The interesting part is the arithmetic, and it starts with a question a tuner has to answer at every key on the instrument: how far apart?

1. Hear the pair

Fifteen keys, three octaves of a five-tone Balinese ladder. Press one. You will hear the lower bar alone, or the upper alone, or the two together, depending on the switch. The sound is synthesised, and the synthesis is described honestly further down: what matters here is that the beat you hear is the number the panel prints.

2. The tuner's choice, and what it sounds like

Switch the policy and press play all fifteen under each. Under a constant rate the wave stays the same speed from the bottom of the instrument to the top: eight beats a second down among the big low keys, eight beats a second up among the small bright ones. Under a constant interval the low keys sway lazily and the top keys buzz, because holding the gap at a fixed number of cents means the gap in hertz grows with the pitch. At the top of these three octaves the constant-interval pair beats at …, which is no longer a shimmer at all. It is roughness.

So the tuner's choice is easy, and Balinese tuners made it a long time ago: they set a rate. The shimmer is the sound of the instrument, and an instrument whose sound changes character between its bottom and its top is not one instrument.

3. What the choice costs

Here is the thing that is not obvious. Ask what the same detuning looks like as an interval. Eight hertz at 125 Hz is a gap of …, which is most of a semitone. Eight hertz at 2,000 Hz is a gap of …, which is a twentieth of one. Constant in hertz means steeply shrinking in cents, and cents are what a scale is made of.

Now take any two keys, low and high. Call the lower bar's frequencies u₁ and u₂, and let the upper bar sit d₁ and d₂ hertz above each. The interval each member of the pair spans between those two keys is:

span(pengumbang) = 1200 · log₂(u₂ / u₁)
span(pengisep)   = 1200 · log₂((u₂+d₂) / (u₁+d₁))

Subtract, and the logarithms collapse into something short and exact:

span(umbang) − span(isep) = 1200 · log₂(1 + d₁/u₁) − 1200 · log₂(1 + d₂/u₂)

Read the right-hand side out loud. It is the ombak at the low key, expressed as an interval, minus the ombak at the high key, expressed as an interval. So the two facts are one fact: the amount by which the lower instrument's scale outruns the upper one's is exactly the amount by which the shimmer narrows as you go up. A tuner who keeps the wave steady is, by that act and without any further decision, stretching one instrument relative to the other.

Two corollaries follow immediately, and the page checks both over ten thousand random pairs rather than asking you to take them:

• Hold the rate (d₁ = d₂ = d > 0) and, because d/u₁ > d/u₂ whenever u₁ < u₂, the pengumbang's span is strictly wider. For every interval on the instrument, not only the octave. The two halves cannot carry the same scale, and they cannot both hold a 2:1 octave.
• Hold the interval (d₁/u₁ = d₂/u₂) and the difference is exactly zero. This is the only policy under which the pair shares one scale. It is also the one that buzzes at the top.

That is the trade in the title. One scale, or one shimmer. There is no third option, because the two conditions are complementary halves of the same equation.

4. Try to escape it

Set the two detunings yourself. Give both members an exact 2:1 octave and watch what the beat rate has to do; give both keys the same beat rate and watch what the octaves have to do. The panel recomputes the identity from your numbers, both ways, and prints the difference between the two ways of computing it, which is the residual left by floating point and nothing else.

5. Measure it, rather than believe it

A page that computes a beat rate and then plays a sound has told you nothing about whether the sound has that beat rate in it. So the meter below takes the audio buffer this page just built, forgets everything it knows about how it was made, and measures the beat rate back out of the samples two independent ways:

• Resolve and subtract. Find the two partials in a high resolution spectrum and take the difference. Two tones d hertz apart need a window of about 4/d seconds to come apart under a Hann taper, so the meter reports the rate its window can actually support and says so when it cannot.
• The envelope. Take the analytic-signal envelope, work in log space where the decay of a struck note is a straight line, subtract that line, and ask the envelope for its own dominant frequency. This method never looks at the carrier at all.

If the two disagree by more than the stated tolerance the meter reports a disagreement rather than an average, because two methods that disagree have not measured anything. Feed it white noise, or a single bar with no partner, and it refuses.

There is one more thing the meter can see, and it is the reason a pair of bronze bars is not the same object as a pair of sine waves. A struck bar rings in several modes at once, and for an ideal free bar those modes sit at 1, 2.76, 5.40 and 8.93 times the fundamental, the squares of the roots of cos x · cosh x = 1. If the two bars of a pair are the same shape at two slightly different sizes, then every mode is scaled by the same factor, so mode j of the pair beats at p_j times the fundamental's rate. An 8 Hz pair beats at 22 Hz in its second mode and 43 Hz in its third. The shimmer is not one wave. It is a stack of them, and the higher ones are already inside the range a listener hears as roughness rather than as pulsing.

