Some deep analysis of one-line music programs.

It is now a month since I posted the YouTube video "Experimental music from very short C programs" and three weeks since I blogged about it. Now that the initial craze seems to be over, it's a good time to look back what has been done and consider what could be done in the future.

The developments since my last post can be summarized by my third video. It still represents the current state of the art quite well and includes a good variety of different types of formulas.

The videos only show off a portion of all the formulas that could be included. To compensate, I've created a text file where I've collected all the "worthy" formulas I've encountered so far. Most of them can be tested in the on-line JavaScript and ActionScript test tools. Some of them don't even work directly in C code, as they depend on JS/AS-specific features.

As I'm sure that many people still find these formulas rather magical and mysterious, I've decided to give you a detailed technical analysis and explanation on the essential techniques. As I'm completely self-educated in music theory, please pardon my notation and terminology that may be unorthodox at times. You should also have a grasp of C-like expression syntax and binary arithmetic to understand most of the things I'm going to talk about.

I've sorted my formula collection by length. By comparing the shortest and longest formulas, it is apparent that the longest formulas show a much more constructivist approach, including musical data stored in constants as well as entire piece-by-piece-constructed softsynths. The shortest formulas, on the other hand, are very often discovered via non-deterministic testing, from educated guesses to pure trial-and-error. One of my aims with this essay is to bring some understanding and determinism to the short side as well.

Pitches and scales

A class of formulas that is quite prominent among the shortest ones is what I call the 't* class'. The formulas of this type multiply the time counter t with some expression, resulting in a sawtooth wave that changes its pitch according to that expression.

A simple example of a t*-class formula would be t*(t>>10) which outputs a rising and falling sound (accompanied by some aliasing artifacts that create their own sounds). Now, if we introduce an AND operator to this formula, we can restrict the set of pitches and thus create melodies. An example that has been individually discovered by several people, is the so-called "Forty-Two Melody": t*(42&t>>10) or t*2*(21&t>>11).

The numbers that indicate pitches are not semitones or anything like that, but multiplies of a base frequency (sampling rate divided by 256, i.e. 31.25 Hz at the default 8 kHz rate). Here is a table that maps the integer pitches 1..31 to cents and Western note names. The pitches on a gray background don't have good counterparts in the traditional Western system, so I've used quarter-tone flat and sharp symbols to give them approximate names.

By using this table, we can decode the Forty-Two Melody into a human-readable form. The melody is 32 steps long and consists of eight unique pitch multipliers (including zero which gives out silence).

The "Forty-Two Melody" contains some intervals that make it sound a little bit silly, detuned or "Arabic" to Western ears. If we want to avoid this effect, we need to design our formulas so that they only yield pitches that are at familiar intervals from one another. A simple solution is to include a modulo operator to wrap larger numbers to the range where simple integer ratios are more probable. Modifying the Forty-Two Melody into t*((42&t>>10)%14), for example, completely transforms the latter half of the melody into something that sounds a little bit nicer to Western ears. Bitwise AND is also useful for limiting the pitch set to a specific scale; for example t*(5+((t>>11)&5)) produces pitch multipliers of 4, 5, 8 and 9, which correspond to E3, G3, C4 and D4.

Ryg's 44.1 kHz formula presented in the third video contains two different melody generators:

((t*("36364689"[t>>13&7]&15))/12&128)

+(((((t>>12)^(t>>12)-2)%11*t)/4|t>>13)&127)

The first generator, in the first half of the formula, is based on a string constant that contains a straight-forward list of pitches. This list is used for the bass pattern. The other generator, whose core is the subexpression ((t>>12)^(t>>12)-2)%11, is more interesting, as it generates a rather deep self-similar melody structure with just three operators (subtraction, exclusive or, modulo). Rather impressive despite its profound repetitiveness. Here's an analysis of the series it generates:

It is often a good idea to post-process the waveform output of a plain t* formula. The sawtooth wave tends to produce a lot of aliasing artifacts, particularly at low sampling rates. Attaching a '&128' or '&64' in the end of a t* formula switches the output to square wave which usually sounds a little bit cleaner. An example of this would be Niklas Roy's t*(t>>9|t>>13)&16 which sounds a lot noisier without the AND (although most of the noise in this case comes from the unbounded multiplication arithmetic, not from aliasing).

