{"text_format":"3","workbook":"xlsx","n_sheets":13,"sheets":["Control","Sun","Pattern","Selected_LS_Cycles","Moon","Computus","Ford","Astro","Eclipse_Cycles","PlotData","Legend","Desc","Factors"],"not_rendered":{"charts":2,"macros":1,"defined_names":73,"comments":4},"read":"whole","date_system":"1904"}

{"sheet":"Control","index":0,"n_rows":33,"n_cols":15,"rows_kept":33,"merged":[[3,11,3,12],[3,13,3,14],[4,2,4,3],[21,2,21,3]]}
["Fixed Arithmetic Calendar Cycles and their Ford Circles",{"$nulls":13},"Version 7.11(187), Revised 2022.06.23"]
["by Dr. Irv Bromberg, University of Toronto, Canada",{"$nulls":9},true]
["http://individual.utoronto.ca/kalendis/leap/#ford",{"$nulls":4},"Circle Factor: ",45000," (hover to see red-tag comments)"]
[{"$nulls":10},"Max","Long Mixer",null,"Short Mixer"]
[null,"Floor of","Mean Year Limits",null,"Range","Days per","Days per","Length of","Months per","Months per","Years/","Leaps/","Years/","Leaps/","Years/"]
["Click Desired Cycle Type","Mean Year","Longest","Shortest","m:ss","Short Year","Long Year","Leap Unit","Short Year","Long Year","Cycle","Cycle","Cycle","Cycle","Cycle"]
[null,"365 days","05:50:00","05:47:30",{"$f":"C7-D7","v":"00:02:30"},365,366,{"$f":"IF(isLeapLong(),G7-F7,F7-G7)","v":{"v":1,"d":"+1"}},12,12,450,{"$f":"Mixers(B7,C7,D7,F7,H7)","v":1},4,6,25]
[null,"365 days","05:50:00","05:47:35",{"$f":"C8-D8","v":"00:02:25"},365,370,{"v":5,"d":"+5"},12,12,1000,{"$f":"Mixers(B8,C8,D8,F8,H8)","v":2},41,1,21]
[null,"365 days","05:50:00","05:47:35","00:02:25",360,366,{"v":6,"d":"+6"},12,12,1100,{"$f":"Mixers(B9,C9,D9,F9,H9)","v":7},8,69,79]
[null,"365 days","05:49:55","05:47:35","00:02:20",364,371,{"v":7,"d":"+7"},12,12,1000,{"$f":"Mixers(B10,C10,D10,F10,H10)","v":8},45,3,17]
[null,"365 days","05:49:50","05:47:35","00:02:15",360,368,{"v":8,"d":"+8"},15,15,1000,{"$f":"Mixers(B11,C11,D11,F11,H11)","v":2},3,17,26]
[null,"365 days","05:50:00","05:47:35","00:02:25",360,370,{"v":10,"d":"+10"},12,12,1500,{"$f":"Mixers(B12,C12,D12,F12,H12)","v":43},82,11,21]
[null,"365 days","05:50:00","05:47:05","00:02:55",{"$f":"19*I13","v":361},{"$f":"19*J13","v":380},{"v":19,"d":"+19"},19,20,2000,{"$f":"Mixers(B13,C13,D13,F13,H13)","v":21},94,2,9]
[null,"365 days","05:49:55","05:47:35","00:02:20",{"$f":"28*I14","v":364},{"$f":"28*J14","v":392},{"v":28,"d":"+28"},13,14,2500,{"$f":"Mixers(B14,C14,D14,F14,H14)","v":2},45,7,158]
[null,"365 days","05:50:00","05:46:55","00:03:05",{"$f":"30*I15","v":360},{"$f":"30*J15","v":390},{"v":30,"d":"+30"},12,13,2500,{"$f":"Mixers(B15,C15,D15,F15,H15)","v":7},40,11,63]
[null,"365 days","05:50:05","05:47:15","00:02:50",{"$f":"I16*ABS(H16)","v":354.36699551569507},{"$f":"J16*ABS(H16)","v":383.89757847533633},{"$f":"(29+FPC/MPC)*SIGN(H7)","v":29.530582959641254},12,13,2000,{"$f":"Mixers(B16,C16,D16,F16,H16)","v":7},19,102,277]
[null,"668 sols","14:20:00","14:04:00","00:16:00",668,669,{"$f":"IF(isLeapLong(),G17-F17,F17-G17)","v":{"v":1,"d":"+1"}},24,24,200,{"$f":"Mixers(B17,C17,D17,F17,H17)","v":3},5,7,12]
