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.

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 interval 8–10\13 is 0.994 times the frequency difference of the interval 0–3\13. (The exactly DR chord 0\13 – 3\13 – 8\13 – 924.159¢ is +1 +? +1, since the 3rd and 4th notes have exactly the same frequency difference as the interval 0–3\13.)

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. 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 only considering intervals between adjacent notes is that the resulting notes within the intervals could psychoacoustically interfere with the beating of the intervals.

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.

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 or linear.

Thus isodifferential/linear (including isoharmonic JI chords) ⊂ fully delta-rational (including all JI chords) ⊂ 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. If we divide every term by the first term to make the first term 1, the result is called a normalized delta signature.
• 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.

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 almost 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.

<adv>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, c or use one fewer letter by writing one set with positive integers without variables: an +a +b +a +b chord can also be written +1 +c +1 +c where c = b/a.</adv>

Fully delta-rational chords always have a delta signature with no irrational ratios between terms.

Finding a tuning of a MOS scale with an exact DR chord

We start by choosing the MOS scale and equave, and the DR chord.

For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord.

Identify the mappings of each of the deltas. For a Meantone mapping, these are g⁴/4 -1, g-g⁴/4, where g is the frequency ratio corresponding to the Meantone fifth generator. This is because in Meantone, 1/1, 3/2, 5/4 are 1, g, g⁴/4 respectively, so the deltas are identified by subtracting the each term with the one before it.

In this case, we want the ratio between our deltas to become 1, so the delta signature will be +1+1.

To achieve this, we take the difference between the two deltas and set it to zero, so (g⁴/4 -1) - (g-g⁴/4) = 0. Put in integer terms, it's g⁴ - 2g - 2 = 0. Solving for g, the only root that makes sense is g≈1.49453, which in cents is 695.630c. And thus, with this generator, we will have a DR ~4:5:6 Meantone chord!

Note that the equation to solve depends on what chord you want to tune as equal-beating. For example, assuming pure octaves, Meantone admits an equation for tuning the 3:4:5 as equal-beating: g⁴ + 2g − 8 = 0. The latter equation has solution g = 1.4960 = 697.3¢.

If instead we chose a Schismic mapping, the deltas would be g⁸/8 -1, 2/g - g⁸/8, which gives a generator of 498.308c for 4:5:6.