Delta-rational chord: Difference between revisions

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Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicing of JI chords stay JI. However, unlike the case with JI chords, there are mos tunings that can hit a DR chord exactly, if the chord's increment signature has two integer entries.

== Notation ==

=== ~~Increment~~ signature ===

=== Linear signature ===

Some, but not all, delta-rational chords can be described with an ''~~increment~~ signature'' which has integer ratios, i.e. 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 (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer dyads. The example 13edo chord is approximately +1 +? +1. Note that this notation only considers dyads between successive notes, what we call ''increments''.

Some, but not all, delta-rational chords can be described with an ''linear signature'' which has integer ratios, i.e. 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 (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer dyads. The example 13edo chord is approximately +1 +? +1. Note that this notation only considers dyads between successive notes, what we call ''increments''.

If you have some sets of increments related to each other but not to other sets of increments, you could write the related sets with variables a, b, c or use one fewer letter by writing one set with positive integers without variables: an +a +b +a +b chord can also be written +1 +a +1 +a.

Fully delta-rational chords always have ~~an increment~~ signature with no ?'s.

Fully delta-rational chords always have a linear signature with no ?'s.

== Mathematical definitions ==

@@ Line 5: / Line 5: @@
 Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicing of JI chords stay JI. However, unlike the case with JI chords, there are mos tunings that can hit a DR chord exactly, if the chord's increment signature has two integer entries.
 == Notation ==
-=== Increment signature ===
+=== Linear signature ===
-Some, but not all, delta-rational chords can be described with an ''increment signature'' which has integer ratios, i.e. 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 (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer dyads. The example 13edo chord is approximately +1 +? +1. Note that this notation only considers dyads between successive notes, what we call ''increments''.
+Some, but not all, delta-rational chords can be described with an ''linear signature'' which has integer ratios, i.e. 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 (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer dyads. The example 13edo chord is approximately +1 +? +1. Note that this notation only considers dyads between successive notes, what we call ''increments''.
 If you have some sets of increments related to each other but not to other sets of increments, you could write the related sets with variables a, b, c or use one fewer letter by writing one set with positive integers without variables: an +a +b +a +b chord can also be written +1 +a +1 +a.
-Fully delta-rational chords always have an increment signature with no ?'s.
+Fully delta-rational chords always have a linear signature with no ?'s.
 == Mathematical definitions ==