Delta-rational chord: Difference between revisions
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Fully delta-rational chords always have a delta signature with no irrational ratios between terms. | Fully delta-rational chords always have a delta signature with no irrational ratios between terms. | ||
== | == Mathematics of DR == | ||
=== 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). | # 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). | ||
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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 === | ||
=== Exact DR tunings of MOSes are abundant === | ==== Exact DR tunings of MOSes are abundant ==== | ||
Let ''a'', ''b'' be positive integers and suppose gcd(''a'', ''b'') = 1. Let ''p'' > 1. Consider a MOS ''a'''''L'''''b'''''s'''{{angbr|p}} with generator range <math>I \subseteq (1, \sqrt{p})</math>, and consider a triple ('''0''', '''u''', '''v''') (representing a triad in the MOS), '''0''' < '''u''' < '''v'''. Let '''p''' represent the period (octave) and let '''g''' represent the generator. Write | Let ''a'', ''b'' be positive integers and suppose gcd(''a'', ''b'') = 1. Let ''p'' > 1. Consider a MOS ''a'''''L'''''b'''''s'''{{angbr|p}} with generator range <math>I \subseteq (1, \sqrt{p})</math>, and consider a triple ('''0''', '''u''', '''v''') (representing a triad in the MOS), '''0''' < '''u''' < '''v'''. Let '''p''' represent the period (octave) and let '''g''' represent the generator. Write | ||