Overview over the differeent numeral systems:

1. The Quettar numerals reported by Christopher Tolkien.
   • These numerals are not attested by Tolkien himself, but have been redrawn by his son Christopher.
   • We do not have any context about the use of these numerals. Since they resemble the DTS 49 numerals, they might be from the early 1950s.
   • These numerals are the most popular on the internet because they already existed when the internet came about. It is also the system you mentioned in your question and the one described by Oni and ibid.
   • Some details about the use are unclear: What is the use of a sign for 12 in a base-12 numeral system? Why are the numerals marked with a dot or a line?
   • These numerals differ from all the tengwar numeral systems attested by Tolkien himself. There is not a single sign that looks like a normal tengwar (or Rúmilian) letter. There are reasons to believe that this dissimilarity came along with the way Christopher Tolkien redrew these numerals and that his father really intended them to look like normal tengwar: The disambiguation dot would only make sense if they looked like normal tengwar, and the similar DTS 49 numerals look like normal tengwar.
2. The DTS 49 numerals, only partially attested (1 short carrier with a dot above, 3 nwalme with a dot above, 4 halla, 6 mirrored unque with a dot above
   • These numerals are similar to the Quettar numerals, but they look like normal tengwar.
   • This could mean that these are the forms Tolkien really intended for the Quettar numerals.
   • The context of these numerals are a rejected epilogue for the Lord of the Rings. It had been worked out much detail, but Tolkien decided against its publication by early 1954. This means the numerals are probably from the early 1950s.
   • See also Per Lindberg’s analysis Tengwar numerals in the King’s Letter.
3. “dwarven numerals (Fëanorian)”, DTS 87 (1 parma, 2 tinco, 3 calma, 4 quesse, 5 umbar, 6 ando, 7 anga, 8 ungwe, 9 unque, 0 stemless vilya with a stroke inside, all said to be used “with a dot”)
   • These numerals are from the mid 1960s, concomitant to explanations about Dwarvish runes and a nice illustration of Bilbo’s contract.
   • The ordering of parmatéma before tincotéma makes sense phonetically, but it differs from the earlier ordering seen e.g. in Appendix E to the Lord of the Rings. Note that the mode descriptions found in the same documents also have this ordering with parmatéma before tincotéma.
   • Tolkien explains that the numerals were used “with a dot”, except for the numerals for 9 which had no use (in this phonemic English mode) and 0. Also, Tolkien says that these were really “no numerals”, but “letters of the alphabet”.
4. “dwarven numerals (Rúmilian)”, also DTS 87
   • This numeral system is from the same mid 1960s source as the “dwarven numerals (Fëanorian)”. The Rúmilian numerals are mentioned after the Feanorian numerals, and Tolkien said that the Rúmilian numerals were “sometimes used”, so it would appear the Feanorian numerals were preferred.
   • These Rúmilian numerals are different from any other Rúmilian numeral system published so far, cf. Helios’s analysis Rúmilian numerals.
5. PE 20 Q10h, Q11j numerals (1 long carrier with dot above, 2 tinco, 3 ando, 4 vala, 5 esse, 6 silme, 7 calma, 8 anga, 9 rómen, 0 úre)
   • These numerals are from the 1930s.
   • Note that the tengwar used in this system resemble our own Western Arabic numerals.
   • Seeing that the later tengwar numerals do not resemble Arabic numerals, it appears that Tolkien abandoned that idea. Of course, these numerals would still be adequate in a tengwar document written in a 1930s style “Qenya alphabet” mode, which is really a phonemic English mode and by far the best attested tengwar mode of them all.
6. PE 20 Q10h, Q11j Arabic numerals in tengwar style (the numerals that look the most like tengwar are: 1 short carrier with a dot above, 3 alda, 6 silme, 9 rómen, 0 úre)
   • These are really just Arabic numerals with a few stylistic modifications so the fit the tengwar better.
   • They are from the same mode descriptions as the other PE 20 numeral system. There is no explanation about how the two systems would be used, but both sources mention this one after the other.

