Just Intonation: Tuning by Pure Ratios

Pythagorean tuning builds every interval from one ratio — the pure fifth, 3:2. Just intonation asks a different question: what if you tuned every interval to its own pure ratio? A fifth at 3:2, a major third at 5:4, a minor third at 6:5 — each interval drawn directly from the harmonic series, each acoustically pure. The result is the most consonant tuning system possible, and the most impractical.

What just intonation is

Established Just intonation (JI) is any tuning system in which intervals are defined as ratios of small whole numbers. Unlike equal temperaments, which divide the octave into equal logarithmic steps, JI specifies each interval individually. There is no fixed grid. A just perfect fifth is exactly 3:2. A just major third is exactly 5:4. These are not approximations — they are the intervals themselves, the same relationships that exist between the partials of a vibrating string.

Established The most common form of JI is 5-limit tuning, meaning every interval can be expressed as a product of powers of the primes 2, 3, and 5. (Pythagorean tuning, by contrast, is 3-limit — it uses only 2 and 3.) The term "limit" was formalized by the American composer Harry Partch, who extended JI to the 11-limit and beyond in the mid-20th century.

Established The standard 5-limit just diatonic scale — historically known as Ptolemy's intense diatonic or the syntonic diatonic — produces the following intervals:

Established Compare the just major third (5:4, 386.31 cents) with the Pythagorean major third (81:64, 407.82 cents) — a difference of about 21.5 cents, the syntonic comma. This is not a subtle adjustment. It represents a fundamentally different conception of what a "third" should be.

Why pure ratios sound pure

Established When two tones are sounded together, their overtones either align or they do not. A vibrating string produces a harmonic series: the fundamental frequency, then twice that frequency, three times, four times, and so on. When two tones form a simple ratio, their harmonic series overlap extensively — the partials coincide rather than clashing.

Established When partials almost but not quite coincide, they produce a pulsing interference pattern called beating — a periodic fluctuation in amplitude that the ear perceives as roughness or instability. The beat frequency equals the difference between the two misaligned partials. For a just major third (5:4), the fifth partial of the lower note coincides exactly with the fourth partial of the upper note. No beating. For a 12-TET major third (400 cents, ratio approximately 1.2599), these partials miss each other by several hertz, producing audible beats that make the interval sound less settled.

Established Hermann von Helmholtz demonstrated this principle systematically in Die Lehre von den Tonempfindungen (1863), showing that consonance and dissonance are not merely cultural judgments but have a measurable acoustic basis in the alignment or misalignment of overtone partials.

A short history

Established Ptolemy (c. 100--170 CE) described the intense diatonic scale using the ratios 9:8, 10:9, and 16:15 as its three step sizes — a major tone, a minor tone, and a diatonic semitone. This was a deliberate departure from the Pythagorean diatonic, which used only two step sizes (9:8 and 256:243). Ptolemy's version introduced the prime factor 5 into tuning theory, producing the pure major third (5:4) that Pythagorean tuning cannot.

Attested The Roman-era theorist Didymus (1st century BCE) is credited with first adjusting the Pythagorean major third downward by the ratio 81:80 to arrive at the pure 5:4 third. This adjustment — the syntonic comma — is accordingly also called the Didymean comma.

Established Gioseffo Zarlino, in Le istitutioni harmoniche (1558), elevated 5-limit tuning from a theoretical option to a practical necessity. Working from what he called the senario — the ratios formed by the numbers 1 through 6 — Zarlino argued that the triad, not the isolated interval, was the basis of harmony. His framework required pure thirds, and therefore required just intonation.

Established Helmholtz's acoustical work in 1863 provided the physical explanation for what Zarlino had advocated on musical grounds: simple ratios produce aligned overtones, and aligned overtones produce the sensation of consonance.

The fundamental problem

Established Just intonation's purity comes at a cost. The 5-limit diatonic scale contains two different sizes of whole tone: the greater tone (9:8, 203.91 cents) and the lesser tone (10:9, 182.40 cents). The difference between them is the syntonic comma — 81:80, approximately 21.51 cents.

