11institutetext: N. Karjanto 22institutetext: Department of Mathematics,
University College, Sungkyunkwan University, Natural Science Campus, 2066
Seobu-ro, Suwon 16419, Republic of Korea 22email<EMAIL_ADDRESS>33institutetext: F. Beauducel 44institutetext: Université de Paris, Institut
de physique du globe de Paris, CNRS, 75005 Paris, France
Institut de recherche pour le développement, Research and Development
Technology Center for Geological Disaster, Balai Penyelidikan dan Pengembangan
Teknologi Kebencanaan Geologi (BPPTKG), Jl Cendana No. 15, Yogyakarta 55166,
Indonesia 44email<EMAIL_ADDRESS>
# An ethnoarithmetic excursion into the Javanese calendar

Natanael Karjanto and François Beauducel

###### Abstract

A perpetual calendar, a calendar designed to find out the day of the week for
a given date, employs a rich arithmetical calculation using congruence.
Zeller’s congruence is a well-known algorithm to calculate the day of the week
for any Julian or Gregorian calendar date. Another rather infamous perpetual
calendar has been used for nearly four centuries among Javanese people in
Indonesia. This Javanese calendar combines the _Saka_ Hindu, lunar Islamic,
and western Gregorian calendars. In addition to the regular seven-day, lunar
month, and lunar year cycles, it also contains five-day _pasaran_ , 35-day
_wetonan_ , 210-day _pawukon_ , octo-year _windu_ , and 120-year _kurup_
cycles. The Javanese calendar is used for cultural and spiritual purposes,
including a decision to tie the knot among couples. In this chapter, we will
explore the relationship between mathematics and the culture of Javanese
people and how they use their calendar and the arithmetic aspect of it in
their daily lives. We also propose an unprecedented congruence formula to
compute the _pasaran_ day. We hope that this excursion provides an insightful
idea that can be adopted for teaching and learning of congruence in number
theory.

##  1 Introduction

Arithmetic and number theory find applications in various cultures throughout
the world. In addition to solving everyday problems using elementary
arithmetic operations, our ancient ancestors also developed and invented
perpetual calendars without any aid of modern electronic calculators and the
computer. A perpetual calendar is a system dealing with periods of time that
occur repeatedly. It is organized into days, weeks, months, and years. A date
of a calendar structure refers to a particular day within that system. A
calendar is used for various civil, administrative, commercial, social, and
religious purposes.

The English word “calendar” is derived from the Latin word which refers to the
first day of the month in the Roman calendar. It is related to the verb
_calare_ (“to announce solemnly, to call out”), which refers to the “calling”
of the new moon when it was visible for the first time (Brown 1993). Another
source mentions that the modern English “calendar” comes from the Middle
English _calender_ , which was adopted from the Old French _calendier_. It
originated from the Latin word _calendarium_ , which meant a “debt book,
account book, register”. In Ancient Rome, interests were tracked in such
books, accounts were settled, and debts were collected on the first day
(_calends_) of each month (Stakhov 2009).

MathematicsEducationAstronomyCultureMEEMMAEAAECEEMEEAEMAEEMAJC Figure 1: A
theoretical framework for an ethnoarithmetic study of the Javanese calendar.
At the first level of the intersection, ME means Mathematics Education, EM
means Ethnomathematics, MA means Mathematics Astronomy, AE means Astronomy
Education, EA means Ethnoastronomy, and CE means Cultural Education. At the
second level of the intersection, EME means Ethnomathematics Education, MAE
means Mathematics Astronomy Education, EMA means Ethnomathematics Astronomy,
and EAE means Ethnoastronomy Education. The third level of the intersection is
the heart of discussion, we have JC as the Javanese calendar.

Generally, the calculation of and periods in the calendrical system are
synchronized with the cycle of the Sun or the Moon. Hence, the names solar,
lunar, and lunisolar calendars, where the latter combined both the Moon phase
and the tropical (solar) year. Our current Gregorian and the previously Julian
calendars are solar-based calendars, as well as the ancient Egyptian calendar.
An example of the lunar calendar is the Islamic calendar. Prominent examples
of the lunisolar calendar are the Hebrew, Chinese, Hindu, and Buddhist
calendars (Richards 1998; Dershowitz and Reingold 2018).

Studying calendar systems from any culture cannot be separated from attempting
to gain insight and understanding not only the arithmetic behind it but also
its interconnectivity with mathematics in general, history of mathematics,
astronomy, mathematics education, and the sociocultural aspect itself. Hence,
the study of perpetual calendars is encompassed by the fields of
ethnomathematics and ethnoastronomy, the research areas where diverse cultural
groups embrace, practice, and develop mathematics and astronomy into their
daily lives.

The theoretical framework for a discussion on the Javanese calendar considered
in this chapter can be summarized in Figure 1. Although the title of this
chapter contains the term “ethnoarithmetic”, indeed that covering a topic on
the Javanese calendar encompasses other aspects beyond the cultural and
mathematical, in this case, arithmetical aspect per se. As we observe at the
third level of the Venn diagram presented in Figure 1, an excursion into the
Javanese calendar intersects and includes four distinct but related
disciplines: Mathematics, Education, Culture, and Astronomy.

Fundamental work in the area of ethnomathematics has been documented by Ascher
(2002). The author elaborated on several mathematical ideas and their cultural
embedding of people in traditional or small-scale cultures, with some emphasis
on time structure and the logic of divination. In particular, the book also
covered an explanation of the Balinese calendar. For discussion of the
Balinese calendrical system and its multiple cycles, consult Vickers (1990);
Darling (2004); Proudfoot (2007); Ginaya (2018), and Gislén and Eade (2019c).

To the best of our knowledge, the only monograph that discusses an
intersection of three components of the theoretical framework is a volume
edited by Rosa et al (2017). The book covers diverse approaches and
perspectives of ethnomathematics to mathematics education, particularly in
several non-Western cultures. Since these topics stimulate debates not only on
the nature of mathematical knowledge and the knowledge of a specific cultural
group but also on the pedagogy of mathematics classroom, the field of
ethnomathematics certainly offers a possibility for improving mathematics
education across cultures.

A collection of essays dealing with the mathematical knowledge and beliefs of
cultures outside the Western world is compiled by Selin and D’Ambrosio (2000).
The essays address the connections between mathematics and culture, relate
mathematical practices in various cultures, and discuss how mathematical
knowledge transferred from East to West. A coverage of calendars in various
cultures is also briefly touched.

Since many calendrical systems are invented based on the movement of celestial
bodies, particularly the Sun and Moon, we cannot dismantle the role of
Astronomy in the study of calendars, including the Javanese one. Various
civilizations incorporate the cyclical movements of both the Sun and Moon into
their calendrical systems (Ruggles 2015). Some examples of this calendrical-
astronomical relationship can be observed amongst others in the Jewish (Cohn
2007), Indian (Dershowitz and Reingold 2009), Chinese (Martzloff 2000, 2016),
mainland Southeast Asia (Eade 1995), and Islamic calendars (Proudfoot 2006).

