Three Pitch Systems Compared: 53-TET, 24-TET, and Just Intonation

Anyone learning Middle Eastern music soon encounters a confusing situation: Turkish sources describe intervals in Holdrian commas out of 53, Arabic sources use quarter tones out of 24, and acoustics textbooks talk about frequency ratios like 3:2 and 5:4. All three systems claim to describe the same music. None of them fully succeeds. Understanding what each system captures — and what it misses — is essential for any student moving between traditions.

The three frameworks

Established Three theoretical systems dominate the description of pitch in Middle Eastern modal music. Each starts from a different premise about how to divide the octave.

53-TET (Holdrian commas) divides the octave into 53 equal steps of approximately 22.64 cents each. This system closely approximates Pythagorean tuning — intervals built from stacked pure fifths (3:2). It became the official framework for Turkish makam theory through the Arel-Ezgi-Uzdilek (AEU) system in the early 20th century.

24-TET (quarter tones) divides the octave into 24 equal steps of exactly 50 cents each. This system extends the familiar 12-tone Western chromatic scale by inserting a pitch halfway between each semitone. It became the standard notational convention for Arabic maqam following its formal adoption in the early 20th century.

Just intonation does not divide the octave into equal steps at all. Instead, it defines intervals as ratios of small whole numbers — 3:2 for a perfect fifth, 5:4 for a major third, 7:4 for a natural seventh. These ratios correspond to relationships in the harmonic series, producing acoustically pure, beatless intervals. No fixed grid is involved; each interval is specified individually.

What each system claims

Established The 53-comma system claims fine-grained precision. With 53 steps per octave, it can distinguish intervals that differ by as little as 22.64 cents — small enough to capture the subtle differences between, say, a 5-comma interval (113.2 cents) and a 4-comma interval (90.6 cents) that are central to Turkish makam. Its Pythagorean basis means that fifths and whole tones are modeled with extraordinary accuracy.

Established The 24-quarter-tone system claims practical simplicity. Every pitch in a maqam can be notated using standard Western notation supplemented by half-flat and half-sharp accidentals. A musician literate in Western notation can read Arabic maqam notation with the addition of just two new symbols. The tradeoff is coarse resolution: 50-cent steps cannot distinguish intervals that fall between quarter-tone positions.

Established Just intonation claims acoustic purity. Its intervals are not approximations — they are the actual ratios that produce consonance. A just perfect fifth (3:2, 701.96 cents) is the interval the ear recognizes as perfectly in tune. Neither 53-TET nor 24-TET produces this interval exactly, though 53-TET comes remarkably close (701.89 cents, an error of only 0.07 cents).

Comparing the numbers

Established The divergence between these systems becomes concrete when you compare cent values for the same interval. The following table shows key intervals as modeled by each system, alongside the Pythagorean and just-intonation reference values. All cent values are mathematically derived and independently verifiable.

Several patterns emerge from this table.

Established For Pythagorean intervals (built from fifths and fourths), 53-TET is extraordinarily accurate. The fifth at 31 commas deviates by only 0.07 cents from just, and the fourth at 22 commas by only 0.07 cents. These errors are roughly 100 times smaller than the threshold of human pitch discrimination (approximately 5-6 cents under ideal conditions).

Established For the "neutral" intervals that are characteristic of much Middle Eastern music — the neutral second near 150 cents and the neutral third near 350 cents — 24-TET offers cleaner round numbers (150 and 350 cents), while 53-TET provides nearby but less tidy values (7 commas at 158.49 cents, or 15 commas at 339.62 cents). Neither matches the just-intonation ratios exactly. In practice, the performed pitch of these intervals varies considerably depending on the maqam, the phrase, and the performer.

Established For just-intonation intervals involving the factor 5 (the major third at 5:4, the minor third at 6:5), 53-TET is reasonably close — a just major third falls at approximately 17.08 commas, meaning 17 commas (384.91 cents) undershoots by about 1.4 cents. By contrast, 24-TET's 400-cent major third overshoots by nearly 14 cents. This is one area where 53-TET's finer grid provides a meaningful advantage.

