The joy of frequency shifting

Frequency shifting, or single sideband (SSB) modulation as it is also called, differs from the more well known ring modulation in that it produces less partials in the resulting spectrum. Whereas ring modulation is the same as the convolution of the carrier's spectrum with the input signal's spectrum, single sideband modulation just shifts the partials up or down in frequency. Harmonic tones become inharmonic, unless the amount of shift happens to equal the frequency of one of the harmonics.

SSB modulation can be carried out with analog as well as digital techniques. There has been much prejudice and heated debate about the advantages of analog vs. digital. In this comparison between an analog eurorack module and several DSP implementations of SSB, one can hear differences in their sound, but that is not to say that everyone will agree on what sounds better or which method would be preferred in a given situation.

On Modular Purism and Sozialrealismus

Among practitioners of modular synthesis, there is today a widespread reverence for the pure modular, where the modular shines on its own as the single sound source, preferably recorded in one take and with minimal post-processing. In improvised music this kind of purism may make sense, but modulars are often used in other ways, e.g. driven by sequencers or set up as a self-generating system more or less nudged in the right direction by the modularist.

Playing live in a concert, overdubs or edits are obviously not a part of the game. Perhaps that spontaneous flow of live performance is taken as the ideal form that even home studio recordings should mimic.

This ideal of purity often serves as an excuse for acquiring a voluminous modular system. Even more so if one wants to achieve full polyphony. However, there is so much to gain from multi-tracking and editing that it is a wonder why anyone with a modular should refuse to deal with that part.

The pieces on the album SOZIALREALISMUS are collages and juxtapositions of a variety of sources. All pieces are centered around recordings of a eurorack modular, often accompanied by field recordings.

In some pieces the electronic sounds were played through loudspeakers placed on the resonating bodies of acoustic instruments and then recorded again. Recordings have been cut up and spliced together in new constellations.

Although the blinking lights and the mess of patch cords of a modular projects an image of complicated machinery leading its autonomous electric life, another aesthetic is possible even in the realm of modular music, an aesthetic of handicraft, allowing the imperfections of improvisation. The included drawings that come with the digital album and the linoleum print cover on the cassette hint in that direction.

Linoleum print © Holopainen. Makes for an interesting stereo image as well.

There has always been a focus on technique and gear in the electronic music community. When modulars are becoming more common, their ability to provoke curiosity may dwindle if the music made with them fails to convince the listeners.

Golomb Rulers and Ugly Music

A Golomb ruler has marks on it for measuring distances, but unlike ordinary rulers it has a smaller number of irregularly spaced marks that still allow for measuring a large number of distances. The marks are at integer multiples of some arbitrary unit. A regular ruler of six units length will have marks at 0, 1, 2, 3, 4, 5 and 6, and it will be possible to measure each distance from 1 to 6 units. A Golomb ruler of length six could have marks at 0, 1, 4 and 6.

Each distance from 1 to 6 can be found between pairs of marks on this ruler. A Golomb ruler that has this nice property that each distance from 1 to the length of the ruler can be measured with it is called a perfect Golomb ruler. Unfortunately, there is a theorem that states that there are no perfect Golomb rulers with more than four marks.

Sidon sets are subsets of the natural numbers {1, 2, ..., n} such that the sums of any pair of the numbers in the set are all different. It turns out that Sidon sets are equivalent to Golomb rulers. The proof must have been one of the lowest hanging fruits ever of mathematics.

An interesting property of Golomb rulers is that, in a sense, they are maximally irregular. Toussaint used them to test a theory of rhythmic complexity precisely because of their irregularity, which is something that sets them apart from more commonly encountered musical rhythms.

There is a two-dimensional counterpart to Golomb rulers which was used to compose a piano piece that, allegedly, contains no repetition and is therefore the ugliest kind of music its creator could think of.

Contrary to what Scott Rickard says in this video, there are musical patterns in this piece. Evidently they did not consider octave equivalence, so there is a striking passage of ascending octaves and hence pitch class repetition.

At first hearing, the "ugly" piece may sound like a typical 1950's serialist piece, but it has some characteristic features such as its sequence of single notes and its sempre forte articulation. Successful serialist pieces would be much more varied in texture.

The (claimed) absence of patterns in the piece is more extreme than would be a random sequence of notes. If notes had been drawn randomly from a uniform distribution, there is some probability of immediate repetition of notes as well as of repeated sequences of intervals. When someone tries to improvise a sequence of random numbers, say, just the numbers 0, 1, they would typically exaggerate the occurrences of changes and generate too little repetition. True randomness is more orderly than our human conception of it. In that sense the "ugly" piece agrees with our idea of randomness more than would an actually random sequence of notes.

When using Golomb rulers for rhythm generation, it may be practical to repeat the pattern instead of extending a Golomb ruler to the length of the entire piece. In the case of repetition the pattern occurs cyclically, so the definition of the ruler should change accordingly. Now we have a circular Golomb ruler (perhaps better known as a cyclic difference set) where the marks are put on a circle, and distances are measured along the circumference of the circle.

Although the concept of a Golomb ruler is easy for anyone to grasp, some generalization and a little further digging leads into the frontiers of mathematic knowledge with unanswered questions still to solve. 

And, of course, the Golomb rulers make excellent raw material for quirky music.

Reptilian Revolution

Almost unquantized music.

Q: Is this a concept album?
A: Yes. Its subject matter is not only derived from those entertaining kooks who see shapeshifting reptilians on u-tube videos with their very own eyes, hence they must exist; there are also references to more serious topics such as the unhealthy state of affairs alluded to in this previous post.