An explanation of the mathematical and astronomical details of how many
calendars function has been covered extensively by Dershowitz and Reingold
(2018). A related series with non-Western ethnomathematics is a book on non-
Western ethnoastronomy that has been edited by Selin and Sun (2000). In
particular, the book also dedicated one chapter on an astronomical feature of
the Javanese calendar, where a season keeper (_pranotomongso_ or _pranata
mangsa_) guides agricultural activities among rural peasants in Java
(Daldjoeni 1984; Ammarell 1988; Hidayat 2000). The readers might be interested
to compare this with the cultural production of Indonesian skylore across
three ethnic groups: Banjar Muslim, Meratus Dayak, and Javanese peoples
(Ammarell and Tsing 2015).

Integrating the Javanese calendar into elementary school education as an
ethnomathematics study has been attempted by Utami et al (2020). Another
attempt is to embed some topics related to calendrical systems into the first-
year seminar on the mathematics of the pre-Columbian Americas (Catepillán
2016). A study from Taipei, Taiwan, on the movements of the Moon at the
primary level, introduced pupils to both the Gregorian and Chinese calendars
in the context of physical classroom environments (Hubber and Ramseger 2017).

Although we propose an intersection of only four disciplines in our
theoretical framework, the list is by no means exhaustive. Another possibility
is to include psychology or the interaction between mathematics and
psychology. For instance, a theoretical foundation on the calendrical system
and the psychology of time has been modeled by Rudolph (2006). In particular,
the author proposed that Balinese (also equally applied to Javanese) time
might be neither ‘circular’ nor ‘linear’, but _profinite_. Since the Javanese
calendrical systems involve five-day _pasaran_ , seven-day
_saptawara/dinapitu_ , and 30 _wuku_ cycles, it could be modeled by the group
of _profinite integers_ $\widehat{\mathbb{Z}}$. This group bundles together
all different sets of the ring of $p$-adic integers $\mathbb{Z}_{p}$ and
various finite sets $\mathbb{Z}/(p)$ of integers modulo $p$, where $p$ denotes
prime integers (Milne 2020).

A discussion on calendar timekeeping scheme in Southeast Asia is covered by
Gislén (2018). An overview of the calendars in Southeast Asia is given by
Gislén and Eade (2019a). In their subsequent papers, they and Lân also
discussed calendars from Burma, Thailand, Laos, and Cambodia (Gislén and Eade
2019b), Vietnam (Lân 2019), Malaysia and Indonesia (Gislén and Eade 2019c),
eclipse calculation (Gislén and Eade 2019d), and chronicle inscriptions
(Gislén and Eade 2019e). See also Eade (1995); Ôhashi (2009), and (Golzio
2012) for basic facts and further explanations of the calendrical systems in
India and (Mainland) Southeast Asia.

This chapter is organized as follows. After this Introduction, the following
section briefly covers the calendars from ancient and contemporary times,
which includes the pre-Gregorian and Gregorian calendars. After that, a
discussion on the Javanese calendar in the context of ethnoarithmetic will be
dedicated exclusively in one section. The final section concludes our
discussion.

##  2 Ancient and modern calendars

This section briefly covers the calendar in the pre-Gregorian era and Zeller’s
congruence algorithm in the Gregorian calendar.

###  2.1 Pre-Gregorian calendars

Before the Gregorian calendar that we are using today, numerous calendars have
been used in various parts of the world. The Egyptian calendar was among the
first solar calendars with its history dated back to the fourth millennium
BCE.

A Mesoamerican civilization developed by the Maya peoples also developed a
calendar system where the year was divided into 18 months of 20 days (Ascher
2002; Stakhov 2009). The Mayan calendar system contains three separate
calendars, and although they are not related mathematically, all of them are
linked in a single calendar system. The first one is called the long count,
used by the Mayans to measure date chronology for history recording. The
second one is the _Haab_ calendar, a non-chronological civil calendar. The
third one is the Mayan religious calendar, also non-chronological, called the
_Tzolkin_ (Ascher 2002; Cohn 2007).

Related to the Maya civilization is the Inca Empire, where the latter also
developed its calendar system, which was based on several different
astronomical cycles, including the solar year, the synodic and sidereal lunar
cycles, and the local zenith period (Urton and Llanos 2010). A variety of
mathematical development among the native Americans from the prehistoric to
present has been compiled by Closs (1996). In particular, one chapter of the
book discusses the calendrical system of the Nuu-chah-nulth (formerly referred
to as the Nootka), one of the indigenous peoples of the Pacific Northwest
Coast in Canada (Folan 1986).

Some calendars are based on lunisolar, and one of them is the Jewish calendar.
Also called the Hebrew calendar, it is still used until today, mainly for
Jewish religious observance. Although the Jewish calendar was developed in its
current format during the Talmud period in the fifth century CE by Rabbi
Hillel II, the up to date counting has accumulated more than 5000 years
(Ascher 2002; Cohn 2007). For our information, AM 5781 will begin at sunset on
Friday, 18 September 2020, and will end at sunset on Monday, 6 September 2021
CE. Here, AM means _Anno Mundi_ , the Latin phrase for “in the year of the
world”.

Although the traditional Chinese calendar is also a lunisolar type, the most
fundamental component is the sexagenary cycle, a cycle of sixty terms marked
by coordination between 12 celestial stems and ten terrestrial branches, which
is also known as the Stems-and-Branches or _gānzhī_ (干支) (Martzloff 2000). A
discussion on the mathematical aspect of the Chinese calendar has been covered
extensively by Aslaksen (2001, 2002, 2006). A historical aspect of the Chinese
calendar is discussed by Sun (2015). Astronomical aspects and the mathematical
structures underlying the calculation techniques of the Chinese calendar are
highlighted by Martzloff (2016).

The _Hijri_ , or Islamic calendar, is a lunar calendar consisting of 12 lunar
(synodic) months in a year of 354 or 355 days. It is still widely used in
predominantly Muslim countries alongside the Gregorian calendar, primarily for
religious purposes. The current counting started from 622 CE, commemorating
the emigration of Prophet Muhammad and his companions from Mecca to Medina
(_hijra_) (Hassan 2017). In the Gregorian calendar reckoning, the current
Islamic year is 1422 AH, which approximately runs from 20 August 2020 until 9
August 2021. AH means _Anno Hegirae_ , the Latin phrase for “in the year of
the Hijra”. In particular, the history of Muslim calendars in Southeast Asia
in the historical and cultural context has been successfully traced by
Proudfoot (2006).

###  2.2 Gregorian calendar

The language of congruences is a basic building block in arithmetic and number
theory. It allows us to operate with divisibility relationships in a similar
way as we deal with equalities. Congruences have many applications, and one of
them is to determine the day of the week for any date for a given perpetual
calendar. In particular, the procedure for finding the day of the week for a
given date in the Gregorian calendar has been discussed by Carroll (1887);
Conway (1973); Rosen (2011). See also Gardner (1996); Cohen (2000).

The Gregorian calendar was promulgated by Pope Gregory XIII and is a
replacement of the Julian calendar, proposed by Julius Caesar in 46 BC.
Starting from 1582, the Catholic states were among the first countries to
adopt the Gregorian calendar by skipping ten days in October, Thursday, 4
October 1582 was followed by Friday, 15 October 1582. Greece and Turkey are
among the last countries that adopted the Gregorian calendar when they changed
on 1 March 1923 and 1 January 1926, respectively.

