Logiweb time

### Time schemes

Logiweb uses the following time schemes

TAI: International atomic time
UTC: Universal coordinated time
MJD: Modified Julian day
GRD: Gregorian date
LGT: Logiweb time

### TAI

TAI (International atomic time) is a 'paper' clock in the sense that it is a computed average of lots of real, atomic clocks located all over the world.

TAI counts seconds, minutes, and hours as regularly as possible. Each TAI day has 24 TAI hours, each TAI hour has 60 TAI minutes, and each TAI minute has 60 TAI seconds. Each TAI second is, as closely as possible, one SI second. An SI second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

TAI time is independent of the rotation of planet Earth.

In Logiweb, TAI hour hh, minute mm, and second ss is written TAI:hh:mm:ss. We have 00 <= hh <= 23, 00 <= mm < 59, and 00 <= ss <= 59. Occasionally, we shall use TAI:24:00:00 to denote TAI:00:00:00 on the following day. Decimal fractions of a second are written after a dot as in TAI:12:23:34.456 which denotes 0.456 seconds past TAI:12:23:34. The notation is compatible with ISO 8601 except that we prepend "TAI:" to emphasize the use of International Atomic Time.

### UTC

UTC is a combination of TAI and yet another time scale named UT1.

UT1 is a measure of the rotation angle of planet Earth relative to the direction from the Earth to the Sun. Each UT1 day has 24 UT1 hours, each UT1 hour has 60 UT1 minutes, and each UT1 minute has 60 UT1 seconds. It is noon in UT1 when Greenwich is under the Sun. At the time of writing, UT1 is around 33 seconds behind TAI. In 1972, UT1 was 10 seconds behind TAI.

As mentioned, UTC is a combination of TAI and UT1. UTC equals TAI plus a politically decided offset. This UTC offset indicates how much UTC lacks behind TAI. At the time of writing, the UTC offset is 33 seconds indicating that UTC is 33 seconds behind TAI.

At any time, the UTC offset is an integral number of seconds, but the UTC offset may be incremented or decremented by decree from the International Earth Rotation Service (IERS). Hence, UTC depends on TAI and IERS, but IERS has the intension to keep the difference between UTC and UT1 below 0.9 seconds, so UTC indirectly depends on UT1.

UTC makes a leap whenever IERS increments or decrements the UTC offset. Such leaps are implemented by irregular UTC minutes.

A regular UTC minute has 60 UTC seconds. An irregular one either has 59 or 61. Apart from that UTC counts like TAI and UT1: days have 24 hours and hours have 60 minutes.

Whenever IERS increments (decrements) the UTC offset, the last minute of the last hour of a particular day has 61 (59) seconds. IERS intends to place irregular seconds at the end of June 30 and December 31 when necessary and intends to announce the leaps in advance.

In Logiweb, UTC hour hh, minute mm, and second ss is written UTC:hh:mm:ss. We have 00 <= hh <= 23, 00 <= mm < 59, and 00 <= ss <= 60. Occasionally, we shall use UTC:24:00:00 to denote UTC:00:00:00 on the following day. Decimal fractions of a second are written after a dot as in UTC:12:23:34.456 which denotes 0.456 seconds past UTC:12:23:34. The notation is compatible with ISO 8601 except that we prepend "UTC:" to emphasize the use of Universal Coordinated Time.

At the time of writing, IERS has never decremented the UTC offset, but has incremented the UTC offset at the end of the following days:

GRD-1972-06-30
GRD-1972-12-31
GRD-1973-12-31
GRD-1974-12-31
GRD-1975-12-31
GRD-1976-12-31
GRD-1977-12-31
GRD-1978-12-31
GRD-1979-12-31
GRD-1981-06-30
GRD-1982-06-30
GRD-1983-06-30
GRD-1985-06-30
GRD-1987-12-31
GRD-1989-12-31
GRD-1990-12-31
GRD-1992-06-30
GRD-1993-06-30
GRD-1994-06-30
GRD-1995-12-31
GRD-1997-06-30
GRD-1998-12-31
GRD-2005-12-31

Before GRD-1972-06-03, the UTC offset was 10 seconds.

### MJD

MJD (Modified Julian Day) is a scheme for counting days in a completely regular fasion. Each day is simply expressed by the number of days since a particular day.

MJD is a regular and reliable day count used by astronomers. Furthermore, it is politically correct in the sense that, even though Julius Caeser was quite controversial in his own time, few people today are offended by a time scale named after him.

MJD is based on yet another time scale named JD (Julian Day). JD expresses the number of days since noon, January 1, year -4712 (year 4713 BC), in the Julian calender.

In ancient times, a day was measured from noon to noon, so people actually counted nights instead of days (as a reminiscence, a period of 14 days is still called a fortnight in the English tongue). Today, we prefer to step our day counters when the sun is on the other side of the planet, which is of course difficult to observe, but which possesses little problem for modern technology.

To get a day count based on JD which steps at midnight, the Modified Julian Day (MJD) is offset from JD by 2400000.5 days. In consequence, MJD counts the number of days since GRD-1858-11-17.

When we combine MJD with UTC, then MJD steps at UTC:00:00:00. When we combine MJD with TAI, then MJD steps at TAI:00:00:00. Hence, MJD/UTC and MJD/TAI are two different day counts, but at the time of writing they merely differ by 33 seconds.

In Logiweb, MJD day d is written MJD-d. As an example, GRD-1858-11-17 equals MJD-0 and MJD-51544 equals GRD-2000-01-01. The day before MJD-0 is named MJD--1 (i.e. Modified Julian Day hyphen minus one). The notation follows ISO 8601 in using a hyphen in connection with day counting, but is otherwise completely unrelated.

Combinations of day and second counting schemes are glued together with a dot. As an example, 0.456 seconds past TAI:12:23:34 on MJD-51544 is written MJD-51544.TAI:12:23:34.456. This follows ISO-8601 in putting the day before the second but does not follow the suggestion of ISO-8601 to separate day and second by a capital "T".

### GRD

GRD (Gregorian Date) is a scheme for counting days in a fasion so complicated that it has taken millennia to screw it up. Furthermore, GRD is "politically incorrect" in that it counts days, not after "Jesus", but after "The Lord" (Anno Domini), which is not completely neutral.

But GRD is widespread, and hence we use it in the Human-Computer-Interfaces of Logiweb. In Logiweb itself, GRD has no place.

In GRD, day 0 of a year is named "January 1", and day 100 is named April 11 (except if the year is divisible by 4, in which case it is named April 10 (except if the year is divisible by 100, in which case it is named April 11 (except if the year is divisible by 400, in which case it is named April 10))).

In Logiweb, Gregorian year Y, month MM, and day DD is written GRD-Y-MM-DD. We have 01 <= MM <= 12 and 01 <= DD <= 31. The notation is compatible with ISO 8601 except for the following: (1) We prepend "GRD-" to emphasize that we label days like the Gregorian calender does (GRD for GRegorian Date). (2) We allow the year to have more than four digits after year 9999 and to have less then four digits before year 1000. (3) We allow the year to be zero and negative. As examples, GRD-0-01-01 and GRD--5-01-01 are January 1 on year 1 BC and 6 BC, respectively.

When we combine GRD with UTC, then GRD steps at UTC:00:00:00. When we combine GRD with TAI, then GRD steps at TAI:00:00:00. Hence, GRD/UTC and GRD/TAI are two different day counts, but at the time of writing they merely differ by 33 seconds.

Combinations of day and second counting schemes are glued together with a dot. As an example, 0.456 seconds past UTC:12:23:34 on GRD-2000-01-01 is written GRD-2000-01-01.UTC:12:23:34.456. This follows ISO-8601 in putting the day before the second but does not follow the suggestion of ISO-8601 to separate day and second by a capital "T".

### LGT

LGT (Logiweb time) is the number of seconds since MJD-0.TAI:00:00:00.

Logiweb time is expressed on the form M*10^(-E) where M is an integer and E is a cardinal (i.e. a non-negative integer). In Logiweb, M is always non-negative, so one could as well say that M is a cardinal.

Logiweb time M*10^(-E) is written LGW-Me-E. As an example, LGW-1083564821686603e-6 equals GRD.2004-05-03.UTC:06:13:41.686603. The e-E may be replaced by the following decadic suffixes:

e-0 U (unit) e-3 m (milli) e-6 u (micro, a Greek mu may be used instead) e-9 n (nano) e-12 p (pico) e-15 f (femto) e-18 a (atto) e-21 z (zepto) e-24 y (yocto)

In a Logiweb time like LGW-1083564821686603e-6 one should not replace the small e by a capital one as that may cause confusion with the decadic suffix E (Exa) which stands for 10^+15.

By the way note the following: Logiweb is a computational system intended for mathematics. In physics, one uses decadic prefixes that glue in front of physical units. In computing systems it is better to use decadic suffixes that glue behind numbers. When needed, Logiweb uses the SI units meter, kilogram, second, etc., and derived units. As an example, font sizes are measured in meters. Period. A font size of twelve typographic points is 4218u, and a printer with a resolution of 600 dots per 0.0254 meters has a distance between pixels of 42.333u. An area of 1m by 1m (one milli meter by one milli meter) is 1m^2 (one square milli) or 1u (one micro) or 1e-6 measured in the derived SI unit of square meters. A weight of 1m is one gram (one milli kilogram). A weight of 1um is one microgram (one micro milli kilogram). This is almost but not completely different from the use of decadic prefixes in the SI system.