Manifesto for self-generating patches

Ideas for the implementation of autonomous instruments in analog modular synths (v. 0.2)

The following guidelines are not meant as aesthetic value judgements or prescriptions as to what people should do with their modulars — as always, do what you want! The purpose is to propose some principles for the exploration of a limited class of patches and a particular mode of using the modular as an instrument.

Self-generating patches are those which, when left running without manual interference, produce complex and varied musical patterns. Usually, the results will be more or less unpredictable. In this class of patches, there are no limitations as to what modules to use and how to connect them, except that one should not change the patch or touch any knobs after the patch has been set up to run. An initial phase of testing and tweaking is of course allowed, but if preparing a recording as documentation of the self-generating patch, it should just run uninterrupted on its own.

A stricter version of the same concept is to try to make a deterministic autonomous system in which there is no source of modulation (such as LFOs or sequencers) that is not itself modulated by other sources. In consequence, the patch has to be a feedback system.

The patch may be regarded as a network with modules as the nodes and patch cords as the links. Specifically, it is a bidirectional graph, because modules usually have both inputs and outputs. (The requirement that there be no source of modulation which itself is not modulated by other modules implies that, e.g., noise modules or LFOs without any input are not allowed.) Thus, in the graph corresponding to the patch, each node that belongs to the graph must have at least one incomming link and at least one outgoing link. The entire patch must be interconnected in the sense that one can follow the patch cords from any module through intervening modules to any other module that belongs to the patch.


Criterion of elegance:
The smaller the number of modules and patch cords used, the more elegant the patch is. (Caveat: modules are not straightforwardly comparable. There are small and simple modules with restricted possibilities, and modules with lots of features that may correspond to using several simpler modules.)

Aesthetic judgement:
Why not organize competitions where the audience may vote for their favourite patches, or perhaps let a panel of experts decide.

Standards of documentation:
Make a high quality audio recording with no post processing other than possibly volume adjustment. Video recordings and/or photos of the patch are welcome, but a detailed diagram explaining the patch and settings of all knobs and switches involved should be submitted. The diagram should provide all the information necessary to reconstruct the patch.

Criterion of robustness:
Try to reconstruct the patch with some modules replaced by equivalent ones. Swap one oscillator for another one, use a different filter or VCA and try to get a similar sound. Also try small adjustments of knobs and see whether it affects the sound in a radical way. The more robust a patch is, the easier it should be for other modular enthusiasts to recreate a similar patch on their system.

Criteria of objective complexity:
The patch is supposed to generate complex, evolving sounds, not just a static drone or a steady noise. Define your own musical complexity signal descriptor and apply it to the signal. Or use one of the existing complexity measures.

Dissemination:
Spread your results and let us know about your amazing patch!

How to patch your own oscillator

The charming world of analog modular synthesis offers many choices regarding how to construct one's instrument from components. There are lots of oscillators, filters, VCAs, LFOs, signal processors and utility modules to choose among. In that setting, it can be very interesting to build something as elementary as an oscillator out of even more basic components. Here is an example of how it can be done with two modules, neither of which functions as an oscillator on its own.

The modules needed are a utility module that mixes, offsets and inverts signals, and a dual slew limiter (or two separate slew limiters). In particular, this example will work with Doepfer's Slew Limiter A-170 SL and wmd's Invert Offset mk II. However, there is nothing magic about these modules, so other modules that offer equivalent functionality may replace them.

Five patch cords are needed to connect the modules as illustrated. Then, with some tweaking of the knobs, slow oscillations should occur. It is possible to influence the frequency by the settings of all the knobs. By adjusting the two lower knobs of A-170, controlling the rise and fall times, the wave shape can also be varied from rising ramp through triangle to falling ramp. The amplitude may be low, and the frequency usually sub-audio, although low bass frequencies in the audio range can be obtained. The effects are best observed if the CV out of the Invert Offset is routed to the frequency input of another oscillator.

What is actually going on in this patch? To a first approximation, the slew limiter can be regarded as an integrator. In fact, it is probably more accurate to think of it as a leaky integrator. The Invert Offset consists of two identical blocks with two signal inputs and two outputs each. Let us introduce the labels x₊, x_-, y₊ and y_- for the output signals, and u_x, u_y, v_x and v_y for the inputs, as shown in the sketch above. The knobs, labeled c_x and c_y, add a constant offset to the signal. Inferring from the user's manual, the following set of equations should describe what the module does.

Expressing the action of the slew limiter as an integral, and following the patch cords that go into the inputs of the Invert Offset module, the system is given by:

After a number of substitutions, and taking derivatives to get rid of the integrals, the system simplifies to:

If the constants are both zero, the eigenvalues of this system are 1±i, indicating that the system is unstable. Clearly something in the model is wrong, since the actual patch does not blow up in any way. As hinted at earlier, the slew limiters do not actually integrate the signal. If they did, there would be infinite gain at dc so any constant signal fed into one of them would keep increasing linearly. What happens in reality is that, starting from a relaxed state and feeding a constant signal into a slew limiter, the output grows from zero until it reaches the level of the input. If one had two true integrators and an inverter, the equations for an harmonic oscillator

could be realized quite easily. 

The moral of this failed attempt at modeling two quite simple modules is that even seemingly simple modules may hide more complex behaviour than one would naively suspect. In any case, it may be surprising to find that five patch cords connecting these modules in the right way are all it takes to turn them into a low frequency oscillator. Although there are more than one way to patch up an oscillator from these two modules, there are many more ways to patch up systems that do not oscillate. Bistable systems with hysteresis is the result in most cases.