[null,"668 sols","14:20:00","14:02:55",{"$f":"C18-D18","v":"00:17:05"},{"$f":"24*28-7","v":665},{"$f":"24*28","v":672},{"$f":"IF(isLeapLong(),G18-F18,F18-G18)","v":{"v":7,"d":"+7"}},24,24,500,{"$f":"Mixers(B18,C18,D18,F18,H18)","v":18},35,1,2]
[null,"668 sols","14:20:00","14:02:55","00:17:05",644,{"$f":"24*28","v":672},{"$f":"IF(isLeapLong(),G19-F19,F19-G19)","v":{"v":28,"d":"+28"}},23,24,1000,{"$f":"Mixers(B19,C19,D19,F19,H19)","v":29},33,7,8]
[]
[null,null,"Maximum Haros Denominator:",99,"<http://en.wikipedia.org/w/index.php?title=Charles_Haros>",{"$nulls":5},"Max","Long Mixer",null,"Short Mixer"]
[null,"Floor of","Mean Month Limits",null,"Range","Days per","Days per","Days per","Months per","Months per","Months /","Full /","Months","Full /","Months"]
[null,"Mean Month","Longest","Shortest","(secs)","Short Yerm","Long Yerm","49 Months","Short Yerm","Long Yerm","Cycle","Cycle","/ Cycle","Cycle","/ Cycle"]
[null,"29 days","12:44:04",{"$f":"191/360","v":"12:44:00"},{"$f":"86400*(C24-D24)","v":{"v":3.999999999997428,"d":"4.0"}},{"$f":"(INT(I24/2)+1)*30+INT(I24/2)*29","v":443},{"$f":"(INT(J24/2)+1)*30+INT(J24/2)*29","v":502},{"$f":"2*G24+F24","v":1447},15,17,2000,{"$f":"Mixers(B24,C24,D24,F24,G24-F24)","v":26},49,191,360]
[]
[]
["§ Set the Mean Month",null,"Full Months","Months",null,"Days","Yerms",{"$nulls":7},{"$f":"G28&\" yerms, mean month = 29d \"&ExactTime(FPC,MPC)&\" = 29+\"&ReduceFraction(C28,D28,FALSE,FALSE,TRUE)&\" days \"","v":"341 yerms, mean month = 29d 12h 44m 2+82/223s = 29+2958/5575 days "}]
["   for Lunisolar Cycles:","Per Cycle: ",2958,5575,null,{"$f":"FPC*30+(MPC-FPC)*29","v":164633},{"$f":"2*F28-59*MPC","v":341},{"$nulls":7},{"$f":"\" or about 29.\"&MID(FPC/MPC,3,20)&\" days or about 29d \"&TEXT(FPC/MPC,\"HH:MM:SS.000 \")","v":" or about 29.530582959641256 days or about 29d 12:44:02.368 "}]
[]
[null,"or click on a preset cycle: "]
[]
[]
["http://en.wikipedia.org/wiki/Ford_circle",null,null,"http://precollegiate.stanford.edu/circle/math/FareyFord.pdf",{"$nulls":10},"<http://mathworld.wolfram.com/FordCircle.html>"]

{"sheet":"Sun","index":1,"n_rows":12,"n_cols":13,"rows_kept":12,"merged":[[10,3,10,4],[10,6,10,7],[10,10,10,11]]}
[{"$nulls":3},"Maximum Years Per Cycle:",1000,null,"Length of…","sols","months"]
[{"$nulls":3},"Count of Cycles Listed Below:",{"$f":"COUNTIF(C:C,\">0\")","v":0},null,"…Short Year:",644,23]
[{"$nulls":3},"Count of Visible Cycles:",{"$f":"CountVisible(\"Sun\",13,3)","v":0},null,"…Long Year:",672,24]
["Fraction marked with x# indicates length is",{"$nulls":5},"...Leap Unit:",28,{"$f":"LYM-SYM","v":1}]
["# times the equivalent skip sol calendar.",{"$nulls":6},"Longest","Shortest"]
[{"$nulls":6},"Mean Year Limits (fraction >668 sols)","14:20:00","14:02:55"]
[]
[]
[]
["The mediant fraction between a/c and b/d = (a+b) / (c+d).  Circles touch only if bc – ad = 1."]
