Delta-rational chord: Difference between revisions
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JI chords and chords that are subsets of [[isodifferential chord]]s (these correspond to all chords of the form α : α + ''k''<sub>1</sub> : ... : α + ''k''<sub>n</sub> for any positive number α and integers k<sub>1</sub>, ..., k<sub>n</sub>) are a special case of delta-rational chords, but in these chords ''all'' dyads are rationally related in frequency space. | JI chords and chords that are subsets of [[isodifferential chord]]s (these correspond to all chords of the form α : α + ''k''<sub>1</sub> : ... : α + ''k''<sub>n</sub> for any positive number α and integers k<sub>1</sub>, ..., k<sub>n</sub>) are a special case of delta-rational chords, but in these chords ''all'' dyads are rationally related in frequency space. | ||
== | == Mathematical definition == | ||
A chord C = α<sub>1</sub>:...:α<sub>n</sub> is ''delta-rational'' (ΔR) or ''partially delta-rational'' (PΔR) 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 (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. When all dyads are linearly related, i.e. when the chord is of the form (α + k<sub>1</sub>):...:(α + k<sub>n</sub>), we call the chord ''fully delta-rational'' (FΔR). | A chord C = α<sub>1</sub>:...:α<sub>n</sub> is ''delta-rational'' (ΔR) or ''partially delta-rational'' (PΔR) 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 (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. When all dyads are linearly related, i.e. when the chord is of the form (α + k<sub>1</sub>):...:(α + k<sub>n</sub>), we call the chord ''fully delta-rational'' (FΔR). | ||