He called it an eighth
The first table of wavelengths that ever existed was printed in 1802, in the Philosophical Transactions, by Thomas Young. Fifteen rows. A colour in the first column, a length in parts of an inch in the second, the number of waves that fit in an inch in the third, and in the fourth the number per second, in millions of millions. Extreme red, .0000266 of an inch. Extreme violet, .0000167. In the units we use now that is 676 nanometres down to 424, against a modern figure of roughly 700 down to 400, which for a first attempt at a quantity nobody had ever named is extraordinary.
Young did not measure any of it. He says so himself, almost cheerfully, and invites the reader to trust the numbers more for it, on the grounds that they came out of experiments done by a man who would have hated the conclusion.
The experiments were Newton’s rings. Press a convex lens onto a flat plate of glass and you trap a wedge of air that thins to nothing at the point of contact. Light it with one pure colour and the contact point is ringed, bright and dark alternating outward, and the dark rings fall where the air has certain thicknesses — one unit, three units, five. Newton measured those rings for the better part of a decade and explained them with a theory of fits of easy transmission and easy reflection that nobody has believed for two hundred years.
What Young did was to re-read the measurement. A dark ring in transmitted light, he said, is the place where a wave that went straight through meets a wave that bounced twice, and the two are exactly out of step. The second wave went an extra distance of twice the thickness of the air. For them to cancel, that extra distance has to be half a wavelength. Two thicknesses make half a wave; one wavelength is four thicknesses. Multiply Newton’s table by four and you have the wavelength of light.
That is the entire derivation. There is no apparatus in it.
But there is a difficulty one level down, and it is the reason for this essay.
Newton did not publish his thicknesses in inches. He published them as ratios — the thickness at the red end against the thickness at the orange boundary against the thickness at the yellow boundary, and so on — and in the whole of the Opticks there is exactly one absolute number to anchor them: one part in a hundred and seventy-eight thousand of an inch, the air at the first bright ring under what he calls a bright citrine yellow, or confine of yellow and orange. One number, and seven ratios.
And the ratios are a tune.
the thicknesses of the Air between the Glasses there, where the Rings are successively made by the limits of the seven Colours, red, orange, yellow, green, blue, indigo, violet in order, are to one another as the Cube Roots of the Squares of the eight lengths of a Chord, which found the Notes in an eighth, sol, la, fa, sol, la, mi, fa, sol; that is, as the Cube Roots of the Squares of the Numbers, 1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/16, 1/2.
An eighth is an octave. Those eight fractions are the lengths you stop a string at to play one, and Newton’s whole scheme of seven colours — the seven we still teach children, indigo and all — is there because a scale has seven steps before it comes back to itself. The claim is that the spectrum is an octave. He says it in that sentence and he says it again in the drawing, where the colour bands sit under a divided line like frets.
Now take the sentence at its word and do the arithmetic.
A string’s frequency is one over its length, so an octave of string is a factor of two in pitch: that is what makes it an octave. But Newton is not using the lengths. He is using the cube roots of their squares, the lengths raised to the two-thirds power, because that is what fitted his rings. And a factor of two, raised to two-thirds, is not a factor of two. It is 1.5874.
Young’s table has the answer in it, because his extremes are the ends of Newton’s chain. Fifty-nine thousand seven hundred and fifty waves to the inch at the extreme violet; thirty-seven thousand six hundred and forty at the extreme red. The ratio is 1.587407. Two to the two-thirds is 1.587401. In the unit musicians use, where an octave is twelve hundred, the spectrum spans eight hundred cents and seven thousandths of a cent.
Eight hundred cents is eight semitones. It is a minor sixth.
The analogy does not survive its own arithmetic. Newton heard an octave in the rainbow, wrote down the lengths of an octave, applied the transformation his measurements demanded, and published a minor sixth — and then named it an eighth, in the same sentence, four words before the fractions that say otherwise.
He had already done it once. Two lines earlier in the same paragraph there is a rougher version for five principal colours instead of seven, and that one uses the string lengths directly rather than their cube-rooted squares, running from one down to three fifths. Five thirds in frequency. A just major sixth, eight hundred and eighty-four cents. Both of Newton’s statements of the spectrum, the rough one and the careful one, come out a sixth. Neither comes out an octave. The word octave is doing no work in either; it is the thing he was reaching for, not the thing he found.
What I want from this is not that Newton was wrong. It is a question about what travels.
Three bodies of theory have now been stacked on these rings. Newton’s fits, which are gone. Young’s longitudinal waves in a universal ether, which are gone. Ours. Through all of it the numbers walk forward without a scratch: 10000, 9243, 8855, 8255, 7631, 7114, 6814, 6300. Young multiplied them by four and got three good figures on a quantity Newton would have denied the existence of. You can rebuild them today from the fractions and a pocket calculator and land within two parts in ten thousand of what Newton printed — and the one place you miss by more than half a unit is a slip of his own, at 8/9, where the arithmetic says 9244.8 and the book says 9243.
The ratios were load-bearing. The music was not. And nothing in the page tells you which is which.
That matters because of where it leaves the ends. Young’s extreme red and extreme violet are not places where he looked and saw the colour give out. They are the first and last terms of a musical scale, bent by a two-thirds power, anchored on a single thickness of air that Newton had attached to a colour he could only describe as a bright citrine yellow — and which Young, needing a row for yellow, quietly decided to call yellow, though Newton had put it at the border between yellow and orange. The visible spectrum in the world’s first table of wavelengths is as wide as it is because of a decision about a scale. It is eight hundred cents wide. The real one is nearer nine hundred and seventy, and it is wider at both ends, and no amount of care inside the table could have revealed that, because the width was never measured. It was inherited.
A borrowed framework comes with parts that are carrying the structure and parts that are ornament, and they are not labelled, and the author does not always know. Newton needed to say these eight thicknesses stand in this particular non-obvious relation, and the system of ratios his readers already had by heart, tabulated and argued over for two thousand years, was music. So he said it in music. The ratios went into the world dressed as a tune, and a century later a doctor who had no use for the tune took the dress off and found three correct significant figures underneath.
But he kept the hem. He kept the ends of the scale, because the ends were where the numbers stopped, and a number that stops looks like a measurement.
I think the lesson is that you can only find this out from outside, and late. Not by being more careful inside the table — the table is beautifully consistent, and I have checked every column of it against every other, and the only internal discrepancies are roundings in the third figure. The octave reveals itself as a sixth only when you stop asking whether the numbers agree with each other and ask what the first one was, and the last one, and who chose them, and what for.
Newton called it an eighth. It is a sixth. It was a sixth on the day he printed the fractions, and it was still a sixth ninety-eight years later when Young multiplied the whole chain by four and published the result as the length of an undulation of light. I do not know who first raised two to the two-thirds power and noticed. I know it was not either of them, because both of them printed the number and neither of them said the word.
Numbers here are from reading/young-wavelength/verify.py in this repository,
twenty-two checks, built from Newton’s Opticks (4th ed., 1730; Project
Gutenberg 33504) and a transcription of Young’s table from the scan of Phil.
Trans. 1802 p. 39 reproduced in Olivier Morizot, The wavelength of light as
Thomas Young invented it, arXiv:2609.24793, which is what sent me here.
