Why Can’t Every Musical Interval Be Perfectly in Tune at Once?

Hearing Mathematics · How Sound Enters the Body and Mind · Article Five

On a string instrument with continuously adjustable pitch, a player can bring one fifth near 3:2 and one major third near 5:4, reducing beats in the chord of the moment. A fixed-pitch keyboard has to decide every key in advance. One C-sharp must serve A major, F-sharp minor, movements towards D-flat and many other relations. Once tuning is complete, the key cannot move for each new harmony.

The situation sounds like a technical problem awaiting a sufficiently accurate tool. Could twelve notes be tuned truly pure, making every fifth, third and octave exact? The answer is not that historical technicians lacked precision. It is that the requirements cannot all be satisfied simultaneously. The conflict is written into the ratios. A temperament does not abolish error; it decides where unavoidable differences will appear, which intervals will bear them, and what musical capacities will be gained in exchange.

Twelve pure fifths do not return to seven octaves

Starting from one frequency, every pure fifth multiplies by 3/2. Twelve successive fifths produce:

(3/2)^12 = 129.746…

Seven octaves multiply the starting frequency by:

2^7 = 128

Within a twelve-note pitch-class system, those routes are expected to arrive at the same class. Twelve fifths from C pass through G, D, A and the rest before returning to a form of C. Seven octaves also travel from C to C. The numbers, however, are unequal. Twelve pure fifths exceed seven octaves by about 1.36 per cent, or about 23.46 cents. The small discrepancy is the Pythagorean comma.

Greater measuring precision cannot remove it. 3^12 cannot become 2^19; a chain built with the prime factor three does not close perfectly on octaves built with two. A mathematical account of the Pythagorean comma expresses it as the difference between twelve 3:2 fifths and seven 2:1 octaves. No instrument has made a mistake. Two valuable relations simply cannot both be preserved without remainder.

The major third introduces another incompatible demand

If fifths are the sole concern, a sequence of pure 3:2 ratios can be folded back by octaves to make Pythagorean tuning. Its fifths are pure, but some thirds differ substantially from a simple 5:4. Just intonation introduces relations involving the fifth harmonic, giving calm major thirds in selected harmonies. The same named pitch can then receive different frequencies depending on the harmonic path used to reach it.

An E obtained from C by 5:4 is not identical to an E obtained through a chain of pure fifths and octave reductions. A keyboard with one E must select a position, unless it gains split keys or additional notes. Whenever a fixed-pitch system asks a finite set of keys to support many tonalities and chords, it compresses a path-dependent continuum into shared pitch classes.

“Purity” is therefore not one objective. Pure fifths, pure major and minor thirds, exact octaves and a closed cycle impose requirements that are partly incompatible. The question “which tuning is most accurate?” remains incomplete until it specifies: accurate for which intervals, tonal regions, modulations and instruments?

Temperament distributes differences that cannot be eliminated

One strategy preserves many pure fifths and concentrates the remaining discrepancy in a badly mistuned “wolf” fifth. Music restricted to selected tonalities can avoid it. Meantone temperaments deliberately narrow some fifths to obtain common major thirds nearer 5:4, while remote regions still deteriorate.

Well temperament is a family, not one tuning. Its members make all major and minor keys usable in principle without making every semitone equal. Fifths are altered by different amounts, so keys retain distinct interval patterns and colours. Twelve-tone equal temperament instead divides the octave into twelve equal logarithmic steps. The comma is distributed so that each interval type has the same numerical size wherever it begins.

None of these arrangements purifies every relation. They optimise different practices: local purity within a limited tonal territory, smoother common thirds, access to distant modulation, or the ability to reproduce the same keyboard pattern from any starting note.

Bach’s “Well-Tempered” title is not proof of modern equal temperament

Each book of Bach’s Well-Tempered Clavier contains twenty-four preludes and fugues spanning all major and minor keys. This is often paraphrased as evidence that Bach wrote for twelve-tone equal temperament, or even invented or proved the modern system. The evidence does not support that conclusion.

The German wohltemperiert means suitably or well tempered and is not a synonym for equal semitones. Scholars do not agree on one specific well temperament intended by Bach. A recent peer-reviewed corpus study of the forty-eight fugues explicitly begins from the position that the work was not conceived with equal temperament in mind and that Bach’s precise intended temperament remains unknown.

That caution does not weaken the work’s importance to the history of tuning. It makes the achievement more interesting. The Well-Tempered Clavier does not demonstrate that all keys had become acoustically identical. It demonstrates a tuning world usable enough for a composer to build complete musical structures in all twenty-four major and minor keys. Listeners can compare the unfolding C-major Prelude with works in sharp- and flat-heavy keys, but they should not attribute every colour on a modern piano to one established Bach-era tuning.

Fixed and flexible ensembles negotiate the conflict differently

A choir or string quartet is not always locked to every pitch of twelve-tone equal temperament. Performers may adjust pitch subtly according to melodic direction, harmonic function, resonance and the need to maintain an overall centre. A sustained major third may move towards 5:4; a leading tone may be treated differently because of its direction. This flexibility is not the simple pursuit of local purity at every instant. Lines must remain singable, repeated adjustments must not allow the ensemble to drift, and collaboration with a piano requires compatibility with fixed pitches.

Organs, pianos and fretted instruments build coordination into construction or tuning before performance. They surrender local flexibility to gain repeatable keys, complex polyphony and rapid modulation. A temperament is therefore not merely a frequency table. It is an agreement among instrument, composition, ensemble practice and perception.

Electronic instruments can relax some old constraints. A system can retune pitches dynamically according to the current chord or divide the octave into nineteen, thirty-one or more parts. New freedom brings new decisions. Who or what identifies the chord? Should a sustained note move when the harmony changes? How will notation and a performer’s motor habits adapt? More mathematical degrees of freedom do not decide how music should use them.

Imperfection is not a defect but a structural choice that makes a system workable

We normally treat error as deviation from a correct answer. In temperament, deviation is necessary when several correct relations cannot coexist. Equal-tempered thirds are not evidence of a careless approximation to a complete natural system. Their local impurity purchases global symmetry, transposability and broad compatibility.

Well temperaments accept a different inequality. They keep keys non-identical and turn the distribution of error into potentially usable difference. Just intonation can achieve striking fusion within selected harmonies but needs more pitches or active retuning to traverse complex tonal space. Every system answers a practical question: which differences should be preserved, and which can yield?

This is temperament’s deeper philosophical lesson. A finite keyboard is not a failed copy of an infinite acoustic reality. It is a structure formed for action. It allows composition, rehearsal, manufacturing, notation and expectation to coordinate, while leaving traces of incompatible relations in the subtle beats of its intervals.

When listening to The Well-Tempered Clavier, we need not turn each piece into a tuning demonstration. It is more revealing to hear how a keyboard that cannot preserve every pure ratio nevertheless accommodates every key, and how Bach turns that institutional capacity into the diversity of prelude and fugue. Mathematics proves that perfect closure is impossible. Music does not stop. It chooses an order in which the irreducible difference can be carried forward.

Primary sources and further listening

• A geometric and mathematical analysis of the Pythagorean comma
• Blake and Albrecht, “Tempering the Clavier”
• Bach, The Well-Tempered Clavier, Book I: work and score entry

Continue reading: Explore the Hearing Mathematics series.

If you would like to bring these ideas about listening, understanding, and practice to the keyboard, you might try ScoreFlow, an app I developed to make score reading and daily practice flow more naturally together.