A guide to the ideas of one of microtonal music’s most intellectually curious practitioners
If you’ve spent any time in music theory, you know that the 12 notes of equal temperament are a compromise. A practical, elegant compromise — but a compromise nonetheless. Microtonal and xenharmonic musicians have spent decades exploring what lies beyond that grid, and Johannes Werpup, a German theorist and composer born in 1982 near Cologne/Bonn and known online as Xen-Gedankenwelt (or XG), is one of the more systematic and deeply curious of those explorers.
His alias says it all. “Xen” signals both “xenharmonic” and “strange/alien.” “Gedankenwelt” is German for “world of thought” or “world of the mind” — a metaphor, as he explains it, for independent thinking. Put together, it describes someone who finds beauty in unusual things and refuses to uncritically accept the norms handed down by tradition.
Here’s what he’s actually up to, broken down for musicians who know their way around a key signature but are new to this territory.
The Tuning System at the Heart of His Work: 31edo
Werpup’s primary focus is 31edo — a tuning system that divides the octave into 31 equal steps instead of 12. Each step is about 38.7 cents wide (a cent being 1/100th of a semitone), so the steps are noticeably smaller than what we’re used to, but not vanishingly so.
To a musician trained in standard theory, the first thing worth knowing about 31edo is that it’s an extension of the meantone tradition you already know. The major third in 31edo is only about 1 cent away from a pure 5:4 ratio — far more in tune than the 14-cent-sharp major third we accept in 12-tone equal temperament. The fifth is about 5 cents flat of pure, which is exactly the meantone quality that Renaissance and Baroque keyboards aimed for. If you’ve ever wondered what a well-tuned harpsichord or early organ actually sounded like — with that richer, more resonant quality to the thirds — 31edo gets you most of the way there.
But the really interesting thing about 31edo isn’t just that it tunes familiar intervals better. It’s that it opens up entirely new harmonic territory. The tuning has a very close fit to the 7:6, 8:7, and 7:5 ratios, which have no approximate fits in 12-tone equal temperament. Wikipedia That last point is crucial: 31edo gives you genuine, usable access to intervals from the seventh harmonic — intervals that simply do not exist in any recognizable form in standard tuning.
What Is “7-Limit” and Why Does It Matter?
Werpup works primarily with what theorists call 7-limit intervals. To understand this, a quick detour into harmonic series thinking is useful.
When a string or pipe vibrates, it produces not just its fundamental pitch but a cascade of overtones: the octave (2:1), the perfect fifth (3:2), the major third (5:4), and so on. Standard Western harmony, at its theoretical core, deals with ratios built from the primes 2, 3, and 5 — what’s called “5-limit” thinking. The minor seventh you know from a dominant 7th chord is actually a fairly crude approximation; its true harmonic version, the ratio 7:4, is about 31 cents flatter than what 12-tone equal temperament gives you. That difference is big enough to hear clearly once you’re listening for it — and in 12-tone equal temperament, you’re simply never hearing the pure version.
7-limit harmony opens the door to this seventh partial and all the intervals that can be built from it. 7-limit harmony can still make beautiful music, but it often has an alien character that many people find unsettling or unapproachable at first. Microtonaltheory That strangeness is partly the point — it’s a sound world that Western ears haven’t been trained on, but one that has deep acoustic justification.
Werpup is drawn to specific 7-limit intervals that are accessible within 31edo. He highlights things like 35/32 (a neutral second), 48/35 (a superfourth), and 105/64 (a neutral sixth). The interesting trick here is that these are all technically 7-limit ratios — derived from combinations of the primes 2, 3, 5, and 7 — but they fall close enough to certain 11- and 13-limit intervals that they can mimic the sound of even more exotic harmonics. It’s a kind of harmonic borrowing: you get the color of higher overtones without needing a tuning system that supports them directly.
Linear Temperaments, MOS Scales, and “Bridge” Temperaments
This is where things get genuinely fascinating for anyone with a theory background.
In standard music, you’re essentially working with one temperament: 12-tone equal temperament, which supports the diatonic and chromatic scales. Within 31edo, there are many different regular temperaments — each one defined by which intervals it prioritizes and which small commas (the microscopic gaps between pure ratios) it eliminates. These temperaments have evocative names: Orwell, Valentine, Zeus, Meantone.
Each temperament generates its own family of scales. These are often MOS scales — “Moments of Symmetry,” a concept developed by the influential American theorist Erv Wilson. A moment of symmetry scale is a periodic scale where every interval formed by ascending a step is either small or large with no in-between. The most widely used MOS scale is the diatonic scale itself — 5 large steps and 2 small ones. Xenharmonic Wiki Wilson’s insight was that this simple pattern — two step sizes, distributed as evenly as possible — is the structural key to most of the world’s musical scales, and that you can generate an enormous variety of them by changing the generator interval.
What makes Werpup’s approach distinctive is his systematic study of the relationships between temperaments. He uses what he calls “bridge” temperaments to modulate smoothly between scale families — for example, moving from Orwell to Valentine via Zeus as a transitional system. Think of it as the microtonal equivalent of pivot chords: finding a harmonic space that belongs to two different tonal worlds at once, allowing a seamless passage between them.
The Other Tunings He Works With
While 31edo is his home base, Werpup maintains an active interest in several other systems:
22edo gets attention as a solid all-around tuning for 11-limit harmony (that is, harmony derived from the first eleven harmonics) with an interesting range of supported temperaments. It’s also notably different from meantone — its small step is actually quite small, which gives the scale a different internal tension.
11edo he describes as sounding “very interesting in an Oriental but weird way.” It’s a subset of 22edo, and its small, compressed intervals genuinely do evoke scales from non-Western traditions — though in a stretched, alien version of them.
