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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!

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.