["nth","Long","Years","Mixers",null,"Cycle Mean","Mean Year >668 sols",null,"Decimal","Sols per","SymLeap",null,"Average Leap"]
["Order","/cycle","/cycle","33","8","Year (sols)","Decimal","Exact Time","Seconds","Cycle","U","K","Interval (years)"]

{"sheet":"Pattern","index":2,"n_rows":0,"n_cols":0,"rows_kept":0}

{"sheet":"Selected_LS_Cycles","index":3,"n_rows":54,"n_cols":17,"rows_kept":54,"merged":[[2,2,2,3],[2,5,2,6],[2,9,2,10]]}
["Selected Lunisolar Cycles"]
[]
["Long","Years","Mixers",null,"Cycle Mean","Mean Year >365 days",null,"Decimal","Months per","SymLeap",null,"Average Leap","Solar : Lunar","Solar, Lunar","Lunar Cycle","Whole"]
["/cycle","/cycle","19","277","Year (days)","Decimal","Exact Time","Seconds","Cycle","U","K","Interval (years)","Months Ratio","Months GCD","Yerms","#Weeks","Comments"]
[734,1993,32,5,"365+484/1993",{"$f":"484/1993","v":0.24284997491219268},"5h 49m 42+474/1993s",{"$f":"42+474/1993","v":42.237832413447066},24650,1192,996,"2+525/734","29 : 1",850,52,{"$f":"IF(MOD(ROUND(B5*(365+F5),2),7)=0,\"Yes\",\"\")","v":""}]
[2901,7877,123,20,"365+426386/1756571",{"$f":"426386/1756571","v":0.2427376974799197},"5h 49m 32+943388/1756571s",{"$f":"32+943388/1756571","v":32.53706226506073},97425,2756,3938,"2+2075/2901","3897 : 223",25,341,""]
[2427,6590,99,17,"365+17915/73808",{"$f":"17915/73808","v":0.24272436592239324},"5h 49m 31+1777/4613s",{"$f":"31+1777/4613","v":31.385215694775635},81507,4263,3295,"2+1736/2427","807 : 56",101,346,""]
[2167,5884,91,15,"365+318483/1312132",{"$f":"318483/1312132","v":0.24272176884642704},"5h 49m 31+52757/328033s",{"$f":"31+52757/328033","v":31.160828331295935},72775,391,2942,"2+1550/2167","2911 : 223",25,341,""]
[2132,5789,86,15,"365+1405/5789",{"$f":"1405/5789","v":0.24270167559163933},"5h 49m 29+2459/5789s",{"$f":"29+2459/5789","v":29.424771117636897},71600,4244,2894,"2+1525/2132","10 : 1",7160,438,"","10 x 20 inex"]
[1433,3891,59,10,"365+210580/867693",{"$f":"210580/867693","v":0.24268952267679927},"5h 49m 28+108392/289231s",{"$f":"28+108392/289231","v":28.374759275458025},48125,3500,1945,"2+1025/1433","1925 : 223",25,341,""]
[2817,7649,111,20,"365+18563/76490",{"$f":"18563/76490","v":0.24268531834226698},"5h 49m 28+88/7649s",{"$f":"28+88/7649","v":28.011504771865603},94605,4206,3824,"2+2015/2817","1113 : 10",85,52,""]
[987,2680,39,7,"365+1691/6968",{"$f":"1691/6968","v":0.2426808266360505},"5h 49m 27+543/871s",{"$f":"27+543/871","v":27.623421354764638},33147,763,1340,"2+706/987","261 : 13",127,101,""]
[1349,3663,47,10,"365+987/4070",{"$f":"987/4070","v":0.2425061425061425},"5h 49m 12+216/407s",{"$f":"12+216/407","v":12.530712530712531},45305,1097,1831,"2+965/1349","533 : 10",85,52,""]
[2561,6954,89,19,"365+5059/20862",{"$f":"5059/20862","v":0.2424983223085035},"5h 49m 11+991/1159s",{"$f":"11+991/1159","v":11.85504745470233},86009,1097,3477,"2+1832/2561","77 : 3",1117,205,""]
[3102,8423,108,23,"365+40849/168460",{"$f":"40849/168460","v":0.24248486287546006},"5h 49m 10+5830/8423s",{"$f":"10+5830/8423","v":10.692152439748309},104178,4760,4211,"2+2219/3102","291 : 20",358,438,null,"291 inex"]
[2287,6210,79,17,"365+6776/27945",{"$f":"6776/27945","v":0.2424762927178386},"5h 49m 9+197/207s",{"$f":"9+197/207","v":9.951690821256038},76807,5113,3105,"2+1636/2287","863 : 9",89,49,{"$f":"IF(MOD(ROUND(B16*(365+F16),2),7)=0,\"Yes\",\"\")","v":""}]
[2129,5781,71,16,"365+18220/75153",{"$f":"18220/75153","v":0.24243875826646974},"5h 49m 6+17754/25051s",{"$f":"6+17754/25051","v":6.70871422298511},71501,2528,2890,"2+1523/2129","563 : 13",127,101,{"$f":"IF(MOD(ROUND(B17*(365+F17),2),7)=0,\"Yes\",\"\")","v":""}]
