While I can accept that, why do you think it is not a, or possibly more than one, beacon?
A number of factors including me not leaping to a conclusion, trying to build a dossier of facts, examine the pattern, look for a logical answer, etc. You seem to be a deliberate thinker and I appreciate that very much.
As soon as I saw Dag insist he was hearing something quite different I had to question the assumption that this was a "beacon", at least in the hobbyist sense. I also don't yet see a reason to assume that this is a deliberate commercial ISM device that is broadcasting as part of a network of linked devices that are relaying some sort of scientific data.
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I also am not convinced this is a hobby beacon, but many of the signals reported in this forum are not hobby beacons.
This is why I asked if you ever drop in #wunclub. The #wunclub IRC channel is a great place to compare real time receptions. For example you could monitor the signal while I was also monitoring, and when you heard a double you post it, if I did not also hear a double then we are not hearing the same thing.
If you don't want to use the more active #wunclub, the #monitor channel is almost silent, I really only keep it up as an alternate NSA IRC bot access location and because it has existed so long.
By the way, many of the ISM devices found in this band have modulation that is incidental to their purposes. In other words, they are most often not data links or anything like that. For example, the RF based inventory control systems at stores often use this frequency range, and some of them send out a simple periodic pip (sound familiar?) to query RF ID tags, others, most, are more complex waveforms.
In contrast we have some known hobbyist pirate beacons in known locations on frequencies that have propagation patterns. This 13,559 signal is pretty much always there, day and night. If it was that powerful of a signal we would all hear the same thing at some point but we don't.
I am not 100% convinced we are not hearing the same thing. Unless we can confirm the same time of reception in some way I don't think we can be sure. At this point I don't feel strongly about it either way but I lean towards the same signal, if not the same source.
For sure we have all heard different things, no doubt there. But if we all listened at the same time would we hear different things? I have heard periods with doubles and without them.
How about this...just a wild suggestion to consider. What if the basic pulses we are all hearing are the same signal, and the doubles are a local variation or addition from a common local but separate source? What is something local is slotting into the basic pulse train intentionally? Notice that the interval of the start of the first pulse in a double to the start of the next single is about the same as if the double did not happen at all, at least in the few I have seen and measured. Of course, that would argue power levels and things like that, but have we looked at that closely enough to dismiss it? Unlikely, for sure, but an interesting thought process.
Or what if it is a local, but common, signal.
Just for reference though, I have heard this signal, or what appears to be this signal, at decent power levels in some pretty remote locations, sometimes dozens of miles from the closest power lines, let alone anything else. For example, I have heard it in Striped Butte Valley, in the Panamint mountain range, I have heard it in Saline Valley, and I have heard it in the Owlshead Mountains.
All those folks share something that all pirate broadcasters want: to be recognized. If no one seems to hear them they "leak" the information to their community of listeners or they pose as a listener and tell of their "discovery". There was a pirate SW broadcaster here in Orange County (Santa Ana) that could not keep his mouth shut and was repeatedly busted because he loved to know that people heard him and made it easy for the FCC to know he was on the air. I won't point out anybody here but when I read some messages I have to say out loud "Exactly how did you know that detail?".
I can think of close to, if not more than, a dozen pirate / hobby beacons that have been active in the past that no one ever stepped forward on.
I put one on 13559 kHz 3 years ago and it ran for almost a year before dieing, I never told anyone about it myself, but saw several reports of it. I never reported it myself.
There are at least 2 probable ditters / beacons on air right now that I have not heard anyone claim. The "third" beacon in the Rainy / Rocky set has not, as far as I know, been claimed, and it has been active for some time. The 2016.26 kHz ditter has been active for many months and has not been, as far as I know, claimed. And it has only been reported by me as far as I know, however I am hearing it over a pretty fair geographical area.
Of course the ISM band is used by many signals, that is the very purpose of it but not a reason to conclude that this is someone's beacon. It is a nice place for hobbyists to transmit flea powered signals but so far we have different signals in different geographic areas that have common characteristics including pretty reliable reception.
Now after all of that I have to say I love the discussion, I love seeing your and Dag's thoughts and it is a lot of fun! No one wants to take any of it personally but that happens when reading text on a forum even if it is meant to be emotionless technical prose. 
Part of my issue with this signal is my direction of arrival and my location. The direction I am receiving it from just does not argue something local or industrial. But it does argue something high on a mountain. Except for private homes there just is not much in that direction from me, but there is the Sierra Nevada's, in fact the bearing is pretty much in line with Whitney or the area around there. On the other hand if I am having consistent reflection or multipath issues my bearing could be completely out to lunch and misleading. Until it can be confirmed I would call that possible direction.
T!