Triangulharmonic series

The triangulharmonic series is a subset of the harmonic series: only the harmonics which are triangular numbers. It begins:

$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ...$

Essentially, first you skip 1 harmonic, then 2 harmonics, then 3, then 4, then 5, etc. So its unreduced intervals are superbiparticular ratios: $\frac{3}{1}, \frac{4}{2}, \frac{5}{3}, \frac{6}{4}...$

The formula for the nth triangular number is $\frac{n^2 + n}{2}$.

Triangulharmonic Modes

Infinite pairs of triangular numbers one of which is 2 times the other occur[1]. If you take a subsequence of consecutive triangular numbers from one of these to the other, you make an octave-repeating scale, like a mode of the overtone series.

The first occurrence of such a pair is near the beginning: 3 and 6. But that doesn't make an interesting scale, because when assuming octave equivalence, it's only a single note.

First Triangulharmonic Mode

The next occurrence is at 105 and 210. This gives us a hexatonic scale, a subset of the 105th harmonic mode:

$\frac{120}{105}, \frac{136}{105}, \frac{153}{105}, \frac{171}{105}, \frac{190}{105}, (\frac{210}{105})$

which reduces to:

$\frac{8}{7}, \frac{136}{105}, \frac{51}{35}, \frac{57}{35}, \frac{38}{21}, (\frac{2}{1})$

You could think of it like a NEJI for 6-EDO. This is the first triangulharmonic mode.

Second Triangulharmonic Mode

The next occurrence of such a pair of triangular numbers isn't until 3570 and 7140, which produces a 35 note scale with step sizes ranging from 40.7¢ at the bottom to 29.3¢ at the top. That would be called the second triangulharmonic mode.

Third Triangulharmonic Mode

The next occurrence is at 121278 and 242556, which produces a 204 note scale. This scale and beyond are decreasingly likely to be of much interest.