[2923,7937,97,22,"365+17317/71433",{"$f":"17317/71433","v":0.2424229697758739},"5h 49m 5+2735/7937s",{"$f":"5+2735/7937","v":5.344588635504599},98167,1803,3968,"2+2091/2923","1103 : 9",89,49,{"$f":"IF(MOD(ROUND(B18*(365+F18),2),7)=0,\"Yes\",\"\")","v":""}]
[3587,9740,119,27,"365+94447/389600",{"$f":"94447/389600","v":0.24242043121149898},"5h 49m 5+61/487s",{"$f":"5+61/487","v":5.125256673511293},120467,1803,4870,"2+2566/3587","673 : 40",179,438,null,"673 / 2 inex"]
[3102,8423,108,23,"365+38792/160037",{"$f":"38792/160037","v":0.24239394639989503},"5h 49m 2+133946/160037s",{"$f":"2+133946/160037","v":2.8369689509300975},104178,4760,4211,"2+2219/3102","291 : 19",358,416,{"$f":"IF(MOD(ROUND(B20*(365+F20),2),7)=0,\"Yes\",\"\")","v":""},"291 inex"]
[1964,5333,62,15,"365+6463/26665",{"$f":"6463/26665","v":0.24237764860303768},"5h 49m 1+2287/5333s",{"$f":"1+2287/5333","v":1.4288393024564034},65960,2843,2666,"2+1405/1964","388 : 5",170,52,""]
[664,1803,22,5,"365+437/1803",{"$f":"437/1803","v":0.24237382140876318},"5h 49m 1+59/601s",{"$f":"1+59/601","v":1.098169717138103},22300,1078,901,"2+475/664","4 : 1",5575,341,"Yes","4 x 25 saros"]
[1047,2843,33,8,"365+689/2843",{"$f":"689/2843","v":0.24234963067182555},"5h 48m 59+23/2843s",{"$f":"59+23/2843","v":59.00809004572635},35163,353,1421,"2+749/1047","9 : 1",3907,239,""]
[3587,9740,119,27,"365+89691/370120",{"$f":"89691/370120","v":0.24232951475197234},"5h 48m 57+2499/9253s",{"$f":"57+2499/9253","v":57.2700745704096},120467,1803,4870,"2+2566/3587","673 : 38",179,416,"","673 / 2 inex"]
[1964,5333,62,15,"365+3877/15999",{"$f":"3877/15999","v":0.24232764547784236},"5h 48m 57+579/5333s",{"$f":"57+579/5333","v":57.10856928558035},65960,2843,2666,"2+1405/1964","136 : 3",485,89,""]
[2463,6688,75,19,"365+12965/53504",{"$f":"12965/53504","v":0.24231833133971292},"5h 48m 56+127/418s",{"$f":"56+127/418","v":56.303827751196174},82719,6335,3344,"2+1762/2463","117 : 8",707,346,""]
[2621,7117,83,20,"365+384578/1587091",{"$f":"384578/1587091","v":0.24231628810194247},"5h 48m 56+202024/1587091s",{"$f":"56+202024/1587091","v":56.127292007830675},88025,2490,3558,"2+1875/2621","3521 : 223",25,341,""]
[2579,7003,77,20,"365+16969/70030",{"$f":"16969/70030","v":0.2423104383835499},"5h 48m 55+4355/7003s",{"$f":"55+4355/7003","v":55.62187633871198},86615,4551,3501,"2+1845/2579","1019 : 10",85,52,""]
[1957,5314,61,15,"365+287127/1185022",{"$f":"287127/1185022","v":0.24229676748617326},"5h 48m 54+261126/592511s",{"$f":"54+261126/592511","v":54.440710805369015},65725,353,2657,"2+1400/1957","2629 : 223",25,341,""]
[636,1727,18,5,"365+1255/5181",{"$f":"1255/5181","v":0.242231229492376},"5h 48m 48+1344/1727s",{"$f":"48+1344/1727","v":48.77822814128547},21360,687,863,"2+455/636","80 : 3",267,49,""]
[1142,3101,32,9,"365+1395/5759",{"$f":"1395/5759","v":0.24222955374196908},"5h 48m 48+3648/5759s",{"$f":"48+3648/5759","v":48.633443306129536},38354,687,1550,"2+817/1142","302 : 13",127,101,""]
[1852,5029,46,15,"365+271449/1121467",{"$f":"271449/1121467","v":0.24204813873257083},"5h 48m 32+1075696/1121467s",{"$f":"32+1075696/1121467","v":32.95918649411887},62200,334,2514,"2+1325/1852","2488 : 223",25,341,""]
[2439,6623,57,20,"365+20840/86099",{"$f":"20840/86099","v":0.2420469459575605},"5h 48m 32+73712/86099s",{"$f":"32+73712/86099","v":32.856130733225704},81915,4304,3311,"2+1745/2439","645 : 13",127,101,""]
