I wrote an article about this a while back. In the early years of spectroscopy in the 19th century they discovered the discrete spectral lines associated with each element, and saw how they could be used to identify each element (many of the names of the elemens so discovered were based upon the observed spectra, in fact, like “Rubidium”. )
The problem was that there was no theory explaining why the lines were where they were. In many cases the spacing seemed completely random. In others, like Hydrogen, they seemed to suggest some sort of order, but nobody could figure out what it was. In the 1860s, various people discovered apparent regularities – Alexander Mitscherlich of the University of Berlin noted a regular progression in the series barium chloride, barium iodide, and barium bromide. Francis Lecoq de Boisboudron (an unaffiliated scientist) noted a geometrical progression in the lines of potassium. The alkali metals, in fact, were tantalizingly similar to hydrogen in some ways, but unlike in others.
But it was George Johnstone Stoney, professor of Natural Philosophy at Queen’s University in Dublin, who claimed to have finally found the sought-after explanation. Or at least part of it*. Like many scientists, he expected the spectral lines to resemble some sort of vibrating system, with a fundamental frequency and a series of overtones. Once you found your fundamental, it was just a matter of foubling it or tripling it, or finding some other multiple to find the overtones, and these ought to correspond to the observed lines. The hitch is, no one had succeeded in finding such a fundamental and overtones for any of the spectra. Not even hydrogen, the simplest and most regular. So how did Stoney succeed?
He observed that the four visible hydrogen lines were at (in round numbers. Please note that Stoney was more careful than this – he corrected for the wavelengths in vacuum) 4102 Angstroms, 4342 A, 4862 A, and 6564A. 4102 isn’t a good fundamental, but Stoney somehow discovered that the first, third, and fourth of these are in a ratio of almost precisely 20:27:32. Maddeningly, the second one doesn’t fit the pattern. But the others do.
What about the “missing” orders? Why aren’t there lines at 21 and 22 and 23 and so forth? Stoney decided that some sort of “interference” – like the interference observed by Thomas Young in his famous experiment – was responsible for making those other orders invisible. For some reason, only selected lines showed strongly.
Seeking to bolster his case, Stoney collaborated with J. Emerson Reynolds to study another system – chromochloric anhydride** – to see if it could be fit. They were able to find a fundamental frequency that, with proper overtones, allowed them to fit an astonishing 31 lines of the spectrum.
The problem was that they could only do this by using absurdly high overtones (one was the 733rd) and ignoring vast numbers of others that didn’t fit. The “interference” theory didn’t seem convincing. Some were convinced, but others, like Franz A.F. Schuster of the University of Manchester, were not. He calculated the probability that a fundamental and overtones chosen at random could fit observed spectra, and found that it generally yielded a pretty good fit – certainly as good as the ones being made with Stoney’s theory. Schuster’s paper, “On Harmonic Ratios in the Spectra of Gases”, pretty much killed any further efforts to fit spectral lines in this way.
But along the way, Stoney’s work, and that of other researchers, had fit the spectral lines for at least four materials and therefore predicted other spectral lines. Of course, the vast bulk of these didn’t fit anything (even by the theory, which didn’t expect them all to fit). But if you used Stoney’s theory to fit, say, the first and third lines of hydrogen, you can get a correct prediction for the fourth. That seems to pretty much fit the OP’s requirements.
The real solution came from the work of Johann Jacob Balmer of Basel a few years later. As the story is usually told, this was a humble teacher of mathematics at a girl’s school who went in to the work without any preconceptions about the nature of the relationship. Thus unhampered by ideas of overtones, he empirically discovered the now-famous Balmer Formula that was later explained by Rydberg and by Bohr theory.
As is usually the case with “the usual story”, this one isn’t really correct. Balmer was, in addition to teaching at a girl’s school, also a lecturer at the University of Basel. Far from being unaware of an uninfluenced by previous work, he explicitly cites Stoney’s work as his inspiration. What Balmer did observe, however, was that those hydrogen line ratios were too damned close to be purely coincidental. He observed that you could express the raios better in fractional form as 9/5, 4/3, and 9/8. These are pretty small numbers – even smaller than Stoney had used. Moreover, if you expressed it as a fraction, and let the numbers get just a shade bigger, you could fit that stubborn second line with 25/21 and get a fit as good as the others. Then he noticed that you could re-express these ratios as:
9/5 16/12 25/21 36/32
Now the numerators were all perfect squares of 3, 4, 5, and 6. And that looked like a proper series of harmonics. Moreover, the denominators were all exactly 4 less than their numerators. And that showed a pattern. He extended the pattern to different values, and found that he could fit newly-discovered lines in the ultraviolet as well.
*Stoney’s real claim to fame is his work on the electron, which almost no one seems to remember these days. But he’s the one who named the particle.
**I haven’t been able to learn why they used such an odd material. Why not one of the alkali metal spectra, which were much simpler than most other spectra? Why not Sodium, already in use as a standard?