How one man squeezed 41 notes onto a single instrument
If you’ve ever felt that standard Western harmony is somehow almost right — that a major chord is close to perfect but never quite there — you’re not imagining things. The 12-note equal temperament system that underpins virtually all Western music is a brilliant compromise, but it is a compromise. And for a growing community of musicians, composers, and instrument builders, that compromise is worth questioning.
This article is an introduction to one of the most musically compelling alternatives: 41-tone equal temperament, or 41edo. Along the way, we’ll meet Kite Giedraitis, the musician and theorist who has done more than almost anyone to make 41edo playable on a guitar.
What’s Wrong with 12-Tone Equal Temperament?
Nothing is wrong with it, exactly. The system we use — 12 equally spaced notes per octave, called 12edo (Equal Division of the Octave) — is so well-designed that it has dominated Western music for centuries. But equal temperament is a tuning of convenience. The octave (a 2:1 frequency ratio) divides cleanly, but almost everything else is an approximation.
The most glaring example is the major third. In just intonation — the tuning system based on pure, mathematically simple frequency ratios — a major third is a 5:4 ratio, which comes out to 386 cents. In 12edo, a major third is 400 cents. That’s 14 cents sharp. To put that in perspective, most trained musicians can hear a tuning difference of around 5–10 cents. So the major third we’ve been using our whole lives is noticeably out of tune relative to its pure ideal.
The minor seventh is even further off. In a pure dominant seventh chord, the seventh is a 7:4 ratio (969 cents). In 12edo, the minor seventh is 1000 cents — a full 31 cents sharp. That’s why a perfectly tuned barbershop quartet sounds so dramatically smoother than a piano playing the same chord: the singers are naturally gravitating toward just intonation.
Enter Extended Just Intonation
Just intonation (often abbreviated JI) is the approach of building musical intervals from simple frequency ratios. A perfect fifth is 3:2. A major third is 5:4. A minor third is 6:5. These are the intervals that ring with physical clarity because their overtones align rather than clash.
The more complex the ratio — for instance 7:6 for a “subminor third,” or 11:8 for a midpoint between a perfect and augmented fourth — the more exotic the sound, but still physically grounded. Music theorists refer to the largest prime number appearing in these ratios as the prime limit. Standard Western music is essentially 5-limit: the ratios it uses can always be reduced to combinations of 2, 3, and 5. Jazz and blues arguably push into 7-limit territory, where the 7th harmonic comes into play and chords take on that warm, relaxed, “barbershop seventh” quality.
The problem with just intonation on instruments like the guitar is practical: you need many more notes to stay in tune across different keys, melodies can drift in pitch over time through “comma pumps” (more on those later), and building instruments to handle all of this is enormously difficult.
This is where equal temperaments larger than 12 come in. By using more notes per octave, you can approximate just intonation much more closely while retaining the key advantage of equal temperament: every key works equally well.
Why 41?
Not all equal temperaments are created equal. 19edo, 22edo, 31edo, and 53edo are all well-known alternatives with dedicated communities. But 41edo has a special set of properties that many microtonal theorists consider nearly ideal.
41edo divides the octave into 41 equal steps of about 29.3 cents each.
Here’s what makes it exceptional:
Its perfect fifth is extraordinarily accurate. The fifth in 41edo (24 steps out of 41, or about 702.4 cents) is only about 0.5 cents away from the pure 3:2 ratio (702.0 cents). For comparison, 12edo’s fifth is about 2 cents flat. This means 41edo handles Pythagorean and traditional Western harmony beautifully.
It tunes the 7-limit exceptionally well. Both the major third (5:4) and the harmonic seventh (7:4) are represented with only moderate errors — and crucially, those errors partially cancel each other out in chords. A 4:5:6:7 dominant seventh chord in 41edo sounds dramatically smoother than in 12edo.
It handles the 11-limit reasonably well too. Intervals based on the 11th harmonic — particularly the 11:8 ratio, which sits midway between a perfect fourth and augmented fourth — are accessible, opening the door to the “neutral” intervals found in Arabic, Turkish, and Persian music.
It is consistent to the 15-odd-limit, which is a technical way of saying that all simple harmonic relationships up to that complexity are represented without ambiguity or conflict.
In short, 41edo is arguably the smallest equal temperament that represents 5-limit, 7-limit, and 11-limit just intonation all reasonably well simultaneously, while also preserving the excellent perfect fifth that Western musicians rely on.
The Notation: Ups and Downs
One of the practical hurdles of microtonal music is notation. How do you write 41 different pitches per octave in a way that musicians can actually read?
The most elegant solution for 41edo is ups and downs notation, developed by Kite Giedraitis. The idea is elegant: keep all the standard note names, sharps, and flats exactly as they are — so everything you already know still applies — and add just two new symbols: an up arrow (^) and a down arrow (v).
In 41edo, a regular sharp raises a note by 4 steps, and a regular minor second is 3 steps. This means there are notes that fall between the conventional sharps and flats. An up arrow raises the pitch by one step (one 41st of an octave, about 29 cents), and a down arrow lowers it by one step.
