In his article "How Music Articulates Liturgical Structure, Meaning, and Perception: The Kaddish" published in the recent volume of the Journal of Synagogue Music (vol. 47 no. 1), Dr. Boaz Tarsi writes the following about [Eastern] Ashkenazi prayer modes for the Chatzi Kaddish:
A significant number of versions of the Hatzi kaddish introduce unique musical material that is neither appropriated afterward nor introduced beforehand. The unique musical content of these renditions of the Kaddish, therefore, is not found anywhere else in the liturgy. Among these are Ma’ariv for the Three Festivals, every Kaddish recited during a Torah service [...], the Hatzi kaddish for High Holiday Musaf, the Minhah service for Yom Kippur, and a few others.
And in the footnote there (no. 19) he elaborates:
Some of these (e.g., Kaddish for High Holiday Torah services and Yom Kippur Minhah, not to mention [...] Yom Kippur Katan) are almost all but forgotten, because within the current practice in the American service they would take on a musical setting not traditionally prescribed for them, in all likelihood using the music for Shabbat Ma’ariv Kaddish, as described below.
I'm one of those American ignorami, but would like to step up my game by learning and preserving-by-using this rich heritage of ours. Can someone help me find the traditional tunes for these parts of the service?
I am not looking for random melodies set to these pieces, nor straight recitatives ("nusach"), even if those tunes may be thematically appropriate. Similarly, I am not looking for tunes from Sefardic or other non-Ashkenazi Jewish heritages. Rather, I'm looking for classical traditional settings (I accept there may be some variants) from roughly the same cantorial milieu as Dr. Tarsi referred to.