Properties and Signatures of the <i>Mazhor Qatan</i>

Properties and Signatures of the Mazhor Qatan

by Remy Landau

The 19 Year Span Day Counts

The fixed Hebrew calendar arithmetic is premised on the notion of a constant lunar period lasting exactly 29d 12h 793p. The calendar then assumes that 235 complete lunar cycles add up to exactly 19 integral solar years.

Within these relationships, Hebrew months can be either 29 or 30 days long, and Hebrew years may contain either 12 months (common years) or 13 months (leap years).

An additionally implied constraint is that all lunar years begin within less than one month after the start of their corresponding solar years. The constraint may be expressed algebraically as

0 <= NumberOfYears * (LunarYearInMonths - SolarYearInMonths) <= 29d 12h 793p

Given the above information, the 19th century mathematician Carl Friedich Gauss determined that the number of months contained by any Hebrew year span can be algebraically expressed as

Yearspan.months = (235 * Yearspan + 12 * Yearspan MOD 19 ) / 19

Moreover, Gauss also noted that if Yearspan began from some Baseyear, then any Targetyear = Baseyear + Yearspan satisfying the relationship

12 * Targetyear MOD 19 < 7

is a 13-month year for any calendar system as defined above.

The fixed Hebrew calendar's Year count is such that

3 = Baseyear MOD 19

Consequently, for the fixed Hebrew calendar, the 13-month years correspond to the Hebrew year counts in accordance with

(12 * HebrewYear + 17) MOD 19 < 7

Consequently, in the fixed Hebrew calendar, 13-month years begin whenever the Hebrew year count, after division by 19, leaves a remainder that is either 0 , 3, 6, 8, 11, 14, or 17. Since the calendar was constructed at a time when no symbol existed for the value 0, this particular remainder was replaced by the value 19.

The positions of the 13-month years over every 19-year period that begins at 1 = HebrewYear MOD 19 is given the Hebrew acronym GUChADZT. The acronym is formed by the Hebrew letters Gimel, Vov, Het (Chet), Alef, Daled, Zayin, and Tet. Each letter respectively represents in Hebrew denumeration the numbers 3, 6, 8, 1, 4, 7, and 9. The number 10 is left out of the acronym.

In the Hebrew calendar, a period of 19 consecutive years is referred to as a Mahzor Qatan, that is, a small cycle.

Historically, there have been several documented Mahzorim Qatanim. However, these all followed the assumptions of the calendar, and really reflected a cycle count starting at a Hebrew year different than 1 = HebrewYear MOD 19.

The only part of the Hebrew calendar that repeats itself in every 19 years, once the year count start has been set, is the position of the 13-month years in the corresponding Mahzor Qatan. The Hebrew calendar does not repeat itself over 19 year periods. This simple idea can easily be determined from the fact that periods of 19 Hebrew years can have either 6938, 6939, 6940, 6941, or 6942 days. The distribution of the day counts is as follows...

19 YEAR SPANS
`      235 months =      6,939d 16h  595p`
M'+/-DAYSMOD 7dOCCURS
-2
`          0d`
0
`        0`
-1
`      6,938d`
1
`   11,263`
0
`      6,939d`
2
`  311,544`
1
`      6,940d`
3
`  250,123`
2
`      6,941d`
4
`  113,011`
3
`      6,942d`
5
`    3,531`
The maximum variance is 4 days

Since the 6939 and 6940 day spans dominate about 81.5% of the years in the full Hebrew calendar cycle of 689,472 years and the Gregorian calendar also can have 19 year spans of both 6939 and 6940 days, it is highly probable that a given Hebrew date and its corresponding Gregorian date will once again coincide in 19 years... but not necessarily every 19 years.

The correspondence between the Hebrew and Gregorian calendars repeats itself in a cycle of
14,389,970,112 Hebrew years which is also 14,390,140,400 Gregorian years!

In days, 14,389,970,112 Hebrew years = 5,255,890,855,047 days = 14,390,140,400 Gregorian years!

It is interesting to note that, in the Hebrew calendar, all 19 year spans which begin with 12-month years have the following day counts...

19 Year Spans Starting at Common Years
`      235 months =      6,939d 16h  595p`
M'+/-DAYSMOD 7dOCCURS
-2
`          0d`
0
`        0`
-1
`          0d`
0
`        0`
0
`      6,939d`
2
`   17,099`
1
`      6,940d`
3
`   13,648`
2
`      6,941d`
4
`    5,246`
3
`      6,942d`
5
`      295`
The maximum variance is 3 days

On the other hand, all 19 year spans which begin with 13-month years have the following day counts...

19 Year Spans Starting at Leap Years
`      235 months =      6,939d 16h  595p`
M'+/-DAYSMOD 7dOCCURS
-2
`          0d`
0
`        0`
-1
`      6,938d`
1
`    1,609`
0
`      6,939d`
2
`   15,195`
1
`      6,940d`
3
`   12,334`
2
`      6,941d`
4
`    7,150`
3
`          0d`
0
`        0`
The maximum variance is 3 days

See Weekly Question Archive for additional details on the day counts of 19 year spans.

