Where did the specific ratios in the Mesopotamian tunings come from? Why these numbers and no others? After playing with these numbers for a while, I realized that all the ratios in all seven of the tunings are based on just one initial ratio: three over two, or 1.5. This is called a fifth interval, and is the entire basis for the seven Mesopotamian tunings.
How could this be? Can the simple fifth account for the ratios 729/512 and 1024/729? It turns out that it can, and that’s what we’ll discuss here. (How the Sumerians, a supposedly primitive people, came up with this scheme is discussed elsewhere.)
We will start arbitrarily with a frequency of 400 Hertz. If this is done in a spreadsheet, any starting frequency can be used. We use a nice round number to illustrate how quickly this roundness disappears. And we will also arbitrarily call this note C, since on a modern piano, the major scale in C can be played on the white keys, thus starting as simply as possible. But again, this is an arbitrary selection; any note name could be used.
The idea is to multiply each frequency by 3/2 or 1.500, thus determining another note of a scale. Since there are seven white notes in an octave (before then next C is reached), we will repeat this multiplication seven times.
So 400 x 1.5 is 600.00 Hertz. (We will print two decimal places, but will calculate with at least six internally to avoid round-off errors.) What note does 450.00 represent? Well, since the interval of 1.500 is a fifth, we can call this new note the fifth in the scale we are constructing, and the name of the fifth in the C scale is G. This is a well-known interval in music, and is technically called the perfect fifth, although as we already know, in the common Equal Temperament tuning used in the West, the fifth is off and is certainly not perfect. (It is actually 599.32, a little flat.)
Next note. 600 times 1.5 is 900 Hertz, but now we need to examine our method and make a revision. Our goal is to determine the frequencies of one octave – the white notes – from one C to the next. Since two notes that are one octave apart have frequencies that differ by a factor of two, our octave runs from 400 Hertz to 800 Hertz, but here we are, already at 900, and we’re in the next higher octave. Clearly we’re going to increase frequencies rather quickly.
The solution is to halve any number over 800 Hertz, thus stepping back down to the original octave. We lose nothing in this process; we are merely collapsing the series of increasing fifths into a single octave, and if we’re in any doubt at all about doing this, we shall see that when we get done, this is exactly what the Sumerians did 5000 years ago. So let us proceed.
We will take 900 Hz., divide by two, and give the third note the frequency 450 Hz, and the name D, since D is a fifth above G (which was a fifth above C, where we started). So this is our program: multiply by 1.5 and divide by 2 whenever the result of the multiplication jumps up into the next octave. Table 1 shows the results for twelve such steps.
Well, the round numbers disappeared pretty quickly, and so far there are no signs of the simple fractions we saw earlier. Furthermore, we have taken twelve steps instead of seven. Why is that? But what about that last number? The notation C’ means the C one octave higher that where we started. But we know that octaves differ by a factor of two, so how come C’ isn’t 800 Hertz?