True developments in the modular world can at times feel fleeting and infrequent, as the nature of modular synthesis lends itself naturally to horizontal iteration. Recreations of classic and canonized tools abound, as do nearly identical offerings for the basics—take a stroll through the endless array of "Quad VCAs" and "Multimode Filters" on offer these days and you're sure to see this trend on full display. This is all part of the fun of course, providing users with plentiful avenues to create their own music making systems using time tested materials, each with its own unique quirks, interface concepts, and other variations that achieve the bespoke designs and signal flows modular is best for. However, every once in a while, modular synthesis takes a turn toward the dialectical, moving the needle of history forward by resolving contradictions between old and new concepts in music technology, resulting in new hardware offerings that can ask new questions.

One recent example of such a development is from Fancyyyyy Synthesis, a unique music label / hardware design outfit based in Manchester, UK. Fancyyyyy the label is co-run by Adam Campbell, based in Berlin, alongside Tristan Clutterbuck who handles the hardware development side of the team. Centered around a cybernetically-inclined, systems-oriented approach, Fancyyyyy's offerings in both music and hardware follow suit, with a keen ear and sensibility for algorithmic structures that arise from bottom-up emergence rather than top-down planning and strict execution. Following up their fantastically flexible and inspiring Rung Divisions, a Hordijk-inspired Rungler concept perfect for creating all sorts of rhythmic chaos and complexity in a patch, Fancyyyyy has teamed up with DSP paragon Graham Wakefield to unleash the phenomenal K-Accumulator, a forward-thinking sound source that elegantly blends cutting-edge digital synthesis techniques with a classic, complex oscillator-inspired topology.

Instantiating in hardware multiple avant-garde practices like Pulsar synthesis, harmonic stretching and wavefolding, and unique phase modulation algorithms, K-Accumulator leverages both Tristan and Graham's energized engagement with computer music techniques while offering an interface that resonates strongly with established patching practices. Despite this nod toward the familiar, K-Accumulator opens up exactly the sort of patching workflows that make modular synthesis so inspiring as a format. Its core structure dances around an eight-node matrix of remapped panel controls and waveshaping destinations, deftly reshaping its internal logic to provide a vast world of sound that can be explored intuitively in each embodied interaction. The onboard Delta-Sigma pattern generator offers an at-hand source for emergent structure and parametric development, and its seamless connectivity with Rung Divisions goes even further to propose a veritable integrated solution for sonic experimentation. In short, the module is so fancy, you're going to start pronouncing all those y's—it just feels right.

Celebrating this new and quite formidable force on the Eurorack scene, we reached out to the duo behind K-Accumulator for more insight on its development and the inspired work that brought it to the world. Read the full interview below!

[All photos in this article by Roberta Pilleri]

Perfect Circuit: Tristan, K-Accumulator isn't your first foray into hardware design, and before that, Fancyyyyy was established as a record label. How did your background running a label lead you towards releasing Rung Divisions and designing hardware in general?

Tristan Clutterbuck: Fancyyyyy was co-founded with my great friend and collaborator Adam Campbell. We built it on a shared design sensibility and a taste for radical music and novel synthesis techniques. Adam has great taste and his judgement has shaped mine.

I've long thought that systems and musical output are not really separate things. A process for generating music and a finished piece of music are two views of the same object. So releasing a collection of music alongside systems to generate it felt natural; the label was already operating in that territory before the hardware existed.

Running a label involves things that matter when designing instruments: presentation, curation, knowing when something is finished. More than procedural details though, it helped me find a network of people whose judgement I trust and who encouraged me to finish projects that might otherwise have stayed half-built. Joe Gilmore, who did the graphic design for a number of our label releases, went on to work with us for the Eurorack side. The label is a kind of container for those relationships and helps sustain them even when we aren't in frequent contact. The hardware grew out of that network.

×

PC: How did the two of you meet, and when did you decide to work together to design K-Accumulator?

TC: Encountering gen~ during my post-grad studies sent me down a certain path. Before gen~ I was learning C++ to write MSP externals; gen~ made that unnecessary and gave me a more immediate relationship with signal processing. A formative experience was re-creating an Agostino Di Scipio "Audible Ecosystem" work in gen~, in a class led by Owen Green (now of FluCoMa). Di Scipio's work treats the feedback between a system and its acoustic environment as the composition itself. There is no fixed signal path, only coupled dynamical relationships producing emergent behavior. That idea, to design systems rather than sequences, has shaped a lot of what I've done since.

