Delta-rational chord: Difference between revisions
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== Mathematical definitions == | == Mathematical definitions == | ||
# A chord C = α<sub>1</sub>:...:α<sub>''n''</sub> is ''delta-rational'' (DR) or ''partially delta-rational'' (PDR) when the chord has two distinct dyads α<sub>''k''<sub>1</sub></sub>:α<sub>''k''<sub>2</sub></sub> and α<sub>''k''<sub>3</sub></sub>:α<sub>''k''<sub>4</sub></sub>, such that the real intervals (α<sub>''k''<sub>1</sub></sub>, α<sub>''k''<sub>2</sub></sub>) and (α<sub>''k''<sub>3</sub></sub>, α<sub>''k''<sub>4</sub></sub>) are disjoint and (α<sub>''k''<sub>2</sub></sub> − α<sub>''k''<sub>1</sub></sub>)/(α<sub>''k''<sub>4</sub></sub> − α<sub>''k''<sub>3</sub></sub>) is rational. Equivalently, a chord is delta-rational if it has a delta signature with some integers showing up. | # A chord C = α<sub>1</sub>:...:α<sub>''n''</sub> is ''delta-rational'' (DR) or ''partially delta-rational'' (PDR) when the chord has two distinct dyads α<sub>''k''<sub>1</sub></sub>:α<sub>''k''<sub>2</sub></sub> and α<sub>''k''<sub>3</sub></sub>:α<sub>''k''<sub>4</sub></sub>, such that the real intervals (α<sub>''k''<sub>1</sub></sub>, α<sub>''k''<sub>2</sub></sub>) and (α<sub>''k''<sub>3</sub></sub>, α<sub>''k''<sub>4</sub></sub>) are disjoint and (α<sub>''k''<sub>2</sub></sub> − α<sub>''k''<sub>1</sub></sub>)/(α<sub>''k''<sub>4</sub></sub> − α<sub>''k''<sub>3</sub></sub>) is rational. Equivalently, a chord is delta-rational if it has a delta signature with some integers showing up. | ||
# When all dyads are linearly related, equivalently when the chord has a delta signature with all entries integers, we call the chord ''fully delta-rational'' (FDR) | # When all dyads are linearly related, equivalently when the chord has a delta signature with all entries integers, we call the chord ''fully delta-rational'' (FDR) | ||
# A chord that has a delta signature with all entries +1 is called ''isodifferential''. | # A chord that has a delta signature with all entries +1 is called ''isodifferential'' or ''linear''. | ||
Due to the aforementioned equivalence of delta signatures under scaling, delta signatures of ''n'' terms are really elements of <math>S^{n-1};</math> this is because they are specifically in the subset that is the image of the all-positive {{w|orthant}} of <math>\mathbb{R}^n.</math> | Due to the aforementioned equivalence of delta signatures under scaling, delta signatures of ''n'' terms are really elements of <math>S^{n-1};</math> this is because they are specifically in the subset that is the image of the all-positive {{w|orthant}} of <math>\mathbb{R}^n.</math> | ||