How 34-equal temperament’s “tetracot” structure improves on standard harmony
If you’ve spent years studying standard Western harmony — intervals, voice leading, chord progressions, scale theory — you already have a strong foundation for exploring microtonal music.
Of all the alternative tuning systems out there, 34 equal divisions of the octave (or 34edo) might be one of the most inviting entry points. It keeps a lot of what you already know intact, while opening doors to harmonies and colors that simply don’t exist in 12-tone equal temperament (12edo, the tuning system on a standard piano).
This article explains what 34edo is, why musicians find it compelling, and how your existing theory knowledge maps onto it.
What Is 34edo?
Equal temperament works by dividing the octave into a fixed number of equal steps. Standard Western tuning uses 12 steps of 100 cents each. 34edo divides the octave into 34 equal steps of about 35.3 cents each — roughly a third of a semitone per step.
One useful way to picture it: 34edo contains two complete copies of 17edo (17 equal divisions of the octave), offset from each other by one step. Think of it as 17edo with a fine-grained in-between layer slotted in. If you’re familiar with 17edo’s “neogothic” sound — its wide major thirds and narrow minor thirds — you’ll find those same qualities here, but now with additional nuance and more harmonic options.
How Does It Sound? The Basics of Its Harmonic Character
The best way to understand 34edo’s character is through its relationship to just intonation (JI) — the ideal, mathematically pure tuning of intervals based on simple frequency ratios.
34edo is exceptionally good at approximating harmonics 3, 5, 13, 17, and 23. In plain terms:
- Harmonic 3 means the perfect fifth (3/2). 34edo’s fifth is about 705.9 cents — a touch wider than the 12edo fifth of 700 cents, and closer to a pure fifth (702 cents) than 12edo is. The sharpness gives chords a slightly bright, resonant quality.
- Harmonic 5 means the major third (5/4 = 386.3 cents) and minor third (6/5 = 315.6 cents). 34edo approximates both of these with exceptional accuracy — better than 31edo in the 5-limit, in fact. This means major and minor triads sound strikingly pure, almost like acoustic beating disappears from them.
- Harmonics 13, 17, 23 add exotic new “xenharmonic” intervals, chords and scales.
The one area where 34edo underperforms is harmonic 7 — the “barbershop seventh” or blues seventh (7/4 ≈ 968.8 cents). The approximation is rough enough (~45% relative error) that composers working primarily in that harmonic territory might prefer systems like 31edo or 22edo. However, there’s an alternate mapping (called the 34d val) that offers a different, sharper approximation of 7/4 which some composers find musically useful in specific contexts.
What Happens to Familiar Intervals?
Here’s where it gets interesting for someone trained in conventional theory. 34edo doesn’t just add new notes between the existing 12 — it redefines the relationships between intervals you already know.
The Whole Tone Splits Into Six
In 34edo, the standard whole tone (9/8) is divided into six equal steps of ~35.3 cents each. This means:
- A chromatic semitone (like C to C# in 12edo, corresponding to the ratio 25/24) spans three steps.
- A diatonic semitone (like E to F, corresponding to 16/15) spans two steps.
In 12edo, both of these are crammed into a single 100-cent semitone. In 34edo, they’re distinct, and you can actually hear and play the difference between a chromatic half-step and a diatonic half-step. This distinction is fundamental to how the system behaves when you start modulating.
Your Major Scale Still Works
Because 34edo does a great job with 5-limit intervals, the diatonic major scale maps cleanly onto it. In fact, you can think of 34edo as supporting two slightly different versions of the major scale:
- The familiar 5L 2s moment-of-symmetry (MOS) diatonic scale — the standard W-W-H-W-W-W-H pattern.
- A “zarlino diatonic” or nicetone scale with slightly adjusted step sizes that more closely tracks pure just intonation.
You can also extend the diatonic into a 12-note chromatic scale by stacking the two different semitone sizes, giving you something that feels familiar but with the sharpened pure-interval quality.
New Interval Flavors
Between your familiar major, minor, and perfect intervals, 34edo adds several new categories:
- Mid intervals (marked ~): A “mid third” at ~353 cents sits between a minor third (340 cents in 12edo) and a major third (400 cents in 12edo). This corresponds to a neutral third, audible in Arabic maqam music, Georgian polyphony, and a lot of 20th-century jazz. Specifically in 34edo this approximates intervals like 11/9 or 16/13.
- Neogothic thirds: Inherited from 17edo, these include a minor third of ~282 cents (slightly narrow) and a major third of ~424 cents (noticeably wide). These give chords a bright, medieval character.
- Tendo and arto thirds: At ~459 cents (13/10) and ~247 cents (15/13) respectively. These sit outside the usual major/minor framework entirely and are useful for more adventurous harmonic writing.
Notation: How Do You Write This Music?
