Generator-offset property: Difference between revisions
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A scale satisfies the '''alternating generator property''', or the '''AG''' property for short, if it satisfies the following equivalent properties: | A scale satisfies the '''alternating generator property''', or the '''AG''' property for short, if it satisfies the following equivalent properties: | ||
* the scale can be built by stacking alternating generators, for example 7/6 and 8/7. | * the scale can be built by stacking alternating generators, for example 7/6 and 8/7. | ||
* the scale is generated by two chains of generators separated by a fixed interval; either both chains are of size m, or one chain has size m and the second has size m-1. | * the scale is generated by two chains of generators separated by a fixed interval; either both chains are of size m, or one chain has size m and the second has size m-1. | ||
[[Diasem]] is an example of an AG scale. | |||
More formally, a cyclic word S (representing a [[periodic scale]]) is AG if it satisfies the following equivalent properties: | More formally, a cyclic word S (representing a [[periodic scale]]) is AG if it satisfies the following equivalent properties: | ||
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These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2. | These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2. | ||
== Theorems == | == Theorems == | ||
=== Theorem 1 === | === Theorem 1 === | ||