[2801,7606,65,23,"365+8284/34227",{"$f":"8284/34227","v":0.24203114500248343},"5h 48m 31+1867/3803s",{"$f":"31+1867/3803","v":31.490928214567447},94073,5621,3803,"2+2004/2801","1057 : 9",89,49,""]
[485,1317,11,4,"365+12749/52680",{"$f":"12749/52680","v":0.2420083523158694},"5h 48m 29+229/439s",{"$f":"29+229/439","v":29.521640091116172},16289,983,658,"2+347/485","91 : 40",179,438,"","91 / 2 inex"]
[1223,3321,29,10,"365+179224/740583",{"$f":"179224/740583","v":0.24200393473790244},"5h 48m 29+3839/27429s",{"$f":"29+3839/27429","v":29.1399613547705},41075,2987,1660,"2+875/1223","1643 : 223",25,341,""]
[2165,5879,47,18,"365+12803/52911",{"$f":"12803/52911","v":0.24197236869460037},"5h 48m 26+2426/5879s",{"$f":"26+2426/5879","v":26.41265521347168},72713,1632,2939,"2+1549/2165","817 : 9",89,49,""]
[1817,4934,41,15,"365+266223/1100282",{"$f":"266223/1100282","v":0.24195887963267598},"5h 48m 25+135995/550141s",{"$f":"25+135995/550141","v":25.24720026320525},61025,3617,2467,"2+1300/1817","2441 : 223",25,341,""]
[2411,6547,53,20,"365+353222/1459981",{"$f":"353222/1459981","v":0.24193602519484841},"5h 48m 23+397957/1459981s",{"$f":"23+397957/1459981","v":23.272576834904015},80975,5564,3273,"2+1725/2411","3239 : 223",25,341,""]
[833,2262,17,7,"365+10215/42224",{"$f":"10215/42224","v":0.24192402425161047},"5h 48m 22+622/2639s",{"$f":"22+622/2639","v":22.235695339143614},27977,1613,1131,"2+596/833","277 : 56",101,346,""]
[485,1317,11,4,"365+12107/50046",{"$f":"12107/50046","v":0.2419174359589178},"5h 48m 21+5559/8341s",{"$f":"21+5559/8341","v":21.666466850497542},16289,983,658,"2+347/485","91 : 38",179,416,"","91 / 2 inex"]
[1536,4171,30,13,"365+1009/4171",{"$f":"1009/4171","v":0.24190841524814194},"5h 48m 20+3700/4171s",{"$f":"20+3700/4171","v":20.88707743946296},51588,3207,2085,"2+1099/1536","12 : 1",4299,263,"Yes"]
[2341,6357,43,20,"365+1025/4238",{"$f":"1025/4238","v":0.2418593676262388},"5h 48m 16+1376/2119s",{"$f":"16+1376/2119","v":16.64936290703162},78625,2224,3178,"2+1675/2341","185 : 2",425,52,""]
[2837,7704,41,25,"365+2069/8560",{"$f":"2069/8560","v":0.24170560747663553},"5h 48m 3+39/107s",{"$f":"3+39/107","v":3.364485981308411},95285,3389,3852,"2+2030/2837","1121 : 10",85,52,""]
[1821,4945,27,16,"365+239/989",{"$f":"239/989","v":0.2416582406471183},"5h 47m 59+269/989s",{"$f":"59+269/989","v":59.27199191102123},61161,926,2472,"2+1303/1821","87 : 1",703,43,""]
[2271,6167,33,20,"365+47474/196463",{"$f":"47474/196463","v":0.24164346467273737},"5h 47m 57+195549/196463s",{"$f":"57+195549/196463","v":57.99534772450792},76275,5241,3083,"2+1625/2271","3051 : 223",25,341,""]
[1684,4573,22,15,"365+65/269",{"$f":"65/269","v":0.241635687732342},"5h 47m 57+87/269s",{"$f":"57+87/269","v":57.32342007434944},56560,3962,2286,"2+1205/1684","10 : 1",5656,346,""]
[1452,3943,18,13,"365+12385/51259",{"$f":"12385/51259","v":0.2416161064398447},"5h 47m 55+32375/51259s",{"$f":"55+32375/51259","v":55.631596402582964},48768,2425,1971,"2+1039/1452","384 : 13",127,101,""]
[893,2425,11,8,"365+5273/21825",{"$f":"5273/21825","v":0.2416036655211913},"5h 47m 54+54/97s",{"$f":"54+54/97","v":54.55670103092783},29993,907,1212,"2+639/893","337 : 9",89,49,""]
[1111,3017,13,10,"365+7289/30170",{"$f":"7289/30170","v":0.2415976135233676},"5h 47m 54+102/3017s",{"$f":"54+102/3017","v":54.03380841895923},37315,2110,1508,"2+795/1111","439 : 10",85,52,""]
[109,296,2,1,"365+143/592",{"$f":"143/592","v":0.24155405405405406},"5h 47m 50+10/37s",{"$f":"50+10/37","v":50.270270270270274},3661,277,148,"2+78/109","7 : 2",523,64,""]
