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When did Sengo Muramasa live? A very difficult question, depending on who you ask. I will start with dates listed by available sources, then try to narrow down a date.
1. Connessiuer's Guide by Nagayama Oei 1394-1428
2. Japanese Sword by Yomoto 1362
3. Earliest dated Muramasa sword 1st year Bunki 1501
4. Koto Mei Zukushi Taizen Joji 1362-1368
5. Kosei Kokon Kaji-mei Hayamidashi Joji 1362-1368
6. Takeya Naomasa records Entoku 1489-1492
7. Ise no Toko by Sato pre-1501
8. Nihonto Koza by Honami Koson Eikyo to Kakitsu 1429-1444
9. Token Bijutsu article by Inagaki Zenji Bunmei 1467-1487
10. Art of the Sword Vol 8 Bunki 1501-1504
So, now we have 10 sources with varying timeframes of activity for Shodai Muramasa.
I believe that the earliest dates can be attributed to later histories trying to connect Muramasa to Masamune. Masamune lived no later than the middle 14th Century. I would guess that these stories could be compared to fictional stories in the United States about Presidents George Washington and Abraham Lincoln; it makes for fun stories to consider two of our most famous presidents talking to each other. George Washington lived from 1732 to 1799, while Abraham Lincoln lived from 1809 to 1865. If we consider that Washington was most famous during the late 1770's, and Lincoln during the mid 1860's, they were at their respective apex some 90 years apart.....
The dates of Muramasa being active post 1500 appear to explain the first dated work, 1501. Also a connection to his students in 1501. What we may consider is the date of 1501 also being the last year of a long life. It was not unheard of for some Nihonto smiths to live 100 years. Even if they started at a very young age, and worked with assistance at the end of life, a smith could feasibly have had an active period of 80 years.
Another important thing to consider is the exact number of generations of smiths who used the Muramasa name. Many of these same sources listed above, suggest that there may have been three generations working before the one who signed with the long, 1501 signature. Another thing to keep in mind was that, according to our Tokugawa notes, Muramasa blades were in the hands of famous Generals and Daimyo by 1535.
In my opinion, the first generation, Shodai Sengo Muramasa was active from the late Eikyo (1441) to the first year of Bunki (1501), a span of 60 years. This is believable. This date is within range of most of the above sources, as well. And, as an interesting note, if we date Masamune to 1350, it would put Muramasa at his height of popularity some 90 years afterwards.