Let $W$ denote the day of the week from Saturday = $0$ to Friday = $6$, $k$
denote the day of the month, $m$ denote month, and $N$ denote year. For the
month, the convention is March = $3$, April = $4$, …, but January = $13$, and
February = $14$. For the year, $N$ is the current year unless the month is
January or February, for which $N$ is the previous year. The relationship
between year and century is given by $N=100C+Y$, where $C$ denotes zero-based
century and $Y$ denotes a two-digit year. Note that the purpose of adopting
this expression is not to confuse $C$ with the standard century number, which
is $C+1$ for the first 99 years. The formula for finding the day of the week
$W$ of day $k$ of month $m$ of year $N$ is given by Zeller’s congruence
algorithm (Zeller 1882, 1883, 1885, 1887):

<EMAIL_ADDRESS>(1)

Here, $\lfloor x\rfloor$ denotes the floor function or greatest integer
function and mod is the modulo operation or remainder after division.

As an example, we are interested in finding the days of the week when a
Javanese heroine and educator Raden Adjeng Kartini was born and passed away.
She was born in Jepara, a town on the north coast of Java, around 80 km north-
east direction from Semarang, the present capital of Central Java province in
Indonesia. Kartini was a pioneer for girls’ education and women’s emancipation
rights in Indonesia, at the time when Indonesia was still a part of the Dutch
East Indies colonial empire. She could be correlated with her European
counterparts, including an English advocate of women’s right Mary
Wollstonecraft (27 April 1759–10 September 1797) or a Finnish social activist
Minna Canth (19 March 1844–12 May 1897) (cf. Kartini 1911, 1920;
Wollstonecraft 1792; Canth 1885).

Kartini was born on 21 April 1879, so we have $k=21$, $m=4$, $N=1879$, $C=18$,
and $Y=79$. She passed away on 17 September 1904, at the age of 25, and we
identify that $k=17$, $m=9$, $N=1904$, $C=19$, and $Y=4$. Using Zeller’s
congruence formula (1), the days when Kartini was born $W_{b}$ and passed away
$W_{d}$ can be calculated as follows, respectively:

$\displaystyle W_{b}$
$\displaystyle\equiv\left(21+\left\lfloor\frac{13(5)}{5}\right\rfloor+79+\left\lfloor\frac{79}{4}\right\rfloor+\left\lfloor\frac{18}{4}\right\rfloor-36\right)\\!\\!\\!\\!{\@displayfalse\pmod{7}}$
$\displaystyle\equiv\left(21+13+79+19+4-36\right)\\!\\!\\!\\!{\@displayfalse\pmod{7}}\equiv
100\,\\!\\!\\!\\!{\@displayfalse\pmod{7}}\equiv
2\,\\!\\!\\!\\!{\@displayfalse\pmod{7}}$ $\displaystyle W_{d}$
$\displaystyle\equiv\left(17+\left\lfloor\frac{13(10)}{5}\right\rfloor+4+\left\lfloor\frac{4}{4}\right\rfloor+\left\lfloor\frac{19}{4}\right\rfloor-38\right)\\!\\!\\!\\!{\@displayfalse\pmod{7}}$
$\displaystyle\equiv\left(17+26+4+1+4-38\right)\\!\\!\\!\\!{\@displayfalse\pmod{7}}\equiv
14\,\\!\\!\\!\\!{\@displayfalse\pmod{7}}\equiv
<EMAIL_ADDRESS>
Hence, Kartini was born on Monday, 21 April 1879 and passed away on Saturday,
17 September 1904. She was buried at Bulu Village, Rembang, Central Java,
around 100 km east of Jepara.

##  3 Javanese calendar

This section features the main characteristics of the Javanese calendar. We
start with the geographical location of the island of Java, the Javanese
people, and a brief historical background of the calendar. After providing a
detailed discussion, we close the section by computer implementation of the
Javanese calendar.

###  3.1 Where is Java?

Java is an island in Indonesia, not a programming language, although the
latter was renamed after some Javanese coffee by its founders. It is located
in the southern hemisphere, around 800 km from the Equator. It extends from
latitude 6∘ to 8∘ South, and longitude 105∘ to 114∘ East. With an area of
150,000 square kilometers, it is about 1000 km long from west to east and
around 200 km wide from north to south. The island lies between Sumatra to the
west and Bali to the east. It is bordered by the Java Sea on the north and the
Indian Ocean on the south (see Figure 2). It is the world’s 13${}^{\text{th}}$
largest island (Cribb 2000).

###  3.2 Who are Javanese people?

The Javanese people are a native ethnic group to the island of Java. They form
the largest ethnic group in Indonesia, with more than 95 million people live
in Indonesia and approximately 5 million people live abroad. Although they
predominantly reside in the central and eastern parts of the island, they are
also scattered in various parts of the country (Taylor 2003; Ananta et al
2015). The Javanese people possess and speak a distinct language from
Indonesian, called the Javanese language, a member of the Austronesian family
of languages written in Javanese script _hanacaraka_ or _dentawyanjana_.
Thanks to its long history and legacy of Hinduism and Buddhism in Java, the
language adopted a large number of Sanskrit words (Marr and Milner 1986;
Errington 1998).

Figure 2: The situation of the Java island, main cities, and present spoken
languages: Javanese (Central and East Java, and a small enclave in North-West
Java), Sundanese (West Java), Betawi (in and around Jakarta metropolitan
area), Madurese (Madura Island and a part of North-Eastern Java, and Balinese
(Bali Island and a small part of Eastern Java). Basemap uses ETOPO5 and SRTM3
topographic data and shaded relief mapping code (Beauducel 2020a). ETOPO5 is a
five arc-minute resolution relief model for the Earth’s surface that
integrates land topography and ocean bathymetry dataset. SRTM3, the Shuttle
Radar Topography Mission, is a three arc-second resolution digital topographic
database of land elevation limited to latitudes from 60∘ South to 60∘ North.

###  3.3 A background of the Javanese calendar

Javanese people use the Javanese calendar simultaneously with two other
perpetual calendars, the Gregorian and Islamic calendars. The former is the
official calendar of the Republic of Indonesia and the latter is used mainly
for religious purposes. Prior to the adoption of the Javanese calendar in 1633
CE, Javanese people used a calendrical system based on the lunisolar Hindu
_Saka_ calendar (Ricklefs 1993).

Figure 3: Table list of 8 _taun_ , 12 _wulan_ , 7 _dinapitu_ , and 5 _pasaran_
names in Javanese Sanskrit, after Warsapradongga (1892).

The Javanese calendar was inaugurated by Sultan Agung Adi Prabu Hanyakraku-
suma (1593–1645 CE), or simply Sultan Agung, the third Sultan of Mataram who
ruled Central Java from 1613 CE until 1645 CE. Although the counting of the
year follows the Saka calendar, the Javanese calendar employs a similar lunar
year as the Hijri calendar instead of the solar year system like the former
(Gislén and Eade 2019a). The Javanese calendar is sometimes referred to as AJ
(_Anno Javanico_), the Latin phrase for Javanese Year. Since 2008, the
difference between the Gregorian and Javanese calendars is about 67 years,
where the current year 2020 CE corresponds to 1953 AJ (Oey 2001; Raffles
1817).

Table 1: Main cycles of the Javanese calendar.

