A4 in Just Intonation (Major)

In Just Intonation referenced to C as the tonic, A4 is tuned to the ratio 5:3 from C4, giving 436.043 Hz rather than 440.000 Hz. This is technically correct — just intonation is key-center-dependent. To use A4=440 as the tonic in just intonation, all ratios would be applied from A directly.

This system was used for Renaissance a cappella choral music and barbershop quartet.

Chromatic Scale at A4=440 Hz in Just Intonation (Major)

The table below shows all 12 chromatic notes at octave 4. A4 is the tuning reference — its frequency is 440.000 Hz in all temperaments at this concert pitch. The other notes show how Just Intonation (Major) tunes each interval relative to A4.

Positive cents = sharper than equal temperament. Negative = flatter. 100 cents = 1 semitone.

Just Intonation (Major): Mathematical Formula

Just intonation (major) uses small-integer ratios from the harmonic series to tune each pitch class relative to C as the tonic. The major triad on C is built from ratios 1:1, 5:4, and 3:2 — the three purest consonances. A4 is assigned the ratio 5:3 from C4, which gives 261.626 × (5/3) = 436.043 Hz rather than 440.000 Hz. This 7.72-cent flat deviation relative to equal temperament reflects the C-centric nature of the data: just intonation is key-center-dependent. An A-tonic just intonation scale would assign A4 = 1:1 = 440.000 Hz, but all other pitches would then shift accordingly.

Formula type: Just intonation ratios (5-limit)

How Just Intonation (Major) Sounds

Just intonation (major) produces the purest chords in the diatonic C-major context — the major third (5:4) and perfect fifth (3:2) beat at zero, creating an almost electronic stillness in held chords. The effect is striking in close-position vocal harmony, where singers can lock resonance and feel the notes "snap" into alignment. However, melodic passages can feel slightly unequal, since whole tones alternate between 9:8 (203.9 cents) and 10:9 (182.4 cents). Modulation to other key centers exposes the system's C-centric structure, making chromatic music impractical without retuning.

Historical Context

Just intonation in the major mode flourished in Renaissance Italy, particularly in the a cappella sacred music of Josquin des Prez, Palestrina, and their contemporaries. Theorist Gioseffo Zarlino codified the system in his 1558 treatise Le istitutioni harmoniche, arguing that only ratios derived from the first six integers (senario) produce true consonance. The system was practical for voices and unfretted strings, which can adjust intonation in real time, but problematic for fixed-pitch instruments. Keyboard builders experimented with split keys to accommodate multiple pure thirds, but the advent of well temperament eventually made such mechanical solutions unnecessary for ensemble practice.

Other Tuning Systems for A4

See A4 in all temperaments →

For a full deep dive into Just Intonation (Major), see the Tunable guide to Just Intonation (Major).