Seven bands you don’t need to see

In his Opticks (1704) Isaac Newton wrote up the experiments in which he proved that white light can be split into colours. A century later (1820) John Keats complained that by “unweaving” the rainbow, Newton had destroyed its poetry: “Do not all charms fly, at the touch of cold philosophy?” It was a popular view among Romantic literary types.
In his 1998 book Unweaving the Rainbow, Richard Dawkins defended Newton, arguing that science can seriously help you to appreciate how cool rainbows and stuff are. He worded it better than that, obviously.
Dawkins has a whole chapter about sound, but he has more interesting things to say about birdsong than about music. He is a zoologist, after all. Instead, it was from Wikipedia that I learned that Newton was aware of, or even obsessed with, a supposed analogy between a rainbow or spectrum, and a musical scale.
In the darkened room where Newton split sunlight with prisms, he projected a spectrum on a piece of paper and asked an assistant to draw boundary lines between the colours. In his accounts of earlier experiments, Newton already seems very sure that there are seven colours, and he probably saw no need to get a second opinion. He probably just told his helper: “here is a list of the seven colours. Mark the place where each one ends and the next begins.”
“Whatever,” replied the lab assistant, probably a surly teenager working for apples. (Not peanuts. We know Newton had an apple tree. But imagine the consequences if he had grown peanuts instead. Peanuts are not nuts, they do not grow on trees, and they are never seen falling. Newton might never have got the idea of gravity.)
Newton is often described as part mystic, part scientist. But, apart from a religious bit at the end, Opticks is written in a scientific tone. Newton writes of the seven colours as though he not only sees them clearly, but believes they might really be seven completely different kinds of light. Remember, this was the great age of empiricism. Scientists were meant to distrust authorities such as Aristotle, and instead seek the evidence of their own eyes. The notion that our eyes often deceive us, and the concept of an optical illusion, came later.
Wishful thinking, combined with number mysticism, may have predisposed Newton to discern seven basic colours. There were, after all, seven notes in a musical scale, and seven known ‘planets’ including the moon and sun (but not the earth). Once he looked at the colour bands drawn by his assistant, he saw that there were five broad bands and two narrow bands. Remind you of anything?
Even if Newton was not already actively looking for parallels between colour and musical pitch, he now made the connection.
He drew a diagram comparing the spacing of the colour bands to the intervals in a musical scale. The best-fitting scale has the interval pattern WHWWWHW,  which you may recognise as the dorian mode.  The half steps (shown as H) correspond to orange and indigo. Newton decided that these were in some way less important than the five ‘principal’ colours red, yellow, green, blue and violet.

Diagram of colour bands from Newton's Opticks
Isaac Newton thought the spectrum was divided into seven bands, which he compared to musical intervals. By ‘chord’ Newton means a string. The ratios are between string lengths, so smaller fractions correspond to shorter soundwaves or higher pitches. Newton matched the longest light waves (red) to the shortest sound waves. This may seem odd, but since Newton did not accept the wave theory of light, it probably did not bother him.
Another odd thing: Newton does not match the notes of the scale to the perceived colour bands, but to the boundaries between them.
All this is recorded as observation. Newton does not attempt to explain this curious connection between colour and music. Clearly he believed he was on the scent of something, but it is not clear whether he thought it was a new, unifying physical law, or a hidden secret of God’s design of the universe. I would argue that, in Newton’s fiercely scientific and yet deeply pious mind, those would actually have been the same thing.
We now know that light does not come in seven different varieties. Most people do not see seven colour bands in rainbows, or even in laboratory spectra. They see six at most. As Isaac Asimov wrote: “It is customary to list indigo as a color lying between blue and violet, but it has never seemed to me that indigo is worth the dignity of being considered a separate color. To my eyes it seems merely deep blue.”
Mood Indigo Victor label
Irving Mills, the lyricist of the Duke Ellington tune Mood Indigo, seems to agree: “I’m just a soul who’s bluer than blue can be”. The whole concept of the song depends on indigo being a shade of blue, rather than a separate colour.
The web app Google Song Maker continues the tradition of fitting seven rainbow colours to a major scale, starting with red. However, indigo is absent. An aqua shade, between green and blue, makes up the seven colours.
I’m not saying that the visible bands in the rainbow are a non-issue. They remain a live subject of discussion and research. Does everyone see the same number of bands? Why do people see bands at all? Some languages have more colour words than others. Are the colours we see affected by the language we speak?
We now understand that if we see, or imagine, bands in a continuous spectrum, they have nothing to do with the physical nature of light. The phenomenon arises in our eyes and brains. It is an example of categorical perception, a concept which originated in phonetics*.

Still, the seven-colours idea refuses to die. I suspect it is widely and uncritically taught in schools and repeated in children’s books.

I think we need to be wary of similar hangovers in music theory: fossil relics of ancient beliefs. Contemporary musical genres such as jazz require a music theory which reflects modern musical sensibilities and practices.
Hugo Riemann’s theory of functional harmony is one of the great simplifying concepts in music theory. Riemann’s three harmonic functions – tonic, dominant and subdominant – provide a convincing account of the 2-5-1 cadence which is ubiquitous in jazz. But there’s an awful lot besides in jazz harmony. Functional analysis begins to creak when the tonal centres start leaping around, or a modal passage intervenes. Without questioning Riemann’s achievements a century ago, one could argue that from the point of view of jazz and other modern genres, he saw too few shades in the harmonic spectrum – the opposite of Newton’s error.
Jazz harmony has many more than seven colours. All of them are beautiful, and I am curious to know why. The more I understand, the more I enjoy the music. This is the rainbow I want to unweave.

* The space of possible speech sounds is partitioned into a finite number of phonemes. Each language does this in a slightly different way. An English speaker may be insensitive to differences that are important in another language such as Mandarin. The psychologist Alvin Liberman called this ‘categorical perception’, and believed it was unique to speech. It soon became clear that other examples of categorical perception exist. Colour is an obvious one. Categorical perception also seems to be at work when we recognise musical intervals, chords and keys.

My four posts on the circle of fifths were really about categorical perception. In Western music, the continuous and featureless audio spectrum somehow crystallises into the twelve pitch classes of the chromatic scale, and the twelve major and twelve minor keys of the circle of fifths. Other musical cultures have different categories.

Notes added 26 November – 7 December 2019

Some readers may have seen the colour wheel that Newton used, in Opticks, to illustrate his theory of colour mixing. Like his diagram of the solar spectrum (scroll up to see), it shows seven colour bands, and once again the boundaries between them are associated with musical notes. But this time the association is different. Now, shorter light waves correspond to shorter sound waves. Which is how most people seeking an analogy between pitch and colour would go about it today.

Circular diagram of colours and musical notes
This time Newton got the wavelength relationship ‘right’. But why are the musical notes between the colour bands and not lined up with them?

I do not know why Newton allowed himself this inconsistency. But it is striking that in both diagrams, scale tones are matched to colour boundaries rather than colours themselves. Why? That is so weird! I can only conclude that Newton was more concerned with revealing a universal mathematical pattern than with making any serious analogy between the sensations of pitch and colour. The guy was a genius. Give him some slack.

In his book Music and the Making of Modern Science, pianist and physicist Peter Pesic has a whole chapter on Newton’s analogy between scales, spectra and the distances of the planets. He mentions the indigo problem and other inconsistencies, pointing out that Newton himself was diffident about the colour correspondence, which he admitted ‘could be constituted somewhat differently’.

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