# `Calendrical.Julian`

Implements the proleptic Julian calendar.

The Julian calendar is the calendar introduced by Julius Caesar in
46 BCE and used throughout Europe until the Gregorian reform of
1582. It is identical in structure to `Calendrical.Gregorian` —
12 months, 365 or 366 days, the same month lengths — but uses a
simpler leap-year rule (every fourth year) and therefore drifts
about three days every 400 years against the tropical year.

A Julian year begun on another day is a `Calendrical.Calendar.Shifted`
calendar with this one as its base: `use Calendrical.Calendar.Shifted,
base: Calendrical.Julian, year_begins: {3, 25}` builds a Julian variant
whose year begins on the given month and day. The pre-built variants
`Calendrical.Julian.Jan1`, `Calendrical.Julian.March1`,
`Calendrical.Julian.March25`, `Calendrical.Julian.Sept1`, and
`Calendrical.Julian.Dec25` correspond to historical "year-style"
conventions used in different periods and regions.

## Options

* `:new_year_starting_month_and_day` is the `{month, day}` on which
  the year begins. The default is `{1, 1}`.

* `:year` is the Julian year, reckoned from 1 January, that a year
  takes its number from: `:beginning`, the one it begins in;
  `:ending`, the one it ends in; or `:majority`, the one most of it
  falls in, which is the first for a year that begins in January to
  June, as for a calendar made by `Calendrical.new/3`. The default is
  `:beginning`.

## The number of a year

A year that begins on another day than 1 January lies across two
Julian years, and the historical styles differ in which of them gives
it its number. A year reckoned from 1 March or 25 March began after
1 January of the same number, and one reckoned from 1 September or
25 December began before it. C. R. Cheney's *A Handbook of Dates*
sets out the reckonings from Christmas Day, from the Annunciation
and from Venice's 1 March, and a Byzantine year of the world begins
on 1 September, four months before the January year that is 5508
less:

| Calendar | Year 1100 begins on | `:year` |
|---|---|---|
| `Calendrical.Julian.March1` | 1 March 1100 | `:beginning` |
| `Calendrical.Julian.March25` | 25 March 1100 | `:beginning` |
| `Calendrical.Julian.Sept1` | 1 September 1099 | `:ending` |
| `Calendrical.Julian.Dec25` | 25 December 1099 | `:ending` |

A variant date carries counted months: month 1 begins on the
new-year day and the months follow in the order of time, so a date's
fields order as its days do and `Date.compare/2`,
`Date.beginning_of_month/1` and `days_in_month/2` hold. A variant
whose year begins on the first of a Julian month has twelve counted
months, each a whole Julian month (`March1`'s month 1 is March and
its month 12 February); one whose year begins part-way through a
month (`March25`, `Dec25`) has thirteen, the split month's later days
as a short month 1 and its earlier days as a short month 13: in
`March25`, month 1 of 2024 is 25–31 March 2024, day 1 being 25 March,
and month 13 is 1–24 March 2025. The Julian month and day that name a
date are answered by `cardinal_month/1` and `cardinal_day/3`, and the
Julian date itself by each variant's `base_date/3`.

The module itself is also a fully-functional calendar that can be
used directly (`~D[1500-03-15 Calendrical.Julian]`), in which case
the year begins on January 1.

# `day`

```elixir
@type day() :: 1..31
```

# `month`

```elixir
@type month() :: 1..12
```

# `year`

```elixir
@type year() :: -9999..-1 | 1..9999
```

# `add`

Adds whole years, quarters or months to the fields of a naive
datetime, keeping its time of day: the `add/9` callback
[`CalendarInterval`](https://hexdocs.pm/calendar_interval) steps an
interval of a year or a month with. The date moves as `shift/6`
moves it, clamping the day into a shorter month.

### Arguments

* `year`, `month` and `day` are the date's fields in this calendar.

* `hour`, `minute`, `second` and `microsecond` are the time of day,
  returned unchanged.

* `precision` is `:year`, `:quarter` or `:month`.

* `step` is the number of years, quarters or months to add, which
  may be negative.

### Returns

* `{year, month, day, hour, minute, second, microsecond}`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {year, month, day, 12, 30, 0, {0, 0}} =
    ...>   Calendrical.Julian.add(date.year, date.month, date.day, 12, 30, 0, {0, 0}, :month, 1)
    iex> {year, month, day} == Calendrical.Julian.shift(date.year, date.month, date.day, :months, 1, [])
    true

# `calendar_base`

Returns what this calendar's year is built of, `:month` or `:week`,
as its `info/0` states.

### Returns

* `:week` for a calendar whose year is built of weeks.

* `:month` otherwise.

### Examples

    iex> Calendrical.Julian.calendar_base() in [:month, :week]
    true

# `calendar_from_cldr_calendar_type`

Returns the calendar module whose CLDR calendar type is the one
given, whichever calendar is asked
(`Calendrical.calendar_from_cldr_calendar_type/1`).

### Arguments

* `calendar_type` is a CLDR calendar type as an atom or a string.

### Returns

* `{:ok, calendar_module}`, or

* `{:error, exception}` for a calendar type no calendar module has.

### Examples

    iex> Calendrical.Julian.calendar_from_cldr_calendar_type(:gregorian)
    {:ok, Calendrical.Gregorian}

    iex> {:error, %Localize.UnknownCalendarError{}} =
    ...>   Calendrical.Julian.calendar_from_cldr_calendar_type(:not_a_calendar)

# `cardinal_day`

Returns the day a date is written with: the day of its
`written_date/3`.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The day of the date's `written_date/3`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar
  does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {_year, _month, written_day} = Calendrical.Julian.written_date(date.year, date.month, date.day)
    iex> Calendrical.Julian.cardinal_day(date.year, date.month, date.day) == written_day
    true

# `cardinal_month`

Returns the CLDR month a month of the year names without a year
(`Calendrical.Month.cldr_month/2`).

### Arguments

* `month` is a month of this calendar.