Bitwise waveforms and harmonies

Another class of formulas that is very prominent among the short ones is the bitwise formula. At its purest, such a formula only uses bitwise operations (shifts, negation, AND, OR, XOR) combined with constants and t. A simple example is t&t>>8 -- the "Sierpinski Harmony". Sierpinski triangles appear very often in plotted visualizations of bitwise waveforms, and t&t>>8 represents the simplest type of formula that renders into a nice Sierpinski triangle.

Bitwise formulas often sound surprisingly multitonal for their length. This is based on the fact that an 8-bit sawtooth wave can be thought of consisting of eight square waves, each an octave apart from its neighbor. Usually, these components fuse together in the human brain, forming the harmonics of a single timbre, but if we turn them on and off a couple of times per second or slower, the brain might perceive them as separate tones. For example, t&48 sounds quite monotonal, but in t&48&t>>8, the exactly same waveform sounds bitonal because it abruptly extends the harmonic content of the previous waveform.

The loudest of the eight square-wave components of an 8-bit wave is, naturally, the one represented by the most significant bit (&128). In the sawtooth wave, it is also the longest in wavelength. The second highest bit (&64) represents a square wave that has half the wavelength and amplitude, the third highest halves the parameters once more, and so on. By using this principle, we can analyze the musical structure of the Sierpinski Harmony:

The introduction of ever lower square-wave components can be easily heard. One can also hear quite well that every newly introduced component is considerably lower in pitch than the previous one. However, if we include a prime multiplier in the Sierpinski Harmony, we will encounter an anomaly. In (t*3)&t>>8, the loudest tone actually goes higher at a specific point (and the interval isn't an octave either).

This phenomenon can be explained with aliasing artifacts and how they are processed by the brain. The main wavelength in t*3 is not constant but alternates between two values, 42 and 43, averaging to 42.67 (256/3). The human mind interprets this kind of sound as a waveform of the average length (42.67 samples) accompanied by an extra sound that represents the "error" (or the difference from the ideal wave). In the t*3 example, this extra sound has a period of 256 samples and sounds like a buzzer when listened separately.

The smaller the wavelengths we are dealing with are, the more prominent these aliasing artifacts become, eventually dominating over their parent waveforms. By listening to (t*3)&128, (t*3)&64 and (t*3)&32, we notice an interval of an octave between them. However, when we step over from (t*3)&32 to (t*3)&16, the interval is definitely not an octave. This is the threshold where the artifact wave becomes dominant. This is why t&t>>8, (t*3)&t>>8 and (t*5)&t>>8 sound so different. It is also the reason why high-pitched melodies may sound very detuned.

Variants of the Sierpinski harmony can be combined to produce melodies. Examples of this approach include:

t*5&(t>>7)|t*3&(t*4>>10) (from miiro)

(t*5&t>>7)|(t*3&t>>10) (from viznut)

t*9&t>>4|t*5&t>>7|t*3&t/1024 (from stephth)

Different counters are the driving force of bitwise formulas. At their simplest, counters are just bitshifted versions of the main counter (t). These are implicitly synchronized with each other and work on different temporal levels of the musical piece. However, it has also been fruitful to experiment with counters that don't have a simple common denominator, and even with ones whose speeds are nearly identical. For example, t&t%255 brings a 256-cycle counter and a 255-cycle counter together with an AND operation, resulting in an ambient drone sound that sounds like something achievable with pulse-width modulation. This approach seems to be more useful for loosely structured soundscapes than clear-cut rhythms or melodies.