Graham Wakefield: Interestingly, Di Scipio’s ecosystemic work—along with other cybernetic influences – was also a formative influence in my doctoral research – which, through a rather convoluted path, ended up at the creation of gen~ (and also my bio-inspired installation work with Haru Ji in Artificial Nature!

TC: I started using gen~ as a way to experiment with hardware instrument design without having to breadboard or spin out PCBs. Among other things, I used it to design and test the logic for Rung Divisions. Adam tipped me off that there was a thread about building Rung Divisions in gen~ on the Cycling '74 Discord. That's how we ended up here.

GW: Yes—I had already been keenly following the Rung Divisions from Tristan's earliest posts about it, well before the final module was released. I’d long been interested in Runglers and other shift register algorithms as pattern generators, which is why we included a section about them in the GO book, and it was in the midst of those conversations on the gen~ Discord channel that Tristan got in touch with me. It blew my mind that the Rung Divisions CMOS logic had been prototyped in gen~! So the meeting was like closing many loops, really. The more we talked the more we found we were on the same page in so many ways; not just techniques, but what synthesis is actually for, and what it could be.

Tristan told me that he had wanted to do a phase modulation oscillator for a long time, and was thinking about building upon some of the PM ideas in the GO book, so we started from an initial oscillator module idea of maybe 10-12hp. Over 18 months we were passing Max patches, videos of ideas, panel designs and endless discussions back & forth, and it grew into the 32hp K-Accumulator. The design kept revealing that it needed more room, not because of feature creep but because of the ways that the algorithms kept opening into each other.

PC: When you started designing this module, what aspects of the complex oscillator archetype were important to keep in place, and what did you feel were worth exploring alternative approaches or extending capabilities? (Or perhaps, in what ways do you feel the final design is similar and dissimilar to a classic "complex oscillator"?)

TC: For me the question was less about what to keep and what to throw away, and more about finding a path from a traditional hardware complex oscillator (two oscillators with AM, FM and sync between them) into digital synthesis techniques that use the same underlying principles. I had been working on an analog PM/pulsar complex oscillator for a long time, but kept bumping against the limits of what's practically feasible in analog circuitry. I originally started working with the Daisy to implement the mod oscillator of that older design as a digital function generator, keeping the carrier core analog. This was before collaborating with Graham, and before seeing how much more we could do if the whole design was digital and every element could talk to itself in 32-bit floats at 96KHz.

GW: In broad terms of course there’s a debt to Buchla (including the sometimes overlooked fact that Buchla developed a lot of digital waveshaping algorithms), and also to Serge in the UFG. But honestly I personally wasn’t focused on any historical designs at all. I wanted to treat the domain and the digital platform from first principles rather than from historical recreation, and try to seriously examine what could be the most interesting things to do with arbitrary slope generation, phase modulation and frequency shifting, and to open them up as much as possible.

I'm hugely indebted to Curtis Roads, my doctoral supervisor, and a lineage of radical thinking about what signal processing can offer that goes back to Iannis Xenakis and early cybernetics. The through-line there is a view of synthesis that is not primarily about producing familiar timbres or reproducing acoustic instruments. Rather, it is a means of exploring dynamical systems, of working with processes whose behavior isn’t prescribed by prior conventions. I think that perspective is more urgent now than it was in Xenakis's time. We don't teach even the basics of system dynamics in schools, but modular synthesis (and environments like Max) offer a way to develop physical intuition for the nature of feedback, tipping points, and the limitations of prediction, not just in the mind but also under the fingers. That kind of understanding matters well beyond music.

TC: Early in the design, Graham's partner Haru Ji lent us this word "bicameral" that we kept coming back to. There is something compelling about the two-chambered coupled oscillator, controlled by two hands, discerned by two lobes. We owe a lot to those original designs, and retain the core principle and some design cues. But the structure of K-A became more fluid and less hierarchical.

