The publication of an Equal Temperament tuning scheme in June 1722 by Johann Mattheson coincided with the appearance of two manuscripts: 

• Johann Sebastian Bach's Das Wohltemperirte Clavier (The Well-Tempered Keyboard), containing a tuning scheme represented as a spiral at the top of its cover page with a Prelude and Fugue in each major and minor key.
• Friederich Suppig's Calculus musicus, containing a temperament for a 'broken keyboard' with an illustrative Labyrinthus musicus modulating through all keys (see: Other Spirals and Designs).

Rediscovery

For centuries, the purpose of the spiral at the top of the cover page of Das Wohltemperirte Clavier was overlooked by musicologists. Given an established tradition of cover sheet decoration in Bach's day, they assumed that the spiral was purely decorative. Lacking the mathematical skills to analyse further, it was left to the mathematician Andreas Sparschuh to realise its significance.

Sparschuh worked as a keyboard tuner to support his mathematical studies at the Technical University of Darmstadt. He had developed an interest in historical tuning manuscripts and had ready access to a library that housed rare originals. On analysing the spiral on the cover sheet of Das Wohltemperirte Clavier, Sparschuh concluded that it was a tuning scheme. In 1999, he published a paper offering an interpretation of the spiral, eventually receiving the Golden Tuning Fork award for his achievement.

Michael Zapf, the then president of the German Clavichord Society, attended one of Sparschuh's talks in 2001. Although convinced that the spiral on the cover sheet of Das Wohltemperirte Clavier did indeed represent a tuning system, Zapf was troubled by two properties of Sparschuh's proposal:

• Commencement of the tuning procedure at 'A'.
• The disparate octaves in which tuning occurred.

Accordingly, he proposed an alternative scheme in which the loops represented seconds per beat, with the tuning procedure starting on 'C'.

Keith Briggs, Senior Mathematician at the BT Research Laboratories, analysed Zapf's proposal and realised that by closing the circle of fifths, harmonic theory would yield a soluble system of equations allowing pitch predictions to be made. He noted that if the final fifth beats once per second, Zapf's scheme predicts a pitch of a=425 Hz, according quite well with the presumed Cammerton pitch standard at Bach's time. However, while the seconds per beat assumption accounted for the first twelve components of the spiral, the thirteenth remained unexplained.

John Charles Francis applied the 'closed-circle' technique of Keith Briggs using symbolic computation software to exhaustively analyse all possibilities arising from the following hypotheses:

• Tuning is performed for some twelve semitone contiguous range.
• The small knot is the basic tempering unit, which denotes one beat per second, while the double knot indicates two beats per second (or equivalently one beat per second in the octave below).
• The ends of the spiral denote the interval closing the tuning circle and two cases are shown beating once per second (left) and twice per second (right).

The Esoteric Keyboard Temperaments of J.S. Bach published in February 2005, demonstrated that all thirteen components of the spiral can be accounted for in a consistent manner and no assumption of starting note or keyboard range is needed. The following results were obtained:

• The spiral contains matching transpositions for Cammerton and Cornet-Ton pitched instruments. The beat rates of the eleven central components transpose from Cammerton to Cornet-Ton pitch, while the beat rate of the interval closing the tuning circle increases under transposition as indicated by the spiral.
• The twelve semitone contiguous range for the Cammerton tuning is Cammerton middle C, C#, D, Eb, F, F#, G, G#, A, Bb, B ('one-line octave').  The twelve semitone contiguous range for the Cornet-Ton tuning is Cornet-Ton middle C, C#, D, Eb, F, F#, G, G#, A, Bb, B.
• The Cammerton tuning scheme starts on Cammerton middle C; the Cornet-Ton transpose starts one tone lower at Cornet-Ton Bb.
• The Cammerton tuning proceeds Cammerton CF, FBb, BbEb, D#G#, G#C#, C#F#, F#B, BE, EA, AD, DG (tuning check GC). Notably, Johann Georg Neidhardt's scheme of 1724 proceeds in identical manner.
• The Cornet-Ton transposition proceeds at Cornet-Ton pitch BbEb, D#G#, G#C#, C#F#, F#B, BE, EA, AD, DG, GC, CF (tuning check FBb).
• For the Cammerton transpose, the best major third occurs at Cammerton G (one sharp). In the Cornet-Ton transpose, the best major third occurs on Cornet-Ton F (one flat).  The Cammerton and Cornet-Ton transposes are symmetric about C.
• For the Cammerton transpose, the worst major third occurs on Cammerton F#. For the Cornet-Ton transpose, this is equivalent to Cornet-Ton E. Of note, F# is a typical position for the widest third, while Sorge Chorton temperament of 1758 has a widest third on E.
• The major and minor thirds increase and decrease in a smooth progressive manner between these extremes (i.e. they are monotonic).
• The temperament displays a minimal asymmetry with respect to the position of best and worse thirds. Accordingly, the variation in the width of thirds traversing around the flats is more gradual than the sharps.
• The Cammerton and Cornet-Ton pitch predictions accord with expectations based on surviving historical instruments (A = ~418 Hz in the Cammerton case).
• The temperament is circular and avoids Pythagorean major thirds.
• The tuning check for the Cammerton case and the eleven central components are drawn by one stroke of the pen. The tuning check for the Cornet-Ton is drawn by a separate stroke of the pen, being a secondary consideration.
• The calligraphy of the letters can be interpreted in a manner consistent with the Cammerton and Cornet-Ton trans; the prominent serif of the first letter indicating the tuning starts at C at the first position; the serif of the 'C' of the third word providing a Cornet-Ton pitch indication.