Equal Temperament: Tuning the Modern Musical World

A Mystery That Begins with the Ears

Before the C-sharp existed, there was the problem of the C-sharp.

There is something every musician who has ever tuned an instrument knows, even if they have not thought much about it: tuning is hard. Not in the sense that it requires skill — it does — but in a deeper sense: tuning perfectly is, strictly speaking, mathematically impossible.

That claim sounds strange. Music seems like something ordered, precise, measurable. Notes have defined frequencies; intervals have exact numerical relationships. How can it be impossible to tune perfectly? The answer lies in a contradiction that the Pythagoreans discovered more than two thousand years ago and that seventeenth-century instrument builders were forced to resolve: the mathematics of musical intervals do not close. No tuning system can be perfectly just in all keys at the same time. For centuries, that imperfection sat at the heart of one of the most complex technical problems in the history of music.

The Problem: The Pythagorean Comma

To understand the problem of temperament, it helps to understand what an interval is and why its mathematics do not add up.

An interval is the relationship between two sound frequencies. The octave, the most fundamental interval of all, has a ratio of 2:1: if the note A vibrates at 440 hertz, the octave above vibrates at 880. This relationship is perfect, universal, recognised by every human ear as consonant.

The problem appears when we try to build the other intervals with the same mathematical rigour. The perfect fifth — the interval between C and G, for instance — has a ratio of 3:2. It is a beautiful, stable, perfectly consonant interval. If we start from C and ascend twelve successive perfect fifths, we should arrive exactly seven octaves above the original C. Mathematics says that is how it should work. But it does not.

When you ascend twelve perfect fifths of 3:2, the result does not coincide exactly with seven perfect octaves of 2:1. The difference is small — roughly a quarter of a semitone — but it is real and audible. Medieval mathematicians called it the Pythagorean comma, and for more than a thousand years it was a kind of secret scandal of music theory: the sonic universe does not close, is not perfectly round, has a crack in it.

The Practical Consequences: The Wolf

This mathematical crack had very concrete consequences for musicians of the fourteenth, fifteenth, and sixteenth centuries.

The keyboard instruments of that era — organs, harpsichords, virginals — were tuned according to what we now call meantone temperament, which attempted to make major thirds perfect (the most important intervals in Renaissance polyphony) at the cost of slightly distorting the fifths. The result was that the instrument sounded splendidly in the most common keys — C major, D major, F major, G major — but produced increasingly distorted intervals as one moved toward keys with many sharps or flats.

The extreme case of this distortion had a name: the wolf interval. It was the interval produced by the ‘leftover’ fifth — the one that appeared when the cycle of fifths failed to close perfectly — and it sounded so bad, so dissonant, so alien to the rest of the instrument, that musicians called it the wolf because it howled. No musician played in the keys that contained it, simply because the result was unbearable. A keyboard tuned in meantone temperament thus had, in practice, a limited number of usable keys.

Equal Temperament: The Mathematical Solution

The solution to the wolf problem was elegant and, in a certain sense, philosophically bold: instead of trying to make some intervals perfect at the expense of others, why not distribute the imperfection evenly across all the intervals?

This is the idea of equal temperament: dividing the octave into twelve exactly equal semitones, each with the same mathematical ratio to the next. The octave remains perfect (2:1), but all other intervals are slightly imperfect. The fifth in equal temperament is not exactly 3:2 but an approximation that falls just two cents below the pure fifth. The major third is slightly wider than it would be in pure theory.

The result? No interval is perfectly consonant, but none of them howls. The wolf disappears because its imperfection has been distributed across all the fifths of the circle. And most importantly: all keys become exactly equivalent to one another. C major and F-sharp major sound with the same character, the same stability, the same palette of colours.

The idea of equal temperament was not new in the seventeenth century: Chinese and European theorists had discussed it long before. What changed was the need to apply it. As the tonal system demanded more and more modulations between distant keys, meantone temperament ceased to be an acceptable solution. The composer who wanted to modulate from C major to F-sharp major found that their keyboard instrument simply could not follow.

