A4 in Kellner

A4 is 440.000 Hz in all tuning systems that use A=440 as their reference. In Kellner, the surrounding chromatic notes are tuned according to Herbert Anton Kellner's 1977 reconstruction distributes the Pythagorean comma across five narrowed fifths and one slightly widened fifth, derived from analysis of Bach's numerical signatures.

This system was used for Bach keyboard works and Baroque German harpsichord music.

Chromatic Scale at A4=440 Hz in Kellner

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 Kellner tunes each interval relative to A4.

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

Kellner: Mathematical Formula

Herbert Anton Kellner's 1977 temperament narrows five fifths — C-G, G-D, D-A, B-F#, and F#-Db — by 1/5 of the Pythagorean comma (4.69 cents each), while slightly widening the fifth Eb-Bb to compensate. The remaining six fifths are pure. This creates a system where C, G, F, D, and Bb major benefit from improved major thirds, while remote key signatures retain Pythagorean-quality fifths for purity. The scheme is deliberately simple and derives from what Kellner argued was a numerological structure embedded in Bach's monogram and proportional signatures.

Formula type: Cent offsets from equal temperament

How Kellner Sounds

Kellner's reconstruction has a warm, settled quality in the near-key signatures and a clear brightening as the music moves toward more distant keys. C major and G major are noticeably smoother than equal temperament — major thirds approach 390-393 cents — while Eb and Ab major have a pure-fifth clarity that gives them a different but pleasing character. The system creates audible key-color variation without the strong contrasts of Kirnberger III. Performers using Kellner for Bach's keyboard works report that it gives the music a sense of tonal hierarchy — certain keys feel like home, while others feel like excursions — that equal temperament does not provide.

Historical Context

Herbert Anton Kellner, a German physicist and amateur harpsichordist, proposed this temperament reconstruction in a 1977 article in the journal Das Musikinstrument. Kellner argued that the system could be decoded from Bach's cabbalistic numerology and monogram patterns — a methodology that musicologists have largely rejected as unverifiable. Despite its contested historical basis, the temperament has been taken seriously as an acoustic system: it is practical, pleasing to the ear, and well-suited to the key signatures that predominate in Bach's keyboard works. The Kellner temperament has been recorded by several harpsichordists and is included in Tunable as one of the standard historical temperament options.

Other Tuning Systems for A4

See A4 in all temperaments →

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