User:FloraC/Fokker analysis of rank-3 scales: Difference between revisions
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Rothenberg propriety is a modal propriety. Likewise, Rothenberg stability is a modal stability. Lumma's refinements take the same principles. They work for scales in the traditional sense but not for what we have here because they disregard domal rotations. | Rothenberg propriety is a modal propriety. Likewise, Rothenberg stability is a modal stability. Lumma's refinements take the same principles. They work for scales in the traditional sense but not for what we have here because they disregard domal rotations. | ||
For rank-3 scales, the assumption is that the domal chroma is smaller than the modal chroma, | For rank-3 scales, since the assumption is that the domal chroma is smaller than the modal chroma, we may let go of an extension of propriety, but stability is really a different story. | ||
The modal–domal stability can be considered in two ways. First, there is the stability concerning the entire pitch spectrum, defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on domal rotations. We will dub it the domal stability. Second, there is the stability concerning the modal coverage, defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on residual rotations. It is equivalent to the sum of the portion which is not covered by those on modal rotations and the portion which is covered by those on domal rotations. We will dub it the residual stability. The following identity holds: | The modal–domal stability can be considered in two ways. First, there is the stability concerning the entire pitch spectrum, defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on domal rotations. We will dub it the domal stability. Second, there is the stability concerning the modal coverage, defined as the portion of the pitch spectrum which is not covered by the interval differences of each class on residual rotations. It is equivalent to the sum of the portion which is not covered by those on modal rotations and the portion which is covered by those on domal rotations. We will dub it the residual stability. The following identity holds: | ||