Established This discrepancy means that certain intervals within the scale are impure even though the system was designed to maximize purity. In a just C major scale, the triad C--E--G is perfectly consonant. But the triad D--F--A is not: the fifth D--A spans 40:27 (680.45 cents) instead of the pure 3:2 (701.96 cents), producing audible beating. This mistuned fifth is a wolf interval — an unavoidable consequence of the system's own logic.

Established The problem compounds when you try to modulate. Moving to a new key requires retuning intervals, because the pure ratios that work in one key do not work in another. Each modulation accumulates a syntonic comma of drift. A passage that moves through several keys and returns to the starting pitch arrives back shifted by one or more commas — audibly out of tune. This is comma drift, and it is the reason just intonation was ultimately abandoned for fixed-pitch Western instruments in favor of temperaments that distribute the comma's burden across all intervals.

Beyond the 5-limit

Established Limiting intervals to factors of 2, 3, and 5 is a choice, not a necessity. The harmonic series continues past the fifth partial. The seventh partial (7:4, 968.83 cents) produces a "natural seventh" that is flatter and more consonant than any version available in 5-limit tuning. The eleventh partial (11:8, 551.32 cents) yields a "neutral" interval that lies between a perfect fourth and a tritone.

Attested Harry Partch (1901--1974) built instruments and composed music using ratios up to the 11-limit, producing scales with 43 tones per octave. His work demonstrated that JI could be extended indefinitely, but each new prime factor multiplied the number of distinct intervals and further complicated the problem of comma drift.

Practice For Middle Eastern music, the higher-limit intervals are not merely theoretical curiosities. The "neutral" seconds and thirds characteristic of many makam and maqam scales fall close to ratios like 11:9 (347.41 cents) and 12:11 (150.64 cents). Whether performers are "aiming at" these ratios or simply producing intervals shaped by modal context and tradition is a question that the theory cannot settle and the practice does not need to.

Relevance to Middle Eastern music

Practice Neither Turkish makam theory nor Persian dastgah theory uses just intonation as its theoretical framework. Turkish theory employs the 53-comma Holdrian system, a Pythagorean approximation. Persian theorists like Hormoz Farhat have described intervals using direct cent measurements, avoiding commitment to any temperament grid. Yet certain intervals in both traditions closely approximate just ratios.

Established The 53-TET system approximates the just major third (5:4, 386.31 cents) at 17 commas (384.91 cents) — an error of only 1.4 cents. The just minor third (6:5, 315.64 cents) falls at 14 commas (316.98 cents), an error of 1.3 cents. These are far closer than 12-TET's approximations (400 cents and 300 cents respectively), meaning that the theoretical grid of Turkish makam can accommodate 5-limit intervals with reasonable accuracy even though it was designed on Pythagorean principles.

Practice In performance, the question is rarely which tuning system a musician is "using." Makam and dastgah are melodic traditions in which intonation is flexible, context-dependent, and guided by the ear. A performer may produce a third that closely matches the just 5:4 in one phrase and shade it toward the Pythagorean 81:64 in another, depending on the seyir of the makam and the musical moment. Just intonation is not the framework — but its ratios describe some of what the ear is doing.

Comparison with Pythagorean tuning and 53-TET

Established Pythagorean tuning and just intonation agree on the perfect fifth (3:2) and fourth (4:3). They diverge on every interval that involves the prime factor 5. The syntonic comma (81:80, 21.51 cents) is the measure of their disagreement — roughly the size of one Holdrian comma (22.64 cents), which is not a coincidence. The near-equality of the syntonic comma and the Holdrian comma is one reason the 53-tone system can approximate both Pythagorean and just intervals.

Established The following table shows the divergence on key intervals:

Established For fifths and whole tones, all three systems agree. For thirds and sixths, 53-TET tracks just intonation more closely than Pythagorean tuning — despite being designed as a Pythagorean approximation. This dual accuracy is why the 53-tone system has proven so durable.