The genealogy shown below is readily available online. Its authority is unclear but it may derive from the Montefiore family source identified here (scroll down to note 13). The 2020 publication of ref 319 allows a partial critique. Thus, the 1707 marriage shown below appears in ref 319 p145 as on 13 Apr and the 1702 marriage appears in ref 319 p155 as on 14 Oct 1712 with the groom's father being Menachem (presumably an equivalent of Manuel), and the bride being Rica (not Ricta shown below). Our incorporation of the new data into a chart of the early Montefiores is shown here and further errors below became apparent eg it seems unlikely that Moses Raphael was the son of Joseph. In addition it is probable that key family branches have been left out, especially those based elsewhere than Livorno eg Ancona. And naming appears inconsistent - Mach Vita Montefiore is also named Israel Vita Montefiore.
This online genealogy shows 3 Joseph Montefiores - none with the correct birthdate to have been the maternal grandfather of Joseph Barrow #238, a possibility we considered here.