Cycle Name |  Length |  Unit |  Comment
---|---|---|---
Pancawara |  5 |  day |  Javanese week
|  |  |  5 pasaran names (see Table 3.4)
Wuku |  7 |  day |  Gregorian/Islamic week
|  |  |  7 dinapitu names (see Table 3.4)
|  |  |  30 wuku names
Wetonan |  35 |  day |  35-day names as Dinapitu and Pasaran
Wulan |  29 or 30 |  day |  day number in a wulan is dina
|  |  |  12 wulan names (see Table 3.4)
Pawukon |  30 |  Wuku |  30 weeks $\equiv$ 210 days
Taun |  12 |  Wulan |  354 or 355 days
|  |  |  taun number starts on 1555 AJ
|  |  |  8 taun names (see Table 3.4)
Windu |  8 |  Taun |  96 wulan $\equiv$ 81 Wetonan $\equiv$ 2,835 days
|  4 |  Windu |  32 _taun_ , 4 _windu_ names
Lambang |  2 |  Windu |  16 _taun_ , 2 _lambang_ names
Kurup |  15 |  Windu |  120 Taun $-1$ day $\equiv$ 42,524 days
|  |  |  4 kurup names until today (see Table 3.4)

###  3.4 Some characteristics of the Javanese calendar

Different from many other calendars that employ a seven-day week cycle, the
Javanese calendar adopts a five-day week cycle, known as _pancawara_.
Amalgamating with the seven-day week cycle of the Gregorian and Islamic
calendars, namely the _saptawara_ cycle, one obtain the 35-day cycle, known as
_wetonan_ (Darling 2004). This foundation cycle interferes with additional
cycles:

  * •

a 210-day cycle of 30 weeks, named as the _pawukon_ ;

  * •

a more complex combination of the lunar month _wulan_ , which has 29 or 30
days, the lunar year _taun_ , which is 12 lunar months, _windu_ , which is
eight lunar years, and finally _kurup_ of 15 _windu_ or 120 lunar years minus
one day, which matches exactly the Islamic calendar cycle (Proudfoot 2007).

There are also additional cycles, but they are no longer used in the Javanese
tradition. To be exhaustive, we list them here (Richmond 1956; Zerubavel
1989):

  * •

a six-day cycle called the _Paringkelan_ : ‘Tungle’, ‘Aryang’, ‘Wurukung’,
‘Paningron’, ‘Uwas’, and ‘Mawulu’;

  * •

an eight-day cycle called the _Padewan_ : ‘Sri’, ‘Indra’, ‘Guru’, ‘Yama’,
‘Rudra’, ‘Brama’, ‘Kala’, and ‘Uma’;

  * •

a nine-day cycle called the _Padangon_ : ‘Dangu’, ‘Jagur’, ‘Gigis’,
‘Kerangan’, ‘Nohan’, ‘Wogan’, ‘Tulus’, ‘Wurung’, and ‘Dadi’;

The list of main cycles and characteristics are summarized in Table 3.3 and
are detailed in the following subsubsections. Figure 3 shows an original table
of the main cycles’ names written in the Javanese script _hanacaraka_.

Table 2: Names of the five _pasaran_ days in the Javanese week and commonly
associated symbols.

Ngoko |  Krama |  Meaning |  Element |  Color |  Direction |  Posture
---|---|---|---|---|---|---
Pon |  Petak |  – |  water |  yellow |  West |  sleep
Wage |  Cemeng |  dark |  earth |  black |  North |  sit down
Kliwon |  Asih |  affection |  spirit |  mixed color |  focus/centre |  stand-up
Lêgi |  Manis |  sweet |  air |  white |  East |  turn back
Pahing |  Pahit |  bitter |  fire |  red |  South |  to face

#### Pancawara and Pasaran

The Javanese five-day cycle is named _pancawara_ and made of five days known
as _pasaran_ : ‘Pon’, ‘Wage’, ‘Kliwon’, ‘Lêgi’, and ‘Pahing’ (or ‘Paing’). The
word comes from ‘pasar’ which means market. Historically, the market was held
and operated on a five-day cycle based on a _pasaran_ day, e.g., ‘Pasar
Kliwon’, or ‘Pasar Legi’. Until today, most of the markets in Java still have
a _pasaran_ name like the Kliwon Market in Kudus, Central Java, although they
usually operate every day (Oey 2001).

Each _pasaran_ is associated with some symbols, in particular, the five
classical elements of Aristotle, colors, cardinal directions (note that the
Javanese culture has five of them including a Center), and human posture
(Pigeaud 1977), (cf. Brinton 1893). See Table 3.4.

The computation of the _pasaran_ day from the Gregorian or Islamic calendar
date can be performed using a similar strategy as Zeller’s congruence
algorithm. Indeed, the formula is based on the observation that the day of the
week progresses predictably based upon each subpart of the date, i.e., day,
month, and year. In this case, we will consider the five-day Javanese week
_pasaran_. Each term within the formula is used to calculate the offset needed
to obtain the correct day.

Let $P$ denote the _pasaran_ day, in the order of Pon $=0$, Wage $=1$, Kliwon
$=2$, Lêgi $=3$, and Pahing $=4$. We now propose the following _pasaran_ day
congruence formula:

<EMAIL_ADDRESS>(2)

where $k$, $m$, $C$, and $Y$ denote the same variables as the ones defined for
Zeller’s congruence formula (see expression (1) in the previous section). Each
term in (2) can be analyzed as follows:

  * •

$k$ represents the progression of the day of the week based on the day of the
month since each successive day results in an additional offset of one;

  * •

the term $+\left\lfloor\frac{3(m+1)}{5}\right\rfloor$ adjusts for the
variation in the days of the month. Indeed, starting from March to February,
the days in a month are {31, 30, 31, 30, 31, 31, 30, 31, 30, 31, 31, 28/29}.
The last element, February’s 28/29 days, is not a problem since the formula
had rolled it to the end. The number of days for the 11 first elements of this
sequence modulo $5$ (still starting with March) would be {1, 0, 1, 0, 1, 1, 0,
1, 0, 1, 1} which basically alternates the subsequence {1, 0, 1, 0, 1} every
five months and give the number of days that should be added to the next
month. The fraction $\frac{3}{5}\equiv 0.6$ applied to $m+1$ and the floor
function have that effect and will add the proper amount of days;

  * •

since there are 365 days in a non-leap year, and
$365\\!\\!\\!\\!{\@displayfalse\pmod{5}}\equiv 0$, there is no need to add an
offset for the normal year;

  * •

for the leap years, $366\\!\\!\\!\\!{\@displayfalse\pmod{5}}\equiv 1$ so a day
must be added to the offset value by the term
$+\left\lfloor{\frac{Y}{4}}\right\rfloor$;

  * •

there are $36,524$ days in a normal century and $36,525$ days in each century
divisible by $400$; the term $4C$ adds
$36,524\\!\\!\\!\\!{\@displayfalse\pmod{5}}\equiv 4$ days for any century, and
$-4\left\lfloor\frac{C}{4}\right\rfloor$ removes these $4$ days for a century
divisible by $400$;

  * •

the overall function, $\\!\\!\\!\\!{\@displayfalse\pmod{5}}$, normalizes the
result to reside in the range from $0$ to $4$, which yields the index of the
correct day for the date being analyzed.

