The calendar was originally fixed by observation, and ultimately by calculation. Up to the fall of the Temple (AD 70), witnesses who saw the new moon came forward and were strictly examined and if their evidence was accepted the month was fixed by the priests. Eventually the authority passed to the Sanhedrin and ultimately to the patriarch. Gradually observation gave place to calculation. The right to determine the calendar was reserved to the Patriarchate. The Jews of Mesopotamia [in Babylon] tried in vain to establish their own calendar but the prerogative of Palestine was zealously defended."
So long as Palestine remained a religious centre, it was naturally to the homeland that the Diaspora looked for its calendar. Uniformity was essential, for if different parts had celebrated feasts on different days, confusion would have ensued. It was not until the 4th century A.D. that Babylon fixed the calendar." Encyclopedia Britannica; Vol 4, article "Calendar"
Encyclopedia Britannica has this to say about the modern “week” and the calendar:
In the 21st century, we take for granted a seven-day week. However, in some ancient cultures, the length of the week differed. According to the Encyclopedia Britannica:
“Among primitive peoples, it was common to count moons (months) rather than days, but later a period shorter than the month was thought more convenient, and an interval between market days was adopted. In West Africa some tribes used a four-day interval; in central Asia five days was customary; the Assyrians adopted five days and the Egyptians, 10 days, whereas the Babylonians attached significance to the days of the lunation that were multiples of seven. In ancient Rome, markets were held at eight-day intervals; because of the Roman method of inclusive numeration, the market day was denoted nundinae (“ninth-day”) and the eight-day week, an inter nundium.” Measurement of time and types of calendars: Standard units and cycles”; Britannica 2002 Deluxe Edition; 1994-2002; Encyclopædia Britannica, Inc.
"The (early) Hebrews employed lunar seven-day weeks, which ended with special observances on the seventh day but none the less were tied to the moon's course."
Hutton Webster in his book Rest Days, page 254-255
"These imported [from Babylon] superstitions eventually led Jewish rabbis to call Saturn Shabbti, 'the star of the Sabbath.' [and] it was not until the first century of our era, when the planetary week had become an established institution, that the Jewish Sabbath seems always to have corresponded to Saturn's Day [Saturday]." Rest Days, p244 Hutton Webster
Talking about the time prior to the Diaspora, Hutton Webster tells us that
an old and still common theory derives the Sabbath institution from the worship of
SATURN after which planet the first day of the astrological week [Saturday] received
its designation. The theory is untenable for more than one reason. In the first place the Hebrews did not name their weekdays after the planets, but indicated them by ordinal numbers. In the second place SATURN'S DAY [SATURDAY] BEGAN THE PLANETARY WEEK, while the Jewish Sabbath was regarded as the LAST DAY of the seven, a suitable position for a rest day. And in the third place neither the Hebrews nor any other Oriental people ever worshipped the planet Saturn as a god and OBSERVED HIS DAY AS A FESTIVAL
(Rest Days, Hutton Webster p. 243).
"...each lunar month was divided into four parts corresponding to the four phases of the moon. The first week of each month began with the new moon, so that, as the lunar month was one or two days more than four periods of seven days, the additional days were not reckoned at all." The Universal Jewish Encyclopedia, Vol 10. p482 Article "week"
The idea of the week, as a subdivision of the month [was found]...in Babylonia, where
each lunar month was divided into four parts, CORRESPONDING TO THE FOUR
PHASES OF THE MOON. The first week of each month BEGAN WITH THE NEW
MOON, so that, as the lunar month was one or two days more than four periods of seven days, these additional days were not reckoned at all. Every seventh day
(sabbatum) was regarded as an unlucky day. This method of reckoning time spread
westward through Syria and Palestine, and WAS ADOPTED BY THE ISRAELITES,
probably after they settled in Palestine
(The Universal Jewish Encyclopedia vol. 10, p. 482. Article "Week.").
"The Hebrew Sabbathon was celebrated at intervals of seven days, corresponding with changes in the moon's phases..." Encyclopedia Biblica, 1899 edit., p.4180
In the years following Clement of Alexandria’s time, an ominous change started to take place that was to radically change the Christian concept of the Sabbath. “This intimate connection,” records the Encyclopedia Biblica, “between the week and the month was soon dissolved. It is certain that the week soon followed a development of its own, and it became the custom -- without paying any regard to the days of the month (i.e. the lunar month) -- ...so that THE NEW MOON NO LONGER COINCIDED WITH THE FIRST DAY OF THE WEEK”
Encyclopedia Biblica (The MacMillan Company, 1899. P. 5290).