Overview over the different numeral systems:

1. The Quettar numerals reported by Christopher Tolkien.
   • These numerals are not attested by Tolkien himself, but have been redrawn by his son Christopher.
   • We do not have any context about the use of these numerals. Since they resemble the DTS 49 numerals, they might be from the early 1950s.
   • These numerals are the most popular on the internet because they already existed when the internet came about. It is also the system you mentioned in your question and the one described by Oni and ibid.
   • Some details about the use are unclear: What is the use of a sign for 12 in a base-12 numeral system? Why are the numerals marked with a dot or a line?
   • These numerals differ from all the tengwar numeral systems attested by Tolkien himself. There is not a single sign that looks like a normal tengwar (or Rúmilian) letter. There are reasons to believe that this dissimilarity came along with the way Christopher Tolkien redrew these numerals and that his father really intended them to look like normal tengwar: The disambiguation dot would only make sense if they looked like normal tengwar, and the similar DTS 49 numerals look like normal tengwar.
2. The DTS 49 numerals, only partially attested (1 short carrier with a dot above, 3 nwalme with a dot above, 4 halla, 6 mirrored unque with a dot above
   • These numerals are similar to the Quettar numerals, but they look like normal tengwar.
   • This could mean that these are the forms Tolkien really intended for the Quettar numerals.
   • The context of these numerals are a rejected epilogue for the Lord of the Rings. It had been worked out much detail, but Tolkien decided against its publication by early 1954. This means the numerals are probably from the early 1950s.
   • See also Per Lindberg’s analysis Tengwar numerals in the King’s Letter.
3. “dwarven numerals (Fëanorian)”, DTS 87 (1 parma, 2 tinco, 3 calma, 4 quesse, 5 umbar, 6 ando, 7 anga, 8 ungwe, 9 unque, 0 stemless vilya with a stroke inside, all said to be used “with a dot”)
   • These numerals are from the mid 1960s, concomitant to explanations about Dwarvish runes and a nice illustration of Bilbo’s contract.
   • The ordering of parmatéma before tincotéma makes sense phonetically, but it differs from the earlier ordering seen e.g. in Appendix E to the Lord of the Rings. Note that the mode descriptions found in the same documents also have this ordering with parmatéma before tincotéma.
   • Tolkien explains that the numerals were used “with a dot”, except for the numerals for 9 which had no use (in this phonemic English mode) and 0. Also, Tolkien says that these were really “no numerals”, but “letters of the alphabet”.
4. “dwarven numerals (Rúmilian)”, also DTS 87
   • This numeral system is from the same mid 1960s source as the “dwarven numerals (Fëanorian)”. The Rúmilian numerals are mentioned after the Feanorian numerals, and Tolkien said that the Rúmilian numerals were “sometimes used”, so it would appear the Feanorian numerals were preferred.
   • These Rúmilian numerals are different from any other Rúmilian numeral system published so far, cf. Helios’s analysis Rúmilian numerals.
5. PE 20 Q10h, Q11j numerals (1 long carrier with dot above, 2 tinco, 3 ando, 4 vala, 5 esse, 6 silme, 7 calma, 8 anga, 9 rómen, 0 úre)
   • These numerals are from the 1930s.
   • Note that the tengwar used in this system resemble our own Western Arabic numerals.
   • Seeing that the later tengwar numerals do not resemble Arabic numerals, it appears that Tolkien abandoned that idea. Of course, these numerals would still be adequate in a tengwar document written in a 1930s style “Qenya alphabet” mode, which is really a phonemic English mode and by far the best attested tengwar mode of them all.
6. PE 20 Q10h, Q11j Arabic numerals in tengwar style (the numerals that look the most like tengwar are: 1 short carrier with a dot above, 3 alda, 6 silme, 9 rómen, 0 úre)
   • These are really just Arabic numerals with a few stylistic modifications so the fit the tengwar better.
   • They are from the same mode descriptions as the other PE 20 numeral system. There is no explanation about how the two systems would be used, but both sources mention this one after the other.

Overview over the differeent numeral systems:

In addition to the numeral systems, the tengwar could also be used for simple enumerations, like the Latin letters A, B, C, D, ..., can be used. The problem is, though, that different sources do not use the same order for the tengwar. The most canonical one certainly is the one from Appendix E.

Overview over the differeent numeral systems: