So, about chords...

[jancivil]

I would say the chances are it sounds just like a major 7th chord. Again, if the fundamental is really there, so is the fifth partial, corresponding with the major third. That's 'real theory'. 'Distortion and feedback'; providing harmonic distortion, ie. the fifth partial comes to the fore... 'chewing up thirds', yes. The M3 your [12-tone ET] guitar gives you is 13.69¢ sharp over that natural resonance (5:4 in ratio to the fundamental), so you have a clash. Minor triads in metal, problematically dissonant out of the same 'theory', actually a fact here.
If you have 'saws', you have harmonic distortion.

"Fifths are the first thing you can omit" is a truism for the same reason, that's the third partial and it tends to be supplied by a good fundamental.

if you wanted exactly the sonority to specifically leave that third out and required another to provide that by their hand, you write "C^7 No 3". In the material you cite it's probably a fine distinction with no useful difference.

[Syncretia]

If I have a CMaj7 chord and I, for example, remove the note E from it, then I will have 3 notes left, but how to call such chords?

It's a 7th chord that's neither minor nor major. Don't worry about it! Who cares what it's called? There's no rules in the book that say you must play every note in a chord. Also, this chord could be considered a G maj 4/11th without the 5th or 7th. Who really gives a crap?

The 3rd usually gives the chord its distinctive minor/major quality, but it's not always that simple. It will also depend on what other instruments are doing such as the bass or the melody. The mode of the part will probably make more of a difference anyway.

It never really makes sense to think of a chord in isolation. People tend to think of chords that way because it makes it easier to break it down for your particular instrument.

At the end of the day, it's your ears that should dictate what notes sound best.

[tapper mike]

But it's not.

7th chords are dominant 7's not major 7's

[chordwizard]

I agree with @JJBiener above. A Cmaj7 chord contains CEGB, where E functions as the 3rd degree of the chord. The choice of 3 vs b3 is what changes a chord from being essentially major or minor.

So by leaving out this note, you still get a chord that can function as Cmaj7, but it's a more ambiguous chord that could also be used as the minor equivalent Cm(maj7).

There's more on this exact topic here:

http://www.howmusicworks.org/707/Playin ... sing-Notes

For a comparison of chord types, have a look at this page:

http://www.chordwizard.com/chords_cwsg.html

You'll see that almost all of them (except for a handful of oddballs like the sus4 chords) contain either a 3 or b3 degree to define either the major or minor character of the chord.

[jancivil]

But it's not.

7th chords are dominant 7's not major 7's

But it is. In general if there is a note a seventh from the root in a chord, it is, in "traditional" harmony class or what-have-you known as a Seventh Chord.

If you want to be particular about it, what you are insisting is the one and only 'seventh chord' is known as a major/minor seventh chord. aka dominant seventh by that particular context; where it is in fact the only M/m7 that occurs by default [in major], yes? in say a James Brown tune it isn't a dominant seventh, it's coincidental with that but is isn't that, it doesn't function.*

In rock or pop it may mean want you want it to mean there, while other 7ths are given more precisely, but in study of harmony the other types are called 'seventh chords' as a matter of course. The distinction is made in more detailed analyses by noting the quality of the triad and the quality of the seventh. In strictly diatonic music that is redundant, everyone knows ii7 is m/m7 in major, in minor ii7 is diminished/minor7 (hence 'hald' diminished), etc.

*<"C7" means "the B is flat"> is what is known as a *convention*. There are other conventions.

I wrote Cm7 as c7 for years, I knew from it, the other people knew from it. Some people know from the convention C-7 for Cm7. Some people know from C^7 as C Maj7. but in harmony class the convention doesn't follow that of a pop lead sheet. Someone noticing a seventh fr. the root is liable to say, 'that's a seventh chord' and no one is alarmed.

[tapper mike]

Theory is situational not universal. Some theories work in certain situations and not in others.