### Returns

* The CLDR month the month is named in the year that holds
  1 January 2000, the month itself in the Hebrew calendar, whose
  names depend on the year, and the month itself where the
  reference year has no such month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2000-01-01], Calendrical.Julian)
    iex> Calendrical.Julian.cardinal_month(1) == Calendrical.Julian.cldr_month(date.year, 1)
    true

# `cardinal_year`

Returns the year a date is written with: the year of its
`written_date/3`.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The year of the date's `written_date/3`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar
  does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {written_year, _month, _day} = Calendrical.Julian.written_date(date.year, date.month, date.day)
    iex> Calendrical.Julian.cardinal_year(date.year, date.month, date.day) == written_year
    true

# `cldr_calendar_type`

Returns the CLDR calendar type whose data names this calendar's
months, days and quarters, as its `info/0` states.

### Returns

* A CLDR calendar type such as `:gregorian` or `:hebrew`.

### Examples

    iex> Calendrical.Julian.cldr_calendar_type() == Calendrical.Julian.info().cldr_calendar_type
    true

# `cldr_calendar_type`

Returns the CLDR calendar type whose data names a date's months and
days: the member in effect on the date in a composite calendar, and
this calendar's own otherwise (`Calendrical.Info.cldr_calendar_type/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* A CLDR calendar type.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> in_effect = Calendrical.Info.in_effect(Calendrical.Julian, date.year, date.month, date.day)
    iex> Calendrical.Julian.cldr_calendar_type(date.year, date.month, date.day) == in_effect.cldr_calendar_type()
    true

# `cldr_month`

Returns the CLDR month of a month of a year
(`Calendrical.Month.cldr_month/3`).

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* A CLDR month number, or `{month, :leap}` where the month's name
  has a leap form, or

* `{:error, exception}` for a month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> cldr_month = Calendrical.Julian.cldr_month(date.year, date.month)
    iex> Calendrical.Julian.written_month_from_cldr_month(date.year, cldr_month) ==
    ...>   Calendrical.Julian.numeric_month(date.year, date.month, date.day)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Julian.cldr_month(date.year, 99)

# `cyclic_year`

Returns the cyclic year of a date (`Calendrical.Era.cyclic_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The date's place in the sexagenary cycle, `1` to `60`, in an East
  Asian calendar, and its year in a calendar whose years run in no
  such cycle, or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> cyclic_year = Calendrical.Julian.cyclic_year(date.year, date.month, date.day)
    iex> cyclic_year in 1..60 or cyclic_year == date.year
    true

    iex> {:error, %Calendrical.MissingFieldsError{}} = Calendrical.Julian.cyclic_year(nil, 1, 1)

# `date_from_cardinal_date`

Returns the date of this calendar that written figures name
(`date_from_written_date/3`), as fields.

### Arguments

* `year`, `month` and `day` are the figures a date is written with,
  as `written_date/3` returns them.

### Returns

* `{year, month, day}`, the date of this calendar, or

* `{:error, %Calendrical.InvalidDateError{}}` where the figures name no date.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {year, month, day} = Calendrical.Julian.written_date(date.year, date.month, date.day)
    iex> Calendrical.Julian.date_from_cardinal_date(year, month, day) == {date.year, date.month, date.day}
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.date_from_cardinal_date(date.year, 0, 1)

# `date_from_day_of_year`

Returns the date of a day of a year, counted from the year's first
day.

### Arguments

* `year` is a year of this calendar.

* `day_of_year` is the day of the year, counted from 1.

### Returns

* A `t:Date.t/0` of this calendar, or

* `{:error, %Calendrical.InvalidDateError{}}` for a day the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> day_of_year = Calendrical.Julian.day_of_year(date.year, date.month, date.day)
    iex> Calendrical.Julian.date_from_day_of_year(date.year, day_of_year) == date
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.date_from_day_of_year(date.year, 1000)

# `date_from_iso_days`

```elixir
@spec date_from_iso_days(integer()) :: {year(), month(), day()}
```

Returns a Julian `{year, month, day}` tuple for the given ISO day
number.

### Arguments

* `iso_days` is an integer count of days since the proleptic
  ISO epoch.

### Returns

* A three-tuple `{year, month, day}` in the Julian calendar.

### Examples

    iex> Calendrical.Julian.date_from_iso_days(740_000)
    {2026, 1, 6}

# `date_from_week_date`

Returns the date of a day of a week of a week-year
(`Calendrical.Week.date_from_week_date/4`).

### Arguments

* `week_year` is a week-year of this calendar.

* `week` is a week of the week-year.

* `day` is the day of the week, numbered from 1 for the day the
  calendar's weeks begin on.

### Returns

* A `t:Date.t/0` of this calendar, or

* `{:error, %Calendrical.InvalidDateError{}}` where the week has no such day.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {week_year, week, day} = Calendrical.Julian.week_date_from_date(date.year, date.month, date.day)
    iex> Calendrical.Julian.date_from_week_date(week_year, week, day) == date
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.date_from_week_date(date.year, 1, 8)

# `date_from_written_date`

Returns the date of this calendar that written figures name: the
inverse of `written_date/3`, the fields themselves where they are a
date.

### Arguments

* `year`, `month` and `day` are the figures a date is written with,
  as `written_date/3` returns them.

### Returns

* `{:ok, {year, month, day}}`, the date of this calendar, or

* `{:error, :invalid_date}` where the figures name no date.

The reason is an atom, not an exception: Localize's date parser reads
it, as `c:Calendrical.date_from_written_date/3` sets out.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {year, month, day} = Calendrical.Julian.written_date(date.year, date.month, date.day)
    iex> Calendrical.Julian.date_from_written_date(year, month, day) == {:ok, {date.year, date.month, date.day}}
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.date_from_written_date(date.year, 0, 1)
    {:error, :invalid_date}

# `date_to_iso_days`

```elixir
@spec date_to_iso_days(year(), month(), day()) :: integer()
```

Returns the number of ISO days for the given Julian `year`,
`month`, and `day`.

