into two groups of six grouped around pivotal grand plenum organum settings (Wir glauben and Auf tiefer Noth). The duets are related by the successive key progression, E minor, F major, G major, and A minor. Clavier-Übung III thus combines many different structures: pivotal patterns; similar or contrasting pairs; and progressively increasing symmetry. There is also an overriding numerological symbolism. The nine mass settings (3 × 3) refer to the three of the Trinity in the mass, with specific reference to Father, Son and Holy Ghost in the corresponding texts. The number twelve of the catechism chorales can be seen as a reference to the usual ecclesiastical use of the number 12, the number of disciples. The whole work has 27 pieces (3 × 3 × 3), completing the pattern. However, despite this structure, it is unlikely that the work was ever intended to be performed as a whole: it was intended as a compendium, a resource for organists for church performances, with the duets possibly accompaniments for communion. comments on the occurrences of the golden ratio in Clavier-Übung III pointed out by various musicologists. The division of bars between the prelude (205) and fugue (117) provides one example. In the fugue itself the three parts have 36, 45 and 36 bars, so the golden ratio appears between lengths of the middle section and outer sections. The midpoint of the middle section is pivotal, with the first appearance there of the first subject against a disguised version of the second. Finally in BWV 682, Vater unser in Himmelreich (the Lord's Prayer), a pivotal point, where the manual and pedal parts are exchanged, occurs at bar 41, which is the sum of the numerical order of letters in JS BACH (using the Baroque convention of identifying I with J and U with V). The later cadence at bar 56 in the 91 bar chorale prelude gives another instance of the golden ratio. 91 itself factorises as 7, signifying prayer, times 13, signifying sin, the two elements—canonic law and the wayward soul—also represented directly in the musical structure. Prelude and fugue BWV 552 The descriptions below are based on the detailed analysis in . BWV 552/1 Praeludium Together with the Toccata in F major BWV 540, this is the longest of Bach's organ preludes. It combines the elements of a French overture (first theme), an Italian concerto (second theme) and a German