…

6. What Balinese tuners actually do

Everything above is arithmetic, and arithmetic cannot tell you what anybody does. So here is the published record, quoted rather than paraphrased, because the exact wording is what carries the claim.

The principle, from the article that assembled Andrew Toth's measurements of 54 Balinese gamelan:

The rate is not random or arbitrary, but is deliberately fixed and consistently applied. A pande or tuner establishes a desired ombak rate, typically about 8 Hz for a gong kebyar, and then keeps the ombak constant throughout the orchestra’s complete multi-octave range. This rate—the distance between paired tones—is called penyorog by Balinese smiths and musicians. Wayne Vitale and William Sethares, Analytical Approaches to World Music 9.2 (2021), p. 6

And the consequence, on the very next page, together with the older statement it credits:

As a result of this “constant ombak” principle, whereby the lowest pair of jegogan tones and the highest pair of kantilan tones should beat at the same rate, unisons and octaves enter into a fascinating interdependence, where exact 2:1 octaves are impossible to achieve on both members of a pair. As Toth (1980) noted, “if a constant beat rate is desired throughout the range of a set, one soon realizes that not all the octaves on the low instruments and on the high instruments can be tuned to a perfect 1200 cents.” Vitale and Sethares 2021, p. 7, quoting Andrew Toth 1980

So the impossibility is not ours. It has been in print since 1980, and Michael Tenzer stated the same either-or in 2000, twenty years before the analysis that formalised it. What this page adds is the exact size of the thing, the fact that it applies to every interval rather than only to octaves, and the sharp boundary condition at the end of this section.

How fast, and how independently do we know it

The rate is not a single number. The Toth archive's own figure caption gives the spread, and an acoustics group in Japan measured about 90 sets over many years without citing Toth, Vitale or Sethares anywhere in its reference list, so the two figures below are genuinely independent of each other:

The ombak (beat rate) of all gamelan in the data set lies between 6 and 10 Hz, and individual gamelans maintain remarkably consistent rates internally, as shown by the error bars. Vitale and Sethares 2021, Figure 2 caption, p. 7

Approximately 90 sets of Gamelan, mainly Gamelan Gong Kebyar commonly used in Bali, Indonesia, were measured and analyzed in Japan and Bali. As a result, it has definitely been shown that their interference beat frequencies were tuned between 5 Hz and 10 Hz and their pitches and intervals were different depending on the regions and periods in Bali. Hiroyoshi Shiokawa, Hideharu Umeda, Koichi Minagawa and I Made Kartawan, Report of the Research Institute of Industrial Technology, Nihon University 100 (2016), abstract

Balinese has its own words for these speeds, and they are finer grained than any figure in the analytical literature. I Made Kartawan, a Balinese tuner and scholar, glosses six of them: pengayun 3 to 4 Hz, pengumbang lambat 5 to 7, pengumbang sedeng about 7 to 8, pengumbang bulus 9 to 10, pengejer 11 to 15, and pengetor, the fastest, 16 to 20 Hz. That last figure is worth holding next to the constant-interval experiment in section 2, whose top key beats at …: the vocabulary of a tradition that has been counting these beats for centuries runs out well before you get there.

One real gamelan, and the identity closing on it

Below is a single scale degree, ding, across all five octaves of the gamelan of Peliatan (Gunung Sari), measured by Andrew Toth in the mid-1970s and printed in Figure 7 of the 2021 article. Ten numbers, quoted. Every other column is computed here, in your browser, by the same engine that made the sounds above.

…

Now the four octave steps that column contains, and the whole argument on real bronze. The last two columns are the same number computed two completely different ways: once from the two instruments' octave sizes, and once from how much the shimmer narrowed.

…

The footnote, and where its boundary actually is

The 2021 article states the always-wider rule in a footnote, and is careful about what kind of statement it is making:

One principle that can be seen visually is that the octaves of the pengumbang (blue lines) are always wider than the octaves of the pengisep (orange lines). If the pengisep octave is stretched, the pengumbang octave is even more stretched; if the pengumbang octave is compressed, the pengisep octave is more compressed. And while it is common for the pengumbang octave to be stretched while the pengisep octave is compressed, the opposite (with the pengumbang octave compressed and the pengisep stretched) is not possible. Vitale and Sethares 2021, p. 19, note 17

Note the words seen visually, and the blue and orange lines: that footnote is describing five idealised tempering strategies drawn in a figure, not a finding about the measurements. Read as a claim about instruments it would need a boundary, and the identity supplies one. The pengumbang's span exceeds the pengisep's exactly when d₁/u₁ > d₂/u₂, so across an octave, where u₂ = 2u₁, the rule holds exactly when

the beat rate less than doubles from one octave to the next.

That is the whole condition, and it is sharp: at exactly double, the two octaves are the same size; past double, the footnote's impossible case happens. It is impossible only in the sense that no tuner would do it, because doubling the rate every octave is the constant-interval policy you can hear buzzing in section 2. In the Peliatan column above the rate rises across every octave, which is already against the idealisation, and it still never gets close: the largest step reaches … of the doubling that would flip the sign.

What we did not do, and why

The article ships supplemental spreadsheets for all 54 gamelan. They are freely downloadable and analysing them is the obvious move, and this page does not do it, because the journal's terms say:

no portion of the article or any of its accompanying media may be modified, transformed, built upon, sampled, remixed, or separated from the rest of the article. Analytical Approaches to World Music, terms of use, printed at the foot of the article (p. 35) and at aawmjournal.com/guidelines.htm

Building upon is what an analysis is. So the page quotes a published figure, with citation, the way any paper would, and leaves the dataset alone. The terms name a route, which is prior written permission from the authors and notice to the editors, and the request for it is filed in this repository at oversight/requests/. Until it is answered, treat the absence of a 54-gamelan analysis here as a licence boundary and not as an oversight.

Three things to be careful about

• Constant is the target, not the outcome. Kartawan, measuring nine sets of gender wayang, is blunt about it: The measurements in the figure make it clear that gender wayang do not have perfect ombak in all note pairs. Perfect ombak is an ideal that is difficult or impossible to achieve. This means that some amount of variation is acceptable. The Peliatan column above shows the same thing: its rate is not flat, it runs from 7.3 to 9.7 Hz. What is flat, to a very good approximation, is the rate compared with the pitch, which moves by a factor of 17 across the same five octaves.
• Nobody in the literature proposes the constant-interval alternative. We looked. The contrast in section 2 is a foil we built to make the choice audible, not a live scholarly dispute, and it should not be read as one.
• One quotation here is second hand. Tenzer's 2000 statement of the same either-or, which is the earliest full version of it we found, reaches us through Kartawan's thesis, pp. 57 to 58, quoting Gamelan Gong Kebyar p. 32. We could not open the book: the Internet Archive copy is lending-only. So we cite it as quoted, and if you have the book on a shelf, page 32 is the place to check us.

7. Real bronze

Everything you have heard so far was synthesised. Here are four recordings of an actual Balinese gong kebyar, dedicated to the public domain, which your browser will fetch, decode and measure with the same engine that made those sounds. Press measure and watch what comes back.

Shorten the window and watch the reading stop rather than degrade. Two partials d hertz apart need roughly 4/d seconds of signal before a Hann-tapered spectrum can tell them apart, so at some point the meter says so instead of returning a number. That is the same limit a companion layer is entirely about, met here in the wild rather than in a demonstration built to show it.

What we think these are, and how sure we are

The measurement is a measurement: two strong partials, that far apart, in those files. The claim that they are a pengumbang and a pengisep is an inference, and here is the case for it. The separations sit at 8.76 to 8.82 Hz on two different instruments a fourth apart, which is where the published rate for a gong kebyar is. In one clip the two peaks are 3 to 1 in strength and in another take of the same key they are nearly equal, which is what two struck objects do and not what one resonance does. And in the pemadé clip the structure repeats one mode up: two peaks near three and a half times the fundamental, one belonging to each member.

Against it: we were not in the room. The description on the recordings says only which ensemble and which year. We cannot rule out that one key was struck and its neighbour rang. What would settle it is a recording made deliberately, one member at a time, and such a set exists. It is Mattie Rynkiewicz's Gamelan Gong Kebyar Sample Library on the Internet Archive, which labels every instrument ngumbang or ngisep by name. Its licence is CC BY-NC-SA, and the non-commercial clause is one we do not accept on this site, so we did not use it. A request to ask its author about a freer licence is filed with the others in oversight/requests/.

One thing the measurement says that the model did not

Section 5 predicted that if the two bars of a pair were the same shape at two sizes, every mode would be scaled by the same factor, so the second mode would beat at its own ratio times the fundamental's rate. In the one clip where both members' second modes are strong enough to resolve, that prediction is close and wrong in an interesting direction, and the page prints the numbers above. The two bars place their second mode at ratios that differ by about a quarter of a percent, and that small difference is enough to change the second mode's shimmer by several hertz. Two bars of a Balinese pair are not scale copies of one another: each one's upper partials were filed to their own place, by hand, and the shimmer they make together carries that.