Oddharmonic series

The oddharmonic series is simply the odds of the harmonic series: 1, 3, 5, 7, ...

It is equivalent to the overtone sequence OS2, or the class ii isoharmonic sequence.

Throddharmonic series

We may say a number divisible by 3 is "threeven" and otherwise it is "throdd", by analogy with "even" and "odd" for divisibility by 2. Therefore, the throddharmonic series is simply all of the throdds of the harmonic series: 1, 2, 4, 5, 7, 8, 10, 11, 13, ...

A "threevenharmonic series" (3, 6, 9, 12, ...) would not be of any interest because it simply reduces to a harmonic series a tritave lower, in the same way a normal evenharmonic series (2, 4, 6, 8, ...) would reduce to a harmonic series an octave lower.

Tritave modes

Modes of the traditional harmonic series are created by selecting a subset that spans an octave, such as from the 5th harmonic to the 10th harmonic, and using this as an octave-repeating scale. In a similar way, we can create modes of the oddharmonic series that span a tritave, and using that as a tritave-repeating scale.

The first tritave mode of the oddharmonic series, as a one-note scale, is not interesting: 1, (3). We can call this 1-TOH. The second tritave mode of the oddharmonic series is: 3, 5, 7, (9). We can call this 2-TOH (Tritave of Odd Harmonics). The third tritave mode of the oddharmonic series is where it gets really nice, a nonatonic scale: 9, 11, 13, 15, 17, 19, 21, 23, 25, (27). We can call this 3-TOH.

Generalization

For the throddharmonic series, we may select subsets spanning a tetrave (the interval 4/1). And for the fouroddharmonic series (all harmonics not divisible by 4) we could select subsets spanning a pentave (5/1). We can always select repeating n-ave segments of the (n-1)-oddharmonic series, because (n² - n)/(n - 1) is always integer.