Some oneliner songs attach a bitwise operation to a melody generator for transposing the output by whole octaves. A simple example is Rrrola's t*(0xCA98>>(t>>9&14)&15)|t>>8 which would just loop a simple series of notes without the trailing '|t>>8'. This part gradually fixes the upper bits of the output to 1s, effectively raising the pitch of the melody and fading its volume out. Also the formulas from Ryg and Kb in my third video use this technique. The most advanced use of it I've seen so far, however, is in Mu6k's song (the last one in the 3rd video) which synthesizes its lead melody (along with some accompanying beeps) by taking the bassline and selectively turning its bits on and off. This takes place within the subexpression (t>>8^t>>10|t>>14|x)&63 where the waveform of the bass is input as x.

Modular wrap-arounds and other synthesis techniques

All the examples presented so far only use counters and bitwise operations to synthesize the actual waveforms. It's therefore necessary to talk a little bit about other operations and their potential as well.

By accompanying a bitwise formula with a simple addition or substraction, it is possible to create modular wrap-around artifacts that produce totally different sounds. Tiny, nearly inaudible sounds may become very dominant. Harmonious sounds often become noisy and percussive. By extending the short Sierpinski harmony t&t>>4 into (t&t>>4)-5, something that sounds like an "8-bit" drum appears on top of it. The same principle can also be applied to more complex Sierpinski harmony derivatives as well as other bitwise formulas:

(t*9&t>>4|t*5&t>>7|t*3&t/1024)-1

I'm not going into a deep analysis of how modular wrap-arounds affect the harmonic structure of a sound, as I guess someone has already done the math before. However, modular addition can be used for something that sounds like oscillator hard-sync in analog synthesizers, although its technical basis is different.

Perhaps the most obvious use for summing in a softsynth, however, is the one where modular wrap-around is not very useful: mixing of several sound sources together. A straight-forward recipe for this is (A&127)+(B&127), which may be a little long-winded when aiming at minimalism. Often, just a simple XOR operation is enough to replace it, although it usually produces artifacts that may sound good or bad depending on the case. XOR can also be used for effects that sound like hard-sync.

Of course, modular wrap-around effects are also achievable with multiplication and division, and on the other hand, even without addition or subtraction. I'll illustrate this with just a couple of interesting-sounding examples:

t>>4|t&((t>>5)/(t>>7-(t>>15)&-t>>7-(t>>15))) (from droid, js/as only)

(int)(t/1e7*t*t+t)%127|t>>4|t>>5|t%127+(t>>16)|t (from bst)

t>>6&1?t>>5:-t>>4 (from droid)

There's a lot in these and other synthesis algorithms that could be discussed, but as they already belong to a zone where traditional sound synthesis lore applies, I choose to go on.

Deterministic composition

When looking at the longest formulas in the collection, it is apparent that there's a lot of intelligent design behind most of them. Long constants and tables, sometimes several of them, containing scales, melodies, basslines and drum patterns. The longest formula in the collection is "Long Line Theory", a cover of the soundtrack of the 64K demo "Chaos Theory" by Conspiracy. The original version by mu6k was over 600 characters long, from which the people on Pouet.net optimized it down to 300 characters, with some arguable quality tradeoffs.

It is, of course, possible to synthesize just about anything with a formula, especially if there's no upper limit for the length. Synthesis and sequencing logic can be built section by section, using rather generic algorithms and proven engineering techniques. There's no magic in it. But on the other hand, there's no magic in pure non-determinism either: it is very difficult to find anything outstanding with totally random experimentation after the initial discovery phase is over.