For example, to find the _pasaran_ of the very first day in the Javanese
calendar (1 Sura 1555 AJ Alip) which corresponds to 8 July 1633 CE, we have
$k=8$, $m=7$, $C=16$, and $Y=33$. Using (2), we obtain $P\equiv
8+4+8+64-16\equiv 68$ $\\!\\!\\!\\!{\@displayfalse\pmod{5}}$ $\equiv 3$.
Hence, 8 July 1633 CE was a ‘Lêgi’. From the previously considered example,
the _pasaran_ days when Kartini was born $P_{b}$ and passed away $P_{d}$ can
be calculated as follows:

$\displaystyle
P_{b}\equiv\left(21+3+19+72-16\right)\\!\\!\\!\\!{\@displayfalse\pmod{5}}\equiv
99\,\\!\\!\\!\\!{\@displayfalse\pmod{5}}$ $\displaystyle\equiv
4\,\\!\\!\\!\\!{\@displayfalse\pmod{5}}$ $\displaystyle
P_{d}\equiv\left(17+6+1+76-16\right)\\!\\!\\!\\!{\@displayfalse\pmod{5}}\equiv
84\,\\!\\!\\!\\!{\@displayfalse\pmod{5}}$ $\displaystyle\equiv
<EMAIL_ADDRESS>
Hence, both 21 April 1879 and 17 September 1904 fall on a Pahing.

Figure 4: An example of the Javanese calendar for December 2020 CE issued by
the Kraton palace in Yogyakarta. It contains information on _Jimakir taun_ ,
_Bakdamulud_ and _Jumadilawal wulans_ , _pasaran_ day, _padinan_ weekday,
_wuku_ , and _paringkelan_ amongst others.

#### Dinapitu, Wuku and Pawukon

_Dinapitu_ literally means ‘day seven’ in Javanese, and corresponds to the day
names in the Gregorian/Islamic calendar week with exact equivalence, i.e.,
from Monday to Sunday: ‘Sênèn’, ‘Selasa’ (or ‘Slasa’), ‘Rêbo’, ‘Kêmis’,
‘Jemuwah’ (or ‘Jumungah’), ‘Sêtu’, and ‘Ngahad’ (or ‘Ahad’). In recent
literature, we often find the name of the days in the Indonesian language
(from Monday to Sunday: ‘Senin’, ‘Selasa’, ‘Rabu’, ‘Kamis’, ‘Jumat’, ‘Sabtu’,
and ‘Minggu’). They are also associated with particular symbolic meanings. See
Table 3.4. The names of the days of the week are similar in both languages as
they are absorbed from Arabic except for Sunday in Indonesian which was
assimilated from the Portuguese ‘Domingo’.

Table 3: List of names of the seven days in a _wuku_ Gregorian week (cf. Rizzo
2020).

Dinapitu |  Padinan |  Week Day |  Symbol
---|---|---|---
Ngahad |  Dite |  Sunday |  silent
Sênèn |  Soma |  Monday |  forward
Selasa |  Anggara |  Tuesday |  backward
Rêbo |  Buda |  Wednesday |  turn left
Kêmis |  Respati |  Thursday |  turn right
Jemuwah |  Sukra |  Friday |  up
Sêtu |  Tumpak |  Saturday |  down

The seven-day cycle is called _saptawara_ or _padinan_ and a week is named a
_wuku_. The period of 30 _wuku_ makes the _pawukon_ cycle, i.e., 210 days
(Proudfoot 2007). There are 30 different names of _wuku_ (Soebardi 1965;
Headley 2004). See Table 4. Figure 4 shows an example of the Javanese calendar
with _padinan_ , _pasaran_ , _wuku_ , and _paringkelan_ information amongst
others.

Table 4: List of names of the 30 different _wuku_ in a _pawukon_. 1 |  Sinta |  7 |  Warigalit |  13 |  Langkir |  19 |  Tambir |  25 |  Bala
---|---|---|---|---|---|---|---|---|---
2 |  Landep |  8 |  Warigagung |  14 |  Mandasiya |  20 |  Medangkungan |  26 |  Wugu
3 |  Wukir |  9 |  Julungwangi |  15 |  Julungpujut |  21 |  Maktal |  27 |  Wayang
4 |  Kurantil |  10 |  Sungsang |  16 |  Pahang |  22 |  Wuye |  28 |  Kulawu
5 |  Tolu |  11 |  Galungan |  17 |  Kuruwelut |  23 |  Manahil |  29 |  Dukut
6 |  Gumbreg |  12 |  Kuningan |  18 |  Marakeh |  24 |  Prangbakat |  30 |  Watugunung

#### Wetonan

The _wetonan_ cycle combines the five-day _pancawara_ cycle with the seven-day
_wuku_ week cycle. Each _wetonan_ cycle lasts for $7\times 5=35$ days, with 35
distinct combinations of the couple ‘ _dinapitu_ _pasaran_ ’ which is called
the _weton_. Figure 5 displays the 35-day _wetonan_ cycle of the dual
_dinapitu pasaran_. The seven-day _wuku_ cycle is arranged clockwise from
Monday to Sunday and the five-day _pasaran_ day progresses from the center of
the disk outwardly from Legi to Kliwon. It shows a spiral pattern that repeats
seven times and sweeping five sectors each, indicated by solid black, solid
blue, dashed-black, dashed-blue, dashed-dotted black, dashed-dotted blue, and
dotted black spirals, respectively.

LêgiPahingPonWagéKliwonMondaySundaySaturdayFridayThursdayWednesdayTuesday$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$15$$16$$17$$18$$19$$20$$21$$22$$23$$24$$25$$26$$27$$28$$29$$30$$31$$32$$33$$34$$35$
Figure 5: A 35-day cycle of _wetonan_ in the Javanese calendar. The regular
Gregorian seven-day cycle is arranged clockwise from Monday to Sunday. The
five-day _pasaran_ cycle is arranged from inside to outside. The white disk,
red, blue, green, and yellow rings correspond to Lêgi, Pahing, Pon, Wagé, and
Kliwon, respectively.

Although the _weton_ can be calculated independently using either equation (1)
or (2), it is also possible to propose a single congruence formula:

$\displaystyle w$
$\displaystyle=k+\left\lfloor\frac{153(m+1)}{5}\right\rfloor+15Y+\left\lfloor\frac{Y}{4}\right\rfloor+19C+\left\lfloor\frac{C}{4}\right\rfloor+5$
(3) $\displaystyle W$ $\displaystyle\equiv
w\,\\!\\!\\!\\!{\@displayfalse\pmod{7}}$ $\displaystyle P$
$\displaystyle\equiv w\,\\!\\!\\!\\!{\@displayfalse\pmod{5}}$

where $k$, $m$, $C$, and $Y$ are the same variables as defined for previous
congruence formulas, $w$ is a 35-day offset congruence, and $W$ or $P$
corresponds to the index of _dinapitu_ or _pasaran_ day, respectively. Each
term of the congruence can be analyzed as follows:

  * •

$k$ represents the progression of the day of the week based on the day of the
month, since each successive day results in an additional offset of one;