From the start we were aligned that the user interface should feel as analog as possible, and that is probably what K-A has most in common with the complex oscillators of the 20th century. The controls are continuous, and nothing is hidden behind menus. As we're both committed to attenuverters on as many parameters as possible (cybernetics and feedback demand attenuation and inversion) it felt important to have a built-in modulation source normalled to all the inputs. The modulation source grew and morphed over eighteen months until it became the UFG and Delta-Sigma. At which point we had three oscillators, not two, and the design needed a different organizing principle.

The root frequency sets how fast the core phasors accumulate, but the arrangement of the three oscillators inside those time increments is flexible; sometimes the UFG or the Mod sets the root frequency and the Main Osc sets an overtone (or undertone). The deep PM and cross-PM can completely blur the line between modulator and carrier. As the design grew, the bicameral metaphor faded in favor of lines and knots, with the trefoil knot as a touchstone for the topology we were trying to achieve.

×

PC: K-Accumulator's panel suggests novel connections to some familiar synthesis and/or DSP techniques. Can you give us a quick rundown of each section (UFG, Delta-Sigma pattern generator, Root, Mod, and OSC), and explain the device's overall signal flow / how these individual sections all fit together?

GW: The signal flow is simpler than the panel might suggest. The UFG, Delta-Sigma, and Modulator were designed as modulation sources for the main oscillator, but as the design developed they became rich circuits in their own right, and you can use them quite independently. Still, the UFG and Delta-Sigma are available as normalled signals to the Oscillator and Modulator; and the Modulator is the primary source of phase modulation in the 2OP and XPM nodes of the main oscillator.

The UFG borrows inspiration of flexibility from the Serge slope generator. Like the Serge, it works equally well as an LFO, an envelope (AD or ASR with different retrigger options), or as an audio-rate oscillator with a variety of sync responses. Unlike the Serge, it holds stable tuning regardless of pulsewidth and shape, and supports through-zero FM. A lot of work went into making it this flexible but still cleanly antialiased at high frequencies. There’s some very powerful capabilities hidden in this – in particular, the TZFM and Trig inputs both respond extremely well right up to audio rates, and with the different sync modes you can get some really unique sounds out of it.

The UFG is normalled to the Damped Sync/Pulsar input of the Oscillator, where it can do hard sync and AM enveloping, becoming Pulsar synthesis at pitch rates. The sync here is a bit novel: you can morph out of it by turning down the knob. This is implemented using filtered phase modulation, so the exit is gradual rather than abrupt, with some interesting PLL-like behaviors and sometimes chaotic subharmonic locking at points in between. It's one of many places in the K-Accumulator where we've used continuous transitions between regimes rather than distinct modes. Those in-between, liminal spots can be remarkably responsive and characterful.

The Modulator is all about harmonics. It can track the OSC, the UFG, or just the Root, and has several ways of selecting harmonics according to the Order knob: standard harmonic and subharmonic ratios, options for blending smoothly between nearest harmonics, even only, odd only, and some more unusual patterns of 3rd- and 4th-order harmonics. It also has a detune knob to introduce beat frequency relationships down to very subtle differences.

×

PC: The labels on the ring of LEDs accompanying the Morph control seem to suggest K-Accumulator can accommodate a large number of distinct-but-related synthesis methods/arrangements. How does this control actually impact K-Accumulator's sound generation structure? (Is there an underlying conceptual commonality between these techniques that K-Accumulator exploits to enable this seemingly multi-modal behavior?)

GW: There are two ways to look at it. On the surface, it is a macro-oscillator: each of the eight nodes around the ring has a distinct behavior and sonic character. Underneath, it is a continuous modulation matrix, routing signals to different aspects of the same core algorithm with different weights and filtering at each point.

What that means is that these nodes are not presets or separate algorithms. Rather, they are extremities on a continuous parameter manifold, and the space between them is as important as the nodes themselves. We were both very clear that it should be possible to morph freely around the ring under CV, which makes Morph function more like a macro-controller than a preset selector. And it’s a lot of fun to modulate it!

TC: The principle of continuity runs through the whole design. We wanted the user to be able to smoothly sweep between audio and sub-audio range, and to simultaneously morph between different algorithms and sync behaviors. The parameters are frequency-aware, so the behavior adapts as you move through that range rather than breaking at a boundary. The structure is more like a folded surface than a set of modes, something you can learn to traverse by feel rather than selecting options from a menu. This was a deliberate choice against the current trend in digital modules toward mode-switching and discrete algorithms. Even within a single morph position, the three waveshaping parameters interact with each other and with the core algorithm; as you move through the morph, those parameters are remapped to different filters and properties of the oscillator core. Every morph position shares the same static center: a sine / cosine pair. But as soon as you leave this, the whole instrument is in motion.

