Delta-rational chord: Difference between revisions

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.)  

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.

# A chord C = α₁:...:α_''n'' is ''delta-rational'' (DR) or ''partially delta-rational'' (PDR) when the chord has two distinct dyads α_''k''₁:α_''k''₂ and α_''k''₃:α_''k''₄, such that the real intervals (α_''k''₁, α_''k''₂) and (α_''k''₃, α_''k''₄) are disjoint and (α_''k''₂ − α_''k''₁)/(α_''k''₄ − α_''k''₃) 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, equivalently when the chord has a delta signature with all entries integers, we call the chord ''fully delta-rational'' (FDR) or ''linear''.

Due to the aforementioned equivalence of delta signatures under scaling, delta signatures of ''n'' terms are really elements of a projective space <math>\mathbb{R}\mathbf{P}^{n-1};</math> they are specifically in the subset that is the image of the all-positive {{w|orthant}} of <math>\mathbb{R}^n.</math>

* 0-3-6-11 (+1 +1 +2)

* 0-6-11-19 (+1 +1 +2)

== Higher-order differences of frequency ==

Generalizing, one could consider chords where differences between its frequency deltas (as Tom Price has called them, '''precessions''') are rationally related, while the deltas themselves may not be. This corresponds to chords where differences between various interference beatings go in and out of sync in a periodic manner. One precession-rational chord is 5:5.4142...:6.8284...:9.2426..., a +(sqrt(2) &minus; 1) +sqrt(2) +(sqrt(2) + 1) chord.

Precession being the second-order difference (Δ²) of frequency, we similarly have the theoretical notions of Δ³-rationality, Δ⁴-rationality, and so on. The practical consequences of higher-order differences are as of yet speculative, though a few people have reported finding precession psychoacoustically meaningful.