Ground Truth · measured, not assumed

Observed, and Assumed

In 1885 Alexander Ellis printed a Javanese gamelan scale twice on one page: once as he had measured it, once as he supposed it ought to be. Eight pages later the supposition had acquired the word properly. Here is a real gamelan, key by key, measured in your browser, and twenty-eight more from the survey that settled the question and was not read.

Ellis is the man who gave us the cent, the hundredth of a semitone that every sentence below is written in. He invented it because he wanted to write down scales that European notation could not hold, and in 1885 he published the results as an appendix to his translation of Helmholtz. Page 518 carries a table of scales from around the world. Two of its rows are these.

Page 518, rows 94 and 95

Nothing sounds until you press a button.

Row 94 is a real instrument. Its five steps are cents, a spread of cents between the widest and the narrowest. Row 95 is five steps of 240. Ellis labelled the second one assumed, which is an honest word, and he was right that it is close.

The word that changed

Turn eight pages. In the same appendix, in Ellis's own commentary, the assumed line comes back, and it is no longer assumed.

Ellis is not the villain of this. He is making a good point and a perceptive one: 240 cents falls between a European whole tone and a European minor third, so an ear trained on the piano hears it as one or the other and writes the scale down wrong. He is defending the gamelan against a European mishearing. But in the sentence where he does it, the model he built to make the defence has quietly replaced the measurement he made, and it has taken the word properly with it.

That is the mechanism this page is about. Not a lie and not an error: a rounding, made for a reason, that outlived the reason. The reference works still carry it. They are careful to say the tunings vary, and then they say the five pitches are "roughly equally spaced within the octave" and stop. Nowhere does the popular record say how roughly. That number is the entire content of the question, so here it is, three ways.

One. Measure a gamelan yourself

Below are fifty-four keys of a real Javanese gamelan, the double set owned by Casa da Música in Porto: the five slendro degrees of each of nine instruments, and each instrument's tone 1 an octave up, which is how the octave gets measured further down. One strike per key, recorded and published by the hall's own musicians under an open licence. Press a key. Your browser fetches the recording, reads the fundamental out of its spectrum, and tells you what it found. Nothing here is a stored answer: the number appears because a Fourier transform ran on the sound you just heard.

The instrument

Press a key.

Against five-equal plays each measured tone and then the tone a five-equal slendro would have put there. Where they are close you hear one note struck twice. Where they are not, you hear a wince.

Doing that for the forty-five scale keys gives this. The five slendro degrees are numbered 1, 2, 3, 5 and 6, which is not an error: Javanese cipher notation numbers slendro's five tones out of the seven positions pelog uses, and 4 and 7 have no slendro key.

Nine instruments, one gamelan, measured here

Every step in that table was computed from a shipped recording by public/_kit/barpitch.js, which is the same file your browser just ran and the same file the verifier runs in Node. If the two ever disagreed, the build would fail.

Is that departure real, or is it the ruler?

A gamelan that measures 253 cents where five-equal wants 240 has either been tuned that way or been measured badly. The nine instruments settle it, because they were tuned to match each other and measured independently. If the departure were noise, they would fall on either side of 240 at random.

The sign test

Nine instruments, spanning 130 Hz to 1831 Hz, made of thick bronze, thin bronze, wood and cast pots. On the first step all nine sit above 240; on the second all nine sit below. Under a fair coin that is a one-in-256 event, twice. The fourth step is the one where the sign test does not reach significance, and it is reported as such.

So the shape is the gamelan's. Its first step is wide, its second narrow, its third wide, its fourth narrow, and nine instruments made of four different materials agree about it. What they do not agree with is the grid.

Two. Twenty-eight named gamelans

In 1972 a team at Gadjah Mada University in Yogyakarta put an electronic frequency counter to the celebrated gamelans of Yogyakarta and Surakarta and published the numbers. Their stated resolution was that "frequency differences of about 0.1 percent were easily detected", which is under two cents. This is the measurement that answers the question, and it has been sitting in a 1972 Indonesian university press book that almost nobody outside the field has read. Twenty-eight of its slendro gamelans are here.

Where the tones actually fall

Two things are true about that picture at once, and they pull in opposite directions.