### GUTC

When presenting Logiweb time to a user, we use GRD/UTC which we shall refer to as GUTC. This section describes GUTC in more detail than the individual sections on GRD and UTC.

GUTC is irregular compared to Logiweb time in that it occasionally includes leap seconds and, furthermore, it counts days in a rather complicated (Gregorian) manner, which includes leap days.

GUTC is built up from the following cycles:

GUTC second. The length of a GUTC second is one SI second (which equals one TAI and one UTC second). Each GUTC second starts at the 'tick' of the TAI 'paper' clock. As an example, the duration from UTC:00:00:00 to UTC:00:00:01 on GRD-2000-03-01 (March 1, year 2000) is a GUTC second.

GUTC minute. Regular GUTC minutes consist of 60 GUTC seconds. Irregular GUTC minutes consist of 61 or 59 GUTC seconds. As an example, the duration from UTC:00:00:00 to UTC:00.01.00 on GRD-2000-03-01 is a regular GUTC minute.

GUTC hour. Regular GUTC hours consist of 60 regular GUTC minutes. Irregular GUTC hours consist of 59 regular GUTC minutes followed by one irregular GUTC minute. As an example, the duration from UTC:00:00:00 to UTC:01.00.00 on GRD-2000-03-01 is a regular GUTC hour.

GUTC day. Regular GUTC days consist of 24 regular GUTC hours. Irregular GUTC days consist of 23 regular GUTC hours followed by one irregular GUTC hour. As an example, the duration from GRD-2000-03-01.UTC:00:00:00 to GRD-2000-03-02.UTC:00.00.00 is a regular GUTC day.

Long GUTC month. A long GUTC month consists of 31 GUTC days. As an example, the duration from GRD-2000-03-01.UTC:00:00:00 to GRD-2000-04-01.UTC:00.00.00 (i.e. March) is a long GUTC month.

Short GUTC month. A short GUTC month consists of 30 GUTC days. As an example, the duration from GRD-2000-04-01.UTC:00:00:00 to GRD-2000-05-01.UTC:00.00.00 (i.e. April) is a short GUTC month.

GUTC dimester. A GUTC dimester (dimester = two months, compare trimester = tres menses = three months and semester = sex menses = six months) consists of a long GUTC month followed by a short one. As an example, the duration from March to April (inclusive) is a GUTC dimester.

GUTC quimester. A GUTC quimester (quimester = five months) consists of a long, a short, a long, a short, and a long GUTC month. In other words, a quimester consists of two regular dimesters followed by an irregular one that ends abruptly at the end of the quimester. As an example, the duration from March to July (inclusive) is a GUTC dimester. The duration from August to December is another quimester.

GUTC Roman year. A GUTC Roman year is the duration from March 1, inclusive, to the following March 1, exclusive. A regular Roman year has 365 GUTC days; an irregular one has one more. As an example, the period from GRD-2000-03-01.UTC:00:00:00 to GRD-2001-02-28.UTC:24:00:00 is GUTC Roman year 2000, which is regular. In contrast, the Gregorian year 2000 is a leap year and, hence, irregular. The difference arises because the Gregorian and Roman years have newyear before and after the leap day, respectively.

A GUTC Roman year consists of three quimesters, the third of which ends abruptly at the end of the year. As a consequence of the conventions mentioned until now, the last month of a regular GUTC Roman year (February) gets 28 GUTC days, and all the other months gets 30 and 31 GUTC days in the pattern prescribed by the Gregorian calender.

GUTC olympiad. A regular GUTC olympiad consists of three regular GUTC years followed by an irregular one. An irregular GUTC olympiad consists of four regular GUTC years. As an example, the period from March 1, 2000 to February 29, 2004 is a regular GUTC olympiad.

GUTC century. A regular GUTC century consists of 24 regular GUTC olympiads followed by an irregular one. An irregular GUTC olympiad consists of 25 regular GUTC olympiads. As an example, March 1, 2000 to February 28, 2100 is a regular GUTC century.

GUTC Gregorian cycle. A GUTC Gregorian cycle consists of three regular GUTC centuries followed by an irregular one.

The rules above allow to convert from LGT to GUTC and back provided one knows the location of leap seconds. A historical 'Roman year' just had 10 months, starting around vernal equinox. 10 month after vernal equinox, the romans stopped counting days and just waited for the next vernal equinox. The dimester/quimester structure described above is accidental. For the sake of Roman political correctness, the month of August (named after Augustus) was extended to 31 days to make it as long as July (named af Julius).