Tibetan Calendar Mathematics by Svante Janson (2007).pdf

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TIBETANCALENDARMATHEMATICS
SVANTEJANSON
Abstract.ThecalculationsoftheTibetancalendararedescribed,us-
ingmodernmathematicalnotationsinsteadofthetraditionalmethods.
1.Introduction
TheTibetancalendarisderivedfromtheIndiancalendartradition;it
hasthesamegeneralstructureasIndiancalendars,see[1],butthedetails
diersignificantly.ThebasisfortheTibetancalendaristheKalacakra
Tantra,whichwastranslatedfromSanskritintoTibetaninthe11thcentury.
(Traditionaldateofthetranslationis1027whenthefirst60yearcycle
starts.)ItisbasedonIndianastronomy,butmuchmodified.Thecalendar
becamethestandardinTibetinthesecondhalfofthethirteenthcentury.
AsinIndiancalendars[1],monthsarelunar(fromnewmoontonew
moon),butnumberedbythecorrespondingsolarmonths,whiledaysare
numberedbythecorrespondinglunardays.Sincethesecorrespondences
arenotperfect,thereareoccasionallytwomonthswiththesamenumber,
inwhichcasethefirstofthemisregardedasaleapmonth,andoccasionally
anomitteddateortwodayswiththesamedate(inthiscaseIamnotsure
whichonetoregardasleapday,soIwillcallthemearlierandlater).Unlike
modernIndiancalendars,therearenoomittedmonths.
Variousimprovementsofthecalculationshasbeensuggestedoverthe
centuries,see[4]and[3,ChapterVI],anddierenttraditionsfollowdierent
rulesforthedetailsofthecalculation.Thereareatleasttwoversions
(PhugpaandTsurphu)oftheTibetancalendarinusetodaybydierent
groupsinsideandoutsideTibet,seeAppendixB.Thedierentversions
frequentlydierbyadayoramonth.
ThedescriptionbelowreferstothePhugpatradition,introduced1447,
whichisfollowedbytheDalaiLamaandisthemostcommonversion;itcan
beregardedasthestandardversionoftheTibetancalendar.Thedierences
intheTsurphutraditionarediscussedinAppendixB,butsomedetails
remainobscuretome.
IwillseparatethediscussionoftheTibetancalendarintotwomainparts.
Inthefirstpart(Sections4–6),themonthsareregardedasunitsandIdis-
cusshowtheyarenumbered,whichimpliesthepartitioningoftheminto
yearsandalsoshowswhichmonthsthatareleapmonths.Inthesecondpart
Date:December31,2007(Sunday23,month11,Fire–Pigyear).
1
2
SVANTEJANSON
(Sections7–10),Idiscussthecouplingbetweenmonthsanddays,including
findingtheactualdayswhenamonthbeginsandendsandthenumbering
ofthedays.Finally,somefurthercalculationsaredescribed(Section11)
andsomemathematicalconsequencesaregiven(Sections13–14).Calcula-
tionsfortheplanetsandotheratrologicalcalculationsaredescribedinthe
appendices.
ThedescriptionisbasedmainlyonthebooksbySchuh[4]andHenning
[3],buttheanalysisandmathematicalformulationsareoftenmyown.For
furtherstudyIrecommendinparticularthedetailedrecentbookbyHenning
[3],whichcontainsmuchmorematerialthanthispaper.
Acknowledgements.ThisstudywasmadepossiblebythehelpofNachum
Dershowitz,EdwardReingoldandEdwardHenning,whohaveprovidedme
withessentialreferencesandgivenmehelpfulcomments.
2.Notation
Mixedradixnotation.TraditionalTibetancalculationsaremadewith
rationalnumbersexpressedinapositionalsystemwithmixedradices.I
usuallyusestandardnotationforrationalnumbers,butwhencomparing
themwiththetraditionalexpressions,Iusenotationsofthetype(ofvarying
length)
a 0 ;a 1 ,a 2 ,a 3 (b 1 ,b 2 ,b 3 ) meaning a 0 + a 1 +(a 2 +(a 3 /b 3 ))/b 2
b 1
.
Formally,wehavetheinductivedefinition
a 0 ;a 1 ,a 2 ,...,a n (b 1 ,b 2 ,...,b n ):=a 0 + a 1 ;a 2 ,...,a n (b 2 ,...,b n )
b 1
forn2,anda 0 ;a 1 (b 1 ):=a 0 +a 1 /b 1 .Wewillusuallyomitaleading
0.(ThisnotationistakenfromHenning[3],althoughheusuallyomits
allormostoftheradicessincetheyaregivenbythecontext.Schuh
[4]usesthesimilarnotation[a 1 ,a 2 ,...,a n ]/(b 1 ,b 2 ,...,b n )meaningeither
0;a 1 ,a 2 ,...,a n (b 1 ,b 2 ,...,b n )ora 1 ;a 2 ,...,a n (b 2 ,...,b n ).)
