Decimals of π in 10TET

The digits of π have been translated into music a number of times. Sometimes the digits are translated to the pitches of a diatonic scale, perhaps accompanied by chords. The random appearance of the sequence of digits is reflected in the aimless meandering of such melodies. But wouldn't it be more appropriate, in some sense, to represent π in base 12 and map the numbers to the chromatic scale? After all, there is nothing in π that indicates that it should be sung or played in a major or minor tonality. Of course the mapping of integers to the twelve chromatic pitches is just about as arbitrary as any other mapping, it is a decision one has to take. However, it is easier to use the usual base 10 representation and to map it to a 10-TET tuning with 10 chromatic pitch steps in one octave.

Here is an etude that does precisely that, with two voices in tempo relation 1 : π. The sounds are synthesized with algorithms that also incorporate the number π. In the fast voice, the sounds are made with FM synthesis where two modulators with ratio 1 : π modulate a carrier. The slow voice is a waveshaped mixture of three partials in ratio 1 : π : π².

Despite the random appearance of the digits of π, it is not even known whether π is a normal number or not. Let us recall the definition: a normal number in base b has an equal proportion of all the digits 0, 1, ..., b-1 occuring in it, and equal probability of any possible sequence of two digits, three digits and so on. ("Digit" is usually reserved for the base 10 number system, so you may prefer to call them "letters" or "symbols".) A number that is normal in any base is simply called normal.

Some specific normal numbers have been constructed, but even though it is known that almost all numbers are normal, the proof that a number is normal is often elusive. Rational numbers are not normal in any base since they all end in a periodic sequence, such as 22/7 = 3.142857. However, there are irrational, non-normal numbers, some of which are quite exotic in the way they are constructed.

Limited edition

When does it make sense to publish something on the internet, let's say a recording, as though it were a limited edition? A de facto limitation in the number of downloads is not impossible to achieve if the item is made to drown in the information deluge and then taken off-line as soon as it has been downloaded some specified number of times. Then of course one cannot guarantee that it will never be uploaded again, although that can be made a bit awkward by having very large files.

Does this strategy really work if the internet never forgets? (The Wayback Machine takes care of that, although they do not necessarily save all audio that floats around out there. Can we hope that some unnamed data centre in Utah or elsewhere stores it for us? Maybe, if you send it as an email attachment.) Usually people do their best to boost the number of views and fight hard for their page ranks. Doing the opposite clearly has its merits if the analog of a limited edition is the goal. Obviously, the idea of limited edition is tied to physical media, to something people can hold in their hands, so a digital file will not easily do as a replacement. The point of limited editions is exactly to make it clear that the resource is scarce, the object you are holding in your hands is a collectors item.

[announcement]

Thinking along these lines, I have published the two hour apocryphal piece Teem Work, which I intend to replace with something else before it has become too widespread. Although it can be downloaded, the following reasons speak against it.

• The composition is merely a sketch. 
• It will occupy more space on your hard disk than strictly necessary. 
• It will steal two hours of your precious time. You'll never get them back.

It should be added that Teem Work features synthesis by cross-coupled feedback FM as described in a previous post.

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Update (December 2015):

A remixed edition is now available as a digital album. Still it will steal two hours of your time, precious or not.

Feedback FM with bells and whistles

Single oscillator feedback FM is a most economic technique for producing rich harmonic tones. However, the technique suffers from parasitic oscillations at the Nyquist frequency when the modulation index is turned up sufficiently high. The most obvious thing to try is to modify the original formula

x[n] = sin(ωn + Ix[n-1])

by lowpass filtering the feedback signal with some filter that has a zero at half the sample rate. A two point average increases the range the index can take before the spurious oscillations set in, but it cannot stop them at sufficiently high modulation indices. A complementary trick is to put a filter outside the feedback loop. Again, it helps to a certain extent, but should not be expected to solve the problem in all cases. Finally, there is the Overkill Solution of oversampling the system. Or maybe it's not so overkill after all. In any case, a high sample rate is recommended.

Feedback FM waveform with spurious oscillations and no attempt to squelch them.

Depending on the sample rate, the spurious oscillations will typically ring at such a high frequency that only domestic animals will notice them and possibly object to their presence. Nonetheless, the waveform will be contaminated by a conspicuous wiggle that may annoy any purist, whether or not they hear it. Of course, it is a spurious or parasitic oscillation since it follows the sampling rate rather than the synthesis parameters. Sometimes the parasitic oscillation happens at a third or fourth of the sample rate, or other subharmonics.

Single oscillator feedback FM is limited to harmonic spectra. Much flexibility is gained  by introducing a second oscillator since there are several ways to connect the two oscillators. Rather than listing all cases separately, we introduce coupling parameters c (cross terms) and b (self-modulation) in the coupled system

      x[n+1] = sin(θ[n] + b₁x[n] + c₁y[n])

(*)   y[n+1] = sin(φ[n] + b₂y[n] + c₂x[n])

where the phases θ and φ are incremented by the modulating and carrier frequencies. Which one is which depends on what signal you send to the output. Of course both signals can be used for stereo, but then it makes less sense to call one of the oscillators the carrier and the other one the modulator.

In FM synthesis, the phase variables usually depend only on their respective frequencies. By introducing an interaction term, phase coupling can be used to synchronize the oscillators. Hardsync may have been used with FM before, but the gentler kind of sync used in the Kuramoto model is useful here, as it is also suitable for synchronizing more than two oscillators. Now, the phases are incremented by the oscillators' frequencies as usual, but to that we add a coupling term with strength K:

      θ[n+1] = θ[n] + ω_c - K sin(θ[n] - φ[n])
(#)   φ[n+1] = φ[n] + ω_m - K sin(φ[n] - θ[n])

Turning up K too much will collapse the two oscillators into a single strong team working in perfect sync. The system (*, #) is just a four-dimensional map with seven parameters and may thus be studied with the appropriate methods of dynamic systems.

As often happens with iterated maps, the feedback FM system exhibits typical behaviour such as period two oscillations, and the period doubling route to chaos. The frequency terms may both be set to zero, which means the system (*) becomes autonomous. Then period doublings can be seen more easily, as shown below. The system has to be seeded with nonzero initial values for any oscillation to occur.

Colour legend: gray, period 1; orange, period 2; blue, period 4, bright yellow, period 3, red, chaos. On the horizontal axis, -1 < c₁ = c₂ < 4, vertical axis: - 0.5 < b₁ < 5 and b₂ = 0.5 (constant). The modulator and carrier frequencies are both 0; hence the coupling term does not influence the dynamics.

We are not done with feedback FM!

Other things to try include modifying the feedback signals by waveshapers and filters. Even the phase coupled signal may be filtered. Each filter adds its state variables to the system and increases its dimension. This is complicated territory; suffice it to say that filters plus strong nonlinearities or high feedback gain equals noise!