Can Mathematics Explain Why Music Moves Us?

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

Bach’s Goldberg Variations begins with an Aria, passes through thirty variations, and ends with the direction Aria da capo: the Aria is to be played again from the beginning. If the performer makes no alteration, the notes of the returning Aria are the same. A recording could even splice in an acoustically identical copy. Few listeners, however, experience the ending as “already heard, therefore zero information”. After canons, dances, slow movements, virtuosity and a quodlibet, the Aria seems to return from a distance. The material is the same; the listener is not.

The moment tests the largest temptation of this series. If frequency, proportion, temperament, beat and expectation can be described mathematically, will we eventually obtain a formula for why music moves us? The answer depends on what “explain” means. Mathematics can reveal structures that make expectation, repetition and difference possible. Psychology and neuroscience can investigate bodily response. The significance of a work to a particular person also includes memory, culture, attention, performance and a life situation. Compressing all these layers into one number does not make the explanation deeper. It removes the question.

In probability, surprise is not a fixed property of a sound

Whether an event is surprising depends on the probability a listener assigns to it at that moment. Information theory often represents the information content of an event in context as -log₂ P(event | context): the lower the probability, the greater the surprise. A cadence that is extremely common within a style carries relatively little information when it arrives as expected. An unusual harmonic turn can produce a larger prediction error while remaining intelligible through what preceded it.

The equation matters because it treats surprise as a relation, not an innate property of a note. One C-major chord has different probabilities and meanings at the opening, after a dominant seventh, following a remote modulation, and in a final reprise. A listener familiar with tonal practice and a listener encountering that style for the first time also bring different internal models.

Computational systems can learn conditional probabilities for notes, harmonies and rhythms from musical corpora. Predictions can be compared with human expectation judgements, and researchers can ask how familiarity reduces uncertainty. Such methods locate relatively expected and unexpected moments in a work. Their probabilities still depend on training data and representation. Encoding music as pitch classes, absolute pitches, chord labels or acoustic features yields different worlds. Probability is not an objective transcript of a work’s secret meaning; it is a testable model with a defined perspective.

Pleasure can come from fulfilled prediction and from meaningful correction

Music that fulfils every prediction may reassure and may also become dull. Music in which no event can be predicted may excite and may also fail to cohere. Many accounts therefore examine the dynamic balance between predictability and surprise. Repetition strengthens a model. Deviation makes error salient. Later events may reveal that the deviation belonged to a larger structure.

A broad review of music in the brain presents prediction as an important link among musical perception, action, emotion and learning. An experiment on musical reward prediction errors found, within a defined learning task, that musical outcomes relative to expectation were associated with reward-related neural activity and subsequent choices. These studies make the proposition “expectation affects pleasure” experimentally tractable.

They should not be compressed into “dopamine is why music moves us” or “the best music has a fixed percentage of surprise”. Experimental tasks, participant samples, repertoire and measurement definitions limit conclusions. Reward, emotion, aesthetic judgement, chills and personal significance are not one dependent variable. A listener may be profoundly moved by grieving music without wanting to call the experience pleasure. Another may admire structure without intense emotion.

Prediction theory is most useful for explaining how music regulates attention and tension, not for declaring one cause of all musical feeling.

Repetition preserves material while changing its place in time

At the signal level, exact repetition adds no new sample. In musical experience, repetition always occurs after “having heard”. A first statement establishes memory. A second invites comparison. Further repetitions can produce habit, ritual, oppression or expectation. The repeated item remains the same while its relations multiply.

The Goldberg Variations makes this distinction unusually clear. The opening Aria is not simply a melodic tune ornamented note by note in thirty later pieces. The variations principally share a bass and harmonic model while developing highly independent characters. Peter Williams’s study of the movements describes the Aria itself as a rich melodic setting of an underlying harmonic or bass model rather than a bare tune-theme. After Variation 30, the score explicitly returns to Aria da Capo.

At the return, a listener may not remember every detail of every variation. They retain something of the journey’s duration, contrasts of speed, the succession of canons, and the weight of local events such as Variation 25. Every pause in the Aria has a new scale because it now occupies an ending. Mathematics can identify the same material in a different formal position. Psychology can describe how prior events alter memory and prediction. Musical experience hears “return” as a meaning unavailable to the opening alone.

Personal memory is not noise outside the work

A piece may become connected with a funeral, a family, a period of study or one person. Years later, a few notes can change emotion by evoking autobiographical memory. Research on the neural architecture of music-evoked autobiographical memories found relations among familiarity, autobiographical salience and affective response, involving multiple memory- and self-related processes.

