Delta-rational chord: Difference between revisions
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Mathematically, 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 (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'' (FDR). In practice these terms can loosely refer to approximations of mathematically PDR and FDR chords. | Mathematically, 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 (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'' (FDR). In practice these terms can loosely refer to approximations of mathematically PDR and FDR chords. | ||
== Finding | == Finding approximate DR chords in edos == | ||
Some heuristics for finding delta-rational chords: | Some heuristics for finding delta-rational chords: | ||
* If a chord with a given step numbers in an edo is delta-rational it or similar numbers will usually also be delta-rational in nearby edos. | * If a chord with a given step numbers in an edo is delta-rational it or similar numbers will usually also be delta-rational in nearby edos. | ||
== List of | == List of approximate DR chords in small edos == | ||
=== 11edo === | === 11edo === | ||
* 0-2-4-7 (+1 +1 +2) | * 0-2-4-7 (+1 +1 +2) | ||