Since there are 19 possible cycles of 19 years, and each may not have the same arithmetical properties as the other, a distinction must be made between each Mahzor Qatan. Therefore, for purposes of the following, it will be convenient to identify a Mahzor Qatan by the remainder when its first Hebrew year is divided by 19.

For example, the first year of the 304th Mahzor Qatan, counting from year 1H, is 5758H. The value 5758 when divided by 19 leaves a remainder of 1. Hence, that Mahzor Qatan is identified as being of the type 1 = HebrewYear MOD 19.

The year 5757H, was the last year of the 303rd Mahzor Qatan. The value 5757 when divided by 19 leaves the remainder 0. Hence, a 19 year span counted from the start year 5757H is is identified as being of the type 0 = HebrewYear MOD 19.

Similarly, a 19 year span counted from the start year 5750H is identified as being of the type 12 = HebrewYear MOD 19, since the value 5750 when divided by 19 leaves the remainder 12 and not 1.

When Rosh Hashana advances to a new day in the Gregorian calendar, it always does so in the 9th year of a Mahzor Qatan of the type 1 = HebrewYear MOD 19.
(See the Additional Notes on The Rosh HaShana Drift for more information on the latest possible occurrences of Rosh Hashana).

The following table shows, over the full and complete Hebrew calendar repetition cycle of 689,472 years, the number of times that any year begins on either Monday, Tuesday, Thursday or Shabbat in the Mahzor Qatan type 1 = HebrewYear MOD 19.

Weekday Starts for All Years in the Mahzor Qatan
YearMonTueThuShbtTOTAL
1 9837 3811 12272 10368 36288
2 10368 3281 12271 10368 36288
3 10368 5184 10368 10368 36288
4 9838 3811 12271 10368 36288
5 10368 3281 12271 10368 36288
6 10368 5184 10368 10368 36288
7 9838 3811 12271 10368 36288
8 10368 5184 10368 10368 36288
9 9838 3811 12271 10368 36288
10 10368 3281 12271 10368 36288
11 10368 5184 10368 10368 36288
12 9838 3811 12271 10368 36288
13 10368 3281 12271 10368 36288
14 10368 5184 10368 10368 36288
15 9838 3811 12271 10368 36288
16 10368 3280 12272 10368 36288
17 10368 5184 10368 10368 36288
18 9837 3811 12272 10368 36288
19 10368 5184 10368 10368 36288
TOTAL 193280 79369 219831 196992 689472

For the Mahzor Qatan type 1 = HebrewYear MOD 19, the table shows that over the 689,472 Hebrew year calendar cycle:-

• Each year of the Mahzor Qatan begins the same number of times on Shabbat. The number 10,368 is exactly 2/(7*19) of the full calendar cycle.
• The number of times that a leap year begins on either Monday, or Thursday is the same as the number of times that these years begin on Shabbat.
• Leap years begin on Tuesdays exactly half as many times as they do on any other day of the week.
• The weekday distribution of Rosh Hashana starts is identical for each of the leap years in the 19 year cycle, which are the 1st, 3rd, 8th, 11th, 14th, 17th and 19th years of the Mahzor Qatan.
• The 16th year has an absolutely unique weekday distribution of Rosh Hashana starts.
• The 1st and 18th years have the same weekday distribution of Rosh Hashana starts.
• The 2nd, 5th, 10th, and 13th years have the same weekday distribution of Rosh Hashana starts.
• The 4th, 7th, 9th, 12th, and 15th years have the same weekday distribution, of Rosh Hashana starts.

Over the full Hebrew calendar repetition cycle, the number of times that any year in the Mahzor Qatan does not begin on Shabbat is 36,288 - 10,368 = 25,920. This number is also equal to the number of halaqim (i.e., parts) in one day which is 24 * 1080 = 25,920.

Signatures of the Mazhor Qatan

Regardless of its type, the qeviyot for any Mahzor Qatan can only be combined and arranged in exactly 61 different ways for that type. See also The Qeviyot.

The signature of any Mahzor Qatan is the particular combination and arrangement of the qeviyot in that 19 year cycle.

For the type 1 Mahzor Qatan, the existence of its 61 signatures was demonstrated by The Rev. Sherrard Beaumont Burnaby beginning in Chapter 6, at page 146 of his book Elements of the Jewish and Muhammadan Calendars published by George Bell and Sons, London (1901). Burnaby noted that his thinking on this phenomenon was based on a paper published in 1843 by Professor Nesselmann. The paper, Beitrage zur Chronologie, was published in Crelle Journal fur die Mathematik, Band 26, page 59, Berlin 1843.