  * •

$+\left\lfloor\frac{153(m+1)}{5}\right\rfloor$ adds the proper amount of days
for each month, considering January and February as the 13th and 14th months
of the previous year, respectively;

  * •

$+15Y$ adds an offset of $365\\!\\!\\!\\!{\@displayfalse\pmod{35}}\equiv 15$
days for the common, non-leap years;

  * •

$+\left\lfloor{\frac{Y}{4}}\right\rfloor$ adds one more day for the leap years
since $366\\!\\!\\!\\!{\@displayfalse\pmod{35}}\equiv 16$;

  * •

$+19C$ adds $36,524\\!\\!\\!\\!{\@displayfalse\pmod{35}}\equiv 19$ days for
any regular century (a century with non-leap year);

  * •

$+\left\lfloor\frac{C}{4}\right\rfloor$ adds one more day for a century leap
year that is divisible by $400$ since
$36,525\\!\\!\\!\\!{\@displayfalse\pmod{35}}\equiv 20$;

  * •

$+5$ adjusts the final offset to fit the indexes defined for _dinapitu_ $W$
and _pasaran_ $P$ after $\\!\\!\\!\\!{\@displayfalse\pmod{7}}$ and
$\\!\\!\\!\\!{\@displayfalse\pmod{5}}$, respectively.

Let take the same example of 8 July 1633 CE with $k=8$, $m=7$, $C=16$, and
$Y=33$. Using (3) we have $w\equiv 8+244+8+304+4+5\equiv 573$, $W\equiv
573\\!\\!\\!\\!{\@displayfalse\pmod{7}}\equiv 6$, and $P\equiv
573\\!\\!\\!\\!{\@displayfalse\pmod{5}}\equiv 3$. Hence, the 8 July 1633 CE
was a ‘Jemuwah Lêgi’. Notice that we are able to replace both the congruence
formulas (1) and (2) using our original single congruence relationship (3).
Obtaining both _dinapitu_ and _pasaran_ days using a tabular method has been
attempted by Arciniega (2020) and a _Java_ application based on the Indian
calendar for calculating the Javanese calendar has been developed by Gislén
and Eade (2019e).

The _wetonan_ cycle is especially important for divinatory systems,
celebrations, and rites of passage as birth or death. Commemorations and
events are held on days considered to be auspicious. In particular, the
_weton_ of birth is considered playing an important role in any individual
personality, in a similar way as a zodiac sign does in Western astrology. The
two _weton_ days of future spouses are supposed to determine their background
nature compatibility and are used to compute the best date of marriage using a
strict arithmetic formula (Utami et al 2019). It also figures in the timing of
many ceremonies of _slametan_ ritual meal and many other traditional
divinatory systems (Utami et al 2020). The eve of ‘Jumat Kliwon’ is considered
particularly popular and auspicious for magical and spiritual matters (Darling
2004; Arciniega 2020).

The anniversary of Javanese birthday occurs every 35 days, so about a thousand
times in a century (exactly 1043 times). For a newborn baby, the first
occurrence of _weton_ , i.e., aged 35 days, is named ‘selapanan’ where the
parents will cut hairs and nails of their child for the first time. For
adults, the _weton_ of birth is considered as an eminent day, not festive but
expressing humility and blissfulness. For example, a person may fast
(sometimes also the day before and the day after), stop his commercial
activity, avoid taking any major decision, or simply be more generous to
surrounding people through philanthropic actions like share a blessed meal or
some ‘jajan pasar’ (early morning fresh sweet snacks from the market).

The _weton_ for the birth and death of Sultan Agung is ‘Jemuwah Lêgi’, which
is also the first day of the Javanese calendar he created. This _weton_ is
therefore one of the recurrent noble days (see subsubsection _Dina Mulya_) and
considered as an important night for pilgrimage. Indeed, the _weton_ of every
King’s birth is a special day; the present Sultan Hamengkubuwana X was born on
2 April 1946 CE, a ‘Selasa Wage’. This _weton_ has been chosen for His
ascension to the throne, on 7 March 1989 CE. Every ‘Selasa Wage’, the animated
touristic center of Yogyakarta, Jalan Malioboro, is closed to motor vehicles,
and the covered sidewalks ‘kaki lima’, where commercial activity usually
abound, is entirely cleaned.

As another especially prominent example, the present palace of Yogyakarta has
been inhabited by Sultan Hamengkubuwana I and His regal suite on the ‘13 Sura
1682 AJ’ (7 Oktober 1756 CE), a ‘Kemis Pahing’. In 2015, the Governor decided
that every ‘Kemis Pahing’, schoolchildren, public servants, and in particular
those working in territorial services of Yogyakarta, must wear the traditional
costume all day long, as a reminder of their regional culture. Merchants from
traditional markets are encouraged to do the same.

Table 5: List of names of the 12 _wulan_ Javanese lunar months and the
associated number of days, depending on the _taun_ and _kurup_ (see Table
3.4).

|  | Wulan length (days)
---|---|---
|  |  Taun 1-4,6-8 | Taun 5
No |  Wulan Name |  All Kurup |  Kurup 1 |  Kurup 2 |  Kurup 3 |  Kurup 4
1 |  Sura |  30 |  30 |  30 |  30 |  30
2 |  Sapar |  29 |  29 |  30 |  30 |  29
3 |  Mulud |  30 |  30 |  29 |  29 |  30
4 |  Bakdamulud |  29 |  29 |  29 |  29 |  29
5 |  Jumadilawal |  30 |  30 |  30 |  29 |  30
6 |  Jumadilakir |  29 |  29 |  29 |  29 |  29
7 |  Rejeb |  30 |  30 |  30 |  30 |  30
8 |  Ruwah |  29 |  29 |  29 |  29 |  29
9 |  Pasa |  30 |  30 |  30 |  30 |  30
10 |  Sawal |  29 |  29 |  29 |  29 |  29
11 |  Dulkangidah |  30 |  30 |  30 |  30 |  30
12 |  Besar |  29/30 |  30 |  30 |  30 |  29
|  Total (days) |  254/355 |  355 |  355 |  354 |  354

#### Wulan

The lunar month is named a _wulan_ and lasts for 29 or 30 days. There are 12
different names: ‘Sura’, ‘Sapar’, ‘Mulud’, ‘Bakdamulud’, ‘Jumadilawal’,
‘Jumadilakir’, ‘Rejeb’, ‘Ruwah’, ‘Pasa’, ‘Sawal’, ‘Dulkangidah’ (or ‘Sela’),
and ‘Besar’. The length of each _wulan_ is attributed as follows (see Table
3.4 and the next subsubsections):

  * •

‘Sura’, ‘Rejeb’, ‘Pasa’, and ‘Dulkangidah’ are always 30 days;

  * •

‘Bakdamulud’, ‘Jumadilakir’, ‘Ruwah’, and ‘Sawal’ are always 29 days;

  * •

‘Sapar’, ‘Mulud’, ‘Jumadilawal’, and ‘Besar’ have lengths depending on the
_taun_ and _kurup_.

