The Hidden Arithmetic Failure of the 72-Steplet Byzantine Tuning System
The 72-steplet (moria) system of traditional Byzantine music is a 19th-century franken-system: a badly put together compromise by a committee, built after Chrysanthos’s failed monochord formalization, doubled for practical resolution, later paired with ratios it does not produce, and taught as fixed pitch theory while functioning as contextual oral practice.
Contrary to the traditionalists’ claims, we now have overwhelming historical and analytical evidence [ … ] that the scale system taught in today’s conservatories is flawed. That the official theory of Byzantine chant, codified in the 19th century and under the Sultan’s rule, is inconsistent and unsound.
(Ari Koufogiannakis, Mastering Byzantine Polyphony, introduction)
Each of these is documented. Each appears in Greek-language sources written from inside the tradition. Together they constitute a century of false precision sold as sacred mathematics. This article makes the case in five cuts.
1. The Origin Was Already a Correction
Modern Byzantine theory traces its quantitative apparatus to Chrysanthos of Madytos, whose Great Theoretical of Music (1832) attempted to give Byzantine music a formal interval structure via the monochord. The attempt failed at the level of basic acoustics. Chrysanthos divided string lengths linearly. He overlooked that pitch is logarithmic in string length. This is why in fretted instruments (a guitar for example) the same interval appears to have different fret lengths across the fretboard. Simply put, as the pitch rises, the same interval requires smaller and smaller string length. This principle in acoustics is taught in high-school physics, which apparently Chrysanthos wasn’t aware of. This was not a subtle error. Broken on arrival. The broken scale was being taught to students for 50 years, two generations of chanters, until a correction was attempted.
The 1881 Patriarchal Committee was convened, in part, to clean up after this and similar problems. The Committee’s founding mandate, set out in the 1881 Patriarchal encyclical, names the threat of European music (its instruments, its tempered scale, its harmonic logic) as the urgent reason for its existence. The motivation was not pure acoustic science. The Committee was tasked with producing a system that could compete pedagogically with Western theory while remaining identifiably not Western.
This matters because the Committee’s choices were constrained from the start. They were not free (or willing) to derive the cleanest tempering from first principles. They had to deliver a system that (a) corrected Chrysanthos’s monochord blunder, (b) preserved the perceived character of the chant tradition, (c) accommodated the contextual attractions (έλξεις) that chanters had always applied in practice, and (d) did not look like 12-tone equal temperament. These constraints are obvious in the system.
2. The Mystery of the 36 → 72 Doubling
The Committee’s diatonic tetrachord, as published, was {6, 5, 4} in “commata”, with the full octave summing to 36. This was the theoretical division. It was then doubled, every interval value multiplied by two, to yield the {12, 10, 8} tetrachord and the 72-steplet octave that appears in nearly every Byzantine theoretical book to this day. No explanation was given for the doubling, other than “for practical reasons.” The doubling has no acoustic justification. The 36-steplet division and the 72-steplet division produce identical pitches. The doubling only adds resolution but not accuracy. And what were those practical reasons, if all intervals were already integers?
Perhaps they wanted to give the attraction system room to operate in integers. Chanters routinely apply pitch alterations depending on context: a sharpened Ga under Di-rotation, etc. At 72-steplet resolution, the same alterations have integer values that read more naturally and admit finer distinctions. This could have been one “practical reason”. There’s also another, which was never disclosed: The ancient Greek enharmonic genus, which contained half-semitones. The committee possibly wanted to allow for those. However, this was never used in practice, not even in the “shade of the Tilt”, which Mastering Byzantine Polyphony recognizes as an ancient Greek Enharmonic relic. The Committee stuck to even steplet values, not to mention that their intervallic definition for the Tilt was completely wrong.
In other words: the unit was sized to fit the practice. The grid was calibrated to the chanters, and the chanters were then taught as if they derived from the grid.
This is the inversion at the center of the pedagogical edifice. The 72-steplet system, presented as a fundamental acoustic structure from which performance practice flows, was created by putting the cart before the horse. Historically, the causation runs the other way. The performance practice came first; the grid was sized to render it in convenient integers.
3. The Arithmetic Does Not Match the Ratios
This is where the system stops being merely awkward and becomes demonstrably wrong.
The 1881 Committee’s underlying ratios for the diatonic scale were 9/8, 800/729, and 27/25. Apply the standard tempering formula:
c = 72 × log₂(f₂ / f₁)
9/8 → 12.23 steplets
800/729 → 9.65 steplets
27/25 → 7.99 stepletsRound to integers and you get {12, 10, 8}. The Committee’s own arithmetic, applied to its own ratios, is internally consistent. So far, nothing wrong, just an idiosyncratic choice of ratios. (Note that 800/729 is not a just-intonation interval in any clean sense; it is 10/9 (major second) lowered (divided) by a syntonic comma 81/80. The Committee’s ratios were a Pythagorean-flavored hybrid, not a derivation from the harmonic series.)
The prestige transfer happens later. In the early 20th century, Konstantinos Psachos, following his acute musical instincts, associated the Byzantine diatonic with the Didymus/Zarlino natural scale (the just-intonation diatonic) built from 9/8, 10/9, and 16/15. His ratios were then copied by many later theorists. Apply the same tempering formula to those:
9/8 → 12.23 steplets
10/9 → 10.94 steplets
16/15 → 6.70 stepletsRound these and you get {12, 11, 7}. Not {12, 10, 8}.
The textbook tradition has, for over a century, been:
- Citing the Didymus/Zarlino ratios as the natural-scale basis of the Byzantine diatonic.
