Delta-rational chord: Difference between revisions
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=== Definitions === | === 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, | # When all dyads are linearly related, equivalently when the chord has a delta signature +k<sub>1</sub> +k<sub>2</sub> ... +''k''<sub>n</sub> with all ''k''<sub>i</sub> integers, we call the chord ''fully delta-rational'' (FDR). | ||
In practice these terms can loosely refer to approximations of mathematically exact PDR and FDR chords, for example in [[edo]] tunings. | In practice these terms can loosely refer to approximations of mathematically exact PDR and FDR chords, for example in [[edo]] tunings. | ||
=== Facts === | === Facts === | ||
==== Finding a tuning of a MOS scale with an exact DR chord ==== | ==== Finding a tuning of a MOS scale with an exact DR chord ==== | ||