Then, on page 4179 of the same encyclopedia, we read: “The introduction...of the custom of
celebrating the Sabbath every 7th day, IRRESPECTIVE OF THE RELATIONSHIP OF THE DAY TO
THE MOON’S PHASES, led to a complete separation from the ancient view of the Sabbath...” Encyclopedia Biblica 1899 p4179
"The four quarters of the Moon supply an obvious division of the month... it is most significant that in the older parts of the Hebrew scriptures the new moon and the sabbath are almost invariably mentioned together. The [lunar] month is beyond question an old sacred division of time common to all the semites; even the Arabs who received the week at quite a late period from the Syrians, greeted the New Moon with religious acclamations. We cannot tell [exactly] when the Sabbath became dissociated from the month."
Encyclopedia Biblica (1899 edit) pp 4178 and 4179
"The Hebrew Month is a lunar month and the quarter of this period-one phase of the moon-appears to have determined the week of seven days."
Encyclopedia Biblica, (1899 edit.) p.4780
"Among all early nations the lunar months were the readiest large divisions of time... (and was divided in 4 weeks), corresponding (to) the phases or the quarters of the moon. In order to connect the reckoning by weeks with the lunar month, we find that all ancient nations observed some peculiar solemnities to mark the day of the New Moon."
The Popular and Critical Bible Encyclopedia )1904 edit) Vol. 3, p. 1497
"In the time of the earliest prophets, the New Moon stood in the same line with another lunar observance, the Sabbath. Ezekiel, who curiously enough frequently dates his prophecies on the New Moon.... describes the gate of the inner court of the (new) temple looking eastward as kept shut for the six working days, but opened on the Sabbath and the New Moon."
Scribner's Dictionary of the Bible (1898 edit) p.521
"The [lunar] calendar was used by all the original disciples of Yeshua. This original Nazarene lunar-solar calendar was supplanted by a Roman "planetary week" and calendar in 135 C.E. when the 'Bishops of the Circumcision' (i.e. legitimate Nazarene successors to Yeshua) were displaced from Jerusalem. This began a three hundred year controversy concerning the true calendar and the correct Sabbath." Shawui Calendar: Ancient Shawui Observance
Early historical records clearly confirms that very early Gentile Christians also kept the same Sabbath as the Nazarenes. This practice was first changed by [Pope] Sixtus in 126 AD, and later officially changed by a royal Roman decree from the emperor Constantine. Observance of the Sabbath day was made illegal and observance of a "sunday" of a fixed week was made mandatory for all except farmers. Previous to this time the Roman saturday was the first day of the Roman week. The veneration of the Sun in the second century AD began to pressure Roman culture to change the first day of their week from Saturn Day to sunday.
Shawui Sabbath: Ancient Sabbath Observance
“Most theologians and some scholars assume that mainstream Jewish society, at the time of Jesus...was practicing a fixed seven-day week which was the same as the modern fixed seven-day week. This is extremely doubtful. The change, from a lunar to a fixed week, was brought about by the power and influence of Rome. As long as the Nazarenes held power in Jerusalem, all Roman practices and customs, including that of THE CONSECUTIVE WEEK, WERE HELD AT BAY”
Shawui Sabbath: Ancient Sabbath Observance
The association of sabbath rest with the account of creation must have been very
ancient among the Hebrews, and it is noteworthy that no other Semitic peoples, even
the Babylonians, have any tradition of the creation in six days. It would appear that
the primitive Semites had FOUR CHIEF MOON DAYS, probably the first, eighth,
fifteenth, and twenty-second of each month, CALLED SABBATHS from the fact that
there was a tendency to end work before them so that they might be celebrated
joyfully. Among the Babylonians these seventh days through astrological conceptions
became ill-omened, while the sabbath in the middle of the month [15th] was made a
day of propitiation, and its name was construed as meaning "the day for ending the
wrath of the gods." The Israelites, on the other hand, made the sabbath the feasts of
a living and holy God. The work of man became symbolic of the work of God, and
human rest of divine rest, so that the sabbaths became preeminently days of rest.
Since, moreover, the LUNAR MONTH had 29 or 30 days, the normal lapse of time
between sabbaths was six days, although sometimes seven or eight; and six working
days were accordingly assigned to the creation, which was to furnish a prototype for
human life. THE CONNECTION OF THE SABBATH WITH LUNAR PHASES, however,
WAS [LATER] DISCARDED BY THE ISRAELITES..."
(The New Schaff-Herzog Religious Encyclopedia pp. 135-136).