Many of the more sophisticated formulas seem to have a good balance between random experimentation and deterministic composition. It is often apparent in their structure that some elements are results of random discoveries while others have been built with an engineer's mindset. Let's look at Mu6k's song (presented in the end of the 3rd video, 32 kHz):

(((int)(3e3/(y=t&16383))&1)*35) +

(x=t*("6689"[t>>16&3]&15)/24&127)*y/4e4 +

((t>>8^t>>10|t>>14|x)&63)

I've split the formula on three lines according to the three instruments therein: drum, bass and lead.

My assumption is that the song has been built around the lead formula that was discovered first, probably in the form of t>>6^t>>8|t>>12|t&63 or something (the original version of this formula ran at 8 kHz). As usual with pure bitwise formulas, all the intervals are octaves, but in this case, the musical structure is very nice.

As it is possible to transpose a bit-masking melody simply by transposing the carrier wave, it's a good idea to generate a bassline and reuse it as the carrier. Unlike the lead generator, the bassline generator is very straight-forward in appearance, consisting of four pitch values stored in a string constant. A sawtooth wave is generated, stored to a variable (so that it can be reused by the lead melody generator) and amplitude-modulated.

Finally, there's a simple drum beat that is generated by a combination of division and bit extraction. The extracted bit is scaled to the amplitude of 35. Simple drums are often synthesized by using fast downward pitch-slides and the division approach does this very well.

In the case of Ryg's formula I discussed some sections earlier, I might also guess that the melody generator, the most chaotic element of the system, was the central piece which was later coupled with a bassline generator whose pitches were deliberately chosen to harmonize with the generated melody.

The future

I have been contacted by quite many people who have brought up different ideas of future development. We should, for example, have a social website where anyone could enter new formulas, listen to the in a playlist-like manner and rate them. Another branch of ideas is about the production of new rateable formulas by random generation or by breeding old ones together with genetic algorithms.

All of these ideas are definitely interesting, but I don't think the time is yet right for them. I have been developing my audiovisual virtual machine, which is the main reason why I did these experiments in the first place. I regard the current concept of "oneliner music" as a mere placeholder for the system that is yet to be released. There are too many problems with the C-like infix syntax and other aspects of the concept, so I think it's wiser to first develop a better toy and then think about a community mechanism. However, these are just my own priorities. If someone feels like building the kind of on-line community I described, I'll support the idea.

I've mentioned this toy before. It was previously called EDAM, but now I've chosen to name it IBNIZ (Ideally Bare Numeric Impression giZmo). One of the I letters could also stand for "immediate" or "interactive", as I'm going to emphasize an immediate, hands-on modifiability of the code. IBNIZ will hopefully be relevant as a demoscene platform for extreme size classes, as a test bed for esoteric algorithmic trickery, as an appealing introduction to hard-core minimalist programming, and also as a fun toy to just jam around with. Here's a little screenshot of the current state:

In my previous post, I mentioned the possibility of opening a door for 256-byte demos that are interesting both graphically and musically. The oneliner music project and IBNIZ will provide valuable research for the high-level, algorithmic aspects of this project, but I've also made some

hands-on tests on the platform-level feasability of the idea. It is now apparent that a stand-alone MS-DOS program that generates PCM sound and synchronized real-time graphics can easily fit in less then 96 bytes, so there's a lot of room left for both music and graphics in the 256-byte size

class. I'll probably release a 128- or 256-byte demo as a proof-of-concept, utilizing something derived from a nice oneliner music formula as the soundtrack.

I would like to thank everyone who has been interested in the oneliner music project, as all the hype made me very determined to continue my quests for unleashing the potential of the bit and the byte. My next post regarding this quest will probably appear once there's a version of IBNIZ worth releasing to the public.

Algorithmic symphonies from one line of code -- how and why?

Lately, there has been a lot of experimentation with very short programs that synthesize something that sounds like music. I now want to share some information and thoughts about these experiments.

First, some background. On 2011-09-26, I released the following video on Youtube, presenting seven programs and their musical output:

This video gathered a lot of interest, inspiring many programmers to experiment on their own and share their findings. This was further boosted by Bemmu's on-line Javascript utility that made it easy for anyone (even non-programmers, I guess) to jump in the bandwagon. In just a couple of days, people had found so many new formulas that I just had to release another video to show them off.

