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, 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'' (FDR). In practice these terms can loosely refer to approximations of mathematically PDR and FDR chords. | # 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'' (FDR). | ||
In practice these terms can loosely refer to approximations of mathematically PDR and FDR chords, for example in [[edo]] tunings. | |||
== Finding approximate DR chords in edos == | == Finding approximate DR chords in edos == | ||