Yaaqov Loewinger noted that at around 1200j, a table containing the 61 signatures, apparently was conceived by the medieval Tosafist Isaac ben Abraham. Loewinger notes that by using this table, all of the qeviyot for any particular mahzor qatan can then be found by entering only the Tishrei molad of the first year in that cycle. He is presently preparing an article outlining this particular Hebrew calendar phenomenon as documented in the classical literature. Once that paper is published, it will be very interesting to see his report of the various classical methologies used to relate the Tishrei moladot to their signatures.

The non-analytical view of the signatures that follows is only a very small and specific part of the much larger problem related to the occurrence of the qeviyot found embedded, not only in any 19 year span, but also in any size of Hebrew year spans. Insights into the far larger problem are provided in questions 229 through 238 found in the Weekly Question Archive beginning with Weekly Questions 231-240.

The Signature Tables

The Signature Tables shown below are constructed relative to the Mahzor Qatan type 1 = HebrewYear MOD 19.

In the Signature Tables, the qeviyot are expressed as an alphabetic, representing the Tishrei 1 weekday, followed by the last two digits of the corresponding Hebrew year length.

The weekdays Shabbat, Monday, Tuesday, and Thursday are coded into the single Roman capitalized letter Z, B, G, H. These letters are the transliteration of the Hebrew symbols Zayin, Bet, Gimel, Heh which are traditionally used to represent these weekdays.

Since all of the possible single Hebrew year lengths the same leading hundreds digit 3, the numeral has been dropped for all of the year length notations. The desired qeviyah is then coded by abutting the two-digit year length code to the right of the weekday letter code. For example, Z85 represents the qeviyah of the 385 day Hebrew year beginning on Shabbat and H54 indicates a 354 day year starting on Thursday.

Explanation of the Columns in the Signature Tables
Column TitleColumn Content
Signaturea sequence number from 1 to 61 reflecting the order in which the signature is first detected in the full Hebrew calendar cycle when viewed starting from year 1H.
Frequencythe number of times that the signature is found in the full Hebrew calendar cycle.
First Timethe Hebrew year at which the signature is first detected in the full Hebrew calendar cycle when viewed from year 1H.
Tishrei Moladthe Tishrei Molad corresponding to the year in the First Time column.
Earliest Moladthe lowest value of the Tishrei Molad found for the first year among all of the occurrences of the corresponding signature.
Latest Moladthe highest value of the Tishrei Molad found for the first year among all of the occurrences of the corresponding signature.
Signature Arrangement of the Qeviyotthe actual sequence of the 19 qeviyot in the signature.
The Qeviyotthe list of the qeviyot in the signature.
The Missing Qeviyotthe list of the qeviyot not in the signature.

The Signatures Ordered by First Occurrence

The first 13 signatures occur exactly 19 years apart. Then incremental gaps in the year counts appear between each of the subsequent entries in the signatures ordered by first occurrence. The 20th and 21st entries form the last pair of entries to be 19 years apart.

