Many people and companies want to keep the details of the note private. The lender is required to have the note in their possession, and foreclosures can be fought over that requirement, but usually the mortgage is the only thing recorded.
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Typically only the mortgage is recorded, but there is nothing preventing the note from also being recorded, usually as an attachment to the mortgage. If the parties don't object, I believe recording the note, especially if it is an original note, is a great way to avoid the "lost note" issue.
On the other hand, recording the note could cause some difficulties in selling the note and mortgage, but only if it is done as the many mortgages/notes were being done in the securitization marketplace. Just because a note is also recorded does not prevent its being sold, and the sale would only require that an Assignment of Mortgage and Note also be recorded to evidence that transaction.
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The mortgage should be recorded to protect that lien interest against others. The note can be recorded, but it's not necessary and very rare to do so.
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