A4 in Werckmeister III

A4 is 440.000 Hz in all tuning systems that use A=440 as their reference. In Werckmeister III, the surrounding chromatic notes are tuned according to an unequal circulating temperament using four narrowed fifths (by 1/4 Pythagorean comma each) among C-G-D-A and B-F#, allowing circulation through all 24 major and minor keys.

This system was used for Late Baroque German keyboard music and Lutheran organ repertoire.

Chromatic Scale at A4=440 Hz in Werckmeister 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 Werckmeister III tunes each interval relative to A4.

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

Werckmeister III: Mathematical Formula

Werckmeister III distributes the Pythagorean comma (23.46 cents) across four fifths: C-G, G-D, D-A, and B-F#, each narrowed by 1/4 comma (5.87 cents). The remaining eight fifths are pure 3:2 intervals. This creates a system where the keys of C, G, D, and A major have nearly pure major thirds (close to 5:4), while more distant keys — Bb, Eb, Ab — have wider, more Pythagorean-sounding thirds. The system fully circulates: every key is playable, though each has a distinct character. The major third on C measures approximately 390 cents, just 3.69 cents from pure.

Formula type: Cent offsets from equal temperament

How Werckmeister III Sounds

Werckmeister III produces a striking palette of key colors. C major and nearby keys sound warm and nearly pure, with major thirds that approach the purity of meantone. As the music moves through the circle of fifths toward the sharp and flat keys, chords become progressively brighter and more tense. F# major and Bb major have a heightened expressive character — not unpleasant, but clearly different from C major. This key-dependent variation was valued by Baroque composers who wrote character pieces and preludes exploiting specific key affects. The system fully supports chromatic music and all modulations, distinguishing it fundamentally from meantone.

Historical Context

Andreas Werckmeister introduced this temperament in his 1691 treatise Musicalische Temperatur, presenting it as a solution to the meantone wolf fifth problem while preserving meaningful key character. Werckmeister was a German organist and music theorist who corresponded extensively with other leading musical figures of his era. His temperament immediately influenced organ builders and keyboard composers throughout northern Germany. Johann Sebastian Bach may have encountered Werckmeister III directly, though the exact temperament Bach used for the Well-Tempered Clavier remains debated. The system was widely adopted in German Lutheran churches through the early 18th century.

Other Tuning Systems for A4

See A4 in all temperaments →

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