If you have a formal ceremony performed at a jinja, a seishiki (or shōden) sanpai, the priest includes your address and name in the norito that is read to the kami. This is so that the kami knows who the prayer is for. After your name, the priest says “eeeeeee”. There is normally a short pause before and after that sound as well.
What’s that about?
As I continue reading around Shinto, I have finally discovered the origin. In ancient Japanese, which is what norito are written in, “ee” (well, in standard transcription it would be written “i”, but the sound is “ee”) was an emphatic particle that could be added after words to draw particular attention to them. This particle is used in norito to emphasise the name of the petitioner.
I have yet to read a direct explanation of why the petitioner’s name is emphasised. Is it to make sure that the kami blesses the right person? Similarly, I am not sure why the “ee” sound is drawn out. But at least I now know what the sound is for.