[3203,8698,35,29,"365+84039/347920",{"$f":"84039/347920","v":0.24154690733501955},"5h 47m 49+2839/4349s",{"$f":"49+2839/4349","v":49.65279374568866},107579,1499,4349,"2+2292/3203","601 : 40",179,438,"","601 / 2 inex"]
[2718,7381,24,25,"365+7129/29524",{"$f":"7129/29524","v":0.24146457119631487},"5h 47m 42+3978/7381s",{"$f":"42+3978/7381","v":42.53895136160412},91290,7085,3690,"2+1945/2718","51 : 4",1790,438,"","12+3/4 x 20 = 255 inex"]
[3203,8698,35,29,"365+79807/330524",{"$f":"79807/330524","v":0.24145599109293123},"5h 47m 41+65909/82631s",{"$f":"41+65909/82631","v":41.79763042925779},107579,1499,4349,"2+2292/3203","601 : 38",179,416,"","601 / 2 inex"]

{"sheet":"Moon","index":4,"n_rows":12,"n_cols":13,"rows_kept":12,"merged":[[3,6,3,7],[10,3,10,4],[10,6,10,7]]}
[{"$nulls":3},"Maximum Months Per Cycle:",2000,null,null,"Length of…","days","months"]
[{"$nulls":3},"Count of Cycles Listed Below:",{"$f":"COUNTIF(C:C,\">0\")","v":0},null,null,"…Short Yerm:",443,15,{"$f":"\" \"&IF(365.25-SYD>LYD-365.25,\"leap\",\"common\")","v":" common"}]
[{"$nulls":3},"Count of Visible Cycles:",{"$f":"CountVisible(\"Moon\",13,3)","v":0},null,null,"…Long Yerm:",502,17,{"$f":"\" \"&IF(365.25-SYD>LYD-365.25,\"common\",\"leap\")","v":" leap"}]
[{"$nulls":6},"Hover here to see Yerms comment"]
[null,"Include:",{"$nulls":6},"Longest","Shortest"]
[null,null,true,true,{"$nulls":3},"Mean Month Limits (fraction >29 days)",0.5306018518518518,0.5305555555555556]
[]
[]
[]
["The mediant fraction between a/c and b/d = (a+b) / (c+d).  Circles touch only if bc – ad = 1."]
["nth","Full","Months","Mixers",null,"Cycle Mean","Mean Month >29 days",null,"Seconds","Days per",null,"Yerm"]
["Order","/cycle","/cycle","49","360","Month (days)","Decimal","Exact Time",">29d 12h 44m","Cycle","Yerms","Eras","Comments"]

{"sheet":"Computus","index":5,"n_rows":9,"n_cols":12,"rows_kept":9,"merged":[[1,3,1,4],[2,3,2,4],[7,3,7,4]]}
["Computus Paschalis Lunisolar Cycles for the following selected Solar Cycle:"]
[]
[null,"Years per","Cycle Mean","Mean Year >365 days",null,"Days per"]
[null,"Solar Cycle","Year (days)","Days","Exact Time","Solar Cycle",null,81]
[null,293,"365+71/293",{"$f":"71/293","v":0.24232081911262798},"5h 48m 56+152/293s",107016]
[]
[{"$f":"\"All of the following lunisolar cycles have a mean year of exactly \"&C5&\" days:\"","v":"All of the following lunisolar cycles have a mean year of exactly 365+71/293 days:"},{"$nulls":4},"(Excel versions prior to 2007 are limited to autofiltering 1000 unique items per column.)"]
["Yerm","Full","Months","Mean Month >29 days",null,"Seconds","Days per","Lunar Cycles","Solar Cycles","Years per","Months per","Days per"]
["Eras","/cycle","/cycle","Decimal","Exact Time",">29d 12h 44m","Lunar Cycle","per Computus","per Computus","Computus","Computus","Computus"]

{"sheet":"Ford","index":6,"n_rows":4,"n_cols":9,"rows_kept":4}
[{"$nulls":8},"03:00:00"]
[0,{"$nulls":6},0.12188365650969529,"02:54:32.727"]
[{"$nulls":7},0.12155260469867211,14.897331496043012]
["1x"]

{"sheet":"Astro","index":7,"n_rows":27,"n_cols":10,"rows_kept":27}
["Astronomical Settings",{"$nulls":3},"http://individual.utoronto.ca/kalendis/leap/#ford"]
[]
[null,"Target Year (-1000 to 4000):",2025]
[]
["Choose your preferred color codes for best and good matches to equinoctial or solstitial mean solar years:"]
["(\"Change\" is number of mean solar seconds per century, absolute value <1 is reasonably stable.)"]
