Recursive structure of MOS scales: Difference between revisions
No edit summary Tags: Mobile edit Mobile web edit |
Tags: Mobile edit Mobile web edit |
||
| Line 443: | Line 443: | ||
The number of chunks is b, and gcd(a%b, (b-a%b)) = gcd(a%b, b) = gcd(a,b) by the Euclidean algorithm. | The number of chunks is b, and gcd(a%b, (b-a%b)) = gcd(a%b, b) = gcd(a,b) by the Euclidean algorithm. | ||
=== Binary generated scales with #L coprime to #s are MOS === | === Binary generated scales with #L coprime to #s within each period are MOS === | ||
By ''generatedness'', we mean that every interval in the scale is of the form ''jg'' + ''kp'' where ''g'' is a generator, ''p'' is the period, and ''j, k'' ∈ '''Z''', and that either ''g'' or ''−g'' occurs on every note. We have shown that the result of chunking and reduction is generated and binary and have gcd(#L, #s) = 1. We need only show that any such scale has ''Myhill's property with respect to p'', which we define to mean that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes. | By ''generatedness'', we mean that every interval in the scale is of the form ''jg'' + ''kp'' where ''g'' is a generator, ''p'' is the period, and ''j, k'' ∈ '''Z''', and that either ''g'' or ''−g'' occurs on every note. We have shown that the result of chunking and reduction is generated and binary and have gcd(#L, #s) = 1. We need only show that any such scale has ''Myhill's property with respect to p'', which we define to mean that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes. | ||