The Mahzor Qatan Type 1 Signatures
Initial Signature Occurrences In Ascending Hebrew Year Order
StatisticsSignature Arrangement of the Qeviyot
1 539 1H
`2d  5h   204p`
`2d  2h   924p`
`2d  5h   374p`
B55Z55H83G54Z55H85H54B83Z55H54B83Z55H54B85B53H54B85B55Z83
2 1,605 20H
`4d 21h   799p`
`4d 18h    24p`
`5d  1h   484p`
H54B55Z83H55G54Z85Z53G84B55Z55H83G54Z55H85H54B53H85H54B83
3 539 39H
`0d 14h   314p`
`0d 14h   154p`
`0d 16h   684p`
Z55H54B85B53H55G84B55Z83H54B55Z85Z53G54Z85Z55H54B83Z55H83
4 829 58H
`3d  6h   909p`
`3d  5h   379p`
`3d  9h   199p`
G54Z55H85H54B53H85H54B85B53H54B85B55Z53G84B55Z55H83G54Z85
5 299 77H
`5d 23h   424p`
`5d 22h 1,074p`
`6d  0h   404p`
Z53G54Z85Z55H54B83Z55H85H54B53H85H54B55Z83H54B55Z83H55G84
6 2,438 96H
`1d 15h 1,019p`
`1d 11h   744p`
`1d 22h 1,049p`
B55Z53G84B55Z55H83G54Z85Z55H54B83Z55H54B83Z55H54B85B53H85
7 1,903 115H
`4d  8h   534p`
`4d  2h   924p`
`4d 11h   714p`
H54B55Z83H54B55Z83H55G84B55Z53G84B55Z55H83G54Z55H85H54B83
8 829 134H
`0d  1h    49p`
`6d 22h 1,074p`
`0d  2h   894p`
Z55H54B83Z55H54B85B53H85H54B55Z83H54B55Z85Z53G54Z85Z55H83
9 535 153H
`2d 17h   644p`
`2d 15h   589p`
`2d 18h    19p`
G54Z55H83G54Z55H85H54B83Z55H54B85B53H54B85B55Z53G84B55Z83
10 1,899 172H
`5d 10h   159p`
`5d  9h   229p`
`5d 17h 1,079p`
H55G54Z85Z55H54B83Z55H83G54Z55H85H54B53H85H54B55Z83H54B85
11 1,065 191H
`1d  2h   754p`
`1d  0h   409p`
`1d  5h   329p`
B53H55G84B55Z53G84B55Z83H55G54Z85Z55H54B83Z55H54B83Z55H85
12 1,903 210H
`3d 19h   269p`
`3d 11h   744p`
`3d 20h   534p`
H54B53H85H54B55Z83H54B85B53H55G84B55Z53G84B55Z55H83G54Z85
13 535 229H
`6d 11h   864p`
`6d 11h   744p`
`6d 14h   174p`
Z55H54B83Z55H54B85B53H85H54B55Z83H54B55Z83H54B55Z85Z53G84
14 1,065 286H
`0d 13h   489p`
`0d  9h   229p`
`0d 14h   149p`
Z55H54B85B53H55G84B55Z83H54B55Z83H55G54Z85Z55H54B83Z55H83
15 534 324H
`5d 22h   599p`
`5d 20h   564p`
`5d 22h 1,069p`
Z53G54Z85Z55H54B83Z55H85H54B53H85H54B55Z83H54B55Z83H54B85
16 833 476H
`6d 10h 1,039p`
`6d  7h   874p`
`6d 11h   714p`
Z55H54B83Z55H54B83Z55H85H54B53H85H54B55Z83H54B55Z85Z53G84
17 833 552H
`3d  5h   179p`
`3d  1h   489p`
`3d  5h   329p`
G54Z55H85H54B53H85H54B83Z55H54B85B53H55G84B55Z53G84B55Z85
18 829 666H
`5d  8h   509p`
`5d  5h   379p`
`5d  9h   199p`
H54B55Z85Z53G54Z85Z55H83G54Z55H85H54B53H85H54B55Z83H54B85
19 535 742H
`2d  2h   729p`
`2d  0h   409p`
`2d  2h   919p`
B55Z55H83G54Z55H83G54Z85Z55H54B83Z55H54B85B53H54B85B55Z83
20 1,903 875H
`6d 22h   574p`
`6d 14h   179p`
`6d 22h 1,049p`