So the note a single step above C is called ^C (up-C), and the note one step below D is called vD (down-D). In this system, “^C” and “vD” are actually the same pitch, just written from different directions. A downmajor third — the pure 5:4 major third — is written vM3, because it’s one step flatter than the Pythagorean major third. An upminor third is ^m3.
This notation extends seamlessly to chord names. A chord built on a downmajor third is called a “downmajor chord” or “Cv” (C-down). A chord with an upminor third is “C^m” (C-up-minor). The dominant seventh chord tuned as 4:5:6:7 is “Cv7” — C with a downmajor third and a downminor seventh.
Remarkably, this entire system requires memorizing just two new symbols. All standard music theory — chord names, Roman numeral analysis, scale degrees — carries over with minimal modification.
The Kite Guitar: Making 41edo Playable
Here’s the obvious problem: a guitar with 41 frets per octave is physically unplayable. Standard guitars have 12 frets per octave and are already at the limit of comfortable spacing. At 41 frets per octave, the frets would be barely 3mm apart near the first position. No human fingers could navigate that.
This is the problem that Kite Giedraitis solved with the Kite Guitar.
The solution is called skip-fretting: instead of placing frets for every step of 41edo, you place frets for every other step. This gives you about 20.5 frets per octave — comparable to a 19edo or 22edo guitar, which are known to be playable. Each fret represents 2 steps of 41edo, or about 59 cents.
The catch is that each string now has access to only half of the 41 pitches. But here’s the ingenious part: the interval between adjacent open strings is tuned to 13 steps of 41edo — an odd number — which means adjacent strings cover complementary halves of the scale. Every one of the 41 pitches is accessible on every adjacent pair of strings.
But does skipping every other fret cause musical problems? Which notes end up being hard to reach? Here’s where 41edo’s mathematical properties become almost magical: the notes that become most remote on a skip-fretted guitar are exactly the notes that are most dissonant in chords. The perfect fifths, major thirds, minor thirds, harmonic sevenths — all easy to reach. The “offperfect” intervals — the ones that sound slightly sharp or flat of a perfect fifth or octave — are the hard-to-reach ones. The geometry of the instrument naturally filters toward consonance.
The standard tuning for a Kite guitar uses a downmajor third (13 steps, or about 380 cents) between each adjacent string, rather than the perfect fourth used on a standard guitar. This tuning is called the downmajor tuning.
One important consequence is that the guitar becomes isomorphic: any chord shape works the same way on any string, in any key. On a standard guitar, a C major chord and a D major chord have entirely different shapes. On a Kite guitar, every major chord — or more precisely, every downmajor chord — has the same shape. This dramatically reduces the number of chord shapes that need to be learned.
Who Is Kite Giedraitis?
Kite Giedraitis (pronounced gih-DRIGH-tiss), also known online as “Tall Kite,” is the Portland, Oregon-based musician, theorist, instrument builder, and software developer who sits at the center of the 41edo world.
His path into microtonality is unusual. Unlike many theorists who come from academic music backgrounds, Kite didn’t start playing an instrument until his mid-twenties. His first instruments were the bowed psaltery and African marimba. He encountered microtonal music through the study of traditional African music — particularly the music of Tanzanian musician Hukwe Zawose — and upon hearing 7-limit just intonation for the first time, was immediately captivated.
His contributions to microtonal music are wide-ranging:
The Kite Guitar (invented April 2019): The skip-fretted guitar described above, which makes 41edo as physically accessible as 19edo or 22edo.
Ups and downs notation: A system for notating any equal temperament using just two added symbols, now widely used across the microtonal community for everything from 17edo to 72edo.
Color notation: A complementary system for just intonation that assigns colors to different prime harmonics (white/grey for 3-limit Pythagorean intervals, “yo” yellow for 5-over, “gu” green for 5-under, “zo” blue for 7-over, “ru” red for 7-under, etc.), allowing any ratio to be described as a color plus an interval quality.
Alt-tuner: A software tool that enables real-time adaptive just intonation via MIDI, complete with a color-coded lattice display and foot-pedal control. This is sometimes called the “holy grail” of retuning software.
The EDOstrobetuner: A free strobe tuner specifically designed for microtonal guitars.
The Pergen system: Co-developed with Praveen Venkataramana, a theoretical framework for categorizing rank-2 and rank-3 temperaments that gives composers clearer language for understanding complex tuning systems.
Kite has also written extensively about how to name notes, chords, key signatures, and scales in 41edo — a surprisingly complex problem when you have 41 pitches to work with, and you want musicians to be able to read music without triple sharps and triple arrows cluttering the score.
What Does 41edo Music Sound Like?
The most immediate impression for most listeners is that the chords sound remarkably calm. A standard dominant seventh chord in 41edo — tuned as a pure 4:5:6:7 ratio — has almost no roughness or beating. Where the same chord in 12edo has a slightly tense, unresolved quality (which is actually essential to how it functions in traditional harmony), the 41edo version is deeply settled. Musicians who hear it for the first time often describe it as “the chord I always imagined but couldn’t quite get.”