[]
["Solar Longitude","Earth Mean Year",{"$nulls":3},"Color Codes",null,"Martian Mean"]
["(ecliptic degrees)","(mean solar days)","Time >365d","Change","Description","for Ford chart",null,"Year (mean sols)","Time >668d"]
[0,{"$f":"MeanEqSolstYear(TargetYear,A10)","v":{"v":365.2423696262849,"d":"365.242369626285"}},{"$f":"MOD(B10,1)","v":"05:49:00.736"},{"$f":"MeanEqSolstYearChange(TargetYear,A10)","v":{"v":0.12521254975581542,"d":"+0.13"}},"Green = Northward Equinoctial","Best","Good",668.5907,{"$f":"MOD(H10,1)","v":"14:10:36.480"}]
[{"$f":"A10+90","v":90},{"$f":"MeanEqSolstYear(TargetYear,A11)","v":{"v":365.24161918587197,"d":"365.241619185872"}},{"$f":"MOD(B11,1)","v":"05:47:55.898"},{"$f":"MeanEqSolstYearChange(TargetYear,A11)","v":{"v":-0.6960731188883074,"d":"–0.70"}},"Red = North Solstitial","Best","Good",{"v":668.588,"d":"668.5880"},{"$f":"MOD(H11,1)","v":"14:06:43.200"}]
[{"$f":"A11+90","v":180},{"$f":"MeanEqSolstYear(TargetYear,A12)","v":{"v":365.2420048324566,"d":"365.242004832457"}},{"$f":"MOD(B12,1)","v":"05:48:29.218"},{"$f":"MeanEqSolstYearChange(TargetYear,A12)","v":{"v":-2.7593263221206143,"d":"–2.76"}},"Brown = Southward Equinoctial","Best","Good",{"v":668.594,"d":"668.5940"},{"$f":"MOD(H12,1)","v":"14:15:21.600"}]
[{"$f":"A12+90","v":270},{"$f":"MeanEqSolstYear(TargetYear,A13)","v":{"v":365.2427302748256,"d":"365.242730274826"}},{"$f":"MOD(B13,1)","v":"05:49:31.896"},{"$f":"MeanEqSolstYearChange(TargetYear,A13)","v":{"v":-1.8513646500650793,"d":"–1.85"}},"Blue = South Solstitial","Best","Good",668.5958,{"$f":"MOD(H13,1)","v":"14:17:57.120"}]
[{"$nulls":7},"\"Best\" Within: ",20,"seconds"]
[null,"Mean Lunar Month","Time >29 days",{"$nulls":4},"\"Good\" Within: ",60,"seconds"]
[null,"(mean solar days)","HH:MM","Seconds"]
["Synodic Month:",{"$f":"SynodicMonth(TargetYear)","v":29.530588798951158},{"$f":"INT(MOD(B17,1)*1440)/1440","v":"12:44:00"},{"$f":"MOD(86400*MOD(B17,1),60)","v":{"v":2.8722293800092302,"d":"2.87"}},"mean lunar conjunction interval","Best"]
[]
["Note:  Earth equinoctial and solstitial years, and Mean Lunar Month vary with Target Year setting."]
["(Martian years lengths should also vary over the years, but formula not found yet.)"]
[]
["Source for Martian equinoctial and solsticial years = last paragraph of the following section:"]
["http://en.wikipedia.org/w/index.php?title=Timekeeping_on_Mars#Martian_year"]
["…which is consistent with Figure 3 of page 226 of: Allison M, McEwen M."]
["\"A post-Pathfinder evaluation of areocentric solar coordinates with improved timing recipes for Mars seasonal/diurnal climate studies.\""]
["Planetary and Space Science 2000; 48: 215-235."]
["http://homepage.mac.com/pclwillmott/GAL/2000_Allison_McEwen.pdf"]

{"sheet":"Eclipse_Cycles","index":8,"n_rows":30,"n_cols":7,"rows_kept":30}
["About Eclipse Cycles",{"$nulls":5},"http://individual.utoronto.ca/kalendis/leap/#ford"]
[]
["The saros cycle has 223 mean synodic months, about 242 draconic months, 239 anomalistic months, or about 18 years 11+1/3 days."]
