The major pentatonic scale is the richest scale that can be rendered above a Ⅰ pedal bass using only consonances (in a sense of “consonance” which I'll elaborate). For example on C,
- C: fundamental/octave, obviously stable.
- D: pure Pythagorean (9:4) ninth, not a consonance in the usual sense but actually a perfectly smooth interval as long as the bass and melody are far enough apart.
- E: major tenth, extremely consonant if rendered in just intonation (5:2).
- G: fifth/tritave 3:1, easily pure.
- A: major thirteenth/sixth. This one actually works particularly good in the same octave as the bass note, namely as the 5:3 major sixth.
- c: yet another octave, duh.
So: those have all frequency ratios with numerator smaller than 10 and denominator <5. In fact, if you list all fractions >1 fulfilling that condition, you find almost only notes belonging to the major pentatonic, and only a few out of scale:
- 1⁄1 — C — fundamental
- 5⁄4 — E — third
- 4⁄3 — F — fourth
- 3⁄2 — G — fifth
- 5⁄3 — A — sixth
- 7⁄4 — B₇♭ — septimal seventh
- 2⁄1 — c — octave
- 9⁄4 — d — ninth
- 7⁄3 — e₇♭ — subminor tenth
- 5⁄2 — e — tenth
- 8⁄3 — f — eleventh
- 3⁄1 — g — twelfth
- 7⁄2 — b₇♭ — septimal fourteenth
- 4⁄1 — c' — 2 octaves
- 9⁄2 — d' — 8ve+9th
- 5⁄1 — e' — 8ve+10th
- 6⁄1 — g' — 8ve2+5th
- 7⁄1 — b₇♭' 8ve2+₇7th
- 8⁄1 — c'' — 3 octaves
- 9⁄1 — d'' — 8ve2+9th
In particular, the range from G to e (which is quite a typical domain for a pentatonic melody) is exactly the pentatonic scale if you leave out those foreign 7-limit intervals. The F would of course be a rather natural candidate too, but observe that in my system it only occurs in the low octaves, so it's a more natural choice for the bass to jump to (which indeed it often does in folk tunes) rather than the melody.
The septimal seventh is somewhat disqualified in Western music because it can't be approximated by many instruments. This is quite different in many styles around the world.
You may ask what's special about “numerator <10 and denominator <5”. Well, those exact numbers are certainly a bit arbitrary, but actually you will find that the ninth overtone is about the highest that can be reliably played as a flageolett on string instruments. (I believe most brass instruments can also play the 9th overtone fairly well but get into trouble if trying to go much higher, but not sure, I don't play any brass.) Also, three octaves are quite a typical range “in which most music happens”, traditionally – anything lower or higher is more of a sonic-effect thing.
That's not to say that anything which can't be written as a <10 fraction is necessarily immediately much more consonant. In particular, the 15:8 major seventh is actually a good contender to the 9:8 ninth – in particular, the major seventh chord sounds very sweet. But that relies on the proxy-quality of the chord construction: the major seventh can readily be constructed as a third over the fifth in a normal major chord. This kind of construction doesn't really work for a more intuitive, improvisational, single-voice-over-pedal approach.
As for intervals like the 7-limit sevenths, and also slightly higher limits like 11 or 13: those are found in some music, in particular Indian, Persian and Arabic music. The problem with these is that if you actually devise tuning systems that allow rendering them on fixed-frequency instruments, you pretty much have to commit to one single key, else it gets very complicated. Western music evolved to also embrace modulations. Those can be nicely incorporated if you only consider the 5-limit intervals, which is probably the reason why we have never found much use of the 7-limit intervals. Except we have, actually: the harmonic seventh is all over Barbershop singing, and the subminor third and 7:5 or 11:8 tritones are arguably the prototypical blue note intervals. Just, you don't get to write down such intervals exactly in the Western system, so they remain considered as more of an embellishment.