Z55H54B83Z55H54B85B53H85H54B55Z83H54B55Z83H55G54Z85Z53G84
21 298 894H
`2d 15h    89p`
`2d 14h   179p`
`2d 15h   584p`
B55Z55H83G54Z55H85H54B83Z55H54B85B53H54B85B55Z53G84B55Z83
22 833 932H
`1d  0h   199p`
`0d 20h   564p`
`1d  0h   404p`
B53H54B85B55Z53G84B55Z83H55G54Z85Z55H54B83Z55H54B83Z55H85
23 539 1,065H
`5d 20h    44p`
`5d 18h     4p`
`5d 20h   534p`
Z53G54Z85Z55H54B83Z55H83G54Z55H85H54B53H85H54B55Z83H54B85
24 833 1,255H
`4d 17h   594p`
`4d 14h   179p`
`4d 18h    19p`
H54B55Z83H54B55Z85Z53G84B55Z55H83G54Z55H85H54B53H85H54B83
25 1,899 1,388H
`2d 13h   439p`
`2d  5h   379p`
`2d 14h   149p`
B55Z55H83G54Z55H85H54B83Z55H54B83Z55H54B85B53H55G84B55Z83
26 1,605 1,464H
`6d  7h   659p`
`6d  0h   409p`
`6d  7h   869p`
Z55H54B83Z55H54B83Z55H85H54B53H85H54B55Z83H54B55Z83H55G84
27 299 1,483H
`2d  0h   174p`
`1d 22h 1,074p`
`2d  0h   404p`
B55Z53G84B55Z55H83G54Z85Z55H54B83Z55H54B85B53H54B85B55Z83
28 535 1,578H
`1d 10h   989p`
`1d  9h   229p`
`1d 11h   739p`
B55Z53G84B55Z55H83G54Z85Z53G54Z85Z55H54B83Z55H54B85B53H85
29 535 1,654H
`5d  5h   129p`
`5d  2h   924p`
`5d  5h   354p`
H54B55Z85Z53G54Z85Z55H83G54Z55H83G54Z55H85H54B53H85H54B85
30 834 1,768H
`0d  8h   459p`
`0d  5h   379p`
`0d  9h   224p`
Z55H54B85B53H54B85B55Z83H54B55Z83H55G54Z85Z55H54B83Z55H83
31 1,065 1,787H
`3d  0h 1,054p`
`2d 20h   564p`
`3d  1h   484p`
G54Z55H85H54B53H85H54B83Z55H54B85B53H55G84B55Z53G84B55Z83
32 535 1,844H
`4d  2h   679p`
`4d  0h   409p`
`4d  2h   919p`
H54B55Z83H54B55Z83H54B85B55Z53G84B55Z55H83G54Z55H85H54B83
33 535 2,167H
`0d 19h 1,074p`
`0d 18h    24p`
`0d 20h   534p`
B53H54B85B55Z53G84B55Z83H54B55Z85Z53G54Z85Z55H54B83Z55H85
34 298 2,319H
`1d  8h   434p`
`1d  7h   874p`
`1d  9h   199p`
B53H55G84B55Z53G84B55Z85Z53G54Z85Z55H54B83Z55H54B85B53H85
35 298 2,395H
`5d  2h   654p`
`5d  1h   489p`
`5d  2h   894p`
H54B55Z83H55G54Z85Z53G84B55Z55H83G54Z55H85H54B53H85H54B85
36 535 2,490H
`4d 13h   389p`
`4d 11h   744p`
`4d 14h   174p`
H54B55Z83H54B55Z85Z53G84B55Z55H83G54Z55H83G54Z55H85H54B83
37 540 2,566H
`1d  7h   609p`
`1d  5h   334p`
`1d  7h   869p`
B53H55G84B55Z53G84B55Z85Z53G54Z85Z55H54B83Z55H54B83Z55H85
38 299 2,585H
`4d  0h   124p`
`3d 22h 1,074p`
`4d  0h   404p`
H54B53H85H54B55Z83H54B85B55Z53G84B55Z55H83G54Z55H85H54B83
39 535 2,680H
`3d 10h   939p`
`3d  9h   229p`
`3d 11h   739p`
H54B53H85H54B55Z83H54B85B53H54B85B55Z53G84B55Z55H83G54Z85
40 535 2,756H
`0d  5h    79p`
`0d  2h   924p`
`0d  5h   354p`
Z55H54B85B53H54B85B55Z83H54B55Z83H54B55Z85Z53G54Z85Z55H83
41 295 2,908H
`0d 17h   519p`
