Enharmonic and Microtonal Scales.
Keely time and again uses the term enharmonic. From what I have read in comments about Keely's work it is clear the term is poorly understood. Most commentators appear to understand the term to mean disharmonic. It means quite the opposite.
The term enharmonic comes from the Greek and means "of one harmony". An enharmonic major third are two notes that have the precise relationship of 5/4 in any key. We are talking of pure natural harmonics of course, not some scale contrived to make playing a tune easier.
In the equal temered scale commonly used today the notes C-sharp and D-flat are the same, both being represented by the black key between C and D. This is achieved by elevating the flat and diminishing the sharp. For most purposes this is an adequate arrangement.
In reality C-sharp and D-flat are not the same note. They are separated by the famous Pythagorean comma or 23 cents (100 cents equals one semitone).
By putting two black keys between C and D and any other two notes that are separated by a full tone we can now play pure enharmonic thirds in any key on the diatonic scale.
For that purpose keyboards were built for organs and other keyboard instruments that used this arrangement.
I have ever only seen two organs who had two black keys for every black key in the conventional layout. Unfortunately I could not obtain permission to play either instrument so I have to rely on musical literature as to the audible difference in playing a major third.
When the music is written in, say the key of B-flat or F-sharp, a further subdivision becomes necessary in order to preserve the correct relationship between notes. The interval C - D is then subdivided into
C - D-double flat - D-flat - C-sharp - C-double sharp - D
Also the notes separated by a semitone, such as B and C, require two subdivisions:
B - C-double flat - B-double sharp - C.
In other words four black keys for notes separated by a full tone and two black keys for notes separated by a semitone, all separated by a Pythagorean comma.
Such a keyboard is capable of playing pure major thirds in all chromatic keys. Instruments with 31 notes to the octave, like this, have actually been built.
So, what would happen if we decided to play in the key of B-double flat?
I am afraid further subdivisions are required, but this would bring us into the realm of microtonal music, a subject that far exceeds the scope of this paper. For those that are interested, there is excellent literature available on the net and in libraries.
So, what is Keely talking about when he talks about "antagonistic enharmonic thirds"?
At this stage I cannot definitively say what "antagonistic" refers to, my best guess is that the chord is antagonistic to certain discordant molecular aggregations within a given chord mass and is capable of aligning these elements for his purposes. Keely leads us to assume this, some further research is evidently needed here.
The knowledge of enharmonic scales is really only important to gain an understanding of what Keely is talking about, for he speaks in those terms. When dealing with precise harmonic relationships in terms of frequency we do not need to know all the ins and outs of enharmonic and microtonal scales. I have only included this here to enable the reader to understand that, difficult though it might be to understand Keely's terminology, the man was talking perfect sense and not the pseudo-scientific gibberish that he was so often accused of.