Delta-rational chord

A delta-rational (DR) chord is a chord that has integer differences between harmonics, but the harmonics are not necessarily integers. DR chords are typically denoted in the form of +a+b..., called the chord's delta signature. 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 ratios between intervals (when measured as absolute frequency differences) 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 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 DR 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 only considering intervals between adjacent notes is that the resulting notes within the intervals could psychoacoustically interfere with the beating of the intervals.

Examples

1. 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).
2. 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).
3. 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 + k_n for any positive (possibly irrational) number α and integers k₁, ..., k_n) 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.

Denoting a delta-rational chord

A delta-rational chord is determined by two things:

• The interval formed by its lowermost two notes;
• Its delta signature which has integer ratios, i.e. a list of (scaled) frequency increases between successive notes, their ratios showing the simple rational relationships, with a + before each increase. Note that it is whether the deltas are rationally related to each other that defines DR, not whether the deltas are related to the frequency of the root.
• Two delta signatures are equivalent if one can be obtained from the other by scaling by a positive real number. For example, +2+e+3 is equivalent to +2φ+eφ+3φ, and both signatures imply a delta-rational chord.
• Fully delta-rational chords always have a delta signature with all terms integers.

For example, a chord with a +1+2+1 delta signature is a:(a + 1):(a + 3):(a + 4) for some possibly irrational a.

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.

Linear stretching

For a frequency ratio $f_1/f_2$ and a (valid) real number $\alpha >0$ we call $(f_1-\alpha )/(f_2-\alpha )$ a linear stretching and $(f_1+\alpha )/(f_2+\alpha )$ a linear compression.

Examples:

• 23:29:35 = (4-1/6):(5-1/6):(6-1/6) is a linear stretching of 4:5:6 (approximately 0-5-9\15)
• 13:16:19 = (4+1/3):(5+1/3):(6+1/3) is a linear compression of 4:5:6

Linearly stretching a chord preserves the delta signature, unlike logarithmic stretching, which preserves the logarithmic ratios between intervals (ratios between cent values) in a chord. All the chords in the examples above are +1+1.

A small logarithmic stretch approximates a small linear stretch. This can be useful when hunting for approximate DR chords in equal divisions: for example, 0-6-11\19 can be logarithmically stretched to 0-6-11\18 and logarithmically compressed to 0-6-11\20. All of these chords are roughly +1+1.

See also

Expert

• Optimizing MOS scales for a DR chord

Technical

• DR error measures

External links

• Inthar's DR chord explorer