The current consensus calendar is called the Gregorian calendar.

Types of calendars, and facts about calendar design

Primary classification of calendars:

• lunar: based on cycles of the moon (about 28 days).
• solar: based on cycles of the earth around the sun.
• lunisolar: some combination of lunar and solar. Generally they are lunar but then have periodic 'leap months' to resync the calendar with the earth's orbit around the sun

Other named calendar attributes:

• [1] Perennial: A calendar which does not change between years (excepting leap years).
• [2] Blank-day: A calendar which have some days standing outside of the usual weekday cycle. A disadvantage compared to the Gregorian is that this causes trouble for people whose religion demands that they have a special day exactly once every 7 days; for these people, the day-of-the-week of their special drifts when using a blank-day calendar, but not with the Gregorian.

Some facts/difficulties:

• The duration of one day is not synched with (does not evenly divide into) the duration of the Earth's orbit around the Sun, so either calendar dates must drift with respect to the seasons, or there must be leap years.
• The cycles of the moon are not synched with (do not evenly divide into) the cycles of the sun, so the beginning of the 'year' on a purely lunar calendar must drift through various seasons.
• The sidereal year, which is the "time taken for the Earth to complete one revolution of its orbit, as measured against" the fixed stars, is about 365.256 days. The mean tropical year is a related quantity relevant for the seasons; it is about 365.24 days. [3]. The prime factorization of 365 is 5*73. The prime factorization of 364 is 2*2*7*13. The prime factorization of 366 is 2*3*61.
• For business accounting purposes, it would be nice to have the year cleanly divided into something like quarters such that each quarter had the same number of days. But this is difficult because of the need for leap years. In addition, it would be nice if each quarter had the same number of weeks and the same number of months and began and ended on weekly and monthly boundaries.

Months

Months are divisions of the calendar roughly as long as a natural period related to the orbit of the moon. A sidereal month, which is the moon's orbital period in a non-rotating frame of reference, is about 27.3 days. However, a cycle of moon phases (synodic month) is about 29.53 days [4].

Weeks

Most calendars appear to segment time into periods of 5 to 13 days called weeks, although a few cultures had weeks as short as 3 or as long as 20.

Often one or two days per week is special, and often ordinary work is interrupted on that day (a sabbath, or vacation, or market day, or unlucky day).

Some ideal design goals

These are not possible to simultaneously achieve; and some are redundant:

• The calendar should be asymptotically roughly synchronized with the seasons (so its long-term mean should be roughly equal to the mean tropical year).
• The calendar should be perennial, eg should have the same form every year
• The year should be divided into 3, or 4, or 5, or 6, or 7 periods (analogous to quarters or seasons)
• The quarters, or quarter-analogs, should each have the same number of days
• Each season or quarter-analog should be divided intod about 3 months
• The months should be synchronized with lunar cycles
• Each month should be divided into about 4 weeks
• Each month should have the same number of days, somewhere around 27-30
• Each month should be cleanly divided into weeks
• Weeks should have duration of roughly 5-13 days
• The number of the day of the year should determine which month and day of the week it is, without knowing which year is meant
• The complexity of memorizing and computing the exceptions to these rules should be minimized
• Scheduling periodic special days for religions to fall upon the same day of the week
• Birthdays should drift in between days

My prioritization of the ideal design goals

I think the primary goals should be:

• synchronization with the seasons
• simplicity of computation
• perennial-ness
• day-of-week being determined by day number in year
• equal number of days in each week, month, quarter, except for some minimally necessary leap-something

Non-goals/lower priorities:

• give up on lunar synchronization of months. In modern life lunar cycles are no longer central, and anyways we're already doing without it in the Gregorian.
• give up on everything regarding months. Without lunar synch, months have no reason to exist, except as a tractable unit smaller than the season. Maybe design everything else then add months in at the end in whatever way is most convenient.
• since synchronization with the seasons is demanded, we must have leap years
• the religious objections against blank-day calendars. My guess is that the vast majority of people would be satisfied simply treating the special blank-days as religious special days, as this provides them with MORE religious special days than the Gregorian, and ensures that AT LEAST one out of 7 days is a religious special days. My guess is that those people who interpret their religion as demanding that the special days be EXACTLY one out of seven days (rather than AT LEAST one out of seven days), or who care about the precise historical continuity of the PHASE of the special days, will be a small minority.
• birthday drift. This is contradictory to each numbered day of the year falling on the same weekday every year, and the latter is more important because it bears on simplicity.

Design implications

• since we must have leap years, either we add the leap days to one of the seasons/quarter-analogs, or we leave them separate. I think we should add them to one of the seasons/quarter-analogs because for business purposes they are going to be counted in one of the financial quarter-analogs in any case, so this avoids having to treat them differently in fiscal calendars and conventional calendars.
• to keep the number of days per season/quarter-analog as even as possible, it would be better to have single leap days rather than leap weeks or leap months
• Most proposals for perennial calendars seem to fall into three classes; either leap-days are added ('blank-day'), leap-weeks are added, or the calendar is allowed to drift with respect to the seasons [5]. Our demand for seasonal synchronization combined with the previous bullet point's preference for single leap days therefore suggests that we focus on blak-day calendars.

