User:VectorGraphics/Augmented and diminished intervals

I find the words "augmented" and "diminished" not that useful.

Let's step through some major thirds:

ratio	edosteps	cents
	4/12	400
5/4		386
81/64		409
24/19		404
	6/19	378

All about the same size, right?

Now let's look at supermajor thirds:

ratio	edosteps	cents
	6/17	423
9/7		435
	11/31	425
	7/19	442

Also about the same size.

Now, let's take a look at the augmented thirds of all the scales we've looked at so far that have them defined, and a few more:

ratio	edosteps	cents
	8/17	564
177147/131072		522
	12/31	464
	7/19	442
125/96		456
	5/12	500
	9/26	415
	14/27	622

Yup. That's right. According to the definition of augmented, all these different intervals - varying by over 200 cents - are augmented thirds. Yes, even that large tritone (which is extra cursed). The theoretical maximum and minimum within diatonic scales vary by 378 cents (interestingly, a major third) and overlap with major thirds, supermajor thirds, and even perfect fifths.

Now, it might be a little unfair to compare augmented thirds specifically given how much thirds can already vary across diatonic scales, so let's instead take a peek at augmented fifths:

ratio	edosteps	cents
	8/17	847
6561/4096		815
	20/31	774
	12/19	757
25/16		772
	8/12	800
	16/26	738
	20/27	888

Not much better - that's still a variation of 150 cents!

"Augmented", along with its partner "diminished", simply don't fit in a brightness-based system. They're scale-specific. And it's also rather hard to know exactly what interval one is talking about when using "augmented" and "diminished" - even between pythagorean and classical here, that's still about a diesis apart. And if we bring in ARCHY, then that comma grows to over 100 cents.

Simply put, I don't find "augmented" or "diminished" useful for naming intervals. Chords, maybe.