Copenhagen, 1654. A young academic is packing his books in rather a hurry. He is a short, rather intense man in his early thirties. Always full of energy, the hasty move is beginning to upset his normally even temper. One item he remembers to pack is a pair of notebooks full of work about the manipulation and use of ratios. But one thing that doesn't find a place in his trunk is the little prospectus he published the previous year, outlining the contents of a book he proposed to write about the same subject. Now it doesn't seem likely that the book will ever be written.

The academic's name was Nicolaus Mercator. Not Gerard Mercator, who made maps and has the Mercator projection named after him. Our Mercator was brought up – in the region of Holstein, in Germany – as Niklaus Kauffman: as an adult he translated his surname (it means 'merchant') into Latin. Not an unreasonable thing for a scholar to do who travelled as much as he did: but he probably wasn't sorry to gain the association with his famous namesake. He studied in Rostock and Leiden, and then got a job at the University of Copenhagen.

In 1654 Mercator left Copenhagen in a hurry because plague had broken out. He had little choice: the epidemic was so bad that the university closed and he was temporarily out of a job. Plague had been spreading north and west from the Balkans since the beginning of the 1650s, and after it reached northern Poland it spread rapidly around the rest of the Baltic coast: when it reached Copenhagen, a fifth of the city's inhabitants died. Meanwhile plague was also spreading from the west: it had never been eradicated in the Low Countries after an epidemic in the 1640s. The outbreak in the '50s abated before reaching central Germany, or northern France or England. But it was to break out again in the Low Countries in 1663, and this time it reached England and caused the Great Plague of London in 1664. Mercator was not to know that, and in 1654 he fled to England.

What did Mercator expect to write about in his book on ratios? The list of contents looks rather curious to us: the book was to contain sections on devising calendars, astronomical predictions, the proper division of the musical scale, and military constructions. Mercator believed that an ingenious application of the theory of ratios could help with the solution of all these kinds of problems. I am particularly interested in the musical material.

In Mercator's day there was a long tradition of considering ratios as things quite different from numbers, and quite different from fractions. They had special properties and combined in an unusual way. For example, the most natural way to combine two ratios was to multiply their terms: a:b and c:d made ac:bd. To take one ratio away from another, you divided the terms: c:d taken from a:b gave (a/c) : (b/d). In fact, these two operations were often called 'adding' and 'subtracting' ratios until the late seventeenth century, terms which can seem quite confusing to us.

If these are 'adding' and 'subtracting', an interesting problem arises: what are 'multiplying' and 'dividing'? It isn't easy to see how to 'multiply' a ratio by another ratio. What you can do, though, is to multiply a ratio by a number, so that for example two times a:b is the same as a:b 'plus' a:b – that is, a²:b². And so on. So we can in fact 'multiply' a given ratio by any whole number, simply by repeatedly 'adding' it to itself.

But what would it mean to 'divide' a ratio by another? This isn't at all easy, and from perhaps the fourteenth century onwards certain mathematicians considered that finding a 'ratio of ratios', or how to 'divide' one ratio by another, was an unsolved problem. The difficulty can be put another way by asking, how many times does the ratio a:b go into the ratio c:d? That is, what number must we 'multiply' a:b by – how many times must we 'add' a:b to itself – in order to get c:d?

You might be able to see how to find the answer. Mercator did too. But first, what do ratios have to do with music?

 

Rather a lot, it turns out. If you have two strings made of the same material, with the same thickness and tension, but different lengths, they will make different musical pitches when you pluck them (like a guitar) or stroke them with a bow (like a violin). The relationship between the two pitches determines whether they blend or not; whether they make a consonance or a dissonance. That relationship in turn is determined by the ratio of the two strings' lengths. If the ratio is 1:1, the two strings make the same pitch: they are in unison. If it is 2:1, they are an octave apart (e.g. from one C to the next C above it). If it is 3:2, they are a perfect fifth apart (e.g. from C to the G above it). If it is 4:3, they are a perfect fourth apart (e.g. from C to the F above it). Finally, if the ratio is 9:8, the two notes are a tone apart (e.g. from C to the D above it).

When mathematical music theory was done in ancient Greece, those were all the ratios and musical intervals that were used. You might notice certain relationships between them. For example, a fifth plus a fourth is an octave, since 3:2 'plus' 4:3 is 2:1. And the difference between a fifth and a fourth is a tone: 3:2 'minus' 4:3 is 9:8.

But what if we want to divide up the octave into small steps to make a scale? How many times, for instance, will a tone go into an octave?

Now you see the connection between music and ratio theory. To divide the octave into tones is the same as 'dividing' 2:1 by 9:8. (That is, how many times must we 'add' 9:8 to itself to get 2:1?) By trial and error we can see that it goes more than five times and less than six. It doesn't go exactly. If we want to know how many fourths or fifths there are in an octave, or how many tones in a fourth or fifth, they don't go exactly either, and we find ourselves needing again and again to be able to perform 'divisions' with ratios.

One solution is an approximation: divide the octave into six, or twelve, exactly equal portions, and use these to make approximations for the smaller intervals. This is what is normally done nowadays: the twelve notes in an octave are exactly equally spaced, a tuning that is called equal temperament. The twelfths of an octave are called semitones. A tone is taken to be two semitones, a fourth five, and a fifth seven. These are good approximations, but not very good ones, and in Mercator's time it hadn't been completely accepted that this was the best thing to do.