Edit 2011-10-10: note that there's now a third video as well! http://www.youtube.com/watch?v=tCRPUv8V22o

It all started a couple of months ago, when I encountered a 23-byte C-64 demo, Wallflower by 4mat of Ate Bit, that was like nothing I had ever seen on that size class on any platform. Glitchy, yes, but it had a musical structure that vastly outgrew its size. I started to experiment on my own and came up with a 16-byte VIC-20 program whose musical output totally blew my mind. My earlier blog post, "The 16-byte frontier", reports these findings and speculates why they work.

Some time later, I resumed the experimentation with a slightly more scientific mindset. In order to better understand what was going on, I needed a simpler and "purer" environment. Something that lacked the arbitrary quirks and hidden complexities of 8-bit soundchips and processors. I chose to experiment with short C programs that dump raw PCM audio data. I had written tiny "/dev/dsp softsynths" before, and I had even had one in my email/usenet signature in the late 1990s. However, the programs I would now be experimenting with would be shorter and less planned than my previous ones.

I chose to replicate the essentials of my earlier 8-bit experiments: a wave generator whose pitch is controlled by a function consisting of shifts and logical operators. The simplest waveform for /dev/dsp programs is sawtooth. A simple for(;;)putchar(t++) generates a sawtooth wave with a cycle length of 256 bytes, resulting in a frequency of 31.25 Hz when using the the default sample rate of 8000 Hz. The pitch can be changed with multiplication. t++*2 is an octave higher, t++*3 goes up by 7 semitones from there, t++*(t>>8) produces a rising sound. After a couple of trials, I came up with something that I wanted to share on an IRC channel:

main(t){for(t=0;;t++)putchar(t*(((t>>12)|(t>>8))&(63&(t>>4))));}

In just over an hour, Visy and Tejeez had contributed six more programs on the channel, mostly varying the constants and changing some parts of the function. On the following day, Visy shared our discoveries on Google+. I reshared them. A surprising flood of interested comments came up. Some people wanted to hear an MP3 rendering, so I produced one. All these reactions eventually led me to release the MP3 rendering on Youtube with some accompanying text screens. (In case you are wondering, I generated the screens with an old piece of code that simulates a non-existing text mode device, so it's just as "fakebit" as the sounds are).

When the first video was released, I was still unsure whether it would be possible for one line of C code to reach the sophistication of the earlier 8-bit experiments. Simultaneities, percussions, where are they? It would also have been great to find nice basslines and progressions as well, as those would be useful for tiny demoscene productions.

At some point of time, some people noticed that by getting rid of the t* part altogether and just applying logical operators on shifted time values one could get percussion patterns as well as some harmonies. Even a formula as simple as t&t>>8, an aural corollary of "munching squares", has interesting harmonic properties. Some small features can be made loud by adding a constant to the output. A simple logical operator is enough for combining two good-sounding formulas together (often with interesting artifacts that add to the richness of the sound). All this provided material for the "second iteration" video.

If the experimentation continues at this pace, it won't take many weeks until we have found the grail: a very short program, maybe even shorter than a Spotify link, that synthesizes all the elements commonly associated with a pop song: rhythm, melody, bassline, harmonic progression, macrostructure. Perhaps even something that sounds a little bit like vocals? We'll see.

Hasn't this been done before?

We've had the technology for all this for decades. People have been building musical circuits that operate on digital logic, creating short pieces of software that output music, experimenting with chaotic audiovisual programs and trying out various algorithms for musical composition. Mathematical theory of music has a history of over two millennia. Based on this, I find it quite mind-boggling that I have never before encountered anything similar to our discoveries despite my very long interest in computing and algorithmic sound synthesis. I've made some Google Scholar searches for related papers but haven't find anything. Still, I'm quite sure that at many individuals have come up with these formulas before, but, for some reason, their discoveries remained in obscurity.

