Finger positions along the neck

Just asked Kevin a related question in a PM, but the full progression of numbers along the neck is 0-1-2-3-3#-4-5-6-7-8-9-9#-10-11-12-13-13# etc.
And my new question is this:
In Jiuta, the numbering is 1-2-3-4#-5-6-7-8-8#-9-X-10-11-12-13-14-14# etc.
(Edit: 1-2-3-4-4#-5-6-7-8-8#-9-X-11-12-13-14-14#)
Compared to Tsugaru, besides the difference in starting number, the fourth entry is a sharp, and the eighth is a sharp (The sharps are also five spots apart, which made me think pentatonic, but I just realized that that is meaningless)
In Tsugaru, the fifth entry is a sharp, along with the twelfth entry, putting eight spots between them, so clearly it’s a different arrangement, even considering the difference of one.
So where in the progression of these two styles, and any other styles with different numbering, did this replacement of sharps come from, and in general why is there a sharp position and not just a new number?

I don’t have any jiuta material at arm’s reach here that I can check this with, but I’m going to assume you left out the 4 between 3 and 4# in your jiuta numbering. Is that right? If that is the case, they fall into place, and the first sharp in both cases is a major third, and the second sharp is a major 7th. Again, I may be mistaken about this, but if that is the case, the sharps are there because those two pitches are rarely used in traditional Japanese music. I’ve got one type of nagauta notation in front of me right now, and in this system the major second (which is rarely used in nagauta) is 1# , for the same reason.
So the question is, did they add the sharps after the fact, once they had to deal with new pitches, or are the sharps there to remind us that these notes are uncommon, or (in the case of the tablature generally used for Tsugaru pieces) do they like the idea of 10 being the octave, so that the second octave pitches have similar numbers to thier lower octave counterparts. I’m guessing it’s the last one, but I don’t know for sure. I’ve wondered about this myself for a long time.

Gerry, you are correct about the “dropped” 4, and Anthony has also doubled the 10s. (It’s either X (on string 3, because the scoring convention only allows for single-digit string numbers) or 十 (on strings 1 and 2). So the jiuta numbering is 1-2-3-4-4#-5-6-7-8-8#-9-[10 or X]-11-12-13-14-14#, etc.

Ah, thanks, Linda, I thought that’s what was going on with the X. That means that in jiuta as well, the numbers in the upper octave match those in the lower octave.
But, now I see Anthony’s point: 8# is not the major 7th, so why did the jiuta players choose to put the sharp on that note in particular?

Yikes, yeah sorry for the typos!
Of course there is the 4, and I just got it mixed up in my head with the 9 X and 11 (It’s written with that 1 with a dot).
Anyway, so while they both lack the major third (Is that why Niagari is a fifth chord? :P), Jiuta has the major sixth singled out with the sharp while Tsugaru has the major seventh…
But so it’s true that Japanese music doesn’t use the seventh and third?
Do you have a good source to learn Japanese music theory from, Gerry? Both traditional, and possibly modern adaptations?
And I just realized that they do go to ten, that seems to be a good reason. It’s strange that it works out that way though-are the two notes left out really the only two intervals that are rarely played? It seems almost magical that we get the nice ten, along with the sharped notes being the two most special notes. But I guess it’s the way it is for a reason.
And Linda, why do the scores have to be one character? And isn’t that messed up when you get to the octave and have the funky dotted numbers?

Sorry if this was a little sloppy, I wrote it somewhat backwards.