A4 in Kirnberger III

A4 is 440.000 Hz in all tuning systems that use A=440 as their reference. In Kirnberger III, the surrounding chromatic notes are tuned according to a circulating temperament concentrating comma adjustments on one highly narrowed fifth (D-A) and one slightly adjusted fifth, keeping most other fifths pure.

This system was used for Classical-era German keyboard music and Fortepiano performance.

Chromatic Scale at A4=440 Hz in Kirnberger III

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 Kirnberger III tunes each interval relative to A4.

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

Kirnberger III: Mathematical Formula

Kirnberger III concentrates its comma adjustments on just two fifths: D-A is narrowed by a full syntonic comma (21.51 cents), making it substantially narrow at 680.45 cents, while E-B is narrowed by one Pythagorean comma minus one syntonic comma. The other ten fifths are pure 3:2 intervals. This creates a system where C, G, F, and Bb major are extremely well-tuned (with nearly pure major thirds close to 5:4), while D, A, and E major still function but have less pure thirds. Kirnberger III is unusual in concentrating the full comma distortion on a single fifth (D-A), making it a least-equal circulating temperament.

Formula type: Cent offsets from equal temperament

How Kirnberger III Sounds

Kirnberger III has a distinctive warmth in the keys of C, F, G, and Bb major, which feature beautifully pure major thirds almost indistinguishable from just intonation. These keys have a lush, singing quality that rewards slow harmonic progressions. The D-A fifth — the most compressed in the system — creates a notably narrow interval that gives the D major and related keys a slightly tense, metallic quality when this fifth appears melodically. Key contrast in Kirnberger III is concentrated around this single narrow fifth, creating an asymmetric pattern of key colors. Johann Philipp Kirnberger promoted this temperament as the one most consistent with the natural harmonic series.

Historical Context

Johann Philipp Kirnberger, a student of Johann Sebastian Bach, introduced this temperament in his 1779 treatise Die Kunst des reinen Satzes (The Art of Strict Musical Composition). Kirnberger argued that his system was the most consonant circulating temperament and claimed it was consistent with Bach's own tuning preferences. This claim has been disputed by scholars, particularly those who favor the Bach-Lehman reconstruction. Kirnberger III found adoption among German keyboard players in the late 18th century and into the early 19th century, particularly for fortepianos. The system is valued today for its unusually pure near-key major thirds.

Other Tuning Systems for A4

See A4 in all temperaments →

For a full deep dive into Kirnberger III, see the Tunable guide to Kirnberger III.