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, 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 the resulting notes within the dyads can psychoacoustically interfere with the beating of the dyads.

A '''delta-rational''' ('''DR''') chord is a chord with dyads which are close to having simple integer ratios between frequency ''differences'' of dyads, called '''deltas''', 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 the resulting notes within the dyads can psychoacoustically interfere with the beating of the dyads.

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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	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 the resulting notes within the dyads can psychoacoustically interfere with the beating of the dyads.		A '''delta-rational''' ('''DR''') chord is a chord with dyads which are close to having simple integer ratios between frequency ''differences'' of dyads, called '''deltas''', 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 the resulting notes within the dyads can psychoacoustically interfere with the beating of the dyads.

	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).		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).