Delta-rational chord: Difference between revisions

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A '''delta-rational''' ('''DR''') chord is a [[chord]] that has integer ratios between frequency ''differences'' of some pair of dyads, called '''deltas''', with the dyads in question assumed to ~~''not'' overlap~~ (Δ, capital delta, is often used to denote "difference").

A '''delta-rational''' ('''DR''') chord is a [[chord]] that has integer ratios between frequency ''differences'' of some pair of dyads, called '''deltas''', with the dyads in question assumed to be between successive notes (Δ, capital delta, is often used to denote "difference").

DR chords generalize JI chords, in which all frequency differences of dyads are exactly integer ratios. But unlike JI chords, a DR chord need not have integer ratios between frequencies of notes. For example, the [[13edo]] chord 0-3-8-10\13 (0¢-277¢-738¢-923¢) is close to being delta-rational, because the frequency difference of the dyad 8-10\13 is 0.994 times the frequency difference of the dyad 0-3\13. (In the exactly DR chord 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes have exactly the same frequency difference as the dyad 0-3\13.)

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

Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple ratios between dyads (when measured as absolute frequency differences) tend to be perceived as 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 is expected to be weaker in chords with wider voicings, 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 the resulting notes within the dyads could psychoacoustically interfere with the beating of the dyads.

Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple ratios between dyads (when measured as absolute frequency differences) tend to be perceived as 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 is expected to be weaker in chords with wider voicings, 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 only considering dyads between adjacent notes is that the resulting notes within the dyads could psychoacoustically interfere with the beating of the dyads.

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 most JI chords, a 2/1-equivalent [[MOS scale]] can tune a DR chord exactly, provided that the chord's delta signature specifies two integer entries.

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-A '''delta-rational''' ('''DR''') chord is a [[chord]] that has integer ratios between frequency ''differences'' of some pair of dyads, called '''deltas''', with the dyads in question assumed to ''not'' overlap (Δ, capital delta, is often used to denote "difference").
+A '''delta-rational''' ('''DR''') chord is a [[chord]] that has integer ratios between frequency ''differences'' of some pair of dyads, called '''deltas''', with the dyads in question assumed to be between successive notes (Δ, capital delta, is often used to denote "difference").
 DR chords generalize JI chords, in which all frequency differences of dyads are exactly integer ratios. But unlike JI chords, a DR chord need not have integer ratios between frequencies of notes. For example, the [[13edo]] chord 0-3-8-10\13 (0¢-277¢-738¢-923¢) is close to being delta-rational, because the frequency difference of the dyad 8-10\13 is 0.994 times the frequency difference of the dyad 0-3\13. (In the exactly DR chord 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes have exactly the same frequency difference as the dyad 0-3\13.)
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 [[JI]] chords and chords that are subsets of [[Delta-rational chord#Isodifferential chord|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 (possibly irrational) 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 either '''fully delta-rational''' (FDR) or '''linear'''.
-Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple ratios between dyads (when measured as absolute frequency differences) tend to be perceived as 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 is expected to be weaker in chords with wider voicings, 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 the resulting notes within the dyads could psychoacoustically interfere with the beating of the dyads.
+Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple ratios between dyads (when measured as absolute frequency differences) tend to be perceived as 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 is expected to be weaker in chords with wider voicings, 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 only considering dyads between adjacent notes is that the resulting notes within the dyads could psychoacoustically interfere with the beating of the dyads.
 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 most JI chords, a 2/1-equivalent [[MOS scale]] can tune a DR chord exactly, provided that the chord's delta signature specifies two integer entries.