Fifty-three violin students, one piece, every note measured

Fifty-three conservatory violin students in Spain sat down in front of a laptop and sight-read the same piece. Fifty-four bars, written for the study, walking through fifteen keys so no interval could settle into a comfortable position or lean on an open string. Each student played it once. Then the researchers measured every note by hand.

They wanted to know how close a violin student sits to the tuner on the music stand.

The fourths prove nothing

The students played fourths within two cents of equal temperament, and the same goes for major seconds and the one minor sixth in the piece. A tuner would have passed them.

Those intervals also sit within two cents of Pythagorean tuning and of just intonation, because all three systems put a fourth in the same place. A fourth in tune says that the player can hear a fourth, and says nothing about which system they have in mind.

The thirds and the semitones are where the three systems part company, and where these students went two directions at once. Major thirds came in nine or ten cents under the tempered size, most of the way to just intonation. Semitones came in narrow, twelve cents narrow going up, which points at Pythagorean tuning instead.

Violin treatises have argued for two centuries over which of those systems a string player uses. These fifty-three students used both, inside a minute of music.

Up and down are different sizes

A semitone played upward measured 88 cents. The same semitone played downward measured 92. Minor thirds did the same thing harder and the other way round: wide going up, close to tempered coming down. Fourths stayed the same size in both directions.

The gap held across fifty-three players who study with different teachers in six schools. The researchers read it as a left-hand problem rather than a musical choice. A finger arriving from below and a finger arriving from above make two different movements, and they stop in two different places.

None of those students chose a narrow semitone. You cannot choose something you cannot hear, and four cents sits under the threshold where an untrained listener hears any difference at all.

The years did not close the gap

The students ranged from second year to the final years of higher education. Across thirteen kinds of interval, one showed a relationship with year of study, and it accounted for about a tenth of the variation. The other twelve showed none.

Years of work move tone, control, rhythm, phrasing. They move what a teacher can hear and name in a lesson. The interval sizes stayed where they were, because a practice room holds no instrument that measures them. You cannot correct a four-cent asymmetry by resolving to correct it.

Each student had their own semitone

The students' intervals tracked each other. A player whose ascending semitones ran narrow had descending semitones that ran narrow too, close enough that one predicts the other. A player who stretched the major third squeezed the minor third to match, and between those two, one accounts for about forty percent of the variation in the other.

Each student carries a semitone of their own and a working idea of how wide a third is, and carries it through the whole piece in both directions. That is a signature, and a single reading cannot show it to you. In one note it amounts to four cents, which is noise. It surfaces once you stack enough notes that the accidents cancel and the habit does not.

The cost of measuring this

That study took a spectrogram and someone marking the start and the end of every note by hand, for months. Few studies like it exist, because the numbers appear once you measure a whole performance, and measuring a whole performance costs a semester.

Your tuner is accurate. It answers where a note sits right now, and then it throws the answer away. The study asked a slower question: where does this note keep going? Answering that one takes the measurements kept.

wabetune listens while you practise and keeps the numbers, session after session, so the note that drifts shows up as a pattern instead of a bad afternoon.

The limits

Fifty-three students is a decent sample and a small one. They came from six schools in one Spanish region, so a limited number of teachers taught them. They were students, and they sight-read the piece once, so some of that spread is reading rather than intonation. They played violins, which leaves the cello and the double bass unmeasured.

The shape survives those limits. Players do not play one system, and direction changes the size of an interval. The years alone move neither.

The researchers closed by suggesting that a standard electronic tuner makes a poor model for learning intonation. We had already built the product on that. Reading it in a journal was still a strange morning.

Source: Caldera and Archilla-Segade, “Equal Temperament and Interval Sizes of Ascending and Descending Intervals: A Descriptive and Inferential Analysis of Fifty-three Violin Students’ Intonation”, Music Perception 42(1), 2024, pp. 19–33.