This creates interesting challenges for translation. A traditional V7–I cadence relies partly on the dissonance of the seventh chord to create forward motion. In 41edo, you sometimes need to deliberately choose a less pure tuning of the dominant seventh to maintain that harmonic tension.
The melodic palette is also substantially expanded. Where 12edo has only two “shades” of most intervals (major and minor), 41edo has four: downminor, minor, upminor, and “mid” (a neutral interval exactly between major and minor), plus downmajor, major, and upmajor. Scales can be constructed that sound like Western major scales, natural minor, Arabic maqamat, Indian ragas, and Indonesian gamelan, all within the same tuning system.
The blues is particularly well-served. The “blue third” — the ambiguous, slightly flat major third that defines so much of blues and jazz tonality — exists as a clear, precise interval in 41edo. What was formerly an expressive inflection becomes a definable pitch.
Comma Pumps: The Trickiest Part
One concept that comes up repeatedly in 41edo music is the comma pump. This is worth understanding even at a basic level.
In just intonation, some common chord progressions — like I–vi–ii–V–I — technically require the tonic to drift slightly in pitch over their course. This drift is called a comma, and the smallest important one is the syntonic comma (81:80, or about 21 cents). In 12edo, this comma vanishes: the system is designed so that a full cycle of such progressions brings you back exactly to your starting pitch.
In 41edo, the syntonic comma (called the "Gu comma" in color notation) maps to a single step — about 29 cents. This means that certain familiar chord progressions, when translated naively to 41edo, create a gradual drift of one step through the key. This is handled with a small "pitch shift" — a deliberate half-fret adjustment at one point in the progression — which experienced Kite guitarists learn to execute smoothly and which casual listeners rarely notice.
Giedraitis has written extensively about strategies for managing these pitch shifts gracefully, including voicing the shifting note in a middle voice, placing it in a different octave from its previous appearance, or timing it to fall on a less prominent beat.
41edo as a Universal Tuning
One of 41edo's most striking properties is how many world music traditions it can approximate. This isn't a claim to replace those traditions — real authenticity requires far more nuance than any equal temperament can provide — but it makes 41edo a uniquely versatile platform for cross-cultural exploration:
- Western classical and folk: The near-pure fifth supports Pythagorean and meantone harmony. The 5-limit major and minor scales are beautifully in tune.
- Blues and jazz: The harmonic seventh and "blue notes" are natural and expressive.
- Middle Eastern music: Neutral seconds and thirds (from the 11-limit) enable authentic-sounding Arabic, Turkish, and Persian maqamat.
- Indian classical music: The Magic temperament's 22-note scale approximates the 22 shrutis of Carnatic music while preserving pure fifths.
- Indonesian gamelan: The Slendro scale is approximated by an 8-step generator in 41edo, and Pelog is accessible through related temperament structures.
- Japanese gagaku: Pythagorean pentatonic scales with narrow semitones are naturally present.
Getting Started
If you're curious about exploring 41edo yourself, here's a practical entry point:
Listen first. The KiteGuitar.com website has audio and video recordings. Aaron Wolf's "12-Bar Blues on Kite Guitar" is a gentle introduction; Kite's own "Evening Rondo" shows the instrument's melodic range. The album Intervallic Prism by Pixel Archipelago, recorded on Kite guitar, is a full artistic statement in the tuning.
Play virtually. The Kite Guitar WebPlayer at KiteGuitar.com lets you try the instrument in a browser, either with a mouse or a MIDI keyboard.
Explore the theory. The Xenharmonic Wiki (XenWiki) has exhaustive documentation on 41edo, ups and downs notation, the Kite Guitar, and related topics. Kite has also written a book, Alternative Tunings: Theory, Notation and Practice, available through TallKite.com.
Try it on an instrument you own. If you have a fretless instrument (violin, trombone, fretless bass), you can begin ear-training in 41edo immediately using the free EDOstrobetuner. For keyboard players, a Lumatone or Linnstrument with a 41edo layout opens up harmonic exploration without requiring a new instrument.
Consider a conversion. Kite Guitar conversions — where a luthier re-frets your existing guitar — are available, as are pre-slotted fretboards. The community meets via a weekly public videochat every Sunday, details on the KiteGuitar.com contact page.
Conclusion
41edo won't replace 12edo any more than jazz replaced classical music, or electric guitars replaced acoustic ones. But it offers something genuinely new: a system where the chords you've always heard almost right can finally be exactly right, where the blue notes of the blues are actual notes rather than inflections, and where the musical traditions of entirely different cultures can share a common tuning framework.
Kite Giedraitis's contribution to all of this is hard to overstate. By designing the skip-fretted guitar, developing an approachable notation system, building the software tools, and tirelessly documenting and sharing his work — mostly for free — he has done for 41edo what very few individuals have done for any alternative tuning: made it practical for ordinary working musicians.
The 41-note octave is waiting. It sounds better than you might expect.
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