`0d 16h   689p`
`0d 17h 1,079p`
Z55H54B85B53H55G84B55Z83H54B55Z85Z53G54Z85Z55H54B83Z55H85
42 5 2,984H
`4d 11h   739p`
`4d 11h   719p`
`4d 11h   739p`
H54B55Z83H54B55Z85Z53G84B55Z53G84B55Z55H83G54Z55H85H54B83
43 534 3,079H
`3d 22h   474p`
`3d 20h   564p`
`3d 22h 1,069p`
H54B53H85H54B55Z83H54B85B55Z53G84B55Z55H83G54Z55H83G54Z85
44 5 3,174H
`3d  9h   209p`
`3d  9h   204p`
`3d  9h   224p`
H54B53H85H54B53H85H54B85B53H54B85B55Z53G84B55Z55H83G54Z85
45 540 3,269H
`2d 19h 1,024p`
`2d 18h    24p`
`2d 20h   559p`
G54Z55H85H54B53H85H54B83Z55H54B85B53H54B85B55Z53G84B55Z83
46 5 9,634H
`0d 20h   549p`
`0d 20h   539p`
`0d 20h   559p`
B53H54B85B55Z53G84B55Z83H55G54Z85Z53G54Z85Z55H54B83Z55H85
47 5 23,124H
`6d 11h   719p`
`6d 11h   719p`
`6d 11h   739p`
Z55H54B83Z55H54B85B53H85H54B53H85H54B55Z83H54B55Z85Z53G84
48 4 35,873H
`5d  5h   364p`
`5d  5h   359p`
`5d  5h   374p`
H54B55Z85Z53G54Z85Z55H83G54Z55H85H54B53H85H54B53H85H54B85
49 5 42,352H
`5d  9h   219p`
`5d  9h   204p`
`5d  9h   224p`
H55G54Z85Z53G54Z85Z55H83G54Z55H85H54B53H85H54B55Z83H54B85
50 5 55,652H
`5d  2h   919p`
`5d  2h   899p`
`5d  2h   919p`
H54B55Z85Z53G54Z85Z53G84B55Z55H83G54Z55H85H54B53H85H54B85
51 4 75,051H
`0d  5h   374p`
`0d  5h   359p`
`0d  5h   374p`
Z55H54B85B53H54B85B55Z83H54B55Z83H55G54Z85Z53G54Z85Z55H83
52 5 75,792H
`0d  2h   899p`
`0d  2h   899p`
`0d  2h   919p`
Z55H54B85B53H54B85B53H85H54B55Z83H54B55Z85Z53G54Z85Z55H83
53 5 82,081H
`1d  9h   204p`
`1d  9h   204p`
`1d  9h   224p`
B55Z53G84B55Z53G84B55Z85Z53G54Z85Z55H54B83Z55H54B85B53H85
54 5 88,541H
`5d 20h   544p`
`5d 20h   539p`
`5d 20h   559p`
Z53G54Z85Z55H54B83Z55H85H54B53H85H54B53H85H54B55Z83H54B85
55 4 108,491H
`1d 22h 1,054p`
`1d 22h 1,054p`
`1d 22h 1,069p`
B55Z53G84B55Z55H83G54Z85Z55H54B83Z55H54B85B53H54B85B53H85
56 4 114,780H
`3d  5h   359p`
`3d  5h   359p`
`3d  5h   374p`
G54Z55H85H54B53H85H54B85B53H54B85B55Z53G84B55Z53G84B55Z85
57 5 161,159H
`2d 14h   154p`
`2d 14h   154p`
`2d 14h   174p`
B55Z55H83G54Z55H85H54B83Z55H54B85B53H54B85B53H55G84B55Z83
58 5 167,448H
`3d 20h   539p`
`3d 20h   539p`
`3d 20h   559p`
H54B53H85H54B55Z83H54B85B55Z53G84B55Z53G84B55Z55H83G54Z85
59 5 213,827H
`3d  5h   334p`
`3d  5h   334p`
`3d  5h   354p`
G54Z55H85H54B53H85H54B85B53H54B85B53H55G84B55Z53G84B55Z85
60 4 246,545H
`0d 18h     4p`
`0d 18h     4p`
`0d 18h    19p`
B53H54B85B53H55G84B55Z83H54B55Z85Z53G54Z85Z55H54B83Z55H85
61 4 305,483H
`6d 22h 1,054p`
`6d 22h 1,054p`
`6d 22h 1,069p`
Z55H54B83Z55H54B85B53H85H54B55Z83H54B55Z85Z53G54Z85Z53G84

