Something that woke me up in the middle of the night, realising that if you take the 12 notes in an octave in western music, and from that you remove all those belonging to a major scale, you are left with 5 notes arranged as a pentatonic scale! (the mode of the 7 note scale or the minor/major quality of the pentatonic is irrelevant here).

How come in twenty years of playing guitar I've never come across this fact? Not that is useful (or is it?), but even now doing a few Internet searches, I can't find any reference to this.

For example, take C major (all white keys in piano). You are left with Eb, Gb, Ab, Bb, Db (all black keys in piano), which is an Eb minor pentatonic.

I'm not very good with music theory, so maybe this is obvious to anyone that goes to music school. I just found very interesting that these two patterns, that are by far the most commonly used, more than any melodic, harmonic, or other exotic scale arrangements (always within western music), and their notes are arranged so that they add up to all twelve notes in the octave without overlapping.

Two questions: Is this just a coincidence? Is there a musical way to use this?

And if someone could point out to any website where this is mentioned, I'd be curious to know.

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    Interesting! And the new key is a tritone away from the original...And the new key is the opposite to a relative - as in Cm is relative to Eb major, or F#m is relative to A major - sort of backwards ! – Tim Dec 2 '15 at 15:57
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    You could come up with a question and then answer it yourself... Separate your post into a question and then answer it, and then it may fit into the guidelines for this forum. – amalgamate Dec 2 '15 at 15:57
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    You just need to ask the question WHY? Otherwise it'll get closed. – Tim Dec 2 '15 at 16:35
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    The changes to the question are good enough for me. – amalgamate Dec 2 '15 at 17:07
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    This is more or less already acknowledged by other comments, but to make it explicit: this is much easier to notice on a piano than on a guitar, which is probably why you didn't notice it before. – Kyle Strand Dec 3 '15 at 0:08

It is known, maybe especially to piano/keyboard players, that the black keys form the Gb major / Eb minor pentatonic scale. Given the structure of the major scale and the structure of the pentatonic scale it's of course no coincidence, but I don't think there's some deeper meaning to it; it's just a result of these structures.

The interesting question (in my opinion) is if there's a "musical way" to make use of this fact. The answer is yes, there is. Many jazz musicians know that the minor pentatonic scale a minor third higher than the root of a dominant seventh chord (or the major pentatonic scale a tritone away from the root of a dominant seventh chord) contains all possible alterations and the minor seventh of that dominant seventh chord. So, e.g., over a C7 chord the Eb minor / Gb major pentatonic scale contains

Eb -> #9 Gb -> b5 / #11 Ab -> b13 Bb -> 7 Db -> b9

So you can use that pentatonic scale over C7 to create altered or outside sounds. As an example, if you play over a II-V-I in C major (Dm7 - G7 - Cmaj7), you can play a different pentatonic scale over each chord. One way to do that is to move up chromatically, which gives you an altered sound over the V chord (G7):

Dm7: A minor pent. (Dm9 sound) G7: Bb minor pent. (altered) Cmaj7: B minor pent. (lydian)

Of course, this trick for playing outside can also be used over other chords, such as minor 7 chords. Check out this example of Chick Corea using a G minor pentatonic scale to step outside over an Em7 vamp.

Perhaps the closest thing to a “reason” for this is that both diatonic and pentatonic scales can be considered as approximate Pythagorean scales. Now observe how the 12 degrees are constructed from circle of fifths: for instance,

E♭  B♭  F   C   G   D   A   E   B   F♯  C♯  G♯  D♯≅E♭
└───B♭-major diatonic───┘   └E-maj pentatonic┘  └temperament

Note that the story doesn't necessarily end there: while 12-edo tuning tempers out the Pythagorean comma that would follow between D♯ and E♭ here, other tunings do not. 17-edo is pretty good as a Pythagorean tuning, here you get an additional pentatonic scale before returning to the start:

G♭  D♭  A♭  E♭  B♭  F   C   G   D   A   E   B  F♯  C♯  G♯  D♯  A♯  E♯≅G♭
└───D♭-major diatonic───┘   └G-maj pentatonic┘ └F♯-maj pentatonic┘ └temperament

There's a fairly intuitive reason for this--"intuitive" in this case meaning that I'm not going to give a formal music-theoretic reason, but rather try to simply provide some intuition about why this might be so based on the circle of fifths and how "close" and "far" particular tonalities are from each other.

