The Story of Equal Temperament — Just Intonation vs. Equal Temperament

The Fate That “a Pure Tuning Doesn’t Close”

Rameau theorized harmony, and the road to “freely sounding chords in all keys” came into view (Article 4). But there was a physical and mathematical obstacle in the way — the problem of temperament.

The Pythagorean comma seen in Article 1 — the fate that stacking twelve perfect fifths does not land on seven octaves. This means “a mathematically pure tuning does not close as a ring of 12 notes.” The history of temperament is the history of where to push this gap that won’t close.

Just Intonation — Beautiful but Unable to Modulate

Just intonation builds every interval from simple integer ratios (octave 2:1, fifth 3:2, major third 5:4). This tuning, theorized by Zarlino (Article 3), lets chords fuse perfectly without “beating,” if played in a specific key.

But it has a fatal weakness: it immediately becomes impure when you modulate. Tune to optimize one key, and the intervals go out of tune in another. A keyboard in just intonation was “heaven in one key, hell in the next.” — Purity (just intonation) and freedom (playing in all keys) cannot coexist. This is the heart of the temperament problem.

Intermediate Solutions — Meantone and Well Temperament

Composers and tuners sought compromise points:

• Meantone — narrows the fifth a little in exchange for keeping the major third beautiful. Widely used on Renaissance-to-Baroque keyboards, but distant keys still break down (an awful “wolf fifth” arises)
• Well temperament — devised by Werckmeister and others. It makes all keys playable while leaving each key a slightly different “character.” The key characters — “bright D major,” “dark E-flat minor” — arise from this uneven tuning

The Symbol of Bach’s Well-Tempered Clavier

J.S. Bach’s Well-Tempered Clavier (Book I 1722 — the same year as Rameau’s Treatise on Harmony! / Book II 1742) is a monumental work of preludes and fugues in all 24 keys (12 major and minor each).

• it loudly proclaimed the possibility of “a tuning playable in any key”
• however, the “wohltemperiert” of the original title, “Das wohltemperierte Clavier,” likely refers not to modern 12-fold equal temperament but rather to well temperament (Bach’s intended specific tuning is still debated)
• in any case, as a symbol of the thought “conquer all keys,” a monument of temperament history

12-Fold Equal Temperament — the Decision to “Spread It Evenly”

What the modern age ultimately chose was 12-fold equal temperament:

• the Pythagorean comma (the gap that won’t close) is spread thinly and evenly across all 12 semitones
• every key becomes perfectly equivalent, and modulation becomes free
• the cost: every interval except the octave deviates slightly from just (the third in particular is a bit impure)

This was a democratic compromise: “every key is a little impure, but every key can be played the same way.” A historic decision to abandon “the purity of one key” and take “the equal freedom of all keys.” It became fully standardized in the 19th-20th centuries, and modern pianos and electronic instruments are nearly all in 12-fold equal temperament.

What Was Gained, What Was Lost

	Just intonation	12-fold equal temperament
Purity of sonority	◎ (fuses perfectly)	△ (slightly impure)
Modulation / all-key playing	× (impure)	◎ (free)
Key character (color per key)	present	lost (all keys equivalent)

Some musicians still lament that equal temperament lost the intrinsic character of each key — “C major is naive, D-flat major is sweet.” On the other hand, without equal temperament, the free modulation of the Romantics and jazz would have been impossible. A clear trade-off between what is gained and what is lost — the temperament problem is a model of “a choice with no correct answer.”

From This Article to the Next

We have traced temperament — the eternal trade-off between just intonation and equal temperament. With the foundation of all-key playing in place, harmonic theory reached its mature period in the 19th century. The next article turns to Riemann, who completed functional harmony reducing chords to the three functions T/S/D, and Helmholtz, who explained consonance and dissonance by physics (acoustics) — the era in which the thread “number vs. ear” is reunited as science.