Rhythmic Canon

Definition, history

The concept of gun is old in music. The specific study of the theory of the rhythmic guns, which is focused on the only rates/rhythms by being unaware of heights, stamps and other parameters of the sound, was really initiated per daN Tudor Vuza in the years 1980. The principle of the gun is simple, several voices play the same reason but shifted in time. For a rhythmic gun one can formalize a voice thereafter moments when a note is played, $A= (a_1, a_2… a_n)$ . One deals with gun if all the voices are relocated one of the other: $A'= (a'_1, a'_2… a'_n) = has + b$ . If one notes $B= (b_1, b_2… b_n)$ the whole of the shifts, the various voices are them $b_i + (a_1, a_2… a_n)$ .

Of course one can make rhythmic guns without another constraint; the problem becomes interesting when that the voices are not recovered, i.e is required that the sum $A+B$ is direct, i.e still that there is not more than one note per moment. It is even more constraining if a note exactly is required, neither more nor less, by moment. A problem of paving then is found (or mosaic).

Various technical theorems establish that a gun which paves every possible moment with a reason finished is necessarily periodic, and that a gun has necessarily a common unit of time. One is thus brought back mathematically speaking to the question about a paving of the unit Z of the relative entireties of the form $Z=a+B+n Z$ or, passing to the quotient modulo the period N of the gun, $Z/n Z = has \ oplus B$ .

Vuza was interested particularly in the cases where neither has, nor B have clean period, which it calls RCMC (Rhythmic Canon off Maximum Category) and that the following researchers rather called " Vuza" guns;. It proves that the results found by Vuza on this subject are equivalent to a difficult classification of the groups cyclic studied by several mathematicians since the years 1950 and completed by Sands.

Mathematical results

Old theorems: Hajos, De Bruijn.

Goods and bad groups (Sands).

Examples.

Recent developments

Enumerations

Fripertinger.

Bond with the conjecture of Fuglede

Amiot

Extensions of the concept of gun rhythmic

Canon by retrogradation

Examples. The theorem of the trichorde of Wild.

Guns by increase

The problem of Johnson with 0 1 4.

Can the theorem of periodicity spread?

Guns modulo p

Their unconditional existence.

Random links: