Delta-rational chord: Difference between revisions

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Fully delta-rational chords can always be written in increment notation with no ?'s.

== Mathematical ~~definition~~ ==

== Mathematical definitions ==

~~Mathematically, a~~ chord C = α1:...:αn is ''delta-rational'' (DR) or ''partially delta-rational'' (PDR) when the chord has two distinct dyads αk1:αk2 and αk3:αk4 such that (αk2 − αk1)/(αk4 − αk3) is rational. When all dyads are linearly related, i.e. when the chord is of the form (α + k1):...:(α + kn), we call the chord ''fully delta-rational'' (FDR). In practice these terms can loosely refer to approximations of mathematically PDR and FDR chords.

# S chord C = α1:...:αn is ''delta-rational'' (DR) or ''partially delta-rational'' (PDR) when the chord has two distinct dyads αk1:αk2 and αk3:αk4 such that (αk2 − αk1)/(αk4 − αk3) is rational.

# When all dyads are linearly related, i.e. when the chord is of the form (α + k1):...:(α + kn), we call the chord ''fully delta-rational'' (FDR). In practice these terms can loosely refer to approximations of mathematically PDR and FDR chords.

# A chord is ''increment-rational'' if it has an increment-notation expression with some integers showing up. Not all PDR chords are increment-rational.

== Finding approximate DR chords in edos ==

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 Fully delta-rational chords can always be written in increment notation with no ?'s.
-== Mathematical definition ==
+== Mathematical definitions ==
-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> &minus; α<sub>k<sub>1</sub></sub>)/(α<sub>k<sub>4</sub></sub> &minus; 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.
+# S 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> &minus; α<sub>k<sub>1</sub></sub>)/(α<sub>k<sub>4</sub></sub> &minus; 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.
+# A chord is ''increment-rational'' if it has an increment-notation expression with some integers showing up. Not all PDR chords are increment-rational.
 == Finding approximate DR chords in edos ==