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 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 {{dash|0, 3, 8, 10|med}}\13 ({{dash|0¢, 277¢, 738¢, 923¢|med}}) 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 {{dash|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.)

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 {{dash|0, 3, 8, 10|med}}\13 ({{dash|0¢, 277¢, 738¢, 923¢|med}}) 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 {{dash|0\13, 3\13, 8\13, 924.159¢|med}}, the 3rd and 4th notes have exactly the same frequency difference as the dyad 0–3\13.)

[[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 : {{nowrap|α + ''k''1}} : ... : {{nowrap|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'''.

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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 be between successive notes (&Delta;, 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 {{dash|0, 3, 8, 10|med}}\13 ({{dash|0¢, 277¢, 738¢, 923¢|med}}) is close to being delta-rational, because the frequency difference of the dyad 8&ndash;10\13 is 0.994 times the frequency difference of the dyad 0&ndash;3\13. (In the exactly DR chord {{dash|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.)
+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 {{dash|0, 3, 8, 10|med}}\13 ({{dash|0¢, 277¢, 738¢, 923¢|med}}) is close to being delta-rational, because the frequency difference of the dyad 8&ndash;10\13 is 0.994 times the frequency difference of the dyad 0&ndash;3\13. (In the exactly DR chord {{dash|0\13, 3\13, 8\13, 924.159¢|med}}, the 3rd and 4th notes have exactly the same frequency difference as the dyad 0&ndash;3\13.)
 [[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 : {{nowrap|α + ''k''<sub>1</sub>}} : ... : {{nowrap|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'''.