Fokker block: Difference between revisions

A Fokker block of rank r has [[maximum variety]] less than or equal to 2^(r-1). For example, a rank-1 Fokker block has max variety <= 2 (hence is a MOS); a rank-2 Fokker block has max variety <= 4.

While the idea generalizes easily to [[Just_intonation_subgroups|just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[Harmonic_Limit|p-limit]] situation with n=pi(p) primes up to an including p.

Suppose we have n-1 commas, which we will assume are greater than 1, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep&gt;, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get w2*e2+w3*e3+...+wp*ep where the w2, w3 ... wp are integers. We interpret this as the [[Vals_and_Tuning_Space|val]] v = &lt;w2 w3 ... wp|. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if w2&lt;0 we reverse sign, and we have a val V which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175 and 385/384, the above procedure gives us V = &lt;22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-limit. We may call the val V the epimorph val, and the n-1 commas, which form a basis for the kernel of V, the chroma basis.