The assumption is an excellent average. Across the internal steps of those twenty-eight gamelans the mean step is cents. Ellis's 240 was not a guess that missed. As a description of slendro-in-general it is very nearly exact, which is presumably why it stuck.

It is nobody's instrument. Of the twenty-eight, the number whose every internal step falls within five cents of 240 is . Within ten cents, . The average is real and no gamelan is the average.

Three. How unequal is that, exactly?

A number alone is hard to feel, so here is a yardstick from a tradition that argued about tuning in public for two centuries. European keyboard music before equal temperament was played on well temperaments: deliberate, published, fought-over schemes in which the twelve semitones are not equal. Werckmeister III, Vallotti, Kirnberger, Young. Nobody has ever called any of them twelve equal steps.

Ask both traditions the same question. Take one tuning; measure how far its steps sit from the equal-step ideal of its own system, as a root-mean-square in cents; then divide by that system's own step, so that 240-cent steps and 100-cent steps can be compared without cheating.

Departure from equal, as a fraction of the system's own step

The best-measured gamelan corpus we have departs from five-equal by per cent of a step. The European well temperaments depart from twelve-equal by per cent. Slendro is the more equal of the two.

Which is the sentence this page exists for. The claim "slendro is five equal steps" is false, and it is false in a specific and small way: by about the amount that Werckmeister III is not twelve equal steps. We do not describe a well-tempered harpsichord as an equal-tempered one. We have described slendro that way for a hundred and forty years, and the difference is not in the data.

Two honest qualifications. First, in cents, which is the unit an ear works in, the gamelan departure is the larger of the two, because its steps are larger: cents against . Second, a temperament is a specification, a set of numbers a theorist wrote down, carrying no measurement error and no drift; a gamelan tuning is a measurement of an object that has been standing in a pavilion. That asymmetry favours the temperaments, and the gamelan still comes out more equal.

The four scatters, in order

There is one more way to ask whether the departure is real, and it is the one that convinced me. Put every kind of disagreement in this study on the same footing, as root-mean-square cents on step sizes, and sort them.

How much do things disagree?

Read that from the top. The instruments inside one gamelan disagree with each other by about as much as different gamelans disagree with each other. Two research teams reading the same named gamelan disagree by slightly more than either. And the departure from five equal steps is larger than all three.

The folklore says every gamelan is unique. On the best single survey, at this precision, it is not: the gamelans of the Javanese courts agree with one another more closely than they agree with the grid. Slendro is not chaos with an average. It is a shared scale, held to within about eight cents across twenty-eight instruments in two cities, and it is not equidistant. Those are different claims from the ones usually made in either direction.

The octave that is not 1200

Something else fell out, and it needed the data to be read slowly. A Scala scale file conventionally ends on the octave, and most of the compiled entries write that last degree as the exact ratio 2/1: the octave was supplied by whoever typed the entry, not measured. Every statistic above therefore runs only on the four internal steps, which that convention cannot have touched.

That is not a quirk of one archive. It is a documented habit of the field:

Like other early foreign investigators, Kunst likely did not measure all individual frequencies, but rather measured only a single octave span and then presumed that other octaves were tuned to 2:1 frequency ratios. Wayne Vitale and William Sethares, "Balinese Gamelan Tuning: The Toth Archives", Analytical Approaches to World Music 9.2 (2021), p. 8.

So we measured it. Every instrument in the Porto set ships its tone 1 an octave up, and Surjodiningrat's table gives the span of all five steps for each of the twenty-eight.

The octave, measured three ways

A gamelan key is a bar, and a bar's overtones are not whole-number multiples of its fundamental, so a tuner matching octaves by ear on bronze is not matching 2:1. This ground has a page on the same effect in a piano string, where it forces the stretched octaves every piano tuner knows about. The direction is the same and the size is larger.

What this is not

It is not a claim that Javanese and Balinese tuners are imprecise. It is close to the opposite: the sign test above says a gamelan's departure from the grid is reproducible across nine of its own instruments, which is what a deliberate tuning looks like and not what carelessness looks like.

