Up to 300000 triplets relating the Nlarge reflections are stored for active use in the Direct Methods phasing process.

Triplets are estimated according to their second representation (i.e. P10 formula, as described by Cascarano *et al*., 1984). The concentration parameter of the von Mises distribution is given by

G = C (1 + q)

where C is the concentration parameter of the Cochran’s distribution

C = 2|Eh||Ek||Eh-k| /√N (Cochran, 1955)

and q is a function (positive or negative) of all the magnitudes in the second representation of the triplet. The G values are rescaled on the C values and the triplets are ranked in decreasing order of G. The top ranked relationships represent a better selection of triplets (with phase value close to zero) than that obtained sorting triplets according to C. Triplets estimated with a negative G value represent a sufficiently good selection of relationships close to 180 degrees. Positive and negative triplets will be actively used in the phase determination process (Giacovazzo *et al*., 1992).

Negative quartets are generated by combining the PSI-0 triplets (relating two reflections with large |E| and one with |E| close to zero) in pairs; those with cross-magnitudes smaller than a given threshold are estimated by means of their first representation, as described by Giacovazzo (1976). These quartets are actively used in the phasing process (Giacovazzo *et al*., 1992).

To improve the efficiency of the tangent formula in Sir2019, the information contained in the PSI-0 triplets has been actively combined with the P10 relationships and negative quartets, according to Cascarano & Giacovazzo (1995).