Holdrian Commas and 53-TET: The Mathematics of Ottoman Tuning

Western music divides the octave into 12 equal steps. Turkish makam theory divides it into 53. This is not arbitrary — 53 equal divisions happen to approximate the acoustically pure intervals of Pythagorean tuning with remarkable accuracy, and this mathematical coincidence became the foundation of modern Turkish music theory.

What is a Holdrian comma?

Established A Holdrian comma is 1/53 of an octave. In cents (the standard unit for measuring musical intervals, where an octave = 1200 cents), one Holdrian comma equals approximately 22.64 cents:

1200 / 53 ≈ 22.6415 cents

Attested The name honors William Holder (1616--1698), an English clergyman and music theorist who noted in his Treatise of the Natural Grounds and Principles of Harmony (1694) that dividing the octave into 53 parts produces intervals closely matching those of just intonation. Holder was not the first to observe this — the Chinese theorist Jing Fang (78--37 BCE) and the medieval Islamic theorist Safi al-Din al-Urmawi (d. 1294) had noted similar divisions — but the Western name stuck.

Why 53?

Established The number 53 is not chosen for convenience. It emerges from a fundamental property of acoustics. If you stack 53 pure perfect fifths (each with a frequency ratio of 3:2) and reduce them back into a single octave, you land almost exactly back where you started. The discrepancy — the amount by which 53 fifths overshoot 31 octaves — is vanishingly small:

(3/2)^53 ≈ 2^31 × 1.00209...

Established This means a circle of 53 acoustically pure fifths nearly closes on itself. The resulting 53-step division of the octave therefore approximates Pythagorean intervals (built from stacked fifths) with extraordinary precision. A Pythagorean major second (9:8 ratio, ≈ 203.91 cents) falls almost exactly on 9 Holdrian commas (9 × 22.64 ≈ 203.77 cents) — an error of only 0.14 cents, well below the threshold of human perception.

By comparison, 12-TET (the Western standard) approximates a major second at 200 cents — an error of nearly 4 cents from the Pythagorean value. The 53-tone system is roughly 28 times more accurate for this interval.

The Arel-Ezgi-Uzdilek system

Established In the early 20th century, three Turkish theorists — Hüseyin Sadettin Arel, Suphi Ezgi, and Salih Murat Uzdilek — formalized the use of 53-TET as the theoretical framework for Turkish makam. Their system, known by the acronym AEU, became the official theory taught in Turkish conservatories and remains dominant today.

Established In the AEU system, intervals within a makam are expressed as sums of Holdrian commas. The key interval sizes are:

Established The intervals of 5 and 8 commas have no close Western equivalents. These are the "between frets" pitches — the microtonal intervals that distinguish makam from any 12-tone system. A 5-comma interval is roughly a quarter tone larger than a Western semitone; an 8-comma interval is roughly a quarter tone smaller than a Western whole tone.

Theory vs. practice

Practice The AEU system is a theoretical model, not a prescription for exact intonation. Performers do not measure commas while playing. In practice, a skilled makam musician adjusts intonation according to the character (seyir) of the makam being performed, the register, the musical context, and aesthetic judgment. The comma values provide a shared vocabulary for discussing and teaching intervals, not rigid targets.

Attested Some scholars and practitioners have criticized the AEU system's strict 53-TET grid as an oversimplification. Ozan Yarman has argued that a 79-tone system better models the intervals performers actually use. Others have noted that historical Ottoman practice did not adhere to any single equal temperament. The debate is ongoing, but the 53-comma framework remains the standard pedagogical tool.

Relationship to other tuning systems

Established The 53-TET system is specifically a Pythagorean approximation — it excels at modeling intervals derived from stacked fifths. It is less accurate for intervals based on pure thirds (the 5:4 ratio of just intonation), though it still approximates these better than 12-TET.

Attested The Persian dastgah tradition does not use the 53-comma framework. Persian theorists have generally described their intervals in other terms — Hormoz Farhat used cent measurements directly, avoiding any commitment to a fixed temperament grid. This difference in theoretical language is one of several ways the Turkish and Persian traditions, despite shared historical roots, have diverged in their modern codifications.