### Arguments

* `year` is any non-zero Julian year as an integer.

* `month` is a month in the range `1..12`.

* `day` is a day-of-month.

### Returns

* An integer count of days since the proleptic ISO epoch.

### Examples

    iex> Calendrical.Julian.date_to_iso_days(2025, 1, 1)
    739630

# `dates_in_gregorian_year`

Returns the dates of this calendar, earliest first, whose month and
day are those given and that fall in a Gregorian year.

### Arguments

* `gregorian_year` is the Gregorian year to search within.

* `month` and `day` are a month and a day of this calendar.

### Returns

* A list of zero, one or two `t:Date.t/0` of this calendar, earliest
  first, since this calendar's years and the Gregorian years drift.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> date in Calendrical.Julian.dates_in_gregorian_year(2024, date.month, date.day)
    true

# `day_numbers`

Returns the days a month of a year has, as runs of their numbers
(`Calendrical.Month.day_numbers/3`).

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* A list of ranges: one from 1 in every month of every calendar but
  a composite, whose month a change of calendar falls in has the
  days that are left, and `[]` for a month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.day_numbers(date.year, date.month) ==
    ...>   [1..Calendrical.Julian.days_in_month(date.year, date.month)//1]
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.day_numbers(date.year, 99)
    []

# `day_of_era`

```elixir
@spec day_of_era(year(), month(), day()) :: {day :: pos_integer(), era :: 0..1}
```

Returns the day-of-era and era for the given Julian date.

### Arguments

* `year` is any non-zero Julian year as an integer.

* `month` is a month in the range `1..12`.

* `day` is a day-of-month.

### Returns

* A two-tuple `{day_in_era, era}` where `era` is `0` or `1`.

### Examples

    iex> Calendrical.Julian.day_of_era(2025, 1, 1)
    {739267, 1}

    iex> Calendrical.Julian.day_of_era(1, 1, 1)
    {1, 1}

# `day_of_week`

Returns the day of the week of a date (`Calendrical.Week.day_of_week/5`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

* `starting_on` is `:default` for the day this calendar's weeks
  begin on, or the name of a weekday, `:monday` to `:sunday`.

### Returns

* `{day_of_week, 1, 7}`, the day numbered from the day the week
  begins on, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for a
  missing field.

### Examples

    # 1 June 2024 is a Saturday
    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.day_of_week(date.year, date.month, date.day, :monday)
    {6, 1, 7}

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.day_of_week(date.year, date.month, 0, :monday)

# `day_of_year`

Returns the day of the year of a date (`Calendrical.Period.day_of_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The day of the year, counted from 1 for its first day, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for a
  missing field.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> first_day = Calendrical.Julian.year(date.year).first
    iex> Calendrical.Julian.day_of_year(date.year, date.month, date.day) == Date.diff(date, first_day) + 1
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.day_of_year(date.year, date.month, 0)

# `days_in_month`

Returns the number of days in a month without a year
(`Calendrical.Month.days_in_month/2`).

### Arguments

* `month` is a month of this calendar.

### Returns

* The number of days where the month has as many in every year, or

* `{:ambiguous, lengths}` where its length depends on the year, the
  lengths a range or, where they are not consecutive, a list, or

* `{:error, %Calendrical.NotDefinedError{}}` for a month the calendar has in no year, or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, where
  the month is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {fewest, most} = Calendrical.Julian.days_in_month_bounds(date.month)
    iex> case Calendrical.Julian.days_in_month(date.month) do
    ...>   days when is_integer(days) -> fewest == days and most == days
    ...>   {:ambiguous, lengths} -> Enum.min(lengths) == fewest and Enum.max(lengths) == most
    ...> end
    true

    iex> {:error, %Calendrical.NotDefinedError{}} = Calendrical.Julian.days_in_month(99)

# `days_in_month`

```elixir
@spec days_in_month(year(), month()) :: 28..31 | {:error, Exception.t()}
```

Returns the number of days in the given Julian `year` and `month`.

### Arguments

* `year` is any non-zero Julian year as an integer.

* `month` is a month in the range `1..12`.

### Returns

* An integer in the range `28..31`, or

* `{:error, %Calendrical.InvalidMonthError{}}` for a month the year
  does not have, or

* `{:error, %Calendrical.MissingFieldsError{}}` where the year or the
  month is `nil`.

### Examples

    iex> Calendrical.Julian.days_in_month(2024, 2)
    29

    iex> Calendrical.Julian.days_in_month(2025, 2)
    28

# `days_in_month_bounds`

Returns the fewest and the most days a month has in any year
(`Calendrical.Month.days_in_month_bounds/2`).

### Arguments

* `month` is a month of this calendar.

### Returns

* `{fewest, most}`, or

* `{:error, exception}` for a month the calendar has in no year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {fewest, most} = Calendrical.Julian.days_in_month_bounds(date.month)
    iex> Calendrical.Julian.days_in_month(date.year, date.month) in fewest..most
    true

    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Julian.days_in_month_bounds(99)

# `days_in_week`

Returns the number of days in a week.

### Returns

* `7`.

### Examples

    iex> Calendrical.Julian.days_in_week()
    7

# `days_in_year`

Returns the number of days in a year (`Calendrical.Period.days_in_year/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* The number of days, or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.days_in_year(date.year) == Enum.count(Calendrical.Julian.year(date.year))
    true

    iex> {:error, %Calendrical.MissingFieldsError{}} = Calendrical.Julian.days_in_year(nil)

# `diff`

```elixir
@spec diff({year(), month(), day()}, {year(), month(), day()}, atom()) ::
  integer() | {:error, Exception.t()}
```

Returns the whole number of `date_part`s from one `{year, month, day}`
to another, as `shift/6` adds them.

`date_part` can be `:years`, `:quarters`, `:months`, `:weeks` or
`:days`. The count is the largest number `shift/6`, clamping the day
into a shorter month, can add to the earlier date without passing the
later one; it is negative when `to` is before `from`. There is no year
zero, so year -1 to year 1 is one year.