The 1932 Cairo Congress

Attested The tension between these systems came to a head at the Congress of Arab Music, convened in Cairo from March 14 to April 3, 1932, under the patronage of King Fuad I. The congress was organized at the suggestion of the French ethnomusicologist Baron Rodolphe d'Erlanger and brought together Arab musicians and theorists alongside prominent European scholars including Bela Bartok, Paul Hindemith, Curt Sachs, and Erich Moritz von Hornbostel.

Attested A central agenda item was the standardization of the Arabic tuning system to 24 equal quarter tones per octave. The proposal had practical appeal: it would allow Arabic music to be notated, printed, and taught using an extension of Western staff notation. The 24-tone grid had already been formalized in writing by the Lebanese theorist Mikhail Mishaqa around 1840, though Mishaqa himself acknowledged that his teacher, Sheikh Muhammad al-Attar (1764-1828), was among many already familiar with the concept.

Attested The congress ultimately endorsed the quarter-tone framework as a notational convention, and it became the standard used in Arabic conservatories and publications. But the decision was pragmatic, not acoustic. The congress also documented 360 live performances (162 of which were commercially released), providing a sonic record that makes clear how much the actual intonation of Arab music diverges from any 24-step grid.

Practice The legacy of the Cairo Congress is double-edged. On one hand, the quarter-tone system gave Arabic music a workable notation and a shared theoretical vocabulary. On the other, it encouraged the misconception — especially outside the tradition — that Arabic music literally uses quarter tones, as if performers aim for pitches at exact 50-cent increments. They do not. The system is a notational convenience, not an intonation target.

Where they agree and disagree

Established All three systems agree on certain fundamental intervals. The perfect fifth and perfect fourth — the structural pillars of modal music across all Middle Eastern traditions — are modeled well by every system. A 700-cent fifth (24-TET), a 701.89-cent fifth (53-TET), and a 701.96-cent fifth (just) are all close enough that no listener would distinguish them.

Established The systems diverge most sharply on the intervals that give Middle Eastern music its distinctive character: the various sizes of second and third that fall between Western categories. A segah third in Turkish makam theory is nominally 5 + 4 = 9 commas above the second degree, landing at a specific point that neither 350 cents (24-TET) nor 386 cents (just 5:4) captures precisely. Similarly, Persian theorists using Farhat's cent-based measurements have documented intervals at positions that fit neatly into neither the Turkish nor the Arabic grid.

Practice The disagreement is most apparent — and least important — in precisely the intervals that matter most musically. The "neutral" seconds and thirds, the moteghayyer (mutable) pitches of Persian dastgah, the expressive inflections within a seyir — these are the intervals that performers adjust by ear, moment to moment, according to modal context. No fixed grid captures this flexibility, and no grid is meant to.

Why no single system wins

Practice Each system optimizes for something different. The 53-comma system offers a fine theoretical grid well-suited to the Pythagorean intervals that dominate Turkish theory. The 24-quarter-tone system offers notational practicality for a broad repertoire. Just intonation offers acoustic truth for isolated intervals but provides no practical framework for describing complete scales.

Attested Some scholars have proposed alternatives that attempt to bridge the gap. Ozan Yarman has argued for a 79-tone system that better models the intervals Turkish performers actually use. Hormoz Farhat avoided temperament grids entirely, measuring Persian intervals in cents and describing them as flexible ranges rather than fixed points. These approaches acknowledge what all three standard systems obscure: that performed intonation in modal music is fluid, context-dependent, and ultimately governed by the ear, not by arithmetic.

For the student

Practice When you encounter interval measurements in your studies, the first question is always: which system is the author using?

• Turkish sources (conservatory textbooks, AEU-based theory) use Holdrian commas. When a source says a segah interval is "5 commas," it means 5 x 22.64 = 113.2 cents.
• Arabic sources (conservatory notation, maqam method books) use quarter tones. When a source marks a note with a half-flat, it implies a 50-cent lowering from the natural pitch.
• Acoustic or comparative sources use cents or ratios. When a source says a neutral third is "355 cents" or "approximately 11:9," it is describing a measured or calculated interval without committing to either tradition's grid.

Practice None of these numbers tell you how the interval should sound. For that, you need to listen — to recordings, to teachers, to the tradition itself. The systems are maps. The territory is the music.