Yigdal

The answer to your question is that we have a special Yigdal we use on Yomim Tovim only (brought in Japhet's notes as Yigdal for Yomim Tovim. And since we have only one day Yom Tov in Eretz Yisro'el (which sums up to 5 Yomim Tovim throughout the year) we stick to that Yigdal every time.
On Shabbos we sing many different melodies of Yigdal, one is done by the choir and is on the site, others will get there eventually IYH.

On this point we were asked in the past - why do we sing Yigdal on Shabbos even though in Frankfurt it was sung only on Yom Tov.
And the reason we do so is as follows:
In the past in FFDM they used to say (not sing) Yigdal after Ma'ariv and Shacharis on Shabbos too, and the minhag faded out since it wasn't sung but only said at the end of the davening. So since the old Minhag in FFDM was to say it, and the minhag until today in the other kehilos following the German rites is to sing Yigdal, we sing it.

Michael