### Arguments

* `from` is the starting `{year, month, day}`.

* `to` is the ending `{year, month, day}`.

* `date_part` is `:years`, `:quarters`, `:months`, `:weeks` or
  `:days`.

### Returns

* The whole number of `date_part`s from `from` to `to`, as an
  integer, or

* `{:error, %Calendrical.InvalidDateError{}}` for a `from` or `to` the
  calendar does not have, `{:error, %Calendrical.MissingFieldsError{}}`
  for one missing a field, or `{:error, %Calendrical.InvalidPartError{}}`
  for a `date_part` other than the five.

### Examples

    iex> Calendrical.Julian.diff({1582, 10, 4}, {1583, 10, 4}, :months)
    12

    iex> Calendrical.Julian.diff({-1, 6, 1}, {1, 6, 1}, :years)
    1

# `era_calendar_type`

Returns the CLDR calendar type that names this calendar's eras, as
its `info/0` states.

### Returns

* A CLDR calendar type such as `:gregorian` or `:japanese`.

### Examples

    iex> Calendrical.Julian.era_calendar_type() == Calendrical.Julian.info().era_calendar_type
    true

# `extended_year`

Returns the extended year of a date (`Calendrical.Era.extended_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The year numbered without a gap at 0, so that in a calendar with
  no year 0 the year before 1 is 0, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have, or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.extended_year(date.year, date.month, date.day) == date.year
    true

    iex> {:error, %Calendrical.MissingFieldsError{}} = Calendrical.Julian.extended_year(nil, 1, 1)

# `info`

Returns the static facts about this calendar, as
`t:Calendrical.Info.t/0`: the Julian calendar has no year 0.

### Returns

* A `t:Calendrical.Info.t/0` struct.

### Examples

    iex> Calendrical.Julian.info().year_zero?
    false

# `iso_week_of_year`

Returns the ISO 8601 week-year and week of the day a date names
(`Calendrical.Week.iso_week_of_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{week_year, week}` of ISO 8601, or

* `{:error, exception}` for a date the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.iso_week_of_year(date.year, date.month, date.day)
    {2024, 22}

# `leap_month`

Returns the position of a year's leap month, or `nil` where the
year has none.

### Arguments

* `year` is a year of this calendar.

### Returns

* The leap month's position in the year, or

* `nil` where the year has no leap month, as for anything but an
  integer year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> leap_month = Calendrical.Julian.leap_month(date.year)
    iex> is_nil(leap_month) or leap_month in 1..Calendrical.Julian.months_in_year(date.year)
    true

# `leap_year?`

```elixir
@spec leap_year?(year :: Calendar.year()) :: boolean()
```

Returns whether the given Julian `year` is a leap year.

Julian leap years occur every four years (every year divisible
by 4 — unlike the Gregorian rule, the centurial exceptions do
not apply).

### Arguments

* `year` is any non-zero Julian year as an integer.

### Returns

* `true` if the year contains 366 days; otherwise `false`.

### Examples

    iex> Calendrical.Julian.leap_year?(2024)
    true

    iex> Calendrical.Julian.leap_year?(2025)
    false

    iex> Calendrical.Julian.leap_year?(2100)
    true

# `lunar_month_of_year`

Returns the traditional month at a position in a year
(`Calendrical.Month.traditional_month/3`), or an error for a place
the year does not have.

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* The month's traditional number, or `{month, :leap}` for a leap
  month, or

* `{:error, %Calendrical.InvalidMonthError{}}` for a month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.lunar_month_of_year(date.year, date.month) ==
    ...>   Calendrical.Julian.traditional_month(date.year, date.month)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Julian.lunar_month_of_year(date.year, 99)

# `month`

Returns the days of a month of a year (`Calendrical.Period.month/3`).

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* A `t:Date.Range.t/0`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a month the year does not have, or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for a
  missing year or month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> date in Calendrical.Julian.month(date.year, date.month)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.month(date.year, 99)

# `month_from_traditional`

Returns the position of a traditional month in a year
(`Calendrical.Month.month_from_traditional/3`).

### Arguments

* `year` is a year of this calendar.

* `traditional_month` is a month's traditional number, or
  `{month, :leap}` for a leap month.

### Returns

* The month's position in the year, or

* `{:error, exception}` where the year has no such month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> traditional_month = Calendrical.Julian.traditional_month(date.year, date.month)
    iex> Calendrical.Julian.month_from_traditional(date.year, traditional_month) == date.month
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Julian.month_from_traditional(date.year, 99)

# `month_numbers`

Returns the months a year has, as runs of their numbers
(`Calendrical.Month.month_numbers/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* A list of ranges: one from 1 in every year of every calendar but a
  composite, whose year a change of calendar falls in has the months
  that are left, and `[]` for a year the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.month_numbers(date.year) == [1..Calendrical.Julian.months_in_year(date.year)//1]
    true

# `month_of_year`

Returns the CLDR month of a month of a year: the index of its name
in this calendar's CLDR data, with `{month, :leap}` where the name
has a leap form (`Calendrical.Month.cldr_month/3`).

### Arguments

* `year`, `month` and `day` are a date of this calendar. The year
  and the day may be `nil`: a month's name needs only the month.

### Returns

* A CLDR month number, or `{month, :leap}`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a day the month does not have, or

* `{:error, exception}` for a month the year does not have, or a
  `Calendrical.MissingFieldsError` where the month is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.month_of_year(date.year, date.month, date.day) == Calendrical.Julian.cldr_month(date.year, date.month)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.month_of_year(date.year, date.month, 0)

# `months_in_leap_year`

Returns the number of months in a leap year: the most months a year
of this calendar has.

### Returns

* A positive integer.

### Examples

    iex> Calendrical.Julian.months_in_leap_year() >= Calendrical.Julian.months_in_ordinary_year()
    true

# `months_in_ordinary_year`

Returns the number of months in an ordinary year: the fewest months
a year of this calendar has.