Standard staff notation can be adapted for 34edo, and the most common approach is called ups and downs notation. It adds two symbols:
- Up (^): raises a note by one step (~35 cents)
- Down (v): lowers a note by one step (~35 cents)
With these tools and the existing sharps and flats (each covering 4 steps in 34edo), you can notate all 34 pitches using ordinary note names with arrow modifiers. For example:
- C = perfect unison
- ^C = one step up from C
- ^^C = two steps up from C
- C# (= D♭) = four steps up from C
- ^C# = five steps up from C
This system has the advantage of familiarity — the seven natural notes (C, D, E, F, G, A, B) still map to their usual positions in the scale. It does produce some dense-looking chord notation for chromatic harmonies, but for diatonic and near-diatonic writing it’s quite readable.
An alternative is Sagittal notation, which uses a specialized set of arrow-like accidentals designed for general use across many microtonal systems. 34edo uses the same sagittal sequence as 41edo and is a superset of the notation for 17edo.
Temperament Theory: What Does 34edo “Temper Out”?
If you’re comfortable with concepts like meantone or equal temperament, here’s a slightly deeper look at what makes 34edo structurally interesting.
A regular temperament is defined in part by the commas it “tempers out” — tiny intervals that are treated as unison, creating equivalences between otherwise distinct just intonation pitches.
Crucially, 34edo does NOT temper out 81/80 (the syntonic comma). This is the comma that meantone (and 12edo) eliminates by making the major third equal to exactly four stacked fifths. In 34edo, the 9/8 and 10/9 whole tones remain distinct — in fact, the difference between them is exaggerated from its just value of 21.5 cents to a full 35.3 cents (one step). This makes 34edo a non-meantone system, which has real compositional implications: stacking four fifths does not give you a major third. The circle of fifths still closes at 17 fifths (since it contains two 17edos), but the thirds work differently.
Some important temperaments supported by 34edo include:
- Tetracot: Four stacked 10/9 whole tones (each mapped to 5 steps) reach the perfect fifth (20 steps). This is a defining structural feature of 34edo.
- Hanson/Kleismic: Based on minor thirds as generators. Six minor thirds nearly reach an octave.
- Würschmidt: Major thirds as generators.
- Srutal/Diaschismic: The 34d val supports this, making 34edo a strong alternative to 46edo or 22edo for this temperament family.
- Pajara: Also supported via the 34d val — 34edo is described as an excellent alternative to 22edo for 7-limit pajara temperament.
The Phi Approximation: A Bonus for Tuning Nerds
As a Fibonacci number, 34 has a special property: 21 steps of 34edo (≈741.2 cents) closely approximate the logarithmic version of the golden ratio phi. Stacking this interval generates moment-of-symmetry scales where the step sizes are nearly in phi ratio to each other — a property with connections to natural growth patterns and certain aesthetic theories of proportion in music.
What Kind of Music Can You Make?
34edo has attracted composers working in remarkably diverse styles. Some examples from the wiki:
- Microtonal jazz and blues: The system’s excellent approximation of 13-limit intervals (overtones 11 and 13) makes it rich territory for extended harmony that goes beyond dominant seventh chords.
- Neogothic and medieval-flavored music: The wide major thirds and narrow minor thirds, inherited from 17edo, lend themselves to organum-style writing and modal music.
- Ambient and electronic music: Artists like Xotla have released full albums of ambient electro, space rock, and electronic rock in 34edo, including works explicitly in tetracot[13] tuning.
- Classical arrangements: Composers have rendered Bach, Chopin, and Joplin in 34edo, sometimes with syntonic-comma adjustments that bring the tuning closer to historical meantone sensibilities.
- Maqam music: The 17-tone structure embedded within 34edo, combined with its neutral thirds, makes it a natural fit for Middle Eastern scales.
Stephen Weigel has specifically recommended 34edo for notating Georgian polyphonic music, which uses neutral intervals not available in 12edo.
Practical Entry Points
If you want to start exploring:
- Start with the diatonic scale. Load a 34edo tuning in a microtonal-capable synthesizer or DAW and play your normal major and minor scales. Notice how much purer the thirds sound.
- Explore the neutral third. Find the ~353-cent “mid third” (step 10 in the interval table) and build chords around it. This is often the most immediately striking new color.
- Try modulating by 10/9 four times. Since four 10/9 steps reach a perfect fifth, you can build a tetracot-based harmonic sequence that feels both logical and completely alien to 12edo ears.
- Use a Lumatone or isomorphic keyboard. 34edo has a well-documented Lumatone layout, and its structure makes it particularly suited to isomorphic instruments where the same fingering patterns work across all keys.
Summary
34edo is a non-meantone, 5-limit-accurate tuning system that sits at a fascinating intersection: familiar enough (diatonic scales work, standard notation adapts, triads sound pure) to be accessible to conventionally-trained musicians, yet different enough to open genuinely new harmonic worlds. Its neutral thirds, neogothic character, phi approximation, and tetracot structure make it musically versatile across styles from ambient to jazz to classical to world music.
If you know your way around a staff and can hear the difference between a major and minor third, you have everything you need to start exploring 34edo.
Sources: 34edo — Xenharmonic Wiki. For related concepts, see also 17edo, Just intonation, MOS scales, Regular temperament, and Ups and downs notation on the Xenharmonic Wiki.
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