The Moon Looks Down at the Earth

A slit drum has two notes. Kele has two tones. So a drum can send the tune of a word and nothing else, and John Carrington, working through the mission dictionary at Yakusu in the 1940s, counted about a hundred and thirty Kele words with the same tune as sango, father. The drummers had solved this long before he arrived, by never sending the word alone. The moon is not songe. The moon is songe li tange la manga, the moon looks down at the earth. Six extra strokes to say one word. This page asks whether six is the right number, using nothing but what Carrington himself printed.

He printed two things you can put together. On page 33 he printed the ambiguity: a hundred and thirty words sharing one tune. Across pages 33 to 39 he printed twenty drum phrases with their tone strings, one letter per stroke, which is a sample of the signal itself. The first tells you how much confusion the drum creates. The second tells you how many bits a stroke carries and how many strokes the drummers actually spend. Multiply and you can ask whether the padding is the right size for the job.

He never did. The book came out in 1949, a year after Shannon, and the word redundancy does not appear anywhere in it. What Carrington wrote instead, on page 37, was Such a duplication of information is to be found in a number of other phrases.

1. Two notes

The Lokele drum is a hollowed log with a slit cut along it. The two lips of the slit sound at two pitches, and that is the entire instrument. Kele has two tones, high and low, and they are lexical: they tell words apart the way vowels do in English. So the drum can follow the tune of a sentence exactly, and nothing else about it survives. Every consonant and every vowel is gone.

Carrington's own worked example of the principle is the word asooya, he has come, drummed as four strokes, low high low high, exactly the tune of the spoken word. Then he raises the objection himself:

The reader will probably have formed one objection to the account so far rendered; namely, that there must be a large number of Kele words with the same tonal pattern as that of sango and others like nyango or wana. If only the tonal patterns are drummed out, how can such words be distinguished on the drum? Working through the Kele dictionary used by the missionaries of the B.M.S. at Yakusu we find that there are some one hundred and thirty words which have the same tonal form as sango, while more than two hundred are tonally like nyango. Carrington 1949, p. 33. His footnote to that sentence: the words are mainly noun forms, and including verbs would raise both counts considerably.

2. The hundred and thirty, in a dictionary you can open

That count comes from a missionary dictionary at a mission station on the upper Congo in the 1940s. We cannot check it. What we can check is whether the scale is right, by doing the same count in a tone language whose dictionary is a download.

Yoruba has three tones rather than two, and a talking drum rather than a slit drum: the dùndún is a tension drum whose pitch slides, so it can render three levels. Wiktionary's Yoruba entries carry a syllabified, tone-marked pronunciation, which gives 5,143 distinct spoken forms, 1,599 of them two syllables long, 1,136 of those nouns. Collapse the three tones to two, as a two-lipped drum would have to, and count the nouns in each of the four possible tunes:

Carrington's 130 sits inside that range and so does his 200. A tone language really does pile up words at this rate, and the drummer's problem is real in the size he reported it. Below you can walk the whole partition: pick a tune, see everything in a modern Yoruba dictionary that a drum would render identically.

Add a syllable and the class collapses. That is the whole mechanism of a drum phrase, and it is why the drummers do not send bare words.

3. What the padding buys

Carrington printed 21 phrases with their tone strings. Together they are 202 drum strokes: 79 high, 123 low. That is a sample of the signal, so it can be priced. A stroke is not a fair coin, because the drum leans low, and the entropy of the printed strings is 0.9655 bits per stroke (bootstrapping over phrases, 95% interval 0.922 to 0.989). Conditioning each stroke on the one before it barely changes anything: 0.9746 bits.

Before any of that could be trusted there was a check to run. Carrington says the drum follows the syllables, so his tone strings should have exactly one letter per syllable of his Kele. Kele syllables are open, so the syllable count is the count of vowel letters. Over all 21 phrases, the two counts agree 21 times out of 21, with no disagreements. That is not a check of Carrington; it is a check of our reading of him, and of the syllable rule, and it is the reason the rest of this section is allowed to exist.

Now the arithmetic. His padded phrases run from 7 to 18 strokes, mean 9.90, median 10. For the 8 words where he prints both the spoken form and the drummed phrase, the padding adds 5.88 strokes on average. Three honest ways to ask how much that padding is worth, which disagree with each other by about a factor of two:

And the demand, from page 33 of the same book: separating 130 words costs 7.02 bits, separating 200 costs 7.64 bits, and 200 is a floor because he wrote more than two hundred.

So the supply is somewhere between 51 and 198 alternatives and the demand is somewhere above 130. The padding is the right size for the job it exists to do, to within about one drum stroke. It could have been out by a factor of a thousand. It is out by less than a factor of three, in a quantity nobody involved was computing.