### Returns

* A positive integer.

### Examples

    iex> Calendrical.Julian.months_in_ordinary_year() <= Calendrical.Julian.months_in_leap_year()
    true

# `months_in_year`

Returns the number of months in a year without a year: the count
where every year has as many, `{:ambiguous, range}` where a leap
year has more, and `{:error, %Calendrical.NotDefinedError{}}` for a composite calendar
(`Calendrical.Info.months_in_year/1`).

### Returns

* The number of months where every year has as many, or

* `{:ambiguous, fewest..most}` where a leap year has more, or

* `{:error, %Calendrical.NotDefinedError{}}` for a composite calendar, whose bounds
  `months_in_year_bounds/0` gives.

### Examples

    iex> case Calendrical.Julian.months_in_year() do
    ...>   months when is_integer(months) -> months == Calendrical.Julian.months_in_leap_year()
    ...>   {:ambiguous, fewest..most//1} -> {fewest, most} == Calendrical.Julian.info().months_in_year
    ...>   {:error, %Calendrical.NotDefinedError{}} -> is_list(Calendrical.Julian.info().members)
    ...> end
    true

# `months_in_year_bounds`

Returns the fewest and the most months a year has
(`Calendrical.Month.months_in_year_bounds/1`).

### Returns

* `{fewest, most}`, which in a composite calendar counts the months
  left in a year a change of calendar falls in.

### Examples

    iex> {fewest, most} = Calendrical.Julian.months_in_year_bounds()
    iex> fewest <= Calendrical.Julian.months_in_ordinary_year() and
    ...>   most >= Calendrical.Julian.months_in_leap_year()
    true

# `naive_datetime_from_iso_days`

Converts ISO days and a day fraction to a naive datetime of this
calendar, as `c:Calendar.naive_datetime_from_iso_days/1` asks.

### Arguments

* `iso_days` is `{days, {parts_in_day, parts_per_day}}`, a day and a
  fraction of it.

### Returns

* `{year, month, day, hour, minute, second, microsecond}` of this
  calendar.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {year, month, day, 12, 0, 0, {0, 6}} =
    ...>   Calendrical.Julian.naive_datetime_from_iso_days({739403, {43200000000, 86400000000}})
    iex> {year, month, day} == {date.year, date.month, date.day}
    true

# `naive_datetime_to_iso_days`

Converts a naive datetime of this calendar to ISO days and a day
fraction, as `c:Calendar.naive_datetime_to_iso_days/7` asks.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

* `hour`, `minute`, `second` and `microsecond` are a time of day,
  the microsecond as `{microseconds, precision}`.

### Returns

* `{iso_days, {parts_in_day, parts_per_day}}`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.naive_datetime_to_iso_days(date.year, date.month, date.day, 12, 0, 0, {0, 6})
    {739403, {43200000000, 86400000000}}

# `named_month`

Returns the days of a named month in a year
(`Calendrical.Month.named_month/3`).

### Arguments

* `year` is a year of this calendar.

* `named_month` is the CLDR month that names the month.

### Returns

* A list of `t:Date.Range.t/0`, one for each counted month that
  carries the name, earliest first, and `[]` where none does.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> cldr_month = Calendrical.Julian.cldr_month(date.year, date.month)
    iex> Calendrical.Julian.named_month(date.year, cldr_month) |> Enum.any?(&(date in &1))
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.named_month(date.year, 99)
    []

# `numeric_month`

Returns the number a date's month is written with in figures: the
month of its `written_date/3`.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The month of the date's `written_date/3`: a month number, or
  `{month, :leap}` for a leap month of a lunisolar calendar, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar
  does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {_year, written_month, _day} = Calendrical.Julian.written_date(date.year, date.month, date.day)
    iex> Calendrical.Julian.numeric_month(date.year, date.month, date.day) == written_month
    true

# `ordinal_month_from_traditional`

Returns the position in a year of a traditional month
(`Calendrical.Month.month_from_traditional/3`), as `{:ok, month}`.

### Arguments

* `year` is a year of this calendar.

* `traditional_month` is a month's traditional number, or
  `{month, :leap}` for a leap month.

### Returns

* `{:ok, month}`, the month's position in the year, or

* `{:error, %Calendrical.InvalidMonthError{}}` for a month or a leap
  month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> traditional_month = Calendrical.Julian.traditional_month(date.year, date.month)
    iex> Calendrical.Julian.ordinal_month_from_traditional(date.year, traditional_month) == {:ok, date.month}
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Julian.ordinal_month_from_traditional(date.year, 99)

# `parse_date`

Parses a date written in this calendar's own notation.

### Arguments

* `string` is a date written `YYYY-MM-DD`, with a leading `-` for a
  year before 1, or, in a calendar of weeks, `YYYY-Www-D`.

### Returns

* `{:ok, {year, month, day}}`, or

* `{:error, :invalid_format}` where the text is not in that form or
  its fields are not digits, or

* `{:error, :invalid_date}` where the text names no date of this
  calendar.

The reason is an atom, not an exception, because `c:Calendar.parse_date/1`
requires one: `Date.from_iso8601/2` reads it.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.parse_date(Date.to_string(date)) == {:ok, {date.year, date.month, date.day}}
    true

    iex> Calendrical.Julian.parse_date("not a date")
    {:error, :invalid_format}

# `parse_naive_datetime`

Parses a naive datetime written in this calendar's own notation.

### Arguments

* `string` is a date as `parse_date/1` reads it, then `T` or a
  space and a time of day.

### Returns

* `{:ok, {year, month, day, hour, minute, second, microsecond}}`, or

* `{:error, reason}`, such as `:invalid_format` or `:invalid_date`,
  where the text is no such datetime.

