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This Wikipedia article is the source for two overtone series charts...

The numbers above the harmonic indicate the number of cents' difference from equal temperament (rounded to the nearest cent).

• Regarding harmonic 11, is it a 'out of tune' Gb or an F?


• Am I reading the charts correctly that there is no A natural in the series?

If I understood this article about cents, -49 on harmonic 11 is about 1/4 step - enough to be a horribly out of tune P4 - and -12 on harmonic 15 may drift past the threshold of perceptible depending on the person and the exactly sound context (it may be close enough.) Roughly the same for harmonic 5 as 15.

So the resulting major scale (based on these very weak harmonics) will have FA horribly out of tune, and MI and TI questionable.

Why would anyone use the harmonic series to explain the origin of the major scale with such bad tuning?

Based on the strength of the overtones the major triad is supported by the series. But the next strongest thing is a diminished triad! Past that everything is weak and out of tune!

Can the harmonic series explain the origin of the major scale?

Emphatically yes, but that doesn't mean the major scale is literally contained in the harmonic series. We seem to be dealing basically with a black-and-white fallacy here: there aren't just two options “the entire major scale is contained in a single harmonic series” and “the major scale can't explain the major scale”. Rather, the scale is constructed from multiple building blocks that use the harmonic series.

As you said, the major triad is so well supported by the harmonic series that this can hardly be controversial. That's basically enough to explain the entire scale too if you just make use of the obvious candidates for a chord sequence: the Ⅰ chord gives the Ⅰ, ⅲ and Ⅴ scale degrees as 4:5:6 ratio, then the chord which has the original Ⅰ as its fifth is the Ⅳ subdominant, which adds the Ⅳ and ⅵ degrees, and finally the major chord on the Ⅴ degree gives you also the ⅱ and ⅶ.

The scale thus derived, using only the first five overtones in the harmonic series (plus extra octaves), is the Ptolemaic diatonic major scale. It could be argued that this is not the historically most relevant one, with the only 3-limit Pythagorean being an alternative. The fact that Pythagorean melody and Ptolemaic harmony can be pretty well approximated by a single tuning system – that's a meantone temperament, of which 12-edo is but one amongst many – is perhaps the centralmost feature of Western harmony.

Why use those particular three chords, Ⅰ, Ⅳ and Ⅴ to derive everything? Well, they are the obvious candidates insofar as you need to move the fundamental only by the simplest ratio – the 3:2 fifth. It is certainly possible to move around major chord by other steps, in particular by the mediant. That gives the ♭Ⅲ or ♭Ⅵ chords, which indeed are quite common in bluesy rock music and sound very tonal. Those songs sure enough tend to use not so much the full Ionian diatonic scale as the more flexible “blues scales” for their melodic content. It so happens that Common-practice music has not gone that road, which probably is mostly because those kinds of chord changes are much more natural for a parallel movement, sliding chords around the guitar neck style which CP so avoids. But I'd say that both approaches are clearly based on the harmonic series.

Similar story for

But the next strongest thing is a diminished triad

– kinda, however 7:6 is significantly narrower than a 6:5 minor third or even a 12-edo minor third. So it's no surprise if a diminished chord sounds dissonant although 5:6:7 are low integer ratios: because it would be way out of tune from that interpretation. A true 4:5:6:7 chord however sounds pretty amazing. Why does the seventh harmonic not feature in standard Western scales? Well, Western music has been to a huge degree shaped by the composers of the Baroque who loved symmetry and modulation. The playfield set up this way from only 5-limit primitives allowed for and enourmous richness that was carved out in the 18th and 19th centuries. Now, if you toss 7-limit intervals into that kind of music, everything would become a heck lot more complicated. 7- or even 11- and 13-limit intervals do feature in other musical traditions, but the Western composers just happen to take the more symmetrical road of 5-limit meantone music.

I do think extending Western counterpoint with 7- and 11 limit intervals is a great way in which music could evolve in the future. Just it's really quite difficult to keep the overview over the fine-grained scales and still keep it melodically graspable, about the only composer who has really been successful in that direction is Ben Johnston. Better computer frameworks, if not AI, might soon help as an assistant.

There are several numerological methods to derive a major scale. Historically, several sets of six notes were used (called "hexachords") from which by various roundabout means, the various modes were derived. (Not every theorist used the same modes nor the same derivations.) The overtone series doesn't quite generate a nice scale (nor mode); neither do any others.

One method is to take combinations of overtones based on the first overtone (octave) and second (fifth). One gets "Pythagorean Tuning" using ratios containing only twos and threes. Tones an octave apart are considered "equivalent" for these constructions and the rations all become between 1.0 and 2.0. Thus the octave has a ratio of 2:1 (I prefer fractions 2/1) and the fifth (not called that yet) with a ration of 3/2. The inverse of the octave is 4/3 (the fourth). One can get twelve tones by noticing that 12 fifths is about the same as four octaves: (3/2)^12 (reduced) is 531441/252144 or 2.027.... which is close to an octave. Nothing ever comes out exact because no power of two is a power of three (except the 0th power which is 1). (Actually they are never too close except 8 and 9). One can take the twelve powers of 3/2 (0 through 11), reduce them to between 1 and 2, then sort. One gets an approximation to the 12-note chromatic scale. The first 7 powers are similar to the 7-notes of the major scale (or any of its modes).

One problem is that the "major third" (interval from C to E in modern terminology) has a ratio of 81/64 which is a bit out of tune (whatever that means) for lots of people. The next step is to expand the multipliers to using 2s, 3s, and 5s. This yield "just tuning" which sounds pretty good for a few chords. (Check in the Wiki: just tuning, temperament, Pythagorean tuning, mean-tone tuning, hexachord theory, etc. Iterate the bibliography operator.)

The fact that (except for the zeroth power) no prime powers are equal means that all system fail. One must make some type of compromise; either some keys are not in tune or all keys are out of tune. But this is all a bit of a digression (people do get unreasonably passionate about tunings though).

The Greeks defined several tetrachords (four-note collections). Medieval theorists used various tetrachord combinations to derive scales. The Greeks used three descentind tetrachords (later theorists used ascending tetrachords). The "diatonic" tetrachord was (in modern notation) A-G-F-E (two tones and a semitone), the "chromatic" tetrachord A-Gb-F-E (a minor third and two semitones), and the "enharmonic" tetrachord A-Gbb-Fq-E (a major third and two quarter-tones). The major scale (or the Greek version) was made of two (descending again) tetrachords joined by a whole tone: E-D-C-B+A-G-F-E (the pattern of modern major and minor scales by permutation, and modes for that matter). That's how it was done harmonically. The Wiki on "tetrachords" gives a nice description.

My conclusion is that the major (and other) scales were not really derived from overtones. Overtones were later used to "explain" (actually, "almost explain") these scales.