Maybe it's just about technological mismatch: to builders of digital musical circuits, things like LFSRs may have been more appealing than very wide sequential counters. In the early days of the microcomputer, there was already enough RAM available to hold some musical structure, so there was never a real urge to simulate it with simple logic. Or maybe it's about the problems of an avant-garde mindset: if you're someone who likes to experiment with random circuit configurations or strange bit-shifting formulas, you're likely someone who has learned to appreciate the glitch esthetics and never really wants to go far beyond that.

Demoscene is in a special position here, as technological mismatch is irrelevant there. In the era of gigabytes and terabytes, demoscene coders are exploring the potential of ever shorter program sizes. And despite this, the sense of esthetics is more traditional than with circuit-benders and avant-garde artists. The hack value of a tiny softsynth depends on how much its output resembles "real, big music" such as Italo disco.

The softsynths used in the 4-kilobyte size class are still quite engineered. They often use tight code to simulate the construction of an analog synthesizer controlled by a stored sequence of musical events. However, as 256 bytes is becoming the new 4K, there has been ever more need to play decent music in the 256-byte size class. It is still possible to follow the constructivist approach in this size class -- for example, I've coded some simple 128-byte players for the VIC-20 when I had very little memory left. However, since the recent findings suggest that an approach with a lot of random experimentation may give better results than deterministic hacking, people have been competing in finding more and more impressive musical formulas. Perhaps all this was something that just had to come out of the demoscene and nowhere else.

Something I particularly like in this "movement" is its immediate, hands-on collaborative nature, with people sharing the source code of their findings and basing their own experimentation on other people's efforts. Anyone can participate in it and discover new, mind-boggling stuff, even with very little programming expertise. I don't know how long this exploration phase is going to last, but things like this might be useful for a "Pan-Hacker movement" that advocates hands-on hard-core hacking to greater masses. I definitely want to see more projects like this.

How profound is this?

Apart from some deterministic efforts that quickly bloat the code up to hundreds of source-code characters, the exploration process so far has been mostly trial-and-error. Some trial-and-error experimenters, however, seem to have been gradually developing an intuitive sense of what kind of formulas can serve as ingredients for something greater. Perhaps, at some time in the future, someone will release some enlightening mathematical and music-theoretical analysis that will explain why and how our algorithms work.

It already seems apparent, however, that stuff like this stuff works in contexts far beyond PCM audio. The earlier 8-bit experiments, such as the C-64 Wallflower, quite blindly write values to sound and video chip registers and still manage to produce interesting output. Media artist Kyle McDonald has rendered the first bunch of sounds into monochrome bitmaps that show an interesting, "glitchy" structure. Usually, music looks quite bad when rendered as bitmaps -- and this applies even to small chiptunes that sound a lot like our experiments, so it was interesting to notice the visual potential as well.

I envision that, in the context of generative audiovisual works, simple bitwise formulas could generate source data not only for the musical output but also drive various visual parameters as a function of time. This would make it possible, for example, for a 256-byte demoscene production to have an interesting and varying audiovisual structure with a strong, inherent synchronization between the effects and the music. As the formulas we've been experimenting with can produce both microstructure and macrostructure, we might assume that they can be used to drive low-level and high-level parameters equally well. From wave amplitudes and pixel colors to layer selection, camera paths, and 3D scene construction. But so far, this is mere speculation, until someone extends the experimentation to these parameters.

I can't really tell if there's anything very profound in this stuff -- after all, we already have fractals and chaos theory. But at least it's great for the kind of art I'm involved with, and that's what matters to me. I'll probably be exploring and embracing the audiovisual potential for some time, and you can expect me to blog about it as well.

Edit 2011-10-29: There's now a more detailed analysis available of some formulas and techniques.