In common usage, “Jewish time,” like “New York time,” indicates the opposite of “sharp” for starting time. However, there really is a different sense of Jewish time orientation that defines our calendar.
You have to let go of the assumptions we normally hold in the western world to appreciate them, just like we have to not take it for granted that everyone orients toward the north for direction with east to the right and west to the north.. In Biblical orientation, the forward direction is east, so what is to right is south and what is to the north is left.
In the Jewish calendar, we count both days and months. As mentioned in http://snaptwig.com/article/spring-and-the-month-of-nissan, the first month of the year is Nissan, the start of the spring season and the month in which we celebrate Pesach [Passover]. It used to simply be called the “first month,” and the name for it, as for the other 11 months came later. But the days of the week in Hebrew remain just “first day,” “second day,” etc. The only exception is the seventh day, which is called Shabbos/Shabbat [Sabbath]. In fact, each of the designated shir shel yom, chapter of Tehillim [Psalms] that is designated for each day of the week, begins with “Today is ___ day of the Sabbath,” filled in with first, second, third, fourth, fifth, or sixth.
This is part of the fulfillment of the injunction to remember the Sabbath day, which is in the portion of the Torah will be reading this Sabbath, Yithro. Every day of the week, we are not only counting up to the Sabbath but thinking about it in the course of our daily activities. We accomplish this by following the principle of Shamai as recounted in Beitza 16a here (even though we normally follow the practice of Hillel) in looking out every day for something special for the Sabbath table. (for more on this topic, see what Rav Moshe Taragin wrote in vbm-torah.org/archive/metho/shabbat/02zakhor.doc)
The idea is if you see a nice piece of meat on Sunday, which you can safely keep in the freezer until you cook it on Friday, then you buy it then. If on a subsequent shopping trip, you see something that looks even better, you can then buy it for you Sabbath dinner and then prepare what you bought earlier in the week for your weekday dinner. You may buy a melon on Wednesday and then see a nicer one on Thursday and so eat it that melon you bought earlier on Thursday while you keep the nicer one for the Sabbath. In that way, every day’s meal becomes a step in honoring the Sabbath.
That brings me to two points on the Sabbath that Rav Goldwicht shared this past Monday night. One is in connection to the days that we always introduce as “of the Sabbath.” He said that according to a Midrash, the week was set to be made of 6 days of 28 hours (to correspond to the 28 year cycle of the sun that is commemorated with a special blessing, as last observed in 2009). Each day gave up 4 hours in order to add on a seventh day, the Sabbath. Consequently, each day of the week has its share in that day.
The second point he made was in connection to food preparation for the Sabbath. It is considered meritorious to taste the food in advance to be sure it is spiced just right. In fact, the reward for this is good health. Rav Goldwich pointed out that this is a Torah-true segula, recorded in venerable sources, unlike the ones that simply sprout up in marketing ploys today, as detailed in What works in Jewish tradition and an open letter.