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

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!

• Hi Michael - I'm a bit confused by the question "Why would anyone use the harmonic series to explain the origin of the major scale with such bad tuning?". Who is it that is claiming a direct link between the notes in the harmonic series and the notes in the major scale? Mar 25, 2019 at 20:11
• (Incidentally I do think that the harmonic series goes a long way towards explaining the major scale, but more because of the dissonance curve observed in figure 3 here: sethares.engr.wisc.edu/consemi.html) Mar 25, 2019 at 20:18
• @topomorto, the accepted answer here music.stackexchange.com/questions/8173/… is what prompted my question Mar 25, 2019 at 20:49
• Ok, comments to the contrary. I guess I;m more concerned about the +19 votes and accepted answer. That's a pretty strong affirmation by this forum for something that is on theoretically shaky ground. Mar 25, 2019 at 21:04
• @ topo morto: When I met the first time SE 3 months ago and I told my friends about it they were just laughing and mentioning exactly this point :) but this site here is even better than “parentology”, “german language” and some others I have seen. Mar 25, 2019 at 23:07

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 harmonic series 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 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 subdominant, which adds the Ⅳ and ⅵ degrees, and finally the major chord on the 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, , and 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 ♭ or ♭ 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 an enourmous richness. That was carved out by composers 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.

• What are "5-limit primitives?" I only know primitives from computer data types. Mar 26, 2019 at 13:50
• @leftaroundabout, thank you, that is very kind of you to say. Personally (and not just with your answer, I have the same problem with the Just Intonation article on Wikipedia), I get stuck with the fractions and names (I do not understand how notes get their names within the JI system, like "minor third", and where the other names come for scales that have >12 notes). All the numbers seem magical, because I fail to understand the context and how the fractions are derived to begin with. Even after reading Kyle Gann's article on the topic, I don't understand the algorithm for a JI octave. Jul 24, 2019 at 7:14
• Not sure what you mean by “algorithm for the octave”. An octave is just two notes, the higher of which has exactly twice the frequency of the lower. Similar with the other intervals, except it's not just a number 2 but some fraction – ³⁄₂ for a fifth, ⁵⁄₄ for a major third, ⁶⁄₅ for a minor third, ⁹⁄₈ for a major second. And it so happens that all those ratios can be approximated by terms of the form 2^(n / 12), with n = 2 for a major second, n = 3 for a minor third, n = 4 for a major third and n = 7 for a fifth. That is the 12-edo approximation. Jul 24, 2019 at 7:58
• The numbers in the fractions, the 12-edo sizes and the interval names are essentially unrelated numbers. They have different historical/mathematical derivations, and it just so happens that they come out to about the same absolute frequencies. That's perhaps the most confusing thing. Jul 24, 2019 at 8:00
• Re "unrelated numbers" and "different historical derivations": what are you talking about? Twelve-tone equal temperament descends directly from just intonation. They're not in the least unrelated. Jan 27, 2021 at 4:41

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.

• If I follow your 2nd paragraph, harmonic 3, the P5 (strong is the series, important musically) is used in Pythagorean Tuning to get 12 chromatic tones (which contain a diatonic series too.) But that series doesn't match up with any part of the overtone series. This more or less restates my question. Or, by your final comment confirms my attitude on the subject... unless someone gives me a much more convincing argument. Mar 25, 2019 at 20:58
• “The fact that (except for the zeroth power) no prime powers are equal means that all system fail.” Something like that is often said, but IMO pretty baseless. All it means is that finite note systems in JI can't allow infinite modulation – but that's hardly surprising, is it? Regardless, for any finite number of keys (possibly more than 12!), you can have a system that covers all of them in perfect JI. Whether that's practical is another question, but it's certainly not a fundamental mathematical dilemma. Mar 25, 2019 at 23:06
• And actually -1, because a) there's clear historical evidence that the ancient Greeks did know about just intonation and the Western major scale was developed much later b) even if a scale wasn't knowingly designed around integer ratios – most of them match way to well too such ratios to put it off as mere coincidence. At least octaves and fifths really feature in so many cultures across the world, you can't seriously claim that the ratio-integerness isn't a real working principle here. Mar 25, 2019 at 23:15
• A good explanation of why tuning can't accomodate both perfect fifths and octaves perfectly: youtu.be/8syA7S_5E3A?t=225 Mar 26, 2019 at 4:21
• @leftaroundabout "for any finite number of keys, you can have a system that covers all of them in perfect JI" the problem is that you can't cover even one key in perfect JI. For example, in C major you need two As: P5 above D and M3 above F. But because you need two As, you need two Ds, and then two Gs, and then two C's. But now that you have two C's, you need three As, and we've just entered an infinite spiral. In a real piece of music you won't spiral very far before the end of the piece, but the finite set of frequencies needed depends on the piece you're playing, not just on the key. Feb 4, 2021 at 3:54

The major scale was generated by stacking 5ths, the strongest/most audible non-octave overtone. If you're a caveman and you're trying to tune two pieces of sabretooth tiger gut strings you can hold your finger 1/3 of the way down the lowest string, and tune the 2nd string to that note until the "beats" are gone. Now you have the 2nd overtone minus an octave as your 2nd string. Rinse and repeat with the other stings. If you repeat this to the 5th string you get a pretty good pentatonic scale. Cavemen don't have frets, electronic tuners, etc so they can only tune by ears. This is how people tuned things for a long time- eliminating the "beats" caused by an out of tune unison. Eventually metal was invented and we could get more tuning forks, frets etc, and then the metronome was invented and we could measure beats per second accurately. Keyboards got popular. Composers got experimental wanted to write music that modulated into keys that were many 5ths away (as you mentioned earlier, the relationship is a little off from equal temperament, so the more 5ths away something is the more out of tune the modulation will be... it's more complicated than that, but let's go with that.) If you "fudge" or "temper" the numbers a little (totally screwing all the 3rds and 6ths but making them kinda ok) then you can make all the intervals equal to one another (and all a little screwy, but "acceptable.") We didn't "get the major scale" from the overtone series, but rather it was a long process, like boiling a frog, and now nobody knows what 3rds sound like except for really awesome choruses. The major scale is derived from pragmatism and social acceptance, not science.