Bach and the Complete Circle

The most celebrated consequence of the adoption of equal temperament was a collection of pieces that Johann Sebastian Bach composed from 1722 onward: The Well-Tempered Clavier (Das Wohltemperierte Clavier), a series of preludes and fugues in all twenty-four possible keys — twelve major and twelve minor — arranged chromatically.

Bach’s message was as much musical as technical: with a properly tempered instrument, it is possible to compose — and have it sound well — in any key. The complete cycle was a demonstration, a proof, a manifesto. Each prelude and fugue had its own character, its own colour, its own personality: the darkness of C minor, the brilliance of D major, the gravity of B minor. Keys that had previously been unreachable became habitable territories with their own expressive qualities. We will speak about Bach in detail in the next post, but his name and equal temperament are inseparably linked.

Instruments and the Problem of Variable Tuning

Equal temperament solved the problem of keyboards, but not that of all instruments.

String instruments — violin, viola, cello, double bass — are instruments of continuous intonation: the musician can adjust the pitch of each note with infinite precision through finger pressure on the string. An expert violinist instinctively tunes certain notes slightly differently depending on the harmonic context: the third of a major chord will be played a little higher than equal temperament would indicate, because the ‘pure’ third sounds cleaner. This means that in a string quartet, musicians are constantly making micro-adjustments that keyboard instruments cannot make.

Wind instruments had their own problems: the natural horns of the Baroque period could only produce the notes of their instrument’s natural harmonic series; flutes and oboes required special fingerings to correct small intonation discrepancies. The transition to equal temperament was, in many cases, a gradual process that took decades to complete in orchestral practice.

What equal temperament guaranteed was not acoustic perfection — which is mathematically unattainable — but an equality of conditions among all keys. And that equality was what made possible the music we know: the symphony, the sonata, the string quartet as forms that traverse multiple keys in the course of a single work.

A Compromise That Transformed the World

Equal temperament is, at its core, a compromise. A collective agreement to accept evenly distributed imperfection in exchange for total freedom of harmonic movement. No fifth is perfectly just, no third is exactly what the physics of sound would produce naturally. But in return, the composer can go anywhere, modulate to any key, build harmonic structures of a complexity that would have been impossible under any other system.

It is tempting to see in this a metaphor for something larger: art as the art of productive compromises, of imperfections consciously accepted because the result justifies them. Not a single note of the piano is perfectly in tune. But the piano can play Beethoven’s Hammerklavier Sonata.

There is something else worth noting: the human ear adapts. Several generations after the widespread adoption of equal temperament, most Western listeners no longer perceive the slightly compressed fifths or the slightly widened thirds as imperfections. They perceive them as music. The standard born of a mathematical compromise became the reference, the norm, what ‘in tune’ sounds like.

Can an imperfection become invisible when everyone shares it? And what happens with the world’s other musical traditions — Arabic maqam, Indian ragas, the microtonal music of the twentieth century — that organise their semitones differently and which, to an ear trained on equal temperament, may at first sound ‘out of tune’? The answer is always the same: there is no single correct way to divide the octave. There are only different ways, each with its own logic, its own beauty, its own world.

“Nature gave humanity the natural scale; music gave it equal temperament.” — Paraphrased from eighteenth-century theoretical debates on the justification of temperament

Listening Suggestions

• Prelude and Fugue in C major, BWV 846 — Johann Sebastian Bach, The Well-Tempered Clavier (Book I) — the first of the twenty-four: listen knowing it was written to prove that all keys are possible
• Prelude and Fugue in C-sharp minor, BWV 849 — Bach, The Well-Tempered Clavier (Book I) — a key that before equal temperament was practically uninhabitable; it now has a gravity entirely its own
• Brandenburg Concerto No. 2 in F major, BWV 1047 — Bach — listen to the natural trumpet playing at the limits of its harmonic series; temperament does not solve every problem
• Keyboard Pieces — Louis Couperin — a contemporary of Bach; compare his harmonic world with that of The Well-Tempered Clavier and you will hear the difference temperament made