Irrational harmonies

I've been asked to put together a concert to mark the 400th anniversary of the publication of John Napier's Mirifici Logarithmorum Canonis Descriptio. My institution is named after John Napier, and the campus in which I teach is built around his birthplace. The Mirifici Logarithmorum Canonis Descriptio contains extensive material on natural logarithms, and I decided that this is going to be the theme of this concert.

For my own contribution, I decided to use natural logarithms of integers as frequency multipliers in a similar way that integers are employed in the harmonic series. This is similar to my practice in other works such as for three (which is still in progress) which are constructed from integer ratios.

This is the first time that I have used irrational numbers in ratios rather than integers and it has been an interesting experience.

I started off by taking a fixed equally tempered pitch as a starting point (I started off with B at the centre of the treble staff - around 494 Hertz) and then used log 2, log 3, log 4, log 5, etc. as multipliers of this base frequency.
This results in:

I've used the term 'otonal' here, borrowed from Partch's definitions, to contrast with the 'utonal' structure below, where I have used the inverse of the same natural logs as above as multipliers (or, if you prefer, I have divided the starting pitch by each natural log in succession).
This results in:

In each case I have rounded the deviation in cents from the equally tempered pitch to the nearest 5 cents. This has some implications, which I will discuss in a later post.

What is immediately interesting to me is that, as one would expect, there are octave doublings between the 2nd, 4th, and 16th natural log 'harmonics' (i.e. 2^1, 2^2, 2^3), and between the 3rd and 9th natural log 'harmonics' (i.e. 3^1, 3^2) [NB the exact octave doublings are blurred by rounding to the nearest frequency integer, although it should be noted that the deviation between the actual octave and the specified octave is so small that although it creates beats and is therefore an audible discrepancy, it is likely that the performer will correct for this instinctively given the way in which each pitch is created. This will be discussed further in a later post].

There are also octave equivalences between the frequency n(log 5) and n/(log 12), and the inverse n(log 12) and n/(log5), which create a bridge between the otonal and utonal structures.

The next stage is to test the performance possibilities of the exact frequencies notated. As noted above, this builds on work already carried out on for three, although problems with the first performance of the work have meant that these ideas have never been tested 'in the field'.

I need to establish if
a) I can distinguish gradations of pitch as small as 5 cents;
b) a performer might reasonably be expected to tune to as fine a standard as 5 cents.

Once this is established, I will be able to continue the pre-compositional working. Rounding to the nearest 10 cents will change things a little, although it won't be the end of the world!