Shacharis for the First Day of Sukkos
"Adon Olam" is said in the Sukkos tune.
"Baruch Sheamar" is said in the Yom-Tov tune.
[...]
The end of the Ofan [piyyut preceding והאופנים in יוצר אור. SC] from פוצחים is said in the tune for רחם מצוקים, in the piyyut for Mussaf Rosh HaShanah.
[...]
Kaddish Tiskabel is recited in the tune for days with Megillah reading. [In א"י, when there is no Shabbos Chol HaMoed, Koheles or Shir HaShirim is read on the first day of Yom-Tov, rather than the last. SC]
The truth is that this is a bad example. The minhagim for Simchas Torah would be better because the niggunim are "The same as XYZ that is said when Chanukah falls on Shabbos Rosh Chodesh" or some such. However, my Hebrew typing skills just aren't up to it.
Suffice it to say that if the Luach is an indicator of anything, these Ashkenazites are very particular that the correct tune be sung at exactly the correct time. To me, it seems unbelievable that a person would remember to say this, that, or the other in the precise time at the precise occasion.
I mentioned this to my mother, who reminded me that once upon a time, Yom-Tov davening was all the entertainment people got. I think it's absolutely true in this context. The המון עם may not have had much of a Jewish education; they may not be able to understand what they were saying in regular davening, let alone all the obscure piyyutim that even a person who knows kol haTorah Kulah wouldn't be able to translate. They may not have known how to work their way from the top to the bottom of a page of Gemara, but by God, they knew what רחם מצוקים sounded like and God help you if you said פוצחים any other way.
