Fokker block: Difference between revisions

Line 11:

The rank of a Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-''n'' Fokker block has {{nowrap| ''n'' - 1 }} chromas: a consequence of this is that a Fokker block of rank ''n'' has [[maximum variety]] at most 2<sup style="white-space: nowrap;">(''n'' − 1)</sup> (since that's the number of combinations of chromas a note can be altered by). For example, a rank-2 Fokker block has max variety at most 2 (hence is a [[MOS scale|mos]]), and a rank-3 Fokker block has max variety at most 4. These results are true for strong and weak Fokker blocks only if we allow negative steps; otherwise they are only true for strong Fokker blocks. In this way, Fokker blocks generalize mos scales.

A Fokker block can be made [[constant structure]] (with negative steps allowed) by moving the generator sizes by an arbitrarily small amount. If the logarithmic sizes of the generators are linearly independent (as happens in JI, for example), the generator sizes need not be moved. The constant structure will have no negative steps if and only if the Fokker block is strong.

Fokker blocks may be used to describe scales within JI subgroups or regular temperaments, or to describe rank-1 regular temperaments – that is, equal temperaments – themselves (by taking the chromas as commas to be tempered out).

@@ Line 11: / Line 11: @@
 The rank of a Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-''n'' Fokker block has {{nowrap| ''n'' - 1 }} chromas: a consequence of this is that a Fokker block of rank ''n'' has [[maximum variety]] at most 2<sup style="white-space: nowrap;">(''n'' − 1)</sup> (since that's the number of combinations of chromas a note can be altered by). For example, a rank-2 Fokker block has max variety at most 2 (hence is a [[MOS scale|mos]]), and a rank-3 Fokker block has max variety at most 4. These results are true for strong and weak Fokker blocks only if we allow negative steps; otherwise they are only true for strong Fokker blocks. In this way, Fokker blocks generalize mos scales.
+A Fokker block can be made [[constant structure]] (with negative steps allowed) by moving the generator sizes by an arbitrarily small amount. If the logarithmic sizes of the generators are linearly independent (as happens in JI, for example), the generator sizes need not be moved. The constant structure will have no negative steps if and only if the Fokker block is strong.
 Fokker blocks may be used to describe scales within JI subgroups or regular temperaments, or to describe rank-1 regular temperaments – that is, equal temperaments – themselves (by taking the chromas as commas to be tempered out).