Shulchan Aruch (619:1) states, "One should not change the custom of his city, not even with the nigunim or the piyutim they say." The Mishnah Berurah explains, כי עלידי זה מתבלבל דעת הקהל, "This confuses the community."
Furthermore, the holy sefarim teach that the traditional tunes of Rosh Hashanah and Yom Kippur have the power to remove the harsh judgment. One of the translations of the word זמירות is to prune, so (Tehillim 47), זמרו לאלקים means the harsh judgment is pruned away with song.
There was once a king whose young son had gone insane and was sent to a mental asylum. The asylum profited from hosting this royal patient because the king upgraded the services of the asylum so his son would receive the best care.
The prince’s condition improved with time, but the institution’s directors, not wanting to lose the king's ongoing financial support, did not inform the king of his progress.
The prince tried to escape and go home, but the doors were always locked. The prince wrote letters to his parents, describing that he was better and wanted to go home, but those letters were discarded and never sent.
Once, the king's son scribbled on a page and gave it to one of the directors, asking him to send it to his father. The director was glad to comply because this letter proved that his son still needed help. What normal child of his age sends a scribbled paper to his father?
The directors didn't realize that the prince concealed a hidden message within the illegible message. When the king received the letter, he was disappointed that his son had fallen to such a state, but then he studied it from all angles and picked up on the hidden message. He came to the asylum and brought his son home.
This parable explains why we use the specific tunes and melodies on Rosh Hashanah and Yom Kippur. We say many tefillos to the King, our Father in heaven, but the angels intercept our tefillos and don't let the requests ascend to heaven. Therefore, we sing melodies. The malachim don't understand their importance, so they allow the songs to go up. Concealed within these tunes are our yearnings, repentances, and requests for the upcoming year. Hashem understands our hidden messages and grants us a good year.