The reason is an atom, not an exception, because
`c:Calendar.parse_naive_datetime/1` requires one:
`NaiveDateTime.from_iso8601/2` reads it.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.parse_naive_datetime(Date.to_string(date) <> "T12:00:00") ==
    ...>   {:ok, {date.year, date.month, date.day, 12, 0, 0, {0, 0}}}
    true

    iex> Calendrical.Julian.parse_naive_datetime("2024-06-01")
    {:error, :invalid_format}

# `parse_utc_datetime`

Parses a UTC datetime written in this calendar's own notation.

### Arguments

* `string` is a date as `parse_date/1` reads it, then `T` or a
  space, a time of day and an offset from UTC, `Z` or `±HH:MM`.

### Returns

* `{:ok, {year, month, day, hour, minute, second, microsecond}, offset}`,
  the datetime in UTC and the offset in seconds it was written with,
  or

* `{:error, reason}`, such as `:invalid_format`, `:invalid_date` or
  `:missing_offset`, where the text is no such datetime.

The reason is an atom, not an exception, because
`c:Calendar.parse_utc_datetime/1` requires one:
`DateTime.from_iso8601/2` reads it.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.parse_utc_datetime(Date.to_string(date) <> "T12:00:00Z") ==
    ...>   {:ok, {date.year, date.month, date.day, 12, 0, 0, {0, 0}}, 0}
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.parse_utc_datetime(Date.to_string(date) <> "T12:00:00")
    {:error, :missing_offset}

# `parsing_calendar`

Returns the calendar a date written for this calendar is parsed in,
as its `info/0` states.

### Returns

* A calendar module.

### Examples

    iex> Calendrical.Julian.parsing_calendar() == Calendrical.Julian.info().parsing_calendar
    true

# `parsing_calendars`

Returns the calendars a date written for this calendar may be
written in: the members of a composite calendar, and this calendar
alone otherwise (`Calendrical.Info.parsing_calendars/1`).

### Returns

* A non-empty list of calendar modules, the members of a composite
  calendar in the order they take effect.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> in_effect = Calendrical.Info.in_effect(Calendrical.Julian, date.year, date.month, date.day)
    iex> in_effect in Calendrical.Julian.parsing_calendars()
    true

# `periods_in_year`

Returns the number of periods in a year: its months.

### Arguments

* `year` is a year of this calendar.

### Returns

* The number of months in the year, as `months_in_year/1` returns
  it.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.periods_in_year(date.year) == Calendrical.Julian.months_in_year(date.year)
    true

# `quadrimester`

Returns the days of a quadrimester of a year
(`Calendrical.Period.quadrimester/3`).

### Arguments

* `year` is a year of this calendar.

* `quadrimester` is the quadrimester, `1` to `3`.

### Returns

* A `t:Date.Range.t/0`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a quadrimester or a year the
  calendar does not have, or

* `{:error, %Calendrical.MissingFieldsError{}}` where the year or the
  quadrimester is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.quadrimester(date.year, 1).first == Calendrical.Julian.year(date.year).first
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.quadrimester(date.year, 4)

# `quarter`

Returns the days of a quarter of a year (`Calendrical.Period.quarter/3`).

### Arguments

* `year` is a year of this calendar.

* `quarter` is the quarter, `1` to `4`.

### Returns

* A `t:Date.Range.t/0`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a quarter or a year the calendar
  does not have, or

* `{:error, %Calendrical.MissingFieldsError{}}` where the year or the
  quarter is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> quarter = Calendrical.Julian.quarter_of_year(date.year, date.month, date.day)
    iex> date in Calendrical.Julian.quarter(date.year, quarter)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.quarter(date.year, 5)

# `quarter_of_year`

Returns the quarter of the year a date falls in
(`Calendrical.Period.quarter_of_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The quarter, `1` to `4`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for a
  missing year or month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.quarter_of_year(date.year, date.month, date.day) in 1..4
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.quarter_of_year(date.year, date.month, 0)

# `related_gregorian_year`

Returns the related Gregorian year of a date
(`Calendrical.Era.related_gregorian_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The Gregorian year the date's calendar year begins in, or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:ok, first_day} = Date.convert(Calendrical.Julian.year(date.year).first, Calendrical.Gregorian)
    iex> Calendrical.Julian.related_gregorian_year(date.year, date.month, date.day) == first_day.year
    true

# `semester`

Returns the days of a semester of a year (`Calendrical.Period.semester/3`).

### Arguments

* `year` is a year of this calendar.

* `semester` is the semester, `1` or `2`.

### Returns

* A `t:Date.Range.t/0`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a semester or a year the calendar
  does not have, or

* `{:error, %Calendrical.MissingFieldsError{}}` where the year or the
  semester is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.semester(date.year, 1).first == Calendrical.Julian.year(date.year).first
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.semester(date.year, 3)

# `shift`

```elixir
@spec shift(
  year(),
  month(),
  day(),
  :years | :quarters | :months | :weeks | :days,
  integer(),
  Keyword.t()
) ::
  {Calendar.year(), Calendar.month(), Calendar.day()} | {:error, Exception.t()}
```

Adds an `increment` of `date_part`s to a Julian `year-month-day`.

### Arguments

* `year`, `month`, `day` form the Julian date to shift.

* `date_part` is one of `:years`, `:quarters`, `:months`,
  `:weeks`, or `:days`.

* `increment` is the integer number of `date_part`s to add (may
  be negative).

### Options

* `:clamp` (boolean, default `true`) — clamps the resulting
  day-of-month to the last valid day of the new month if the
  original day overflows, so adding one month to 31 March yields
  30 April. With `clamp: false` the day is kept, which can name a
  day the month lacks.

### Returns

* A three-tuple `{year, month, day}` representing the shifted
  Julian date, or

* `{:error, %Calendrical.InvalidDateError{}}` where `year`, `month` and
  `day` are no date of this calendar, or

* `{:error, %Calendrical.MissingFieldsError{}}` where a field is `nil`.