["One saros later an eclipse repeats under similar conditions but 1/3 of the way (120°) further west around the globe and about 1000/3 km further north"]
["or further south, depending on whether the series started at the south pole (ascending lunar node) or north pole (descending lunar node), respectively."]
["The exeligmos cycle is 3 saros cycles, containing almost an integer number of days (19756)."]
["One exeligmos later an eclipse repeats approximately 1000 km further north or further south, depending on whether the series started at the south or north pole, respectively."]
["The inex cycle has 358 mean synodic months, almost exactly 388+1/2 draconic months, about 383+2/3 anomalistic months, or 30+1/2 draconic years."]
["One inex later an eclipse repeats under similar conditions but at the opposite lunar orbital node (hence opposite terrestrial latitude)."]
["Inex eclipse series are much longer-lasting than saros series.  A triad cycle is 3 inex cycles, containing nearly an integer number of anomalistic months (1151)."]
[]
["The \"Lunar\" worksheet can show which cycles are multiples of the saros or inex, but the shortest such cycle, when comprised of a smoothly spread mix of"]
["full 30-day and deficient 29-day months, is the 19-saros cycle of 4237 months. The number eclipse cycles found increases with the maximum denominator:"]
[]
["Maximum","Lunar","Saros","Exeligmos","Inex"]
["Denominator","Cycles","Cycles","Cycles","Cycles"]
[2000,76,0,0,0]
[4000,238,0,0,0]
[5000,365,2,0,0]
[6000,519,3,0,1]
[8000,914,7,0,12]
[10000,1422,8,0,12]
[20000,5644,26,8,32]
[30000,12673,58,16,70]
[]
["A Ford chart with lunar cycles up to a maximum"]
["denominator of 8000 months can be plotted without"]
["errors, but it will probably take more than a minute,"]
["the resulting plot will be extremely dense, and many"]
["of the cycle legends will wrap onto a second line."]

{"sheet":"PlotData","index":9,"n_rows":0,"n_cols":0,"rows_kept":0}

{"sheet":"Legend","index":10,"n_rows":0,"n_cols":0,"rows_kept":0}

{"sheet":"Desc","index":11,"n_rows":0,"n_cols":0,"rows_kept":0}

{"sheet":"Factors","index":12,"n_rows":27,"n_cols":4,"rows_kept":27}
["Factorization of Integers"]
[]
["It may be helpful to find the integer divisors of the number of years per leap cycle, number of days per cycle, etc."]
[]
["For that purpose, this spreadsheet include a macro Divisors( ) function."]
[]
["Syntax and how to use the Divisors macro function:"]
[]
["=Divisors( TheNumber {, FlagPrimes } )"]
[]
["The values for the optional FlagPrimes parameter are:"]
[1,"Return all Divisors, with flagged primes (default)."]
[0,"Return all Divisors without flagging primes"]
[-1,"Return primes divisors only (factorize) and flag all as primes."]
[-2,"(or any value <-1) Return prime divisors only (factorize) but don't flag as primes."]
[]
["See examples below:"]
[]
["Number","FlagPrimes","Syntax","Result"]
[3150,0,{"$f":"\"=Divisors(\"&A20&\",\"&B20&\")\"","v":"=Divisors(3150,0)"},{"$f":"divisors(A20,B20)","v":"2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 25, 30, 35, 42, 45, 50, 63, 70, 75, 90, 105, 126, 150, 175, 210, 225, 315, 350, 450, 525, 630, 1050, 1575, 3150"}]
[3150,1,"=Divisors(3150,1)",{"$f":"divisors(A21,B21)","v":"2', 3', 5', 6, 7', 9, 10, 14, 15, 18, 21, 25, 30, 35, 42, 45, 50, 63, 70, 75, 90, 105, 126, 150, 175, 210, 225, 315, 350, 450, 525, 630, 1050, 1575, 3150"}]
[3150,-1,"=Divisors(3150,-1)",{"$f":"divisors(A22,B22)","v":"2', 3', 5', 7', 3150"}]
[3150,-2,"=Divisors(3150,-2)",{"$f":"divisors(A23,B23)","v":"2, 3, 5, 7, 3150"}]
[293,1,"=Divisors(293,1)",{"$f":"divisors(A24,B24)","v":"293'"}]
[1803,1,"=Divisors(1803,1)",{"$f":"divisors(A25,B25)","v":"3', 601', 1803"}]
[5575,1,"=Divisors(5575,1)","5', 25, 223', 1115, 5575"]
[22300,1,"=Divisors(22300,1)","2', 4, 5', 10, 20, 25, 50, 100, 223', 446, 892, 1115, 2230, 4460, 5575, 11150, 22300"]