This use of continuous parameters has a specific design goal behind it. Some of the most musically interesting behavior on the instrument tends to be at thresholds, between audio and sub-audio rate, between linear and chaotic feedback, between locked and drifting harmonic relationships, and patterns at the edge of predictability. We wanted those thresholds to be reachable easily, with the path back to stability built into the controls that take you across them. This shows up everywhere on the module: damped sync that morphs to hard sync rather than breaking into it; harmonic shifting that blends through integer harmonics rather than skipping; feedback paths with damping filters so the chaotic regimes are playable.

GW: One thing that helps here is that the entire oscillator core runs in quadrature. So it is also a “complex oscillator” in the mathematical sense: all of the processing stages, the harmonic shift, the stretch, the phase modulation, the waveshaping, are entirely computed using complex numbers (quadrature sine/cosine signals). This opened up several things that would not have been possible otherwise, and it is part of what gives the transitions between nodal morph points of the manifold their particular character.

PC: Could you briefly explain the parameters in the OSC section? How do terms like damping, Stretch, Shift, Depth, and Shape relate to the sound generation?

GW: What the Shift, Depth and Shape knobs control depends on the Morph position, but there are common threads.

In most modes, Shift engages a blended harmonic frequency shift, picking out higher harmonics of the oscillator core. It can sound like multiphonic effects on a wind instrument, or in some cases like a very glassy sync. The Modulator also has this kind of blended harmonic shifting capability, and the two combined in 2OP can be really rich.

Depth is primarily the phase modulation index. Mostly this is applied using filtering rather than attenuation. The difference is a bit like the difference between a VCA and an LPG: the filtering produces some very organic sounds, especially in the self-feedback and cross-feedback PM. What drives the PM is a mix that varies according to Morph.

In many modes the Shape control controls a wavefolder, but this wavefolder is a little different to a typical design: it’s created from two wavefolders crossfading between each other, where each wavefolder is a precise harmonic multiple of the fundamental. This has an interestingly different character when the folding depth is being modulated, in some cases more glassy, less ripply than a typical wavefolder, and in other cases, it introduces a kind of harmonic distortion. Like the rest of the oscillator core, the wavefolder runs in quadrature, and borrows a little bias from the Detune knob to induce spatial rotation.

Stretch is perhaps the most unusual parameter. It uses a frequency shifting effect to stretch the spacing between harmonics used by the wavefolder and harmonic shift processes. What’s unusual is that it doesn’t make the oscillator fall out of tune. Normally, frequency shifting will stretch or compress the entire spectrum, knocking everything out of tune. Instead, on the K-Accumulator, we can stretch the spacing between harmonics relative to the fundamental frequency, so the sense of tuning remains stable; the fundamental doesn't move. This also means Stretch usually only has an apparent effect once the harmonic shift or wavefolder is engaged. The knob is biased toward near-integer stretch ratios, where harmonics line up closely with the regular harmonic series but perhaps very slightly detuned. This is a bit like how the strings of a piano are not tuned to perfect harmonic ratios; it can add a lot of richness.

PC: Can you explain a bit more about the Delta-Sigma pattern generator? How does it work, and how do you envision it being used?

GW: The Delta-Sigma (Δ–Σ) section was primarily designed as a pattern generator, but it varies between stepped and smooth, and changes its behavior at audio rates to become a more useful modulation source for the oscillator. There’s also a lot of behind-the-scenes care taken to make this really effective as a pitch control for the oscillator.

We're both interested in pattern generation algorithms like shift registers and sample-and-hold interference patterns, but we were looking for something different here. The generator is loosely inspired by the delta-sigma encoding algorithm used in PT2399 chips, but it's really a hybrid between an analog shift register and a modulo remainder operator. The specifics matter less than the behavior: it functions similarly to many analog shift register pattern generators, with a clock trigger and a data source, along with chance variation and length control. The clock source is the UFG, or an external clock; the data source is the Modulator. You can also get quite deterministic with it if you want to by having the Modulator track the UFG. The delta-sigma method means that even with a static modulator, the pattern continues to change. And where binary shift registers like the Turing Machine can sometimes produce recognisable ascending and descending patterns, this algorithm tends to produce more diverse variations.