First, some observations:

  • Major and minor pentatonic scales are the "nicest" notes that can be grouped together (to Western ears). This is a fuzzy concept, of course, but it's easily formalized: notes (and keys/chords/scales) that are closer together on the circle of fifths sound more similar and are "nicer" together than notes that are further apart on the circle, and any five consecutive notes on the circle form a pentatonic scale.
  • The major scale is a pentatonic plus the two "next-nicest" notes--i.e., any seven consecutive notes on the circle of fifths form a major scale. (Note that in order to take a pentatonic scale and form a major scale of the same key, you must add the two notes directly before and after the pentatonic scale group on the circle of fifths. More about this issue of "directionality" below.)
  • Notes (/keys/chords/scales) that are a tritone apart are as dissimilar as possible; they are diametrically opposed (i.e. maximally far apart) on the circle of fifths. This typically means they sound the least "nice" together (hence the term "Devil's interval").
  • This concept of "similarity" and "dissimilarity" is transitive. For example, C neighbors G on the circle of fifths, and G neighbors D, so C and D are fairly similar. On the other end of the spectrum, C and F♯ are a tritone apart and are thus maximally dissimilar; F♯ is next to C♯ on the circle of fifths, and so C and C♯ also clash (though not as badly as F♯ and C).

Now, sticking with our example of using C as the basis for our analysis (but, as mathematicians say, "without loss of generality"), we can see that:

  • Notes in the C major pentatonic scale are "similar to" C (i.e. they are the next 4 notes going up the circle of fifths).
  • Notes in the F♯ major pentatonic scale are "similar to" F♯, and F♯ is maximally "dissimilar from" C, so, transitively, we've now divided up the keyboard between "notes that are similar to C" and "notes that are similar to F♯ and maximally dissimilar from C."
  • We've got two notes left over: B and F (E♯). These belong to both the C major scale and the F♯ major scale. Thus the property you noticed is (of course) reflexive: if we're in F♯ major, the unused notes are the C pentatonic scale, and if we're in C major, the unused notes are in the F♯ pentatonic scale.

This leads to an interesting thought: we can evenly partition the keyboard into two 6-note "major-ish" scales. Presumably we'd want to give B to C major and F (E♯) to F♯ major to ensure that we have leading tones for both scales.


The biggest "fuzzy" element of this analysis is that "similar" (as used above) doesn't precisely correlate with "nearness" on the circle of fifths, because going backward from C (i.e. moving up by 4ths or down by 5ths) quickly leads to B♭ (A♯), which is part of F♯ major pentatonic. But there's no particular reason why "closeness" on the circle of fifths shouldn't be considered reflexive (i.e. there's no reason to consider D closer to C on the circle than B♭ is). This part of the analysis could be made more rigorous by introducing a concept of directionality into the concept of "similarity" with which we've built the scales in question, i.e., by formalizing a reason for going up by 5ths when constructing our scales rather than going up by 4ths. This is well understood by applying the concept of the tonic-dominant relationship, but that's beyond the scope of this answer.


Can we use this when playing music? Matt L. has already given one use-case, unsurprisingly coming from the world of jazz where the concepts of "nearness" and "farness" are bent pretty far from how they're used in Western classical music.

But even in more "traditional"-sounding music, and especially in pop songs and Broadway-style show tunes, there's another use-case: dramatic modulations. If you're in C major and you want to have a sudden radical shift that still sounds "right", why not modulate as far away from C as possible? If you start playing the F♯ pentatonic scale, suddenly, most of the notes you're using would have been (somewhat) out of place in C major, but in their new context post-modulation, they're as close as possible to your new tonic (F♯). The shift is so dramatic, and the pentatonic scale is so harmonically "clear" or "obvious" sounding, that composers will often simply modulate without any real transition chords; you can jump right from C Major to F♯ Major without preparing the listener, and the chordal structure will still remain clear.

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    B♭ is not part of the F♯ major pentatonic. – leftaroundabout Dec 3 '15 at 22:57
  • @leftaroundabout Obviously I meant enharmonically. I've added a parenthetical to clarify, but the meaning is unchanged; B♭ is part of the F♯ major pentatonic, it just happens not to be called "B♭" in that context. – Kyle Strand Dec 3 '15 at 22:59
  • Well, no, B♭ and A♯ are just not the same note with different names, but two different notes which happen to be so close that some instruments can get away with approximating them both as the same frequency. Just because they're close however doesn't mean they're similar, and your answer actually explains quite well the reason. – leftaroundabout Dec 3 '15 at 23:04
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    @leftaroundabout That's a common misconception. Not only are the physical wavelengths identical with the (practically ubiquitous) equal temperament, there are not actually any "absolute" versions of notes that we "approximate" with particular temperaments. There are particular intervals that we define in absolute terms using the harmonic series, of course, and it's true that constructing A♯ by going up in true perfect fifths from C results in a different pitch-class than constructing B♭ by going down in true perfect fifths from C. – Kyle Strand Dec 3 '15 at 23:48
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    But in standard Western harmony (which is what I'm discussing here, and which is implied by the premise of the question), once a temperament has been selected, there are exactly 12 pitch-classes available for harmony, as represented by the 12-tone scale. This doesn't preclude the possibility of treating B♭ differently from A♯ in other harmonic systems, though of course that's not possible on most keyboard instruments. – Kyle Strand Dec 3 '15 at 23:49

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