#### Taun

A _taun_ is a cycle on 12 _wulan_ and corresponds to the Javanese lunar year.
There are eight different _taun_ names: ‘Alip’, ‘Ehé’, ‘Jimawal’, ‘Jé’, ‘Dal’,
‘Bé’, ‘Wawu’, and ‘Jimakir’, formed by different _wulan_ length sequences (see
Table 3.4). For seven of the _taun_ , the sequence alternates monotonically
between 30 and 29-day lengths. For the final _wulan_ ‘Besar’, it can be either
29 or 30 days, depending not only on the _taun_ but also on the _kurup_ , the
120 lunar year cycle, which has different day length sequences for the fifth
_taun_ ‘Dal’. As a result, the total day length of a _taun_ varies from $354$
(short or normal year, named ‘Taun Wastu’) to $355$ (long or leap year, named
‘Taun Wuntu’), as described in Table 3.4:

  * •

‘Alip’, ‘Jimawal’, and ‘Bé’ are always normal years, with a _wulan_ ‘Besar’ of
29 days;

  * •

‘Jimawal’ and ‘Wawu’ are always leap years, with a _wulan_ ‘Besar’ of 30 days;

  * •

‘Jé’ and ‘Dal’ are normal or leap depending on the _kurup_ ;

  * •

‘Jimakir’ is a leap year for the 14 first _windu_ , but becomes a normal year
for the final _windu_ of a _kurup_ cycle.

Table 6: List of names of the eight _taun_ Javanese lunar years forming a
_windu_ , and the associated number of days, an alternate between 354 (short
or normal year) and 355 (long or leap year, gray-shaded cells) days, depending
on the _kurup_.

|  |  |  | Taun Length (days)
---|---|---|---|---
No |  Name |  Krama |  Meaning |  Kurup 1 |  Kurup 2 |  Kurup 3 |  Kurup 4
1 |  Alip |  Purwana |  intention |  354 |  354 |  354 |  354
2 |  Ehé |  Karyana |  action |  355 |  355 |  355 |  355
3 |  Jimawal |  Anama |  work |  354 |  354 |  354 |  354
4 |  Jé |  Lalana |  destiny |  354 |  354 |  355 |  355
5 |  Dal |  Ngawanga |  life |  355 |  355 |  354 |  354
6 |  Bé |  Pawaka |  back and forth |  354 |  354 |  354 |  354
7 |  Wawu |  Wasana |  orientation |  354 |  354 |  354 |  354
8 |  Jimakir |  Swasana |  empty |  355 |  355 |  355 |  355

Each _taun_ is assigned by a monotonic increasing number, based on the Indian
calendar ‘Saka’. The reason was Sultan Agung decided to continue the counting
from the Shalivahana era, which was 1555 at the time when inaugurating the
Javanese calendar (cf. Nuraeni and Azizah 2017). Thus, the Javanese calendar
began on ‘1 Sura Alip 1555 AJ’, which corresponds to 8 July 1633 CE.

#### Windu and Lambang

Eight _taun_ make a _windu_ (Proudfoot 2006). Despite the variability of each
_taun_ length, the total length of a normal _windu_ is constant since it
always contains both five short and three long _taun_ , which is a total of
2,835 days (about 7 years 9 months in the Gregorian/Islamic calendar). This
corresponds to exactly 81 _wetonan_. This means that each ‘New Windu’ day,
dated as ‘1 Sura Alip’, falls on the same _weton_.

There is an exception to that rule: the final _windu_ of a _kurup_ cycle (see
the next subsubsection) is always shortened by one day, with a 29 days _wulan_
‘Besar’ during the final _taun_ ‘Jimakir’. This induces a shift in the
_wetonan_ cycle at each _kurup_.

Furthermore, there are four different names of _windu_ : ‘Adi’, ‘Kuntara’,
‘Sengara’, and ‘Sancaya’ that compose a 32 _taun_ cycle. Another cycle of 16
_taun_ is combined using two different names of _windu_ , called _lambang_ :
‘Kulawu’ and ‘Langkir’. These two cycles are summarized in Table 3.4.

Table 7: List of names of the four _windu_ and two _lambang_.

Windu Name |  Lambang Name
---|---
Adi |  Langkir
Kuntara |  Kulawu
Sêngara |  Langkir
Sancaya |  Kulawu

#### Kurup

The longest cycle in the Javanese calendar is called a _kurup_ , formed by 15
_windu_ , which is equivalent to 120 _taun_ or 1440 _wulan_ (Gislén and Eade
2019c). But the very last _wulan_ of the cycle, i.e., the twelfth _wulan_
‘Besar’ of the eighth _taun_ ‘Jimakir’ of the fifteenth _windu_ , has only 29
days, such as the total length of a _kurup_ is $2,835\times 15-1=42,524$ days
(about 116 years and 6 months in the Gregorian/Islamic calendar) (Rosalina
2013). This is the same number of days as in 120 lunar years of the Tabular
Islamic calendar. This is a rule-based variation of the Islamic Hijri
calendar. Although the number of years and months are identical, the months
are determined by arithmetical rules instead of observation or astronomical
calculations.

Moreover, each _kurup_ determines:

  * •

which of the _taun_ ‘Jé’ or ‘Dal’ has a long _wulan_ ‘Besar’,

  * •

the sequence of _wulan_ lengths in the _taun_ ‘Dal’,

as given in Tables 3.4 and 3.4. Hence, the full date sequences in the calendar
vary between _kurup_.

As the _weton_ of the first day of a _kurup_ repeats at each first day of the
_windu_ , a _kurup_ is named using the corresponding _weton_ falling on ‘1
Sura Alip’. Table 3.4 lists the first five _kurup_.

Meanwhile, the Sultanate of Mataram was divided under the Treaty of Giyanti
between the Dutch and Prince Mangkubumi in 1755 CE. The agreement divided
ostensible territorial control over Central Java between Yogyakarta and
Surakarta Sultanates. The former was ruled by Prince Mangkubumi, also known as
Raden Mas Sujana or Hamengkubuwono I (1717–1792 CE) and the latter was
administrated by Sinuhun Paliyan Negari, who was known as Pakubuwana III
(1732–1788 CE) (Ricklefs 1974; Soekmono 1981; Frederick and Worden 1993; Brown
2004).

Table 8: List of names of the five first _kurup_ Javanese 120 lunar year
cycles, their short names (a contraction of the _weton_ at each new _windu_ ,
i.e., on ‘1 Sura Alip’), the first and last _taun_ , the total amount of
_taun_ , and the starting dates in the Gregorian calendar.

