A4 in Just Intonation (Minor)

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 minor-key vocal music and historically-informed string chamber music.

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

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 (Minor) tunes each interval relative to A4.

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

Just Intonation (Minor): Mathematical Formula

Just intonation (minor) applies 5-limit ratios optimized for the natural minor mode, with C as the tonic. The minor triad on C uses ratios 1:1, 6:5, and 3:2 — the minor third is pure at 315.64 cents, narrower than the equal-tempered 300 cents. A4 falls at the ratio 5:3 from C4 (the same position as in just-intonation-major), giving 436.043 Hz. This placement is acoustically correct for C-minor tonality: A appears as the sixth degree of the C natural minor scale, and its 5:3 ratio produces a pure minor third with F. The 7.72-cent deviation from 440 Hz is an accurate representation of A's role in a C-tonic just system.

Formula type: Just intonation ratios (5-limit)

How Just Intonation (Minor) Sounds

Just intonation (minor) imparts a warm, dark quality to minor-mode music. The pure minor third (6:5) has a subtly different color from the equal-tempered minor third — less neutral, with a slightly hollow resonance that evokes the acoustic timbre of Renaissance viols or gut-string lutes. The perfect fifth remains as pure as in the major version, anchoring the harmony. Minor triads in this system have an expressive gravity that performers in historically-informed practice find emotionally compelling. Like the major version, chromatic passages and modulations expose key-center limitations.

Historical Context

Just intonation in the minor mode was applied particularly in Renaissance music featuring the Dorian, Phrygian, and Aeolian modes. Theorists such as Francisco de Salinas (De musica libri septem, 1577) explored minor-mode just intonation as a distinct tonal system. The system was well-suited to the polyphonic vocal music of Lassus and Victoria, where the emotional character of the minor third was central to text expression. Unfretted string instruments and voices could naturally adapt their intonation to achieve pure minor thirds. The system fell into disuse as tonal music increasingly demanded smooth modulation, a capability that meantone and well temperaments provided far more practically.

Other Tuning Systems for A4

See A4 in all temperaments →

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