He then gains new pitches by using common-tone transpositions or inversions. The hash table implementing the transposition table can have other uses than finding transpositions. This follows because an arbitrary transposition can be expressed as the product of adjacent transpositions. Any scale having 12 different transpositions is not a mode of limited transposition. These can include cuts, inserts, supplemental rehearsal markings, transpositions and doublings. Every permutation of a finite set can be expressed as the product of transpositions. This term covers both chromatic and diatonic transpositions as follows. Any permutation matrix can be constructed as a product of no more than transpositions. The representation of a permutation as a product of adjacent transpositions is also not unique. Minor changes, transpositions, and alterations to recitative sections are not shown. Because formula 22 contains all disjoint transpositions in formula 14, formula 86. They are each closed under transpositions by 3, 6, or 9 semitones. It entails additions, omissions, and transpositions; it is not significantly different in size from B. Another algorithm for non-coprime dimensions, involving multiple subsidiary transpositions, was described by Catanzaro et al. The actual notes, leaving aside transpositions and differences in accompaniment and articulation, are unchanged.