Some proposals

Blank-day proposals include:

12-month:

• World Calendar ("favored by the UN in the 1950s" [6]). Inspired by the older Invariable Calendar proposal.

13-month:

• Positivist ("The main reason that his suggestion [for calendar reform] failed to find favor with many people seems to have been that he insisted on naming the months for various notable persons" quoted in [7])
• International Fixed ("popular among economists between the World Wars" [8])
• Tranquility; afaict this is similar to International Fixed except out of phase (the first month starts on Gregorian July 21 instead of Gregorian Jan 1; International Fixed has a month ("July") starting on July 16, not July 21.
• Ap-Iccim's (13-month)

Monthless:

• Asimov's World Season (4 seasons of 91 days each)

There are also leap week calendar proposals, such as Hanke-Henry, Pax, Symmetry454, the 4-4-5 calendar, but these do worse than blank-day proposals on the criterion of keeping the number of days per quarter as close to equal as possible.

Winnowing these down

Given the priorities above, I think either the World Calendar or International Fixed or World Season are best, because the other blank-day proposals have different names that causes a marketing problem for the proposals (it may make them sound fanciful). All of these are perennial blank-day calendars. I'll list these again with more details:

• World: 12 months. 4 91-day quarters, each with 13 weeks/3 months. In each quarter, the three months have 31, 30, 30 days respectively. Adds a blank day at the end of the year. On leap-years, a leap-day is added at the end of the second quarter.
• International Fixed: 13 months, each with 28 days/4 weeks (the extra month is month 7 and is called "Sol"). The main part of the year can also be divided into 4 13-week quarters (which don't occur on monthly boundaries). Extra day is added at the end of the year. On leap-years, a leap-day is added just before the beginning of Sol.
• World Season: No months. 4 seasons/quarters of 91 days each, divided into 13 weeks. Extra day at the end of the year. On leap-years, a leap-day is appended to the end of season 2.

Note that all of these proposals have the same quarterly structure (4 91-day quarters, plus an extra day at the end of the year, plus leap-days at the end of the second quarter), but with different month structures. World Season is the simplest, having nothing else besides this common quarterly structure. World Calendar has 12 months, which align with quarters, but which have different lengths and which do not align with weeks (except at quarterly boundaries). International Fixed has 13 months, each with the same length, and each aligned with weeks, but is not aligned with quarters.

As noted above, i would modify each of these proposals slightly to consider the extra days part of the preceding months/quarters rather than standing outside of them.

Some (novel?) proposals

Now i'll present some other proposals that i haven't heard before.

The prime factorization of 365 is 5*73.

Proposal 1: Instead of having 4 quarters/seasons, have 5 fifths/seasons of 73 days each. Stop there; since 73 is prime, this has the disadvantage that the fifths can't be further evenly divided into weeks or months.

Proposal 2: Instead of having 4 quarters/seasons, have 5 fifths/seasons of 73 days each. Declare the last day of each season to be extra/special. Now we have 72 other days. The prime factorization of 72 is 2*2*2*3*3. Divide the 'non-extra' part of each season into 12 weeks of 6 days each (with 24 days = 4 weeks forming a month, so 3 months per season).

Proposal 3: Like Proposal 2, except instead divide the regular 72-day portion of each season into 9 weeks of 8 days each (with 3 months of 24 days = 3 weeks each per season).

Proposal 4: Have 3 extra days at the end of the year. Divide the rest of the year into 10 weeks, organized as 5 fifths of 14 days = 2 weeks each. This proposal has the advantage of 7-day weeks over #1-3, but I don't like this as much as proposals 1 thru 3 because the 3 extra days, when added to the last fifth, makes the fifths more uneven than necessary.

Note that 'fifths' is hard to pronounce, so call them 'cinqs' (pronounced "sinks") instead.

History

The current consensus calendar (Gregorian) was introduced in 1582, replacing the older Julian calendar. The Julian calendar was introduced in 46 BC, replacing the earlier Roman calendar(s).

Apparently the World Calendar, a blank-day calendar, made some headway but was not adopted due to blank-day religious objections [9] World Calendar.

https://en.wikipedia.org/wiki/Invariable_Calendar

Tangentially related: The Antikythera mechanism, thought to be the first analog computer, was a system of gears for a detailed computation of various calendrical and related astronomical quantities.

Conclusion

I think i like any of the following better than the current Gregorian calendar: World Calendar, International Fixed Calendar, World Season Calendar, novel(?) proposals #1-#3.

I haven't decided yet which of these three proposals i like best. World Season Calendar and novel(?) proposal 1 are appealing for their simplicity. International Fixed Calendar and novel(?) proposals 2 and 3 are appealing for having months which fall on weekly boundaries and for having the same number of days in each month. World Calendar and novel(?) proposals 2 and 3 are appealing for having months with quarterly alignment. World Season Calendar, International Fixed Calendar, and World Calendar are appealing for having 7-day weeks.

Proposal 2 is additionally appealing because if 2 days per 6-day week were weekends, then the workweek would be only 4 days instead of 5, about 16% shorter, and also because each cinq has about 12 weeks (the last cinq has an extra day), and 12 is a nice divisible number.

Of course, i suppose that ideally one would have an algorithm that would start with 365 (the rounded mean tropical year) and spit out a calendar system.