Delta-rational chord: Difference between revisions

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JI chords and chords that are subsets of [[isodifferential chord]]s (these correspond to all chords of the form α : α + ''k''1 : ... : α + ''k''n for any positive number α and integers k1, ..., kn) are special cases of delta-rational chords, but in these chords ''all'' dyads are rationally related in frequency space, which we call '''fully delta-rational''' (FDR).

== Mathematical definition ==

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.

A chord can be described as a list of increments between successive notes, their ratios showing the simple rational relationships and ? for no relationship (or no obvious one). For example saying that a tetrad is "+1 +? +1" means the first two notes and the last two notes have almost equal frequency difference, but the middle two notes are not in any simple relationship with the two outer dyads. The example 13edo chord is approximately +1 +? +1.

== Finding delta-rational chords in edos ==

== List of delta-rational chords in small edos ==

=== 11edo ===

* 0-2-4-7 (+1 +1 +2)

=== 13edo ===

* 0-2-4-7 (+1 +1 +2)

* 0-3-8-10 (+1 +? +1)

* 0-5-9 (+1 +1)

* 0-3-9-11 (+1 +? +1 or +2 +5 +2)

=== 14edo ===

* 0-3-9-11 (+1 +? +1)

=== 21edo ===

* 0-6-10 (+1 +1)

* 0-6-10-19 (+1 +1 +2)

[[Category:Chords]]

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 JI chords and chords that are subsets of [[isodifferential chord]]s (these correspond to all chords of the form α : α + ''k''<sub>1</sub> : ... : α + ''k''<sub>n</sub> for any positive number α and integers k<sub>1</sub>, ..., k<sub>n</sub>) are special cases of delta-rational chords, but in these chords ''all'' dyads are rationally related in frequency space, which we call '''fully delta-rational''' (FDR).
 == Mathematical definition ==
 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.
+A chord can be described as a list of increments between successive notes, their ratios showing the simple rational relationships and ? for no relationship (or no obvious one). For example saying that a tetrad is "+1 +? +1" means the first two notes and the last two notes have almost equal frequency difference, but the middle two notes are not in any simple relationship with the two outer dyads. The example 13edo chord is approximately +1 +? +1.
+== Finding delta-rational chords in edos ==
+== List of delta-rational chords in small edos ==
+=== 11edo ===
+* 0-2-4-7 (+1 +1 +2)
+=== 13edo ===
+* 0-2-4-7 (+1 +1 +2)
+* 0-3-8-10 (+1 +? +1)
+* 0-5-9 (+1 +1)
+* 0-3-9-11 (+1 +? +1 or +2 +5 +2)
+=== 14edo ===
+* 0-3-9-11 (+1 +? +1)
+=== 21edo ===
+* 0-6-10 (+1 +1)
+* 0-6-10-19 (+1 +1 +2)
 [[Category:Chords]]