94edo is of interest for more academic reasons: it’s consistent through the 23-odd-limit, meaning it approximates a very wide range of harmonic ratios reliably. It’s useful for studying what happens at the extreme upper edges of the harmonic series.
15edo appeals to him partly for practical reasons. He notes that it can be understood as an equal-tempered version of a specific MOS scale called Valentine[15], and that a 15edo guitar would have a 5edo perfect fourth between adjacent strings — a quirky, fun tuning challenge.
His Theoretical Influences
Werpup’s intellectual lineage is worth understanding because it shows the depth of the tradition he’s working in.
His primary theoretical inspiration is German music theorist Martin Vogel, whose work on 7-limit just intonation gave Werpup his harmonic foundation. Vogel’s approach to harmonic dualism — the idea that major and minor are mirror images of each other, one built from overtones upward and the other from undertones downward — is an old 19th-century idea (associated with theorists like Hugo Riemann) that Vogel revived and grounded in acoustic physics. Vogel’s Tonnetz is a graphical and mathematical representation of just intonation that adds a third dimension for just sevenths to the familiar two dimensions of fifths and thirds. Wikipedia Werpup also credits Vogel for an insight he found particularly useful: that the 7:4 harmonic seventh can be closely approximated by the augmented sixth (225:128), which is the theoretical basis of marvel temperament — a way of treating these two otherwise-distinct intervals as the same pitch.
The second major influence is Erv Wilson, whose work on Moments of Symmetry and the regular mapping paradigm gives Werpup the structural framework for how to organize scales and navigate between them. Wilson’s ideas went on to influence almost every xenharmonic composer and theorist who came after him. Large swathes of modern xenharmonic music and theory make use of MOS scales or constant structures, which are now the predominant way of approaching equal-tempered tunings. Xenharmonic Wiki
He also credits musicians Deja Igliashon, Ron Sword, and Paul Erlich — practitioners who opened his ears to neutral seconds and the small EDOs (like 15, 16, and 22) that make those intervals available in practical, playable form.
Notation: Solving a Real Practical Problem
One of the less glamorous but deeply important challenges in microtonal music is notation. How do you write down music in 31edo in a way that’s readable and practical?
Werpup’s solution is to work from the notation system musicians already know. He uses 12-note notations, reinterpreting the 12 steps of a standard chromatic scale as steps of an uneven 12-note MOS in the target tuning. Then he adds chroma accidentals to indicate deviations from that base scale. The result is a system that guitar players can use to adapt existing 12-edo tablature into microtonal territory without starting from scratch.
For 94edo, he proposes an alternative approach based on Garibaldi[12] as the reference scale — a good choice for fretless instruments because it allows fairly accurate notation of 13-limit intervals using just a single accidental pair. He’s also implemented this in software, writing a Job Plugin for Vocaloid that transposes all base notes to a maximally even 12-note scale for a selected EDO, with accidentals embedded in the song lyrics.
His Instruments and Studio Setup
Werpup owns a 31edo 8-string baritone electric guitar, based on an Ibanez RGA8 and refretted by luthier Nick Page. An 8-string baritone in 31edo is not a minor undertaking — the instrument has 31 frets per octave instead of 12, and each of those frets is physically placed at slightly different positions than on a standard guitar. The extended range of an 8-string baritone suits the rich, low-register harmonics that 7-limit harmony particularly rewards.
He works with the DAW Reaper for its microtonal capabilities and script support, and has planned to use Vocaloid/IA for pitch-perfect microtonal vocals — using the custom scripts mentioned above to bring the software in line with his tuning systems.
A Composer With Named Temperaments to His Credit
It’s a mark of standing in the xenharmonic community that you get to name things. Werpup has coined the names for three temperaments: Cuckoo, Oracle, and Necromanteion. These names don’t describe the mathematics directly but they do reflect the character of the sonic spaces these temperaments carve out — which is very much in the spirit of the xenharmonic tradition, where music theory and composition are treated as a unified practice.
Music Tastes and Other Interests
The musical genres Werpup is drawn to — metal and rock, electronic music, video game and chiptune music, classical, and non-Western traditions — give you a sense of the range of application he has in mind for these tuning systems. Xenharmonic music isn’t a single genre; it’s a set of tools that can serve everything from ambient electronics to shred guitar.
Outside music, his other interests include non-Euclidean (particularly hyperbolic) geometry, programming, the Japanese language and culture, vegan cooking, and board and video games — a profile that’s quite characteristic of the mathematically inclined, intellectually omnivorous corner of the music world that xenharmonics tends to attract.
Where to Go From Here
If you want to explore the ideas discussed here, the Xenharmonic Wiki is the best starting point — it’s a deep, technically rigorous resource that covers temperament theory, specific EDOs, and the broader regular mapping paradigm in as much depth as you could want.
For the tuning systems themselves, 31edo is widely considered the most accessible entry point into non-12 equal temperament for musicians coming from a Western theory background, precisely because its diatonic scale still feels familiar while its extended harmonic palette is so much richer. If you can get time with a retuned synthesizer or a microtonally-aware plugin like Surge XT or Vital with a Scala tuning file loaded, even a few minutes of playing in 31edo will make the appeal of Werpup’s work immediately clear.
Sources: Johannes Werpup’s page on the Xenharmonic Wiki (January 2025 revision); Xenharmonic Wiki entries on 31edo, MOS scales, Erv Wilson, 7-limit, and related topics; Wikipedia on 31 equal temperament, Martin Vogel’s Tonnetz, harmonic dualism, and just intonation.
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