A4 in Neidhardt

A4 is 440.000 Hz in all tuning systems that use A=440 as their reference. In Neidhardt, the surrounding chromatic notes are tuned according to Johann Georg Neidhardt's third-circle system distributes the Pythagorean comma nearly equally across twelve fifths, approaching equal temperament while retaining subtle key-color differences.

This system was used for 18th century German keyboard music and Near-equal temperament research.

Chromatic Scale at A4=440 Hz in Neidhardt

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 Neidhardt tunes each interval relative to A4.

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

Neidhardt: Mathematical Formula

Johann Georg Neidhardt proposed multiple temperament systems in his 1724 and 1732 treatises, categorized by their degree of equality. The Neidhardt system most commonly used today is his "third-circle" (Dritter Kreis) temperament, which narrows most fifths by 1/12 of the Pythagorean comma — identical in principle to equal temperament — while leaving one or two fifths with small deviations that introduce barely perceptible key variation. The result is indistinguishable from equal temperament in most practical contexts. Major thirds cluster very close to equal temperament's 400 cents, with deviations of just 1-3 cents across the chromatic scale.

Formula type: Cent offsets from equal temperament

How Neidhardt Sounds

Neidhardt's near-equal system sounds virtually identical to equal temperament to most ears. The subtle deviations — a few cents difference across the most extreme key comparisons — are only audible in carefully controlled listening conditions with sustained chords. Musicians using Neidhardt for Baroque repertoire report that it provides the acoustical neutrality of equal temperament while remaining historically grounded. For pieces that range across many key signatures without dwelling in any single tonal center, Neidhardt's system is among the most practical historical alternatives because it imposes no key-specific compromises. Its near-equal quality was in fact its design goal: Neidhardt explicitly sought the greatest possible equality.

Historical Context

Johann Georg Neidhardt (1680-1739) was a German music theorist and violinist who published two important treatises on temperament: Sectio canonis harmonici (1724) and Gäntzlich erschöpfte mathematische Abtheilungen (1732). He was the first major theorist to propose multiple temperament systems ranked by their degree of equality, from "village church" (simpler, unequal) to "court" (near-equal) temperament. His near-equal systems anticipated equal temperament's eventual dominance and were widely discussed by German theorists. Neidhardt corresponded with Johann Mattheson and other prominent figures, and his work influenced subsequent theorists including Marpurg. His systems are primarily of historical and scholarly interest today.

Other Tuning Systems for A4

See A4 in all temperaments →

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