Turn Chance right down and you get a steady loop. Bring it up a little and variations drop in from time to time. Unlike a typical shift register, these variations are not permanent; turn Chance back down and the original loop returns. The only thing that permanently changes the stored data is the Mutate switch, and even then you can revert with Undo. So it acts more like a live looper in that way.

Smooth applies glide to certain steps in the loop, always the same steps if Chance is at zero, so you can get 303-style patterns. Bring Smooth all the way up for a continuously wavering modulation. Another approach is to sequence changes to Length: expanding the length wanders into different sections of the loop, contracting it focuses on a subsection. You can get quite interesting wandering, repeating melodies this way.

TC: Δ–Σ reflects a logic that runs through the whole instrument, the ability to drift away from a stable position and find your way back. You can take a locked loop, walk away from it through chance and length changes, and bring the original pattern back. The Δ–Σ also illustrates the audio/sub-audio threshold idea; at sub-audio rate it's a stepped-and-glided pattern generator clocked by the UFG, so together they form an envelope-and-pattern structure. As Time on the UFG crosses into audio rate, the Δ–Σ changes character, becoming a shifting harmonic oscillator better suited as a modulation source for audio rate FM.

PC: How about the Root section? How does this specifically relate to the signal generation structure? Also, how do you handle/define Just Intonation with respect to K-Accumulator's functionality?

GW: The Root section defines a pitch center and quantization structure, which can be applied to the main oscillator, the UFG, both, or none. It works with regular equal temperament notes and common Western scales if you want it to, but it doesn't have to. The pitch center tracks 1V/octave, and the main knob can function as an octave switch, fine tune, free coarse tune, or semitone coarse tune. Setting fine tune to either extreme gives you a C reference.

The Scale knob picks out different scales in TET or JI modes. At both extremes you always get octaves; moving toward noon gives more divisions of the octave, reaching semitones at noon in TET mode. For TET in particular, 1V/octave CV to Root is like transposing or changing key, whereas 1V/octave CV to OSC is like arpeggiation or melodic exploration within the key. Sequencing both can produce quite elaborate behavior.

The JI side goes deeper, especially since there are harmonic relationships running through the synthesis behaviors already. The left side of the knob picks out mostly low-order harmonic ratios, well-suited for FM/PM, perhaps also clock divisions, etc. Noon is the most dense, with 32 divisions of the octave including all harmonics up to denominator 12. The right side of the knob has a selection of microtonal JI scales that include higher-order denominators including Pythagorean, 11-note Partch Primary Ratios, and some JI scales that are close to common TET scales, and so on. The Modulator, the oscillator shift, stretch, and wavefolder all use harmonic ratio blending in different ways, and the JI quantization interacts with all of them.

×

PC: Graham, as mentioned in our previous interview with you, you're the author of Max's gen~ environment and Oopsy—a package for exporting gen~ patches to Electrosmith Daisy hardware. Is it safe to assume that gen~ and Oopsy were part of the development workflow for K-Accumulator?

GW: Yes, gen~, Oopsy, and libDaisy were essential: we developed all of the algorithms in gen~ patches.

TC: The build output for K-Accumulator is really chimeric; we extended both Oopsy and libDaisy to get everything running on the hardware. But beyond the build chain, Max was how we actually designed the instrument. Being able to split off a smaller section of the design, model it in a gen~ patch, and send it back and forth was how many of the decisions were made. It gave us a shared workspace for experimentation that would have been impossible working directly in C++.

PC: What was the division of responsibilities like during the development process? Was Graham focused more on developing the software and functionality of K-Accumulator, while Tristan worked on the PCB layout and other hardware concerns? Or were the lines a bit more blurred than that?

TC: The creative direction was genuinely shared; we passed patches, videos and ideas back and forth constantly, and the design thinking is blurred to the point where it's hard for me to tell where certain ideas began. Where the lines are clearer: the DSP architecture and the gen~ code are Graham's. That’s his domain, and K-Accumulator would not sound the way it does without it. Several of the techniques in the module are, to my knowledge, new: blended harmonic frequency shifting, harmonic wavefolding, the harmonic stretch algorithm, the filtered cross-coupled phase modulation, and a number of other neat touches have not been implemented in hardware before. Some of them haven't been implemented anywhere before. The work extending Oopsy and libDaisy to support what we needed was largely shared. The instrument design sits between all those things and belongs to both of us.

