Casals, measured note by note

Pau Casals used to tell his students not to worry about being out of tune with the piano. The piano was the one out of tune, he said, and its tempered scale was a compromise.

He meant it as more than a joke. Intonation was a question of conscience to him, each note a link in a chain, important in itself and as a connection between what came before and what comes next. He argued against the fixed, equidistant semitones of equal temperament and asked for smaller ones, above all in the steps leading to the first, fourth and fifth degrees of the scale.

Decades later somebody measured him. Every note of his recording of the Prelude from Bach's fourth cello suite, extracted by hand from a spectrogram, compared against equal temperament, against just intonation, and against Pythagorean tuning.

The system he argued against describes him as well as any

Average distance between what Casals played and each of the three theoretical systems, measured from each note to the one before it, in cents:

The gap between the first two is a third of a cent. Nobody can hear a third of a cent. By this measure the cellist who called the piano a compromise plays something the piano describes as well as anything else does.

Just intonation, the system built out of pure ratios and no beating, is five cents worse than either.

It loses in the arpeggios too

This one cuts against what the theory predicts.

The prelude is built from broken chords. If just intonation were going to explain any part of a solo cello performance, it would be there: these are harmonies, spelled out one note at a time, and the pure third is the whole point of the system. Measured on the arpeggios alone, just intonation still comes last.

The reason is that a broken chord is a harmony on the page and a melody in the ear. Beating happens when two notes sound at the same moment and their partials collide. In an arpeggio nothing overlaps. There is no roughness to remove, and removing roughness is what a pure third buys you over a tempered one.

A study of double bass intonation sets out the mechanism. Two notes sounding together are compared side by side, and the untuned version costs the listener more work, so the pure interval wins. Two notes in sequence are a different task: the first one survives as a fading trace, and the learned category takes over from the acoustics.

And the place where Casals did reach for pure intervals, by his own account, was double stops. Two notes, one bow, at the same instant.

The melodic rule does not hold either

The old formula says just intonation for harmony and Pythagorean tuning for melody, and it has been repeated since the 1860s. Casals fits its second half: his major thirds come out wide, near the Pythagorean size, and the leaps that push his intonation highest land on the third and seventh degrees of the most important chords in the piece.

Fifty-three conservatory violin students, sight-reading a melodic piece with no accompaniment at all, played their major thirds nine or ten cents narrow. That is the just intonation direction, in the most melodic situation you could construct. The paper reporting it says the famous attachment of violinists to Pythagorean tuning looks weaker than its reputation.

One cellist and fifty-three students disagree, which means the music being melodic or harmonic is not what decides it.

What decides it

Ask what is sounding alongside you.

Practising alone, one note at a time, nothing overlapping: a pure third has nothing to remove, and you are choosing between two systems you cannot tell apart at a third of a cent. Pythagorean tuning gives you the leading notes Casals wanted. Equal temperament gives you the grid your accompanist will use.

Double stops, a drone, a piano, an orchestra, other people: now partials collide and beating is audible to anybody. Just intonation is the setting where the beats go away, and it is the one situation where a few cents are the difference between a chord that rings and a chord that grinds.

In the app

You declare a tuning system before you play, and wabetune measures every note against that system rather than against the piano.

It matters most when you play with others. Spectrum shows every note sounding at once and names the chord they make, and it grades each note twice: against the system you declared, as the map does, and against a pure interval over the chord's root as that root was played. A quartet's major third sits fourteen cents under the piano's, and the second figure says whether yours does.

Equal temperament is the default because it is what can be said before you have declared anything. It is a starting point and not a recommendation.

What this rests on

One recording of one cellist, with every note read off a spectrogram by hand. The violin figures come from fifty-three students who played their piece once, at sight. The account of how simultaneous and sequential notes are judged comes from a doctoral thesis that assembled the psychology literature and measured nobody.

None of the three tells you which system to declare. They agree on something smaller and more useful: nobody plays one system, and the choice belongs to the person holding the bow.

Sources: Ana Llorens, “Understanding expressive intonation: Casals’ Bach beyond conscience”, Quodlibet 76 (2021). Caldera and Archilla-Segade, Music Perception 42(1), 2024. John Michael Priester, “Double Bass Intonation: A Systematic Approach to Solo and Ensemble Playing”, doctoral thesis, Indiana University, 2015.