Angularunits.Itwillbeconvenient,althoughsomewhatunconventional,
toexpresslongitudesandotherangularmeasurementsinunitsoffullcir-
cles.Toobtainvaluesindegrees,thenumbersshouldthusbemultiplied
by360.(ATibetanastrologerwouldprobablyprefermultiplyingby27to
obtainlunarmansions(=lunarstation=naksatra)andfractionsthereof.A
Westernastrologermightprefermultiplyingby12toobtainvaluesinsigns.
Amathematicianmightprefermultiplyingby2(radians).)
Forangularmeasurements,fullcirclesareoftentobeignored(butsee
below);thismeanswithourconventionthatthenumbersareconsidered
modulo1,i.e.,thatonlythefractionalpartmatters.
TIBETANCALENDARMATHEMATICS
3
Booleanvariables.ForaBooleanvariable`,i.e.avariabletakingoneof
thevaluestrueandfalse,weuse`:={P}todenotethat`=trueifand
onlyiftheconditionPholds;wefurtherlet[`]bethenumberdefinedby
[`]=1when`=trueand[`]=0when`=false.
Juliandaynumber.TheJuliandaynumber(whichweabbreviatebyJD)
foragivendayisthenumberofdaysthathaveelapsedsincetheepoch1
January4713BC(Julian);fordaysbeforetheepoch(whichhardlyconcern
theTibetancalendar),negativenumbersareused.TheJuliandaynumbers
thusformacontinuousnumberingofalldaysby...,−1,0,1,2,....Such
anumberingisveryconvenientformanypurposes,includingconversions
betweencalendars.Thechoiceofepochforthedaynumbersisarbitrary
andformostpurposesunimportant;theconventionaldate1January4713
BC(−4712withastronomicalnumberingofyears)wasoriginallychosenby
Scalingerin1583astheoriginoftheJulianperiod,a(cyclic)numbering
ofyears,andwaslateradaptedintoanumberingofdays.Seefurther[2,
Section12.7].
DershowitzandReingold[1]useanotherdaynumber,denotedbyRD,
withanotherepoch:RD1is1January1(Gregorian)whichisJD1721426.
Consequently,thetwodaynumbersarerelatedbyJD=RD+1721425.
TheJuliandaynumberwasfurtherdevelopedintotheJuliandate,which
isarealnumberthatdefinesthetimeofaparticularinstance.Thefractional
partoftheJuliandateshowsthefractionofadaythathaselapsedsince
noonGMT(UT);thus,ifnisaninteger,thentheJuliandateisn.0(i.e.,
exactlyn)atnoonGMTonthedaywithJuliandaynumbern.
ItisimportanttodistinguishbetweentheJuliandaynumberandthe
Juliandate,eveniftheyarecloselyrelated.Bothareextremelyuseful,
butfordierentpurposes,andmuchconfusionhasbeencausedbymixing
them.(Wefollow[2]inusingslightlydierentnamesforthem,butthatis
notalwaysdonebyotherauthors.)Forthisstudy,andmostotherworkon
calendars,theJuliandaynumberistheimportantconcept.Notethatthe
Juliandaynumberisaninteger,whiletheJuliandateisarealnumber.(A
computerscientistwouldsaythattheyhavedierenttypes.)Moreover,the
Juliandaynumbernumbersthedaysregardlessofwhentheybeginandend,
whiletheJuliandatedependsonthetimeofday,atGreenwich.Hence,to
convertaJuliandatetoaJuliandaynumber,weneedinpracticeknowboth
thelocaltimethedaybeginsandthetimezone,whiletheseareirrelevant
forcalculationswiththeJuliandaynumber.Forexample,1January2007
hasJD2454102,everywhere.ThustheJuliandate2454102.0is1January
2007,noonGMT(UT),andthenewyearbeganatJuliandate2454101.5
inBritain,butatotherJuliandatesinothertimezones.TheTibetanday
beginsatdawn,about5amlocaltime(seeRemark6below),butwedonot
havetofindtheJuliandateofthatinstance.
Othernotations.Weletmoddenotethebinaryoperationdefinedby
mmodn=xifxm(modn)and0x<n(weonlyconsidern>0,

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