### Examples

    iex> Calendrical.Julian.shift(2025, 1, 1, :years, 1)
    {2026, 1, 1}

    iex> Calendrical.Julian.shift(2025, 1, 1, :months, 1)
    {2025, 2, 1}

    iex> Calendrical.Julian.shift(2025, 1, 1, :weeks, 1)
    {2025, 1, 8}

    iex> Calendrical.Julian.shift(2025, 1, 1, :days, 1)
    {2025, 1, 2}

    iex> Calendrical.Julian.shift(1936, 1, 31, :months, 1)
    {1936, 2, 29}

# `shift_date`

Shifts a date by a duration: its years and months together, as
`shift/6` adds months, and then its weeks and days.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

* `duration` is a `t:Duration.t/0`.

### Returns

* `{year, month, day}` of this calendar.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {year, month, day} = Calendrical.Julian.shift_date(date.year, date.month, date.day, Duration.new!(day: 7))
    iex> Date.new!(year, month, day, Calendrical.Julian) == Date.add(date, 7)
    true

# `shift_naive_datetime`

Shifts a naive datetime by a duration: its date as `shift_date/4`
shifts it, and its time as `Calendar.ISO` does.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

* `hour`, `minute`, `second` and `microsecond` are a time of day,
  the microsecond as `{microseconds, precision}`.

* `duration` is a `t:Duration.t/0`.

### Returns

* `{year, month, day, hour, minute, second, microsecond}` of this
  calendar.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> next_day = Date.add(date, 1)
    iex> Calendrical.Julian.shift_naive_datetime(date.year, date.month, date.day, 23, 0, 0, {0, 0}, Duration.new!(hour: 2)) ==
    ...>   {next_day.year, next_day.month, next_day.day, 1, 0, 0, {0, 0}}
    true

# `shift_time`

Shifts a time by a duration, as `Calendar.ISO` does.

### Arguments

* `hour`, `minute`, `second` and `microsecond` are a time of day,
  the microsecond as `{microseconds, precision}`.

* `duration` is a `t:Duration.t/0` of time units.

### Returns

* `{hour, minute, second, microsecond}`, wrapped around midnight.

### Examples

    iex> Calendrical.Julian.shift_time(23, 0, 0, {0, 0}, Duration.new!(hour: 2))
    {1, 0, 0, {0, 0}}

# `solar_term`

Returns the day of a solar term of a Gregorian year, reckoned at the
place this calendar's `info/0` names, or at the Chinese calendar's
meridian, the traditional reference, where it names none.

### Arguments

* `index` is the solar term's number, `1` (lichun, 315° of solar
  longitude) to `24`, every 15° on.

* `gregorian_year` is a Gregorian year.

### Returns

* `{:ok, date}`, a `t:Date.t/0` in `Calendrical.Gregorian`, or

* `{:error, %Calendrical.InvalidSolarTermError{}}` where `index` is not
  `1` to `24`.

### Examples

    # Qingming, the fifth solar term
    iex> Calendrical.Julian.solar_term(5, 2025)
    {:ok, ~D[2025-04-04 Calendrical.Gregorian]}

    iex> Calendrical.Julian.solar_term(25, 2025)
    {:error, %Calendrical.InvalidSolarTermError{index: 25}}

# `traditional_leap_month`

Returns the traditional number of the month a year's leap month
follows, or `nil` (`Calendrical.Month.traditional_leap_month/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* The traditional number of the month the leap month follows, or

* `nil` where the year has no leap month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> is_nil(Calendrical.Julian.traditional_leap_month(date.year)) ==
    ...>   is_nil(Calendrical.Julian.leap_month(date.year))
    true

# `traditional_month`

Returns a month of a year as this calendar names it
(`Calendrical.Month.traditional_month/3`).

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* The month's traditional number, or `{month, :leap}` for a leap
  month, or

* `{:error, exception}` for a month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.traditional_month(date.year, date.month) ==
    ...>   Enum.at(Calendrical.Julian.traditional_months(date.year), date.month - 1)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Julian.traditional_month(date.year, 99)

# `traditional_months`

Returns a year's months in order, named traditionally
(`Calendrical.Month.traditional_months/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* A list of traditional months, the nth naming the year's nth month,
  with `{month, :leap}` for a leap month, and `[]` for a year the
  calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> length(Calendrical.Julian.traditional_months(date.year)) ==
    ...>   Calendrical.Julian.months_in_year(date.year)
    true

# `valid_date?`

```elixir
@spec valid_date?(Calendar.year(), Calendar.month(), Calendar.day()) :: boolean()
```

Returns whether the supplied `year`, `month`, and `day` is a valid
Julian date.

### Arguments

* `year` is any Julian year as an integer. The year `0` is not
  valid (the Julian calendar has no year zero).

* `month` is a month in the range `1..12`.

* `day` is a day-of-month appropriate for the month.

### Returns

* `true` if the date is valid; otherwise `false`.

### Examples

    iex> Calendrical.Julian.valid_date?(2024, 2, 29)
    true

    iex> Calendrical.Julian.valid_date?(2023, 2, 29)
    false

    iex> Calendrical.Julian.valid_date?(0, 1, 1)
    false

# `week`

Returns the seven days of a week of a year (`Calendrical.Week.week/3`).

### Arguments

* `year` is a year of this calendar.

* `week` is a week of the year.

