Throughout his writings Keely continually talks about the necessity of tuning his resonators to "thirds" in order to achieve the effects he is after.

So what are the "thirds"?
Unfortunately here is where it gets complicated.
In music the term can apply to three different conditions:

1) Third octave

2) Third harmonic
3) Major Third, Minor Third

So, what are they and which one is he referring to?

Third Octave

Third octave is clear. This is simply fundamental frequency x 2 x 2

C = 256 (1st octave)
C = 512 (2nd octave)
C =1024 (3rd octave)
He often refers to first, third, sixth and ninth octaves.

Third Harmonic

He also refers to frequencies relating to each other in the ratio of 33 1/3 : 100.

Clearly here he means harmonic thirds.
The third harmonic is the fundamental frequency multiplied by 3 or one third the wavelength of the fundamental.

Major Third

A Major third is a musical interval that is the fundamental frequency multiplied by 5 and divided by 4. The two notes combined constitute a pure harmonic chord.

A pure harmonic chord in many keys can only be produced by using an enharmonic keyboard or on a violin where there are no fixed positions for the notes. He talks about the use of enharmonic scales.
He also talks about the use of E double flat, a clear indication that he is talking about a major enharmonic fifth interval in the scale of B flat, which he mentions also.
This is where it gets confusing. Keely uses the term thirds in conjunction with all three assuming that the reader has sufficient musical knowledge to tell which one is meant.
Since we often have only short quotes of what Keely actually said we can only guess in what context the word third applies.

Increasing Amplitude

As if that is not enough in some of his writings the term applies to amplitude rather than frequency.

One of the first things we notice by studying Keely's machines in the large number of resonators he is using.
Far too many to construct say a chord in three octaves,
For that he would have needed only nine, three for each octave.
So why did he use so many?
Keely worked with acoustic resonators to generate his waveforms.
Say he was using the chord C major, i.e. C - E - G. The interval between C and E is a major third, the interval between E and G is a minor third the interval between C and G a perfect fifth. E3 and G3 are the 3rd and 5th harmonics of C1.
The three notes being struck simultaneously results in each note contributing harmonically to the sound created, achieving overtones further into the spectrum (i.e. of higher frequencies) than that Note C could have achieved on its own regardless of amplitude (volume).
Keely was after very high frequencies, far higher than the note could have achieved on its own with effect. It is believed overtones stretch into infinity albeit with much reduced volume.
So how do you increase the volume of overtones far into the spectrum?
In order to increase the amplitude of the overtones he had to increase the amplitude of the notes contributing to it.
There was only one way he could do it with the technology of the day. He did what organ builders had been doing for centuries.
As I said earlier, an organ pipe either goes full blast or is silent, there are no in-betweens.
There is only one way and that is to add more pipes of the same construction and pitch. That is why large organs have over 5000 pipes. No typing error, they can really have that many. Keely did the same with his resonators.
Here something interesting occurs.
If you add one more pipe you increase the volume by 100%. Or by a factor of two. Adding one more you increase the original volume by a factor of three and so on.
See what I mean?
The volume increase follows the same arithmetic progression as the harmonics.
This where Keely parted company with contemporary acoustics. His machines exhibited not only precise mathematical relationships between the notes, but also in their relative amplitude.
I consider this a vital aspect in the overall design of his machines.
I have in all my research never come across a comment that mentions that fact.