Nor is it a claim about what anyone hears, and this is the qualification that matters most, because a serious scholar has made exactly the opposite point from the same kind of data:

the measured unequal intervals of the salendro tone system coexist with the ‘musical fact’ that, in practice, salendro is treated as an equidistant system. In a cognitive sense, the Sundanese salendro tone system is equidistant. Wim van Zanten, "Encounters in the Context of Inspiring Sundanese Music and Problematic Theories", in Recollecting Resonances (KITLV Press, open access). Van Zanten's fieldwork is Sundanese and his instruments are zithers, not Javanese court gamelan; the point is about what a tuning is for, and it carries.

That is the strongest form of the other side and this page does not touch it. Ellis's row 95 may be an excellent model of the percept and remain a poor description of the object. Nothing here measures perception. Whether a listener raised on slendro hears 253 and 227 as the same interval is a question for an experiment we did not run, and the answer may well be yes.

What this is: the observation that a measurement and its idealisation were printed eight pages apart by the same careful man in the same book, that the idealisation is the one that survived, that the survey which could have settled it in 1972 was published in Yogyakarta and largely not read, and that a hundred and forty years of measuring have not brought a single instrument up to the tidy line.

Show the check

What was verified, and how

Ellis, against Ellis, four ways. The two entries in our compilation that cite page 518 were compared with the page itself, from the full text of the third English edition (1895, a reprint of the second of August 1885) at archive.org item onsensationsofto00helmrich, and with a second, independent database's transcription of the same table.

Ellis had four measurements, not one. His fuller 1885 paper carries four salendro scales, no two alike. The Helmholtz appendix printed one of them, and then row 95.

The estimator checking itself. Two functions run on every key: one takes the loudest peak in the spectrum, one takes the lowest peak within 6 dB of the loudest. On a struck bar those can differ, because a bar can put more energy into its second mode than its first. They agree on 53 of the 54 keys. Here is the one where they do not.

That ratio is worth a second look. An ideal uniform free-free bar has its second mode at times its first, from the roots of cos(x)cosh(x) = 1, which this page solves by bisection rather than quoting: . The key that fooled the naive estimator has its loudest partial at times its fundamental. The trap and the theory are the same phenomenon.

A check that came out against us and stayed. Every compiled entry's label was tested against its own shape: a slendro should have all steps between 160 and 320 cents, a pelog pentatonic should have both a step above 290 and one below 200. One entry fails.

Its source calls it slendro, so it stays in the corpus and the sensitivity table reports the answer with and without it.

Nearest-neighbour matching is not identity, and we nearly used it. Matching each compiled entry to its closest published scale by sorted step vector produced matches as tight as two cents between a Sundanese zither and a Javanese court gamelan. With thirty entries against thirty-three candidates, a two-cent nearest neighbour is what extreme-value statistics hands you for free. Every identity claimed on this page comes from a name, never from a distance. One name-matched pair then turned out to agree to half a cent, which means it is not a second reading at all but the same measurement copied, and it is excluded from the reproducibility figure.

Does the headline survive a different corpus? Nine ways of drawing the line around "a measured slendro", and what each does to the answer.

What we did not do. We did not reach Surjodiningrat 1972 itself: no scan is online, HathiTrust serves its copy at 403, and its numbers here are the DaMuSc database's transcription of its Table 8, in cents, with the original hertz not preserved. We did not read Kunst's Music in Java (1949) or McPhee's Music in Bali (1966) directly; both are in copyright and neither was reachable, so entries credited to them rest on a compilation. We did not measure any instrument in Indonesia; the gamelan we measured ourselves stands in Portugal and we do not know who made it or where. We took no position on perception, and none on what a tuning ought to be.

One number we could not reconcile. A secondary source reports the Gadjah Mada slendro average octave as about 1219 cents; the Scala archive's entry credited to the same study says 1208; our own arithmetic over the twenty-eight transcribed gamelans gives . All three say stretched and they do not say the same amount. Without the book we cannot settle it, so all three are printed.

Reproduce it. Everything is in research/gamelan-tuning/: the 241 scale files as downloaded with the archive's SHA-256, the DaMuSc extract with its source row verbatim, the hand curation with a recorded reason on every one of its 241 rows, the 54 audio clips and the script that cut them, analyse.mjs, and verify-observed-and-assumed.mjs, which re-derives every number on this page, re-measures all 54 keys, and checks that the data block embedded in this file still equals findings.json exactly.

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