User:Astaryuu/15edo Notes/Notes and chords

The semitone

In 12edo, an "edostep" is 100 cents exactly - this is the definition of a cent, in fact. However, 15edo's edostep has edosteps of 80¢. This is initially noticeably distinct from the semitone, which in standard harmony is usually defined as 1\12 exactly. Note the notation: two numbers on either side of a back slash, in the form X\Y, mean X steps of Y, so 1\12 is one step in 12edo.

While it's not quite the 100¢ "standard" semitone size, 1\15 is similar enough to the 12edo semitone that one can get used to it as the new semitone, especially if used in the right context. It's also a close approximation of the valentine semitone, which is why 15edo's chromatic scale is also called "valentine".

However, not all semitones are created equal. Even if it's literally the same size, as 1-edostep intervals are, if your scale of choice is intended to use a semitone that isn't valentine, then it may result in some gaps or missed notes when adapting the scale into 15edo. The 15edo zarlino scale I talked about previously is meant to approximate the 12edo heptatonic diatonic scale, and so ends up using two "different" semitones (one 2 edosteps and one 1 edostep). Technically, the 2-edostep one is our next interval...

The neutral second

You've probably heard of a "minor second" (a semitone) and a "major second" (a whole tone). A "neutral second" is in between the two, too sharp to be minor but too flat to be major. Usually it's defined as exactly 150 cents.

As it turns out, 2\15 is very close to 150¢ - it's 160¢, to be exact. This does mean our neutral seconds lean a little major. However, since it's still quite close to 150¢, scales built on it, like onyx, end up sounding quite ambiguous.

These neutral seconds could also serve as "chameleon intervals" - they'll take on the majoricity of the note before them. So for example, if you're using a minor chord for tension, stepping it up a neutral second can maintain that tension so a following major chord sounds even more satisfying. However, this will work even if going the other direction too.

Now of course, neither of these can really be used in chords well. So let's get on to the real meat and potatoes.

Did we truly change edos?

At 320¢, 400¢, and 480¢, the minor third, major third, perfect fourth sound relatively normal. 20¢ is a rather large comma, but it's still technically a comma, so you can theoretically hear the interval like the 12edo equivalent.

...wait. We skipped 240¢, didn't we? Well, that's because it's in a weird spot. It's right in between a major second and a minor third. We call this interval "wolf", because it's too far away from any 12edo position to sound like it should be one. This wolf third can function as a major tone, but there it's probably going to sound noticeably off. Major tones just are somewhat difficult in 15edo. 560¢, likewise, can be considered a "wolf fourth", but it's honestly better to consider 7\15 to be a diminished fifth - that is, a tritone.

This, of course, is not the only interval this happens to. Now we talk about the star of the 15edo show...

Inside you are two fifths (and two tritones)

15edo... doesn't really have a fifthy fifth. 12edo's fifth sounds the way it does because it's an extremely good estimation of a so-called "perfect fifth". The best fifth in 15edo is 9\15, but that sounds notably sharp. Again, you can probably get used to it, but it'll mean 12edo strategies might sound a bit off in it. That's not what it means to be "wolf", though.

The actual wolf fifth, is the valentine semitone below that 720¢ interval. At 640¢, meet 8\15, the wolf fifth. You could think of this as four neutral seconds stacked on top of each other, but at that point the margin between the neutral second and 2\15 goes beyond the diesis. This is actually also a tritone. Since 15 is odd, there is no true position halfway between octaves - but 7\15 and 8\15 bracket that midpoint.

It might honestly be better to treat the "wolf/perfect" fifths as "minor/major" - they are a semitone apart, like the major and minor second. Technically, this would make the neutral second a "perfect second" and the tritones "minor/major" too. All of these sound incredibly cursed to the 12edo brain, but that's kinda the point of xenharmony, so I'm just gonna do it.

This means we end up calling 9\15 the major fifth, 8\15 the minor fifth and major tritone, 7\15 the minor tritone, and 2\15 the perfect second. After all, our "neutral second" isn't really neutral either. Usually neutral seconds are defined as 150¢ specifically. Thirds and fourths are still somewhat normal relative to 12edo, so they don't flip even though there are audible differences.

Of course, chords are named after the third, so we can't just call them major and minor based on which fifth they use. Instead, we call chords built using the "major" fifth natural, while ones built with the "minor" wolf fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone.