Inserting Melodies in the Tefiloh

In general every place that has "Nigun Nivchor" can have a new melody, and every place that has "Nigun Yodua" has it's own melody which should not be changed. The Divrei Kehillos writes on every melody if it is Yodua or Nivchor.
Also Niggunim Nivchorim sometimes become Niggunim Yedu'im, if in a community a specific melody was done for generations, and there could be a problem to change it, as mentioned in the Rm'o (תרי"ט ס"א).
Besides that - Harav Hirsch Zt'l had very specific requirements from the niggunim (even Nivchorim), and it may be unlikely that a new melody would answer these requirements.
You can see more about this subject in this posting , and there is a long article from R' Sho'ul Schneller about it in Yerushoseinu #3.

Michael FRBSH

Minhag FFDM is to start Musaf at the second day of Rosh Hashono in the Shabbos melody (Kaddish before Shmone Esrei, and the first two Brochos, excluding Zochreinu Lechayim and Mi Chomoucho). The first two and last three Brochos of Chazoras Hashatz in Mincho of Rosh Hashono are done in the weekday melody (unless it falls on Shabbos, which in that case the last three Brochos are done in the Shabbos melody).
Psukei Desimroh and Birkos Hashachar do not have a special melody for Yomim Nouro'im, and (except for Adoun Oulom, Netilas Yodayim, Lehisatef Batzitzis, Boruch Sheomar and Hamelech and onwards) are done like on every Shabbos.
Besides that, the rest of the davening is done in Yomim Nouroim melody.
This applies to things which are Niggunim Yedu'im. Whatever is Niggun Nivchor could come from Shabbos theoretically, but then the question is if it has the right effect and if the melody suits these special days. For example - according to Minhag Ashkenaz, Houtzo'oh and Hachnoso, Ein Keiloukeinu, and even An'im Zemirous, have a special melody for Yomim Nouro'im, and the regular Shabbos melody is not used, even though this officially is done in a Niggun Nivchor.

Michael

Any Piyyut which has a set melody, should definitely not be changed to any other melody (like Eder Vehod, A'apid etc.). The Piyyutim that don't have a set melody could theoretically use any melody, even a Shabbos melody, but again the question is if the Shabbos melody suits the words and the feeling one is supposed to have in those special days. Most of the Piyyutim have old Ashkenaz melodies, so it is always best to cling to those niggunim.

Michael

Steven wrote: If we really want to maintain minhogim and the nusach as it was practiced over the last 2000 years then shouldn't we not introduce any new melodies and why was it OK to do so 200 years ago? This ensures we don't lose any of the most ancient ones?

In principal one can introduce new melodies, but it needs supervision of a Posek and Talmid Chochom like Rav Hirsch.

MPerlman wrote: Where are the requirements of Rav Hirsch written. I am interested in perusing them in their original, or translated, form.

You can find them in the Haskomo he gave to Japhet's notes. They were translated to Hebrew in Yerushoseinu #3, pg 257/

Michael FRBSH