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...

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. The example 13edo chord is approximately +1 +? +1.

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.

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