The other thing his tone strings buy is a number that gets quoted without a source. The claim in circulation is that a drummed message runs about eight times as long as the spoken one. It is not in Carrington. The word eight occurs once in the book, in the phrase seven or eight miles away, and no ratio appears anywhere. What he printed instead, on page 54, is one message in both forms. Counting them: 15 spoken words and 32 syllables become 54 drummed words and 127 syllables, a factor of 3.6 by word and 4.0 by syllable. Before any repetition. Multiply by the three or four repetitions he describes and you land in the neighbourhood of eight, which may well be where the number came from, but the stereotyped phrasing on its own costs a factor of four, not eight.

4. The other way to throw speech away

On La Gomera in the Canary Islands, shepherds whistle Spanish across ravines. Spanish has no lexical tone, so a whistler cannot do what a drummer does. What the whistle carries instead is a single frequency line tracing a squashed version of the vowel space, with consonants realised as the way the line moves into the vowel and whether it breaks. The question is the same as the drummer's: how much is left.

Here the field disagrees with itself, in print, and has done for seventy years.

Trujillo's reduction is the most drastic and the most widely repeated. It is the one in the Canarian government's school manual, and it is the one in the UNESCO listing, whose description reads two distinct whistles replace the five Spanish vowels, and there are four whistles for consonants. It is also the one Meyer published a statistical rebuttal of in 2008. And it is contradicted thirty pages later inside the same government manual, where the section written from the practice of the whistling teacher Isidro Ortiz Mendoza opens En el silbo gomero las vocales quedan reducidas a cuatro, reduced to four.

A phonological reduction is also a claim about a lexicon, and that claim is checkable. Run each account over the words a whistler actually uses. Meyer and Gautheron put the common whistled vocabulary of La Gomera at about two thousand words; the whistlers' own estimate during the revitalisation programme is more than four thousand. Both are below.

Trujillo's own book makes a prediction we can test against our reading of his table. On page 33 he writes that whistled, nada, lana, llano and roña all sound the same. Under our implementation of his four consonants and two vowels they do: all four map to the identical signal. Under every other account they separate. Try them in the bench.

What each account costs, over the two thousand commonest words

Blind accuracy is the simplest thing you can ask: a listener hears the signal, knows nothing else, and picks a word from its ambiguity class at random. Because every word in a class is as good as any other, that number is exactly the count of distinct signals divided by the count of words, which is why it is worth so little on its own and worth something as a comparison. It is a floor. Every account of whistled speech, Trujillo's included, insists that context does work this number cannot see.

Set beside it: the one measurement of isolated whistled words in the literature. Busnel tested five whistlers of Turkish at Kuşköy in 1970 on a word list built to induce confusions, and got 69%, against 96.6% for the same list spoken. That is a different language and a hostile list, so it is a yardstick and not a verdict. But it lands at the top of the range the four accounts predict, where Meyer puts it (69.9%) and Classe puts it (74.0%), and not where Trujillo puts it (43.6%). Widen the vocabulary to four thousand words and every number falls but the ordering does not: 39.9, 59.6, 66.3, 70.7.

This does not settle the phonetics. Meyer's case rests on acoustics and on perception tests with real whistlers, not on lexical statistics, and Trujillo could answer that his two vowels are what a listener reliably recovers while the rest is context. What the arithmetic does is give the disagreement a price. The reduction that UNESCO adopted is the one that makes whistled Spanish hardest to understand, by a wide margin, and nobody appears to have checked what it would cost.

5. Why nobody drums Spanish

Two channels, two languages, one table. Hold the word length fixed at two syllables so the comparison is fair, and ask how many bits each channel delivers per syllable.

The pitch line carries 3.5 times more in Yoruba than in Spanish, because in Yoruba pitch is lexical and in Spanish the only thing modulating it is which syllable is stressed. A Spanish drummer would be sending the number of syllables and the position of the stress, and almost nothing else. The formant line runs the other way: Spanish gets more out of it than Yoruba does, having more consonants and closed syllables to spend.

That is the whole design principle of a speech surrogate, and it was never designed. Each community found the channel that its own language happens to load. Where pitch carries the lexicon, you can drum. Where it does not, you whistle the mouth instead. Nobody drums Spanish because there is nothing in a Spanish word for a drum to hold on to.

6. Three things the record gets wrong

The title. The book is Talking Drums of Africa. It is cited nearly everywhere, including by Gleick, as The Talking Drums of Africa. The definite article belongs to Carrington's 1971 Scientific American article, which is where it probably leaked from.

Redundancy. The 1949 book does not contain the word. Searching its text for redundancy, redundant, Shannon, Hartley, information theory and Bell System returns nothing. Carrington's term is duplication of information. The redundancy framing is real but it is dated 1971, in the deck of his Scientific American piece, twenty-two years later: the drummers of Africa may well have been the first to utilize the principle of redundancy in their communications. Magazine decks are usually written by editors, and we have not read the body of that article, so treat it as the framing that appeared over his byline rather than as his sentence.

Eight times as long. Gleick states it without a citation; Dyson restates it in a different form. It is not in Carrington, in any wording, and his own printed example gives 4.0.

None of these three change what Carrington found. They are the kind of drift that happens to a good book that becomes a famous anecdote, and they are worth fixing because the anecdote is now more read than the book.