The Signatures Ordered by Tishrei Molad of First Occurrence

The most surprising observation in the ordered arrangement is that all of the Tishrei moladot are in precise ascending order.

The next observation to be made is that all of the entries in the Earliest Molad column are exactly 5 halaqim more than the entries in their immediately preceding Latest Molad column.

Since the moladot cycle from 6d 23h 1079p to 0d 0h 0p, the very first Earliest Molad shown, namely, 6d 22h 1,074p, does indeed precede the very first Latest Molad, namely, 0d 2h 894p.

The Mahzor Qatan Type 1 Signatures
Initial Signature Occurrences Ordered by Their First Year Tishrei Molad
StatisticsSignature Arrangement of the Qeviyot
8 829 134H
`0d  1h    49p`
`6d 22h 1,074p`
`0d  2h   894p`
Z55H54B83Z55H54B85B53H85H54B55Z83H54B55Z85Z53G54Z85Z55H83
52 5 75,792H
`0d  2h   899p`
`0d  2h   899p`
`0d  2h   919p`
Z55H54B85B53H54B85B53H85H54B55Z83H54B55Z85Z53G54Z85Z55H83
40 535 2,756H
`0d  5h    79p`
`0d  2h   924p`
`0d  5h   354p`
Z55H54B85B53H54B85B55Z83H54B55Z83H54B55Z85Z53G54Z85Z55H83
51 4 75,051H
`0d  5h   374p`
`0d  5h   359p`
`0d  5h   374p`
Z55H54B85B53H54B85B55Z83H54B55Z83H55G54Z85Z53G54Z85Z55H83
30 834 1,768H
`0d  8h   459p`
`0d  5h   379p`
`0d  9h   224p`
Z55H54B85B53H54B85B55Z83H54B55Z83H55G54Z85Z55H54B83Z55H83
14 1,065 286H
`0d 13h   489p`
`0d  9h   229p`
`0d 14h   149p`
Z55H54B85B53H55G84B55Z83H54B55Z83H55G54Z85Z55H54B83Z55H83
3 539 39H
`0d 14h   314p`
`0d 14h   154p`
`0d 16h   684p`
Z55H54B85B53H55G84B55Z83H54B55Z85Z53G54Z85Z55H54B83Z55H83
41 295 2,908H
`0d 17h   519p`
`0d 16h   689p`
`0d 17h 1,079p`
Z55H54B85B53H55G84B55Z83H54B55Z85Z53G54Z85Z55H54B83Z55H85
60 4 246,545H
`0d 18h     4p`
`0d 18h     4p`
`0d 18h    19p`
B53H54B85B53H55G84B55Z83H54B55Z85Z53G54Z85Z55H54B83Z55H85
33 535 2,167H
`0d 19h 1,074p`
`0d 18h    24p`
`0d 20h   534p`
B53H54B85B55Z53G84B55Z83H54B55Z85Z53G54Z85Z55H54B83Z55H85
46 5 9,634H
`0d 20h   549p`
`0d 20h   539p`
`0d 20h   559p`
B53H54B85B55Z53G84B55Z83H55G54Z85Z53G54Z85Z55H54B83Z55H85
22 833 932H
`1d  0h   199p`
`0d 20h   564p`
`1d  0h   404p`
B53H54B85B55Z53G84B55Z83H55G54Z85Z55H54B83Z55H54B83Z55H85
11 1,065 191H
`1d  2h   754p`
`1d  0h   409p`
`1d  5h   329p`
B53H55G84B55Z53G84B55Z83H55G54Z85Z55H54B83Z55H54B83Z55H85
37 540 2,566H
`1d  7h   609p`
`1d  5h   334p`
`1d  7h   869p`
B53H55G84B55Z53G84B55Z85Z53G54Z85Z55H54B83Z55H54B83Z55H85
34 298 2,319H
`1d  8h   434p`
`1d  7h   874p`
`1d  9h   199p`
B53H55G84B55Z53G84B55Z85Z53G54Z85Z55H54B83Z55H54B85B53H85
53 5 82,081H
`1d  9h   204p`
`1d  9h   204p`
`1d  9h   224p`
B55Z53G84B55Z53G84B55Z85Z53G54Z85Z55H54B83Z55H54B85B53H85
28 535 1,578H
`1d 10h   989p`
`1d  9h   229p`
`1d 11h   739p`
B55Z53G84B55Z55H83G54Z85Z53G54Z85Z55H54B83Z55H54B85B53H85
6 2,438 96H
`1d 15h 1,019p`
`1d 11h   744p`
`1d 22h 1,049p`
B55Z53G84B55Z55H83G54Z85Z55H54B83Z55H54B83Z55H54B85B53H85
55 4 108,491H
`1d 22h 1,054p`
`1d 22h 1,054p`
`1d 22h 1,069p`
B55Z53G84B55Z55H83G54Z85Z55H54B83Z55H54B85B53H54B85B53H85
27 299 1,483H
`2d  0h   174p`
`1d 22h 1,074p`
`2d  0h   404p`
B55Z53G84B55Z55H83G54Z85Z55H54B83Z55H54B85B53H54B85B55Z83
19 535 742H
`2d  2h   729p`
`2d  0h   409p`
`2d  2h   919p`
B55Z55H83G54Z55H83G54Z85Z55H54B83Z55H54B85B53H54B85B55Z83
1 539 1H
`2d  5h   204p`
`2d  2h   924p`
`2d  5h   374p`
B55Z55H83G54Z55H85H54B83Z55H54B83Z55H54B85B53H54B85B55Z83
25 1,899 1,388H
`2d 13h   439p`
`2d  5h   379p`
`2d 14h   149p`
B55Z55H83G54Z55H85H54B83Z55H54B83Z55H54B85B53H55G84B55Z83
57 5 161,159H
`2d 14h   154p`
`2d 14h   154p`
`2d 14h   174p`
B55Z55H83G54Z55H85H54B83Z55H54B85B53H54B85B53H55G84B55Z83
21 298 894H
`2d 15h    89p`
`2d 14h   179p`
`2d 15h   584p`
B55Z55H83G54Z55H85H54B83Z55H54B85B53H54B85B55Z53G84B55Z83
9 535 153H
`2d 17h   644p`
`2d 15h   589p`
`2d 18h    19p`
G54Z55H83G54Z55H85H54B83Z55H54B85B53H54B85B55Z53G84B55Z83