Such significance cannot be calculated from a score alone. Two listeners can receive closely similar acoustic input and respond oppositely because of life associations. Personal memory is not merely contamination of “pure music”. Music can carry mourning, celebration, religion, identity and intimacy in social life precisely because it can be stored and recalled together with context.

It does not follow that explanation lies only in private stories. Reprise, cadence, crescendo and beat systematically shape temporal experience for many listeners. A work and performance establish possibilities. Personal history does not create meaning from nothing; it changes salience, association and value within structures the music provides.

Why “major is happy, minor is sad” is not enough

Computational descriptions of musical emotion often use mode, tempo, level, timbre and rhythm. Some features correlate statistically with arousal or positive–negative judgements, but no single correspondence is dependable. Fast minor-key music can be fierce, comic or ecstatic. Slow major-key music can be solemn, remote or grief-stricken. Harmonic context, melodic direction, articulation, cultural learning and titles alter interpretation.

More fundamentally, four questions are being confused when music is said to be “sad”: does the music express sadness, does a listener recognise sadness, does the listener personally feel sad, and does the listener like the sad music? A data set that asks participants to select happy or sad labels teaches a model about classification in that task, not a complete theory of feeling.

Mathematical models need variables, and variables require operational definitions. Mature quantitative research does not forget scope merely because it can calculate. Numbers allow experiments to be compared while potentially excluding mixed feeling, bodily sensation and personal significance that do not enter the scale. Exclusion can be necessary for research. It must not be disguised in the conclusion as non-existence.

Performance makes the same structure carry different commitments

The returning Aria changes with performer, instrument and tempo. Harpsichord and modern piano have different onsets and decays. A pianist can shape return through touch, voice balance and pause. Decisions about sectional repeats alter the proportion of the whole work. The score specifies many relations without fixing every audible detail.

This is why neither score analysis nor signal analysis is complete by itself. A score displays relations across performances. A recording contains one realised time and timbre. A live event adds space, shared attention and the irreversibility of performance. A listener may be moved by the compositional design, or because one performance makes the design newly audible.

We do not need to choose between “the work determines everything” and “the listener projects arbitrarily”. Musical experience is a relational event. Written possibilities, sounding action, bodily prediction, existing memory and present circumstances meet during a time that cannot be replayed as the same occasion.

How far mathematics can explain

We can now give a definite answer.

Mathematics explains many conditions of music. It shows how vibrations superpose, frequency ratios form intervals, tuning requires trade-offs, beats establish periodicity, auditory voices correlate, and probabilities represent expectation and surprise. These are not peripheral facts. They are part of the framework within which sound can be organised as music.

Mathematics also helps test mechanisms. A model can predict which rhythms are more readily grouped, which spectra are rougher, or which events are rare within a style corpus. Experiments can then ask whether those predictions correspond to behaviour and physiology.

Mathematics alone cannot supply a piece’s meaning for a person. Meaning is not one more variable hidden in the waveform, waiting for a stronger algorithm to extract it. It is formed as structure is lived by a listener with a body, a history, relationships and a culture. We can continue to study regularities in that process without mistaking a regularity for a substitute for individual experience.

This does not reserve music as a mystical domain beyond inquiry. It requires stricter inquiry. Every model should state its object, scale and omissions. A system might predict average pleasure ratings effectively without thereby explaining mourning. Brain imaging can identify associated activity without speaking for a person about why the music belongs to a particular relationship.

To hear mathematics is finally to hear the limits of relations

This series began with the formation of pitch and travelled through octave, harmonics, consonance, temperament, piano tuning, rhythm and polyphony before returning to a listener. Its numbers are real: 440 Hz, 2:1, 3:2, the twelfth root of two, beat rates and probabilities. They make structures comparable, manufacturable and performable.

The numbers did not make themselves into music. The cochlea selects frequencies. Neural systems organise objects. Bodies predict beats. Attention separates voices. Memory alters repetition. Culture supplies learnable grammars. A person carries meaning in a particular moment. Every layer depends on earlier conditions without being completely replaceable by them.

When the Aria returns in the Goldberg Variations, we can say that it is the same as the opening and that it is different. Both claims are accurate because they measure different objects. Mathematics preserves identity of relation. Time creates a new position. Music often moves us precisely where those truths meet.

Primary sources and further listening

• Vuust et al., “Music in the Brain”
• Research on musical reward prediction errors and learning
• Research on the neural architecture of music-evoked autobiographical memories
• Williams on the Aria and movements of the Goldberg Variations
• Bach, Goldberg Variations: scores and recordings

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.