Delta-rational chord: Difference between revisions

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A '''delta-rational''' ('''DR''') chord is a chord with dyads which are close to having simple integer ratios between frequency ''differences'' of dyads (Δ, capital delta, is often used to denote "difference"), but not necessarily integer ratios between frequencies of notes, as in JI chords. For example, the [[13edo]] chord 0\13-3\13-8\13-10\13 (0¢-277¢-738¢-923¢) is close to being delta-rational because the dyad 8\13-10\13 in the chord has a frequency difference 0.994 times the frequency difference of the dyad 0\13-3\13. (In 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes would have exactly the same frequency difference as the dyad 0\13-3\13.) Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple ratios between dyads are more concordant than other chords. This acoustic effect is thought to be caused by synchronized interference beating among the fundamentals and among lower harmonics of the fundamentals; the effect may be more or less pronounced depending on register, timbre, the complexity of the linear relationship, etc. For example, the delta-rational acoustic effect would be weaker in chords with very spaced-out voicing, as well as chords played in timbres with loud higher harmonics (because the higher harmonics would make the delta-rational relationships less obvious).

A '''delta-rational''' ('''DR''') chord is a chord with dyads which are close to having simple integer ratios between frequency ''differences'' of dyads, with the dyads assumed to not overlap (Δ, capital delta, is often used to denote "difference"), but not necessarily integer ratios between frequencies of notes, as in JI chords. For example, the [[13edo]] chord 0\13-3\13-8\13-10\13 (0¢-277¢-738¢-923¢) is close to being delta-rational because the dyad 8\13-10\13 in the chord has a frequency difference 0.994 times the frequency difference of the dyad 0\13-3\13. (In 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes would have exactly the same frequency difference as the dyad 0\13-3\13.) Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple ratios between dyads are more concordant than other chords. This acoustic effect is thought to be caused by synchronized interference beating among the fundamentals and among lower harmonics of the fundamentals; the effect may be more or less pronounced depending on register, timbre, the complexity of the linear relationship, etc. For example, the delta-rational acoustic effect would be weaker in chords with very spaced-out voicing, as well as chords played in timbres with loud higher harmonics (because the higher harmonics would make the delta-rational relationships less obvious). The justification for ignoring overlapping dyads is that notes within a dyad can psychoacoustically interfere with the beating.

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).

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== Mathematical definitions ==

# 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.

# 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 the real intervals (αk1, αk2) and (αk3, αk4) are disjoint and (αk2 − αk1)/(αk4 − αk3) 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 + 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'' (IR) if it has a delta signature with some integers showing up. Not all PDR chords are increment-rational.~~

== Finding approximate DR chords in edos ==

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-A '''delta-rational''' ('''DR''') chord is a chord with dyads which are close to having simple integer ratios between frequency ''differences'' of dyads (Δ, capital delta, is often used to denote "difference"), but not necessarily integer ratios between frequencies of notes, as in JI chords. For example, the [[13edo]] chord 0\13-3\13-8\13-10\13 (0¢-277¢-738¢-923¢) is close to being delta-rational because the dyad 8\13-10\13 in the chord has a frequency difference 0.994 times the frequency difference of the dyad 0\13-3\13. (In 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes would have exactly the same frequency difference as the dyad 0\13-3\13.) Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple ratios between dyads are more concordant than other chords. This acoustic effect is thought to be caused by synchronized interference beating among the fundamentals and among lower harmonics of the fundamentals; the effect may be more or less pronounced depending on register, timbre, the complexity of the linear relationship, etc. For example, the delta-rational acoustic effect would be weaker in chords with very spaced-out voicing, as well as chords played in timbres with loud higher harmonics (because the higher harmonics would make the delta-rational relationships less obvious).
+A '''delta-rational''' ('''DR''') chord is a chord with dyads which are close to having simple integer ratios between frequency ''differences'' of dyads, with the dyads assumed to not overlap (Δ, capital delta, is often used to denote "difference"), but not necessarily integer ratios between frequencies of notes, as in JI chords. For example, the [[13edo]] chord 0\13-3\13-8\13-10\13 (0¢-277¢-738¢-923¢) is close to being delta-rational because the dyad 8\13-10\13 in the chord has a frequency difference 0.994 times the frequency difference of the dyad 0\13-3\13. (In 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes would have exactly the same frequency difference as the dyad 0\13-3\13.) Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple ratios between dyads are more concordant than other chords. This acoustic effect is thought to be caused by synchronized interference beating among the fundamentals and among lower harmonics of the fundamentals; the effect may be more or less pronounced depending on register, timbre, the complexity of the linear relationship, etc. For example, the delta-rational acoustic effect would be weaker in chords with very spaced-out voicing, as well as chords played in timbres with loud higher harmonics (because the higher harmonics would make the delta-rational relationships less obvious). The justification for ignoring overlapping dyads is that notes within a dyad can psychoacoustically interfere with the beating.
 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).
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 == Mathematical 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 (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.
+# 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> &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. 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). In practice these terms can loosely refer to approximations of mathematically PDR and FDR chords.
-# A chord is ''increment-rational'' (IR) if it has a delta signature with some integers showing up. Not all PDR chords are increment-rational.
 == Finding approximate DR chords in edos ==