- Pairing those ratios with the Committee’s {12, 10, 8} steplet values.
- Failing to notice (for a hundred years) that those ratios do not produce {12, 10, 8} under any standard tempering. They produce {12, 11, 7}.
The Committee itself was aware of the difference. A footnote in the 1881 publication explicitly distinguishes its diatonic scale from the Didymus/Zarlino “natural” scale. (Remember the instruction they were given, “don’t look too Western”). Chrysanthos before them likewise treated the natural scale as European and contrasted it with his own made-up diatonic. The conflation was introduced afterward, by theorists who wanted the Byzantine system to inherit the prestige of the just-intonation tradition without doing the arithmetic to check whether the inheritance was real.
It is not real. The numbers do not match. They have never matched. And the published theoretical tradition, with rare exceptions, has not noticed, or has not said.
The claim is not “the system is approximate” or “the system is descriptive.” The claim is: the published steplet values do not equal the tempering of the published ratios. That is arithmetic, and arithmetic does not negotiate.
4. The System Cannot Modulate Without Contradiction
Set the arithmetic mismatch aside, grant the Committee its own ratios and its own {12, 10, 8} tempering. The system still cannot govern modulation without an unwritten reset.
In the Committee’s diatonic scale, the interval {Pa, Vu} is 10 steplets. The interval {Di, Ke} is 12 steplets. These are not the same.
Now consider a modulation in which Pa takes the function of Di: The diatonic Di is cast on the diatonic Pa, a typical operation the Byzantine music modulators (phthores) prescribe in standard practice. Go up a note, find Ke (would have been Vu prior to the modulation). You have gone up 12 steplets. Now modulate-back to the original scale: Place the diatonic Vu modulator on Ke, things are now back to normal. Or, are they? Go down a step, {Vu, Pa}, 10 steplets. Congratulations, you have drifted 2 steplets above your original pitch. You went up 12, and down 10, in the same genus just by applying a plain modulation and resolving it soon after.
The 72-steplet grid cannot be both fixed and modulation-stable. If it preserves absolute pitch, it contradicts itself. If it retunes contextually, it is not a fixed pitch system. In either case, the published grid cannot function as the absolute acoustic foundation it is often claimed to be.
This is why direct theoretical modulations are often softened in real repertoire by preparatory descent, cadential confirmation, melodic detour, or notational management. The composer or chanter avoids crashing into the contradiction and has to walk around it. The survival of the repertoire is therefore not proof that the theory is closed. It is evidence that practical musicians learned how to avoid exposing the theory’s fatal flaw.
The standard defense is that a phthora is not a relabeling but a reorganization: the entire grid rebuilds around the new center, and the chanter retunes accordingly. This is correct, and it is exactly the concession that should not be elided. The phthora moves the chanter. After the modulation, returning to the original key requires a cadential reset that explicitly unmoves the chanter. The reset is real, and it is acknowledged by serious theorists. It is also routinely omitted from pedagogical presentations of the steplet grid.
The steplet grid, as taught, is not a closed system. It is a local map that requires unwritten transition rules to combine across modulations. There is nothing wrong with such a system in itself. Pythagorean tuning, just intonation, and meantone are all open in this sense and all useful. What is wrong is teaching the steplet grid as if it were closed: presenting integer values with the visual authority of a quantized pitch space, while the operations the system itself prescribes route around that quantization. And what is inexcusable is teaching that “traditional byzantine intervals are the best.”
The grid is a notational fiction calibrated to a contextual practice. The teaching pretends otherwise. And frankly, they have no other option, because otherwise they’d have to invalidate the core of the system and admit that the entire operation was just an elusive pursuit of just intonation.
5. The Chanters Do Not Sing It Anyway
The final embarrassment is that, even on its own terms, the 1881 system does not describe what living chanters actually do. This is acknowledged openly in Greek-language Byzantine music scholarship and in working psaltic forums. I’m not saying the chanters are right and that the Committee was wrong. I’m saying that they are both wrong. The chanters, because each has pretty much his own intervals and applies his own “melodic gravity” to taste. The Committee, because they failed to build a robust system that accurately encodes the practice.
This is not, by itself, scandalous; every notational system idealizes the practice it notates. The scandal is the asymmetry. When students are taught the system, they are taught the integers as if they were specifications. When chanters are recorded contradicting the integers, the contradiction is treated as the chanters’ license rather than as the system’s failure to describe its referent. The integers receive the prestige of “the theory.” The chanters receive the prestige of “the oral tradition.” The gap between them is pathologically praised as a feature, instead of recognized as a problem.
What Remains
Mastering Byzantine Polyphony takes a clear and strict position against the microtonal intervals, after careful and in-depth examination of all tuning systems. MBP proudly uses equal temperament as the most suitable for both melody and harmony. There’s no point wasting decades in transition from Chrysanthos to just intonation and even more decades in transition from just intonation to equal temperament because we just realized it’s a pain to modulate from one scale to another without losing our way back. That’s exactly why equal temperament was created and gained popularity. Because it was practical and honest.
The 72-steplet system was useful for its time but it was an ugly hack. The hundred-year habit of confusing and misleading students with inaccuracies, is even uglier. If the ratios are not natural, then the natural-intervals defense of the Committee’s tempering disappears, therefore nothing of theoretical substance is lost by going to Equal Temperament. The harshness of Equal Temperament thirds becomes a question of taste rather than a question of acoustic fidelity to a tradition, because there was no acoustic fidelity to begin with, only a notational pretense of it.