Just Intonation

Historical context

Just Intonation (JI) is any musical tuning in which the frequencies of notes are related by whole number ratios. It is one of the oldest approaches to understanding pitch relationships, rooted in the observation that simple string length ratios produce consonant intervals.

Historically, JI has evolved by expanding its “prime limit”—the largest prime number used in its generating ratios.

Scale Building and the 3-Limit

The earliest documented tuning systems, such as Pythagorean tuning, operated strictly within the 3-limit. In this system, scales are built entirely by stacking pure perfect fifths (the 3:2 ratio) and reducing them by octaves (the 2:1 ratio).

Starting from a base frequency, moving up a fifth multiplies the frequency by 3/2. Doing this 12 times produces a sequence of 12 notes (the “Circle of Fifths”). Mathematically, this yields (3/2)¹², which equals exactly 129.746.

However, moving up exactly 7 octaves gives a multiplier of 2⁷, which is exactly 128.

Because 3¹² does not equal 2¹⁹ (or any power of 2), a cycle of pure 3:2 fifths never closes perfectly at an octave.

The Pythagorean Comma and the Wolf Fifth

The discrepancy between 12 pure fifths and 7 octaves is known as the Pythagorean comma — a microtonal interval of about 23.46 cents.

If an instrument like a harpsichord is tuned strictly by stacking 11 pure fifths, the final “leftover” interval needed to close the 12-note cycle will be horribly out of tune — 23.46 cents flatter than a pure fifth. This severely dissonant, howling interval became known as the Wolf Fifth. Any music modulating into keys that relied on this Wolf Fifth would sound jarring and broken.

The 5-Limit and the Syntonic Comma

As Western music evolved towards triadic harmony in the Renaissance, the 5-limit (Ptolemaic tuning) was introduced. This added the pure major third (5:4) and minor third (6:5).

While pure 5-limit thirds sound incredibly resonant and beatless, they introduce a new mathematical contradiction: the Syntonic comma. If you stack four pure 3-limit fifths (e.g., C -> G -> D -> A -> E) and reduce them by two octaves, the resulting major third (81:64) is notably sharper than the pure 5-limit major third (5:4, which equals 80:64). The geometric difference between the two (81/80) is the Syntonic comma, approximately 21.5 cents.

You cannot have a geometric tuning system that maintains both pure 3:2 fifths and pure 5:4 major thirds across all keys.

Temperament: The Compromise

These mathematical contradictions—the inability of primes 3 and 5 to cleanly map onto prime 2—mean that a fixed-pitch instrument (like a piano) tuned to pure Just Intonation can only effectively play in one key. Modulating to distant keys leads to severe dissonance as the commas accumulate.

To solve this, musicians invented temperaments (like Meantone, Well Temperament, and eventually 12-Tone Equal Temperament). Temperament intentionally detunes (“tempers”) the pure prime ratios slightly to close the geometric gaps, distributing the comma across multiple intervals so that no single interval becomes a “Wolf.”

Role in Prime Period Theory

In Prime Period Theory (PPT), Just Intonation is not merely a historical tuning system; it is the pure mathematical geometry of pitch.

PPT posits that musical relationships are fundamentally ratio relationships between periodic signals. Just Intonation represents these ratios in their pure, uncompromised form, prior to any temperament being applied.

The prime families of PPT map directly to the prime limits of JI:

• 2-prime: The octave equivalence (2:1).
• 3-prime: The structural scaffolding of fifths and fourths.
• 5-prime: The “colour” layer of major/minor thirds.
• 7-prime: The harmonic seventh and subminor intervals.
• 11-prime: The neutral intervals, representing the perceptual ceiling of deliberate harmonic intent in PPT.

While musicians rarely perform in pure, unyielding JI across multiple keys, JI remains the descriptive anchor. All temperaments and microtonal grids (such as 31 EDO and 72 EDO) in PPT are evaluated based on how effectively they represent or approximate these pure prime-ratio relationships.