GW: I think that the damped sync method is new to hardware too—at least as far as I’m aware. But this all emerged through the collaboration, I also can’t really remember where different ideas came from, at least in the gen~ patching realm. But I can safely say that the analog circuitry, PCB and panel layout is 100% Tristan—and so was the amazing development work getting the two Daisy boards to synchronize and share signals with each other!

PC: As the development process and scope of K-Accumulator evolved, what prompted the choice to leverage a dual Daisy design?

GW: We both wanted the algorithms running at 96kHz and extremely low latency (8 or 24 samples). The oscillator core has a latency of just 0.25ms, which is fantastic for feedback patching. The UFG core’s latency is even quicker, at 0.08ms. The high sample rate is about sonic fidelity. A lot of work went into anti-aliasing the synthesis algorithms in the K-Accumulator, but there is no substitute for what higher sampling rates give you.

We pushed code optimisation to extremes, but there was simply no way to get those sample rates and latencies on a single Daisy and still keep all the behaviors we wanted. The dual design was a necessity more than a choice!

TC: The dual Daisy architecture was one of the biggest engineering challenges in the project. Getting two Daisy processors to synchronise tightly enough that they behave as a single instrument, sharing audio with sample-accurate timing, required substantial work extending the platforms we were working on. It's not something you get for free, and brought with it a host of unexpected challenges. But 96kHz and low latency were not negotiable.

×

PC: Assuming that K-Accumulator was initially prototyped within Max on a computer, what were the biggest or most unique challenges in migrating over to embedded hardware with the Daisy?

TC: Performance. I didn't think it would be possible to run at 96kHz with a block size of 24 samples on the OSC side until a few days before we shipped hardware. Beyond that, the STM ADCs are notoriously noisy, and this introduces real design challenges when porting code from the computer to the Daisy. There are ways to mitigate this, but they all have their own costs.

GW: Analog noise is especially challenging if you are coming from a primarily digital background. Things you design in the mathematically pure realm of the computer can break in unexpected ways when analog signals get involved. A lot of our work was resolving these without losing what makes modular patching interesting!

PC: We've got to ask…what's the meaning behind the name? Obviously, "k" has a special meaning in certain branches of mathematics—but how does it relate to the concept(s) behind K-Accumulator?

TC: A good name suggests rather than explains – it’s hyperstitional.

PC: Tell us about the connectivity with Rung Divisions. What sort of modular magic is unlocked using these two modules together?

TC: Rung Divisions and Delta-Sigma are two different approaches to a similar idea: a deterministic but unpredictable pattern generator with clock and data inputs. The connection behind the panel sends the UFG gate output as the clock source for Rung Divisions and the Mod oscillator as the data source, mirroring how they're patched to the Delta-Sigma. This gives you another set of related, clock-divided CV outputs from Rungs to break back into K-A. There are a lot of interesting cross-patching possibilities between the two modules; Graham has made a fantastic video exploring them:

The original design had normalled feedback from Rungs outputs to K-A inputs. As K-A grew, we ended up adding internally normalled connections to almost all of its CV inputs to allow for more interesting standalone behavior. That meant we couldn't implement the normalled Rungs feedback, but it was the right decision. There are things that would have been impossible without the internal normals, like the random walk on the UFG Time CV.

PC: Graham, as you're well aware, a handful of us at Perfect Circuit are long-time Max users, and I think it's safe to say that some of our personal work/music has been pretty hugely impacted by Generating Sound and Organizing Time, your book about DSP concepts in gen~. We have to ask—how is the second volume coming along?

GW: It’s going great, and should be available later this year. It took longer than expected, but I think it is better for it! A lot of things developed for both GO books have found their way into the K-Accumulator (and some things discovered along the way have informed book 2). There’s also more new ideas developed in book 2 that are supporting some of the next modules Tristan & I are designing together, which I’m very excited about!

PC: What's the best way for folks to check out more information about K-Accumulator?

TC: Beyond the quick start guide, a full written manual and a video manual series are in progress, going into more depth on the topics covered in this interview. Be sure to sign up to the mailing list to be notified when these are released.