The theory behind western pentatonic scales

Question

I'm curious about a more in-depth explanation as to why the major and minor pentatonic scales are prevalent in the western music tradition (Yes, i make a claim here that some might consider debatable. Please hear me out).

I did a brief googling and the most common explanations were: "because it's easy" or "because it sounds good", or the likes. It's not that i disagree, but i feel that there probably is more to the theory than that.

For instance, why are the fourth and seventh notes omitted in a major pentatonic scale (in reference to a major diatonic scale)? It makes me wonder since both the fourth and the fifth are perfect intervals, but only the latter is kept. Is it simply because it goes along the vein of a musical tradition revolving around the I-V relationship?

I'm assuming one could argue that the seventh is left out because of the dissonant quality in relation to the root, no?

Could anyone shed some more light on why this is? I like to hear some of your thoughts on this!

Answer 1

A common "academic" description of these two pentatonic scales is anhemitonic. Major and minor scales are hemitonic, meaning they contain at least one half step (or "hemitone"). Meanwhile, these pentatonic scales are anhemitonic because they do not contain any half steps.

If you think about the C-major collection:

     C D E F G A B C

We see half steps between the third and fourth degrees (E and F) and the seventh and final degrees (B and C). So we simply remove the offenders; by removing scale-degrees 4 and 7 (F and B), we create an anhemitonic scale of C D E G A C.

As for the minor pentatonic, let's just rotate that early collection to start on A instead of C, giving us an A-minor collection:

     A B C D E F G A

We again have half steps between B and C and E and F, so we again remove B and F (although this time they're scale-degrees 2 and 6) to create the A C D E G A minor pentatonic collection.

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We can get a bit more advanced here and look at the interval-class content of these two (same) collections. Without getting too specific, when we track all of the different intervals between every single pitch, we see that this particular anhemitonic collection is made up of 4 perfect fifths/fourths, 3 major seconds, 2 minor thirds, and one major third. Note that there are precisely zero half steps (as expected) or tritones. From a set-theory standpoint, the interval vector is <032140>; Hanson's chemical analysis would be p4mn2s3.

Answer 2

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 — B_₇♭ — septimal seventh
• ²⁄₁ — c — octave
• ⁹⁄₄ — d — ninth
• ⁷⁄₃ — e_₇♭ — subminor tenth
• ⁵⁄₂ — e — tenth
• ⁸⁄₃ — f — eleventh
• ³⁄₁ — g — twelfth
• ⁷⁄₂ — b_₇♭ — septimal fourteenth
• ⁴⁄₁ — c' — 2 octaves
• ⁹⁄₂ — d' — 8ve+9th
• ⁵⁄₁ — e' — 8ve+10th
• ⁶⁄₁ — g' — 8ve²+5th
• ⁷⁄₁ — b_₇♭' 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.

Answer 3

Probably it has to do with the Pythagorean tuning as it was used in the development of medieval music.

The tuning is based on the 3:2 ratio (or the perfect fifth), undoubtedly consonant for human ears. The process of stacking fifths upon a fundamental frequency four times generates the scale:

C G D' A' E''

Which can be rearranged with octave equivalency to form a pentatonic scale:

C D E G A

It works well for the first five notes. Maybe the third would sound considerably higher to our equal-tempered ears, but it was probably fine before equal-temperament was invented (which was not long ago).

The real problem with Pythagorean tuning is that the nice consonance we get for the first five notes is lost for the next two notes (which would make the diatonic scale):

C G D' A' E'' B'' F''' or C D E F G A B

The relationship of 3:2 between B and F is actually consonant, but from C to F we hear a tritone (as in C - F#), as the fourth is too high (even higher than the third).

So it was probably more common to use pentatonic scales with instruments tuned in Pythagorean method during medieval times, as all intervals would sound consonant.

This might have a direct relationship with Renascence music, as they tend to consider octaves, fifths, sixths and thirds as consonances, but seconds, fourths and sevenths as dissonances. Renascence modality was the basis for baroque's tonality and so on.

Answer 4

You are operating under a standard-harmony bias. The pentatonic scale is not a major scale with notes missing it is simply a scale with 5 notes, Penta means 5. It is useful for modern music because its wide intervals make for greater harmonic leeway, but on the other hand, its bare-bones harmony makes it in one sense naked. This is why it often has notes added to it.

It is relevant to the modern genres because, with each new passing of the musical era in the west, there comes a need for the theorist to reinvent the wheel. Now with the pentatonic scales limits in mind, we should still respect it for the big paradigm shift it enabled.

Although I mainly operate under the guise of standard harmony I must say the paradigm shift in music in the 20th century, that was enabled by this scale was a rather brilliant. Jazz, Rock, Blues... all these things take from this piece of harmony. We should all be glad it exists.

Answer 5

If you start at C and go up in 5ths and stop after 5 notes (before the M7 interval), that is all of the notes of the pentatonic scale out of order. C>G>D>A>E 5ths are the most pleasing interval (other than the octave) so you just have one pleasing interval and then it's most pleasing interval and so on til you get 5 notes. Then you can reduce the above notes down by octaves so all notes are within the roots octave range so you get a scale C D E G A (C). That's why it sounds so freakin good, because there isn't an unpleasing note in the bunch.

Answer 6

I am no expert but I'm surprised no one has mentioned this....

Most articles I have read explain that pentatonic scales are elementary starting scales for students to quickly adapt to any instrument.

Once the student has acquired skillful pentatonic scales, they can complete their training with incorporating the seven note scales.

I believe this method for education purposes would keep their student attentive and practicing, without initially feeling lost.

Answer 7

Besides analyzing the theory in an abstract way, it may be useful in this case to take a look at history.

1. Western classical music doesn't really use pentatonic scales at all.

2. Current Western pop music use of pentatonic scales derives from rock music.

3. Rock music's use of pentatonic scales derives from blues.

In other words, so far as Western music is concerned, today's trend of massive use of pentatonic scales in popular music actually originated with the music of American slaves, which lead to various forms of blues (beginning of 20th century) , which in turn heavily influenced rock 'n' roll (50s etc.), which led to the rock music of the 60s and especially the 70s. As rock spread everywhere, pop absorbed more and more elements from it. And one of the key elements that were passed on through all these eras was the use pentatonic scales, especially in instrumental solos and bass lines, and to a slightly lesser extent, song melodies.

Because of this direct line from slaves' songs to modern rock and pop music, I feel that any theoretical analysis of pentatonic scales in themselves runs the risk of being sterile and arbitrary, because it is completely separate from the context in which the tradition was born and grown.

Outside of the Western world, pentatonic scales have a lot of history. Chinese classical music, in particular, is almost completely based on pentatonic scales, and has been so possibly for thousands of years.