Before returning to Mercator and seeing how he went about finding better approximations and exact answers to these musical questions, you might like to experiment with the interactive applet. It shows a musical string and a modern keyboard, and you can see how lengths on the string correspond with positions on the keyboard. You can change either the string position or the keyboard position by dragging its slider, or by typing an exact value into the box. 

For example, you can see that half of the string gives a note an octave - twelve semitones - above the whole string. What happens if you take two-thirds of the string, an interval of a fifth above the whole string? It is about seven semitones above the bottom note on the keyboard, but it is not a whole number of semitones. Experiment with other ratios, and see how they come out on the keyboard: or see what ratios correspond to particular whole numbers of semitones.

 

Oxford, 2006. I've got Mercator's two notebooks in front of me. The paper is yellow with age, but it's intact and the writing is perfectly legible. The text is in Latin throughout, like that of Mercator's published prospectus, Rationes mathematicae ('mathematical ratios'). The first notebook is a real mess: it's been used from both ends, and although it started out with a fairly clear line of thought, Mercator kept getting distracted and jotting down odd pages of completely unrelated material. It's obvious that this was a working notebook, meant to be tidied up and sorted out fairly extensively before it could be published. The musical use of ratios is by far the most prominent topic in these notebooks, although the other subjects Mercator had listed in Rationes mathematicae are also represented here and there.

The other notebook seems at first to be the sorted-out version. It's in a neat hand, probably that of a scribe rather than Mercator himself. It's on good quality paper, and it's very clear and easy to read: even after three hundred years it's in good condition. But towards the middle of the book it gets less and less neat, and the calculations start to have rough working down the side of the page. Eventually this notebook, like the other one, trails off into a lot of rough notes on various different ideas, calculations which don't have much to do with the text, and so on. So at the time he came to England Mercator seems to have been having trouble putting this material in order.

But that's not all the manuscripts we have. It seems that Mercator made some other versions of the musical material, after he came to England. There are two versions; both are in English (slightly imperfect English, which suggests that the translation was done by Mercator himself, perhaps not that long after coming to England). There are manuscript copies of each in the Bodleian Library, Oxford; one of them was also copied out for Robert Hooke by one his assistants, and the copy is still in London. What seems to be the same text gets a mention in a book on music by the Fellow of the Royal Society William Holder in 1694: so it seems to have been passed around quite a bit. The thing that is most prominent in these later, English versions of the material is the use of logarithms to measure musical ratios.

 

Logarithms were invented in Scotland in 1614; and they held the solution to the problem of 'dividing' one ratio by another.

We were interested in how many times one ratio goes into another, that is how many times it must be 'added' to itself – really multiplied by itself – to give the other ratio. That is,

A^x = B,

where A and B are ratios, and we are to find x. If we take logarithms of both sides:

x log A = log B.

So, x = log B / log A.

What that means is that we can 'divide' ratios by (really) dividing their logarithms. (We can also 'add' ratios by adding their logarithms, and 'subtract' ratios by subtracting their logarithms: you might like to think about why this works.) The exact number of tones (9:8) in an octave (2:1) is given by log(2/1)/log(9/8), which is about 5.88.

Mercator did calculations like this in his notebooks, and in his later English manuscripts. Thus he was able to work out the exact relationship between different musical intervals: 5.88 tones in an octave, 3.44 tones in a fifth, and so on.

To get an approximation that could be used for dividing up the octave to make a useable scale, he didn't divide the octave into twelve. Instead he tried to find a division of the octave that would give a good approximation for the fourth and fifth within it. He found that the ratio of the octave to the fifth is about 1.710, which is close to 53/31. So he proposed dividing the octave into 53 parts, of which a perfect fifth would be 31. This is a closer approximation than the equal tempered scale we saw above, where the fifth is made to equal 7/12 of an octave.

Mercator went through the whole of one of the popular tuning systems in use in the seventeenth century – the meantone temperament – and showed how each of its notes could be approximated by one of the 53 steps in his octave. Better and better approximations could be found, but only by dividing the octave into more and more small parts, which would be impractical for real musical instruments.

Mercator proved his point, even if it was only in manuscripts that circulated privately, that ratios could still be useful for studying music, and indeed that musical problems could be a stimulus for new developments in the handling of ratios. It is a shame that he didn't publish his musical work – there is a good deal more to it than I have written about here – he was certainly planning to at one stage, and it seems as though the plague was to blame for disrupting that plan. He re-used some of the same ideas in the 1660s to provide a commentary on some new musical theories that were causing a stir at the Royal Society: but again, nothing was published.

Only one copy is now left of Mercator's prospectus, the Rationes mathematicae, published in Copenhagen in 1653. It's in Paris, and I've never seen the actual book. But these days it's rather easier to get things across the English Channel, and the Bibliothèque Nationale mailed page images to me on a CD. It's a pity that Mercator didn’t have that option.

 

You can find out a little more about the manipulation of musical ratios in my article on music and Euclid’s algorithm:

‘Music and Euclid’s Algorithm’, Plus online mathematics magazine, Sep. 2006:
http://plus.maths.org/issue40/features/wardhaugh/index.html

For a good general introduction to music and mathematics, see:

John Fauvel, Raymond Flood, and Robin Wilson (eds), Music and Mathematics: From Pythagoras to Fractals (Oxford, 2006)

Mark Lindley, Lutes, Viols and Temperaments (Cambridge, 1984).

For more information on Nicolaus Mercator, see

Christoph J. Scriba, ‘Mercator, Nicolaus (1620?–1687)’, Oxford Dictionary of National Biography, Oxford University Press, 2004.