No |  Kurup Name |  Short Name |  First Taun (AJ) |  Last Taun (AJ) |  Taun |  Start Date (CE)
---|---|---|---|---|---|---
1 |  Jamingiyah |  A’ahgi |  Alip 1555 |  Jimakir 1674 |  120 |  8 July 1633
2 |  Kamsiyah |  Amiswon |  Alip 1675 |  Ehé 1748 |  74 |  11 December 1749
3 |  Arbangiyah |  Aboge |  Jimawal 1749 |  Jimakir 1866 |  118 |  28 September 1821
4 |  Salasiyah |  Asapon |  Alip 1867 |  Jimakir 1986 |  120 |  24 March 1936
5 |  Isneniyah |  Anenhing |  Alip 1987 |  Jimakir 2106 |  120 |  26 August 2052

During the second _kurup_ (1749–1821 CE), some experts realized that the
Javanese calendar was still one day behind compared to the Islamic Hijri
calendar. Hence, the King of Surakarta, Susuhunan Pakubuwana V (1784–1823 CE),
decided to end the _Kurup_ ‘Amiswon’ in the year 1748 AJ, even though it had
only been running for nine _windu_ and two _taun_. So, the _taun_ ‘Ehé’ 1748
AJ, which was supposed to be a leap year, was made only 354 days and the third
_kurup_ ‘Aboge’ started on the _taun_ ‘Jimawal’ 1749 AJ. But some noticed that
it would be more appropriate if the incrementation of _kurup_ should have been
carried out two lunar years before, namely on the _taun_ ‘Alip’ 1747 AJ. As a
consequence of this delay, the third _kurup_ ‘Aboge’ is only 118 _taun_ long.
However, the Sultanate of Yogyakarta did not make a similar decision and
pursued the second _kurup_ normally, so that the calendar in the two
concurrent regions was different during 46 years (see Table 3.4). On _taun_
‘Jimakir’ 1794 AJ, the Sultan of Yogyakarta, Hamengkubuwana VI (1821–1877 CE),
finally agreed and decided that the third _kurup_ ‘Aboge’ will also end with
_taun_ ‘Jimakir’ 1866 AJ, reconciling the two calendars.

Table 9: List of dates of the second and third _kurup_ in the Sultanate of
Yogyakarta.

No |  Kurup Name |  Short Name |  First Taun (AJ) |  Last Taun (AJ) |  Taun |  Start Date (CE)
---|---|---|---|---|---|---
2 |  Kamsiyah |  Amiswon |  Alip 1675 |  Jimakir 1794 |  120 |  11 December 1749
3 |  Arbangiyah |  Aboge |  Alip 1795 |  Jimakir 1866 |  72 |  16 May 1866

The fourth and present _kurup_ ‘Asapon’ is planned to last a normal length of
120 lunar years, and will end on ‘29 Besar Jimakir 1986 AJ’, which is 25
August 2052 CE. The following day will start the fifth _kurup_ ‘Anenhing’
(‘Alip Senen Pahin’) on ‘1 Sura 1987 AJ Alip’, but the sequence of leap years
has not yet been decided, so it is formally impossible to calculate the exact
_dina_ , _wulan_ , and _taun_ for such a distant future date. Nevertheless,
there is no obstacle with other strictly monotonic cycles like _weton_ ,
_wuku_ , and _windu_.

#### Dina Mulya

_Dina mulya_ are the noble days in the Javanese calendar. Excepted for the
‘Siji Sura’ which is the new lunar year and falls on the first day of the
first _wulan_ every _taun_ , others are associated with a specific _weton_ and
specific _taun_ or _wuku_ (see Table 3.4).

Table 10: List of the noble days _dina mulya_.

Dina Mulya |  Weton |  Wuku |  Dina |  Wulan |  Taun |  Occurrences
---|---|---|---|---|---|---
Siji Sura |  – |  – |  1 |  Sura |  – |  1 every 354/355 days
|  |  |  |  |  |  (new lunar year)
Aboge |  Rêbo Wage |  – |  – |  – |  Alip |  10 during the Taun
|  |  |  |  |  |  every 7/8 years
Daltugi |  Sêtu Legi |  – |  – |  – |  Dal |  10 during the Taun
|  |  |  |  |  |  every 7/8 years
Kuningan |  Sêtu Kliwon |  Kuningan |  – |  – |  – |  every 210 days
Hanggara Asih |  Selasa Kliwon |  Dukut |  – |  – |  – |  every 210 days
Dina Mulya |  Jemuwah Kliwon |  Watugunung |  – |  – |  – |  every 210 days
Dina Purnama |  Jemuwah Lêgi |  – |  – |  – |  – |  every 35 days

###  3.5 Computer implementation of the Javanese calendar

The computation of the full Javanese calendar has been implemented using _GNU
Octave_ scientific language with a single function ‘weton.m’ (Beauducel
2020b). When using a computer, however, the determination of the weekday or
_pasaran_ day from a date in the Gregorian calendar does not require the
congruence formula (3). In fact, most computer languages are able to calculate
the exact number of days that last from any reference date, correctly taking
into account leap years. In the case of the Javanese calendar, the linear
timeline will be the number of days counted from 8 July 1633 CE, which falls
on ‘Jemuwah Lêgi 1 Sura Alip 1555 AJ Jamingiyah’, the first day of the first
_kurup_. Using that facility, calculation of the 7-day week cycle can be made
by a simple modulo 7, the _pasaran_ cycle by a modulo 5, the _wetonan_ cycle
by a modulo 35, and the _pawukon_ cycle by a modulo 210 functions. Moreover,
since any modern digital calendar is able to set a periodic event, a specific
_weton_ date repeated every 35 days will smoothly give the corresponding
_wetonan_ cycle over the whole calendar.

On the other hand, the computation of _dina_ , _wulan_ , _taun_ , _windu_ ,
and _kurup_ is more complicated since the exact sequences vary throughout
history following human decisions, making these cycles not exactly cyclic nor
monotonic. Hence, the proposed computing strategy is to construct, for each
_kurup_ , a _windu_ table as a $8\times 12$ matrix of rows _taun_ versus
columns _wulan_ , containing the day length of that specific _wulan_. This
matrix is repeated 15 times to form a complete _kurup_ , or less for the
second and third _kurup_. Then, all the matrices are concatenated. The
cumulative sum of this table elements, in the row order, gives the total
number of days that lasts from the origin at the beginning of each lunar
month, and can be compared to the linear timeline described above. Thus, a
simple ‘table lookup’ function will give the corresponding _kurup_ , _windu_ ,
_taun_ , and _wulan_ indexes, and the _dina_ will be given by the remainder.

##  4 Conclusion

In this chapter, we have discussed the cultural, historical, and arithmetic
aspects of the Javanese calendar. Along with the internationally acknowledged
Gregorian and the majority-embraced religiously Islamic Hijri calendars, the
Javanese calendar has its unique place among the heart of many Javanese people
in Indonesia as well as Javanese diaspora overseas. Although many Javanese
people have adopted modern lifestyle, the Javanese calendar is still utilized
in various daily affairs, including to choose the best possible time for
arranging a wedding day.

While determining the day of the week for any given date can be computed using
Zeller’s congruence algorithm of modulo seven, the _pasaran_ day of the
Javanese calendar can be calculated using a new congruence formula of modulo
five. Additionally, we have also proposed a unique and combined congruence
formula for calculating both the day of the week and _pasaran_ day for any
given date in the Gregorian calendar. Furthermore, using a computer program
_GNU Octave_ ‘weton.m’ (Beauducel 2020b), all the cycles of the Javanese
calendar, i.e., _wetonan_ , _wuku_ , _wulan_ , _windu_ , _lambang_ , and
_kurup_ can also be determined straightforwardly.

## Acknowledgment

The authors would like to thank Matthew Arciniega (Vortx, Inc., Ashland,
Oregon, United States of America) for sharing the contents of his old website
and Roberto Rizzo (University of Milano–Bicocca, Italy) for pointing to the
article written by Proudfoot (2007). NK dedicated this chapter to his father,
who introduced to and taught him about calendars during his early childhood.
FB warmly thanks Alix Aimée Triyanti for her cultural influence and
inspiration, and all his Javanese friends for their enthusiastic support.

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