Delta-rational chord
A delta-rational (DR) chord is a chord that has integer differences between harmonics, but the harmonics are not necessarily integers. That is, a chord is DR if it contains frequencies a, b, c, d, ... where one difference between two frequencies is a rational number times another difference between two frequencies. DR chords are typically described using the notation +a+b... showing relative frequency increments between adjacent chord notes, called the chord's delta signature. By relative, it is meant that delta signatures are considered equivalent under scaling: +2+4+2 is the same delta signature as +1+2+1.
For example, the chord 0\13 – 3\13 – 8\13 – 924.159¢ is an exactly DR chord (with delta signature +1 +? +1), since the 3rd and 4th notes have exactly the same frequency difference as the interval 0–3\13. The actual 13edo chord 0 – 3 – 8 – 10\13 (0¢ – 277¢ – 738¢ – 923¢) is close to being delta-rational, because the frequency difference of the interval 8–10\13 is 0.994 times the frequency difference of the interval 0–3\13.
Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicings of JI chords stay JI.
Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple (i. e. low-number) delta signatures tend to be perceived as more concordant than other chords, even when the ratios between the notes themselves are not rational.
Acoustics
The delta-rational acoustic effect is believed to result from synchronized interference beating occurring both among fundamental tones and among their lower harmonics. The strength of this effect varies based on factors such as register, timbre, and the complexity of the linear relationship involved. For instance, the effect tends to be weaker in chords with wider voicings, as well as in timbres where higher harmonics are more prominent thus obscuring the delta-rational relationships. The reason for focusing only on intervals between neighboring notes is that the tones within those intervals might psychoacoustically interfere with the beating patterns of the intervals themselves.
Examples
- The chord 0c-400c-724.7c is a +1+1 chord (approximately 3.8473:4.8473:5.8473) and so is isodifferential (hence fully DR). It is close to the 15edo triad (0-5-9)\15 (0c-400c-720c).
- The chord 0c-281c-734.7c-923.6c is a +1+2+1 chord (approximately 5.675:6.675:8.675:9.675), and so is fully DR (but not isodifferential). It is close to the 13edo tetrad (0-3-8-10)\13 (0c-276.9c-738.5c-923.1c).
- The chord 0c-258.3c-771.7c-944.7c is a +1+?+1 chord (approximately 6.214:7.214:9.704:10.704), and thus a partially (not fully) DR tetrad. It is close to the 14edo tetrad (0-3-9-11)\14 (0c-257.1c-771.4c-942.9c).
Definitions
- JI chords and chords that are subsets of isodifferential chords (these correspond to all chords of the form α : α + k1 : ... : α + kn for any positive (possibly irrational) number α and integers k1, ..., kn) are special cases of delta-rational chords, but in these chords all intervals are rationally related in frequency space, which we call fully delta-rational.
- If all notes are equally spaced in frequency, the chord is called isodifferential.
Thus all isodifferential chords (including isoharmonic JI chords) are fully delta-rational, and all fully delta-rational chords (including all JI chords) are delta-rational.
Deltas that are free, i.e. not required to be related to any other deltas are indicated with +?. For example, saying that a tetrad is "+1 +? +1" means the first two notes and the last two notes have equal frequency difference (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer intervals.
If you have some sets of deltas related to each other but not to other sets of increments, you could write the related sets with variables a, b, ... or use one fewer letter by writing one set with positive integers without variables: a delta signature +a +b +a +b can also be written +1 +c +1 +c where c = b/a.
