Where does transposition collide?

In music, transposition is the operation of shifting an entire piece or passage up or down by a fixed interval, so that every note moves by the same amount and the music sounds in a different key while keeping the same pattern of relative pitches. The idea also underlies transposing instruments, which are written in one key but sound in another, so a player reads a part that has been shifted to suit the instrument. Kostka and Payne's Tonal Harmony (2013) sets out how a melody or chord progression is rewritten at a new pitch level without altering its internal structure. Here the referent is a uniform pitch shift of sounding music, not a swap of two items in a set, the relocation of a gene, or the enactment of a directive into law.[1]

Unit

One rewriting of a passage at a new pitch level, by a fixed interval

Recovery means

Shifting back by the same interval returns the music to its original key

Measured by

Given by the interval of transposition, so that every pitch and the key signature move by that same amount

In the mathematics of permutations, a transposition is a rearrangement that exchanges exactly two elements of a set and leaves every other element fixed. Transpositions are the simplest non-trivial rearrangements, and a central result of group theory is that every permutation can be written as a product of them, which makes the transposition a basic building block of the symmetric group. Dummit and Foote's Abstract Algebra (2004) develops this decomposition and uses it to define the sign, or parity, of a permutation. The referent is a two-element swap within an abstract set, unrelated to shifting music to a new key, moving a gene along a chromosome, or writing a directive into national law.[2]

Unit

A single swap of two elements, leaving all others fixed

Recovery means

Applying the same transposition again restores the original arrangement, since a swap is its own inverse

Measured by

Identified by the two elements it exchanges, and combined with others to build any permutation

In genetics, transposition is the movement of a transposable element, or transposon, from one location in a genome to another, sometimes leaving the original site and sometimes copying itself into a new one. Because these mobile segments can insert into or beside genes, their movement can switch genes on or off and rearrange a chromosome. Barbara McClintock's 1950 study The origin and behavior of mutable loci in maize reported the mobile controlling elements in maize that revealed this behaviour, work later recognised with a Nobel Prize. The referent is the relocation of a physical stretch of DNA within a living genome, not a pitch shift in music, a swap of two elements in a set, or the enactment of European law.[3]

Unit

One relocation of a transposable element to a new genomic site

Recovery means

Not applicable

Measured by

Detected by mapping an element's changed position and its effects on nearby genes, historically through patterns of pigmentation and mutation

In European union law, transposition is the process by which a member state incorporates an EU directive into its own national law, passing or amending domestic legislation so that the directive's goals take legal effect at home. A directive binds each state as to the result to be achieved but leaves the choice of form and method to national authorities, so transposition can be timely or delayed, complete or partial. Kaeding's 2006 study Determinants of transposition delay in the European Union examined why states differ in how quickly they carry directives into national law. The referent is a legislative and administrative act of the state, not a pitch shift in music, a swap of two elements in a set, or the movement of a gene.[4]

Unit

One member state's enactment of a directive into national law

Recovery means

Not applicable

Measured by

Tracked by whether and when national measures are notified, and by their completeness and correctness against the directive