Pythagorean Tuning: How Stacked Fifths Built an Octave

Every note in a makam scale can be traced back to a single interval: the pure perfect fifth, a frequency ratio of 3:2. This is the oldest known method of building a musical scale, and it remains the acoustic foundation beneath the 53-tone system that modern Turkish makam theory uses to describe its intervals.

The method: stacking fifths

Established Pythagorean tuning constructs all intervals from one ratio: the pure perfect fifth (3:2). Starting from a note, you go up by a fifth, then another, then another, bringing each result back down into the same octave by halving its frequency as needed. Each new pitch is acoustically pure — every fifth vibrates in the simplest possible relationship to the one before it.

Established The first five fifths above a starting note C produce: G, D, A, E, B. Continuing the chain and reducing to a single octave generates every pitch in the scale. The intervals that result are themselves ratios of powers of 3 and 2:

These intervals are not approximations. They are exact consequences of the 3:2 ratio. A Pythagorean major second is two fifths stacked (3/2 x 3/2 = 9/4) and reduced by one octave (9/4 / 2 = 9/8). A major third is four stacked fifths reduced by two octaves (81/64). The system is entirely self-consistent — and entirely derived from one building block.

The problem: the Pythagorean comma

Established If you stack twelve pure fifths, you expect to arrive back at your starting note, seven octaves higher. You do not. Twelve fifths overshoot seven octaves by a small but definite amount:

(3/2)^12 / 2^7 = 3^12 / 2^19 = 531441 / 524288 ≈ 1.01364

Established This discrepancy is the Pythagorean comma, approximately 23.46 cents — roughly a quarter of a semitone. It is not a flaw in the method; it is arithmetic. No number of pure fifths will ever exactly equal any number of octaves, because no power of 3 can equal a power of 2. The circle of fifths does not close.

Established Euclid was the first to describe this ratio (531441:524288) in mathematical terms. Chinese theorists documented the same discrepancy by 122 BCE in the Huainanzi. The problem is universal — anyone who tunes by pure fifths will encounter it.

For Western music, this led to centuries of compromise tunings — meantone, well temperament, and eventually 12-tone equal temperament (12-TET), which splits the comma equally across all twelve fifths, making every fifth slightly narrow (701.96 cents becomes 700 cents) but allowing free modulation between keys.

The accident: why 53 fifths nearly close the circle

Established Twelve fifths miss closure by 23.46 cents. But if you keep stacking fifths past twelve, the overshoot does not grow uniformly. At certain points the accumulated chain comes remarkably close to an exact number of octaves. The most striking near-miss occurs at 53 fifths:

(3/2)^53 ≈ 2^31 x 1.00209

Established Fifty-three pure fifths overshoot 31 octaves by only about 3.615 cents — the interval known as Mercator's comma, after Nicholas Mercator (c. 1620--1687), who calculated the value precisely. This is more than six times smaller than the 12-fifth Pythagorean comma.

Established This means that if you divide the octave into 53 equal steps — 53-tone equal temperament (53-TET) — each step absorbs only 1/53 of Mercator's comma, about 0.068 cents. A 53-TET fifth (31 steps = 701.89 cents) differs from a pure fifth (701.96 cents) by only 0.07 cents. Human pitch discrimination bottoms out around 2--5 cents; 0.07 cents is inaudible. For all practical purposes, 53-TET is Pythagorean tuning.

Established The Chinese theorist Jing Fang (78--37 BCE) was the first to observe this property, calculating the 53-fifth overshoot with remarkable precision. Isaac Newton noted it independently in unpublished manuscripts around 1664--1665. William Holder published on it in 1694, and his name eventually attached to the resulting unit: the Holdrian comma, which is 1/53 of an octave (approximately 22.64 cents).

From Greece through Baghdad to Istanbul

Attested The historical chain linking ancient Pythagorean tuning to modern Turkish theory passes through the Islamic theorists who inherited, extended, and systematized Greek acoustic science.

Attested Al-Farabi (c. 870--950), writing in Baghdad, used Pythagorean ratios as the framework for his analysis of the oud's intervals in the Kitab al-Musiqi al-Kabir (The Great Book of Music). He mapped fret positions on the oud's fingerboard according to ratios derived from stacked fifths, grounding the instrument's practice in Pythagorean arithmetic.

Attested Safi al-Din al-Urmawi (c. 1216--1294) carried this further in his Kitab al-Adwar, systematizing the octave into 17 steps derived from a chain of Pythagorean fifths. His 17-note framework encompassed both the Arab and Persian modal traditions of his time, and the Kitab al-Adwar became the most copied, translated, and commented-upon music treatise in the Islamic world for centuries — surviving in Arabic, Persian, and Ottoman Turkish versions.

Attested The step from 17 notes to 53 was not a single leap. Ottoman theorists gradually extended the Pythagorean chain, recognizing finer interval distinctions within the 17-note framework. By the early 20th century, Rauf Yekta, Hüseyin Sadettin Arel, Suphi Ezgi, and Salih Murat Uzdilek formalized the 53-comma system as the official theoretical language of Turkish makam — the Arel-Ezgi-Uzdilek (AEU) system that remains standard in Turkish conservatories today.

Pythagorean tuning vs. just intonation

Established Pythagorean tuning and just intonation agree on the perfect fifth (3:2) and fourth (4:3), but they differ sharply on thirds. A Pythagorean major third (81:64, 407.82 cents) is built from four stacked fifths; a just major third (5:4, 386.31 cents) comes from the fifth partial of the harmonic series. The difference — about 22 cents, nearly a Holdrian comma — is called the syntonic comma, and it is audible.

Established For harmonic music, where thirds must sound consonant in chords, this is a serious problem. Pythagorean thirds sound harsh when sustained in vertical harmony. Western music's turn toward triadic harmony in the Renaissance drove the abandonment of Pythagorean tuning in favor of meantone and eventually equal temperament.

Practice For melodic music — and makam and dastgah are fundamentally melodic traditions — the situation reverses. Pythagorean intervals excel as horizontal, sequential steps. A Pythagorean major second (203.91 cents) is wider and more propulsive than an equal-tempered one (200 cents); the sharper leading tones and wider whole tones give melodic lines a clarity and directional energy that performers in these traditions prize. The "harsh" Pythagorean third, heard as a fleeting melodic step rather than a sustained chord, is not a deficiency — it is a feature.

Why this matters for makam

Practice Understanding Pythagorean tuning clarifies something that otherwise seems arbitrary: why Turkish theory uses 53 divisions, why those divisions work so well, and why performers trained in this tradition produce intervals that sound "out of tune" to ears calibrated to 12-TET but are in fact acoustically purer.

Established The 53-tone system is not an exotic invention. It is what happens when you take the oldest tuning method in recorded history — stacking pure fifths — and follow it to its logical conclusion. The Holdrian comma, the AEU interval table, the comma counts that define each cesni and makam: all of it traces back to the 3:2 ratio and the mathematical accident that 53 applications of that ratio nearly close the circle.