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").

DR chords ~~are approximations of~~ JI chords, in which ~~the~~ frequency differences of dyads are ''exactly~~'' simple~~ 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 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes would have exactly the same frequency difference as the dyad 0-3\13.)

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 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes would have exactly the same frequency difference as the dyad 0-3\13.)

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 either '''fully delta-rational''' (FDR) or '''linear'''.

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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").
-DR chords are approximations of JI chords, in which the frequency differences of dyads are ''exactly'' simple 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 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes would have exactly the same frequency difference as the dyad 0-3\13.)
+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 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes would have exactly the same frequency difference as the dyad 0-3\13.)
 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 either '''fully delta-rational''' (FDR) or '''linear'''.