### Returns

* A `t:Date.Range.t/0` of the week's seven days, which may begin in
  the year before or end in the year after, or

* `{:error, %Calendrical.InvalidDateError{}}` for a week the year does not have, or

* `{:error, %Calendrical.MissingFieldsError{}}` where the year or the
  week is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {week_year, week} = Calendrical.Julian.week_of_year(date.year, date.month, date.day)
    iex> days = Calendrical.Julian.week(week_year, week)
    iex> {Enum.count(days), date in days}
    {7, true}

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.week(date.year, 0)

# `week_date_from_date`

Returns the week-year, the week and the day of the week of a date
(`Calendrical.Week.week_date_from_date/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{week_year, week, day_of_week}`, the day numbered from the day
  the calendar's weeks begin on, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {week_year, week, day} = Calendrical.Julian.week_date_from_date(date.year, date.month, date.day)
    iex> {week_year, week} == Calendrical.Julian.week_of_year(date.year, date.month, date.day) and day in 1..7
    true

# `week_of_month`

Returns the month and the week of the month a date is in
(`Calendrical.Week.week_of_month/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{month, week}`, where the month is the one before or after where
  the date falls in that month's last or first week, or

* `{:error, exception}` for a date the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {month, week} = Calendrical.Julian.week_of_month(date.year, date.month, date.day)
    iex> month in (date.month - 1)..(date.month + 1) and week >= 1
    true

# `week_of_year`

Returns the week-year and the week of the year a date is in
(`Calendrical.Week.week_of_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{week_year, week}`, where the week-year is the year before or
  after where the date falls in that year's last or first week, or

* `{:error, exception}` for a date the calendar does not have: a
  `Calendrical.MissingFieldsError` for a date missing a field, and a
  `Calendrical.InvalidDateError` otherwise.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {week_year, week} = Calendrical.Julian.week_of_year(date.year, date.month, date.day)
    iex> date in Calendrical.Julian.week(week_year, week)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Julian.week_of_year(date.year, date.month, 0)

# `weeks_in_week_year`

Returns the number of weeks in a week-year
(`Calendrical.Week.weeks_in_week_year/2`).

### Arguments

* `week_year` is a week-year of this calendar.

### Returns

* The number of weeks, or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {week_year, week, _day} = Calendrical.Julian.week_date_from_date(date.year, date.month, date.day)
    iex> week <= Calendrical.Julian.weeks_in_week_year(week_year)
    true

# `weeks_in_year`

Returns the number of weeks in a year (`Calendrical.Week.weeks_in_year/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* The number of weeks, or

* `{:error, exception}` for a year the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {week_year, week} = Calendrical.Julian.week_of_year(date.year, date.month, date.day)
    iex> week <= Calendrical.Julian.weeks_in_year(week_year)
    true

# `weeks_of_month`

Returns the weeks of a month of a year in order
(`Calendrical.Week.weeks_of_month/3`).

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* A list of `t:Date.Range.t/0`, one for each week `week_of_month/3`
  names for the month, the first of which may begin in the month
  before and the last run into the month after, or

* `{:error, exception}` for a month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.weeks_of_month(date.year, date.month) |> Enum.all?(&(Enum.count(&1) == 7))
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Julian.weeks_of_month(date.year, 99)

# `written_date`

Returns the figures a date is written with: its own fields, in a
calendar whose dates are written as its own.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{year, month, day}` as the date is written, where the month of a
  lunisolar calendar is its traditional number, `{month, :leap}`
  for a leap month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {year, month, day} = Calendrical.Julian.written_date(date.year, date.month, date.day)
    iex> Calendrical.Julian.date_from_written_date(year, month, day) == {:ok, {date.year, date.month, date.day}}
    true

# `written_month_from_cldr_month`

Returns the month a CLDR month is written with in a year
(`Calendrical.Month.written_month_from_cldr_month/3`).

### Arguments

* `year` is a year of this calendar.

* `cldr_month` is a CLDR month number, or `{month, :leap}`.

### Returns

* The month as `written_date/3` writes it, or

* `{:error, :invalid_month}` where the year has no month of that
  name. The reason is an atom, not an exception: Localize's date parser
  reads it, where `Calendrical.Month.written_month_from_cldr_month/3`
  answers with an exception.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> {_year, written_month, _day} = Calendrical.Julian.written_date(date.year, date.month, date.day)
    iex> cldr_month = Calendrical.Julian.cldr_month(date.year, date.month)
    iex> Calendrical.Julian.written_month_from_cldr_month(date.year, cldr_month) == written_month
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.written_month_from_cldr_month(date.year, 99)
    {:error, :invalid_month}

# `year`

Returns the days of a year (`Calendrical.Period.year/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* A `t:Date.Range.t/0`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for
  no year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> date in Calendrical.Julian.year(date.year)
    true

    iex> {:error, %Calendrical.MissingFieldsError{}} = Calendrical.Julian.year(nil)

# `year_of_era`

Returns the year of era and era of a year (`Calendrical.Era.year_of_era/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* `{year_of_era, era}`, the era in effect on the year's first day,
  or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> Calendrical.Julian.year_of_era(date.year) ==
    ...>   Calendrical.Julian.year_of_era(date.year, date.month, date.day)
    true

# `year_of_era`

```elixir
@spec year_of_era(year(), month(), day()) ::
  {year :: Calendar.year(), era :: 0..1} | Calendrical.date_error()
```

Returns the year and era for the Julian date given by `year`,
`month`, and `day`.

### Arguments

* `year` is any non-zero Julian year as an integer.

* `month` is a month in the range `1..12`.

* `day` is a day-of-month.

### Returns

* A two-tuple `{year_in_era, era}` where `era` is `0` or `1`.

* `{:error, %Calendrical.MissingFieldsError{}}` when `year` is not
  an integer, as for a partial date with no year.

### Examples

    iex> Calendrical.Julian.year_of_era(2025, 1, 1)
    {2025, 1}

    iex> Calendrical.Julian.year_of_era(-50, 1, 1)
    {50, 0}

# `years_in_cycle`

Returns the number of years after which this calendar's years
repeat, as its `info/0` states, or `{:error, %Calendrical.NotDefinedError{}}`.

### Returns

* The number of years in the cycle, or

* `{:error, %Calendrical.NotDefinedError{}}` where the calendar states none.

### Examples

    # A year has as many days as the year one cycle later
    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Julian)
    iex> case Calendrical.Julian.years_in_cycle() do
    ...>   {:error, %Calendrical.NotDefinedError{}} -> true
    ...>   years -> Calendrical.Julian.days_in_year(date.year) == Calendrical.Julian.days_in_year(date.year + years)
    ...> end
    true

---

*Consult [api-reference.md](api-reference.md) for complete listing*