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As you can see, except for seconds, the main diatonic major/minor dichotomies remain intact, albeit with the occasional exception. Specifically, for thirds, fourths, fifths, and sixths, you find "wolf intervals" on two of the fourteen modes for each, which for fifths are between perfect and diminished, and for thirds, they are below minor. Each of these types of wolf intervals is special because they correspond to the prime harmonics we have access to in 15edo, but not in 12edo: the wolf sixth, for example, represents the harmonic seventh ratio of 7/4 (and the wolf third represents its complement 8/7) (but note that these are also our largest seconds and smallest sevenths), and the wolf fourth represents the undecimal tritone, 11/8 (and of course, the wolf fifth represents its complement 16/11.)
As you can see, except for seconds, the main diatonic major/minor dichotomies remain intact, albeit with the occasional exception. Specifically, for thirds, fourths, fifths, and sixths, you find "wolf intervals" on two of the fourteen modes for each, which for fifths are between perfect and diminished, and for thirds, they are below minor. Each of these types of wolf intervals is special because they correspond to the prime harmonics we have access to in 15edo, but not in 12edo: the wolf sixth, for example, represents the harmonic seventh ratio of 7/4 (and the wolf third represents its complement 8/7) (but note that these are also our largest seconds and smallest sevenths), and the wolf fourth represents the undecimal tritone, 11/8 (and of course, the wolf fifth represents its complement 16/11.)
== The torus ==
In 12edo, we often arrange intervals on a "circle of fifths", which outlines a lot of the harmonic structure we use in that tuning. In 15edo, a circle of fifths doesn't end up covering all the intervals (as it repeats after 5 steps), so we might think to use a circle of major thirds, the next simplest interval in terms of JI, but as it turns out, that ALSO doesn't cover all the intervals, repeating after only 3 steps. And with harmonic sevenths, we're back to looping after 5 again. As such, 15edo can be described as contorted in 2.3.7 and 2.5. So, the only solution is to use a two-dimensional "circle", which forms a '''torus'''-shaped "harmonic table":
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We call this a torus because it wraps around from one side to the other, like if you were to print this on the surface of a donut.
Here,
* fifths are found by stepping 1 step up the Y-axis.
* harmonic sevenths are found by stepping 2 steps down the Y-axis.
* major thirds are found by stepping 1 step right along the X-axis.
Instead of two types of semitones, 15edo has four:
* the diatonic semitone, or limma, found by stepping down and to the left by 1 on the table. This separates intervals belonging to different classes.
* the chromatic semitone, or chroma, found by stepping down 1 and right 2 on the table. This separates intervals with the same class, but different qualities.
* the ptolemaic semitone, or syntonic comma, found by stepping up 4 and left 1 on the table. This separates standard intervals from "wolf" intervals.
* the compound semitone, or diaschisma, found by stepping up 4 and right 2 on the table. This separates fifth-generated "diatonic" intervals and their augmented or diminished zarlino counterparts.
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