Metallic MOS: Difference between revisions

Line 263:

This property is not unique to metallic generators, though — it is attributable to their being irrational numbers. A rational generator’s scale sequence eventually terminates, hitting bedrock when the period has been divided up into equal steps, i.e. where the notion of large steps and small steps no longer applies because <math>L = s</math> and <math>L{:}s = 1</math>. For example, the generator

<math>\qquad 5/12 = 0.41\overline{6}

<math>\qquad \frac{5}{12} = 0.41\overline{6}

</math>

generates scales with cardinality 2, 3, 5, 7, but when it reaches cardinality 12, it has generated 12edo. This occurs exactly at the moment when the generator has been repeated until it has returned exactly to from where it started (because 5/12 * 12 = 5, which is a multiple of the period, 1); if we repeated the generator any more, we’d just go over the ground we already trod.

generates scales with cardinality 2, 3, 5, 7, but when it reaches cardinality 12, it has generated 12edo. This occurs exactly at the moment when the generator has been repeated until it has returned exactly to from where it started (because <math>\frac{5}{12} · 12 = 5</math>, which is a multiple of the period, <math>1</math>); if we repeated the generator any more, we’d just go over the ground we already trod.

Irrational generators will never be able to return to exactly from where they started, so they will continue to divide the period up into smaller and smaller steps forever. At some point, however, the scales they generate will cease to be musically practical, because their steps will have become so small.

@@ Line 263: / Line 263: @@
 This property is not unique to metallic generators, though — it is attributable to their being irrational numbers. A rational generator’s scale sequence eventually terminates, hitting bedrock when the period has been divided up into equal steps, i.e. where the notion of large steps and small steps no longer applies because <span><math>L = s</math></span> and <span><math>L{:}s = 1</math></span>. For example, the generator
-<math>\qquad 5/12 = 0.41\overline{6}
+<math>\qquad \frac{5}{12} = 0.41\overline{6}
 </math>
-generates scales with cardinality 2, 3, 5, 7, but when it reaches cardinality 12, it has generated 12edo. This occurs exactly at the moment when the generator has been repeated until it has returned exactly to from where it started (because 5/12 * 12 = 5, which is a multiple of the period, 1); if we repeated the generator any more, we’d just go over the ground we already trod.
+generates scales with cardinality 2, 3, 5, 7, but when it reaches cardinality 12, it has generated 12edo. This occurs exactly at the moment when the generator has been repeated until it has returned exactly to from where it started (because <span><math>\frac{5}{12} · 12 = 5</math></span>, which is a multiple of the period, <span><math>1</math></span>); if we repeated the generator any more, we’d just go over the ground we already trod.
 Irrational generators will never be able to return to exactly from where they started, so they will continue to divide the period up into smaller and smaller steps forever. At some point, however, the scales they generate will cease to be musically practical, because their steps will have become so small.