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In the taunting of children the 7th harmonics is used according to Leonard Bernstein in his Harvard lectures. We sometimes notate this as sol-mi-la-sol-mi but...it is not correct according to Bernstein although it is the best we can come when playing it on a piano. There is no la in this chant according to him. What we have is the 7th harmonics which is a little higher than la he said. In salve regina (simple tone) we have do-mi-sol-la-sol according to sheet music. In this case the 7th harmonic isnt used but the major sixth above do if we listen to chat theory. Some pentatonic melodies would use the 7th harmonic I've heard some say but salve regina (simple tone) is based on hexachords I am being told so from Do-La there is a perfect sixth and not Do-7th harmonics. What do you experts think of this?

The video with Bernstein:

Salve Regina: http://gregorian-chant-hymns.com/_Media/salve-regina-simple-jpeg_med_hr.jpeg

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    That 7th harmonic is lower than b7 and higher than M6, but when taunting children, I always use M6..! – Tim Mar 6 at 9:04
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    Do you have some evidence that children sing a 7th and not the 6th. I could imagine that children of today are singing so false that they aren‘t even able to sing a major second. – Albrecht Hügli Mar 6 at 11:36
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    The taunt song shares so many notes with "Ring Around the Rosie" (or at least the version I'm familiar with) that I'd imagine that children would be able to sing both with the same tune portion consistently (although never underestimate the incompetence of average people as singers--my experience is that the average 4 people cannot sing the same song in tune with each other). – Dekkadeci Mar 6 at 12:12
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    While the 7th harmonic turns up in all kinds of places, I can't recall hearing it in childrens' taunts, but I wouldn't be terribly surprised. A pretty good example of a 7th harmonic is in the "it's a snake" part of "badger badger badger". – Scott Wallace Mar 7 at 10:03
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    The video you link to does not contain the portion of the lecture where he claims that the childhood tune employs the harmonic seventh. – phoog Jun 17 at 19:28
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To my ear, the tune is not sol-mi-7-sol-mi. The 7 is too high. This accords with the fact that the harmonic seventh is more than twice as far from the equal tempered major sixth (a difference of 68.83 cents) as it is from the equal-tempered minor seventh (a difference of 31.17 cents).

One could make a case for it being 7-sol-do-7-sol, however. This makes the first interval, the descending minor third, a 7:6 ratio (or 6:7 if you prefer, since it's descending). That's 266.87 cents, while a 6:5 minor third is 315.64 cents, and an equal-tempered minor third is of course 300 cents. It also makes the interval between the higher two pitches a very large 8:7 step (231.17 cents) instead of a 9:8 step (203.91 cents) or 10:9 (182.40 cents).

Tuning the A of Salve regina to the harmonic seventh also sounds horribly wrong to me. If it was ever sung that way, it would have had to have been long before the early 11th century, when Guido d'Arezzo codified the basis of hexachord-based solmization. But according to Wikipedia, it seems to have been composed in the 11th century, but the "solemn tone" given there is completely different from the tune you link to, which I suppose is even more recent than that. Perhaps you could find evidence for this hypothesis in an older chant.

It does seem fairly well established that medieval keyboards (organs, for example) were tuned in untempered fifths, though I don't know as much about this as I know about baroque keyboard temperaments. Untempered fifths work if you're not playing music with consonant thirds, and your keyboard only has one or two "black" notes (B♭ and F♯). This puts the A at a ratio of 27:16 over C, or 905.87 cents. That's a lot closer to equal tempered A at 900 cents than it is to B♭ at 1000 cents.

Furthermore, the harmonic seventh is closer to C than it is to G, but the note is written using the lower of the two diatonic steps between those two notes. If the harmonic seventh had really been in use, it would seem much more likely that the note would have been written as a B♭ than an A.

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First off, I cannot claim expertise, but:

As Tim correctly pointed out, the 7th harmonic sits betwixt our even-tempered M6 and m7. If this melody is in fact based on the harmonic series, perhaps the reason that the top tone is more commonly sung as a M6 is due to the ear's natural "correction" of leading tones up to their tonic: aural skills students often perceive themselves as singing the tonic when to others they are perceptibly singing the leading tone.

Although reducing this melody to an unfolding of several levels of harmonic overtones may be a theoretically attractive proposition, I would view the M6 as a neighbor tone to 5, perhaps accessed via the subdominant overtone series (taking tonic as a dominant).

In my experience the descending minor third is one of the easiest intervals to sing and hear, so it makes sense for it to be featured in a children's song. But the neighbor tone of the M6 speaks to a deeply developed cultural grammar of tonal retention that belies its presence in such a "simple" song.

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