45 540 3,269H
`2d 19h 1,024p`
`2d 18h    24p`
`2d 20h   559p`
G54Z55H85H54B53H85H54B83Z55H54B85B53H54B85B55Z53G84B55Z83
31 1,065 1,787H
`3d  0h 1,054p`
`2d 20h   564p`
`3d  1h   484p`
G54Z55H85H54B53H85H54B83Z55H54B85B53H55G84B55Z53G84B55Z83
17 833 552H
`3d  5h   179p`
`3d  1h   489p`
`3d  5h   329p`
G54Z55H85H54B53H85H54B83Z55H54B85B53H55G84B55Z53G84B55Z85
59 5 213,827H
`3d  5h   334p`
`3d  5h   334p`
`3d  5h   354p`
G54Z55H85H54B53H85H54B85B53H54B85B53H55G84B55Z53G84B55Z85
56 4 114,780H
`3d  5h   359p`
`3d  5h   359p`
`3d  5h   374p`
G54Z55H85H54B53H85H54B85B53H54B85B55Z53G84B55Z53G84B55Z85
4 829 58H
`3d  6h   909p`
`3d  5h   379p`
`3d  9h   199p`
G54Z55H85H54B53H85H54B85B53H54B85B55Z53G84B55Z55H83G54Z85
44 5 3,174H
`3d  9h   209p`
`3d  9h   204p`
`3d  9h   224p`
H54B53H85H54B53H85H54B85B53H54B85B55Z53G84B55Z55H83G54Z85
39 535 2,680H
`3d 10h   939p`
`3d  9h   229p`
`3d 11h   739p`
H54B53H85H54B55Z83H54B85B53H54B85B55Z53G84B55Z55H83G54Z85
12 1,903 210H
`3d 19h   269p`
`3d 11h   744p`
`3d 20h   534p`
H54B53H85H54B55Z83H54B85B53H55G84B55Z53G84B55Z55H83G54Z85
58 5 167,448H
`3d 20h   539p`
`3d 20h   539p`
`3d 20h   559p`
H54B53H85H54B55Z83H54B85B55Z53G84B55Z53G84B55Z55H83G54Z85
43 534 3,079H
`3d 22h   474p`
`3d 20h   564p`
`3d 22h 1,069p`
H54B53H85H54B55Z83H54B85B55Z53G84B55Z55H83G54Z55H83G54Z85
38 299 2,585H
`4d  0h   124p`
`3d 22h 1,074p`
`4d  0h   404p`
H54B53H85H54B55Z83H54B85B55Z53G84B55Z55H83G54Z55H85H54B83
32 535 1,844H
`4d  2h   679p`
`4d  0h   409p`
`4d  2h   919p`
H54B55Z83H54B55Z83H54B85B55Z53G84B55Z55H83G54Z55H85H54B83
7 1,903 115H
`4d  8h   534p`
`4d  2h   924p`
`4d 11h   714p`
H54B55Z83H54B55Z83H55G84B55Z53G84B55Z55H83G54Z55H85H54B83
42 5 2,984H
`4d 11h   739p`
`4d 11h   719p`
`4d 11h   739p`
H54B55Z83H54B55Z85Z53G84B55Z53G84B55Z55H83G54Z55H85H54B83
36 535 2,490H
`4d 13h   389p`
`4d 11h   744p`
`4d 14h   174p`
H54B55Z83H54B55Z85Z53G84B55Z55H83G54Z55H83G54Z55H85H54B83
24 833 1,255H
`4d 17h   594p`
`4d 14h   179p`
`4d 18h    19p`
H54B55Z83H54B55Z85Z53G84B55Z55H83G54Z55H85H54B53H85H54B83
2 1,605 20H
`4d 21h   799p`
`4d 18h    24p`
`5d  1h   484p`
H54B55Z83H55G54Z85Z53G84B55Z55H83G54Z55H85H54B53H85H54B83
35 298 2,395H
`5d  2h   654p`
`5d  1h   489p`
`5d  2h   894p`
H54B55Z83H55G54Z85Z53G84B55Z55H83G54Z55H85H54B53H85H54B85
50 5 55,652H
`5d  2h   919p`
`5d  2h   899p`
`5d  2h   919p`
H54B55Z85Z53G54Z85Z53G84B55Z55H83G54Z55H85H54B53H85H54B85
29 535 1,654H
`5d  5h   129p`
`5d  2h   924p`
`5d  5h   354p`
H54B55Z85Z53G54Z85Z55H83G54Z55H83G54Z55H85H54B53H85H54B85
48 4 35,873H
`5d  5h   364p`
`5d  5h   359p`
`5d  5h   374p`
H54B55Z85Z53G54Z85Z55H83G54Z55H85H54B53H85H54B53H85H54B85
18 829 666H
`5d  8h   509p`
`5d  5h   379p`
`5d  9h   199p`
H54B55Z85Z53G54Z85Z55H83G54Z55H85H54B53H85H54B55Z83H54B85
49 5 42,352H
`5d  9h   219p`
`5d  9h   204p`
`5d  9h   224p`
H55G54Z85Z53G54Z85Z55H83G54Z55H85H54B53H85H54B55Z83H54B85
10 1,899 172H
`5d 10h   159p`
`5d  9h   229p`
`5d 17h 1,079p`
H55G54Z85Z55H54B83Z55H83G54Z55H85H54B53H85H54B55Z83H54B85
23 539 1,065H
`5d 20h    44p`
`5d 18h     4p`
`5d 20h   534p`
Z53G54Z85Z55H54B83Z55H83G54Z55H85H54B53H85H54B55Z83H54B85
54 5 88,541H
`5d 20h   544p`
`5d 20h   539p`
`5d 20h   559p`
Z53G54Z85Z55H54B83Z55H85H54B53H85H54B53H85H54B55Z83H54B85
15 534 324H
`5d 22h   599p`
`5d 20h   564p`
`5d 22h 1,069p`
Z53G54Z85Z55H54B83Z55H85H54B53H85H54B55Z83H54B55Z83H54B85
5 299 77H
`5d 23h   424p`
`5d 22h 1,074p`
`6d  0h   404p`
Z53G54Z85Z55H54B83Z55H85H54B53H85H54B55Z83H54B55Z83H55G84
26 1,605 1,464H
`6d  7h   659p`
`6d  0h   409p`
`6d  7h   869p`
Z55H54B83Z55H54B83Z55H85H54B53H85H54B55Z83H54B55Z83H55G84
16 833 476H
`6d 10h 1,039p`
`6d  7h   874p`
`6d 11h   714p`
Z55H54B83Z55H54B83Z55H85H54B53H85H54B55Z83H54B55Z85Z53G84
47 5 23,124H
`6d 11h   719p`
`6d 11h   719p`
`6d 11h   739p`
Z55H54B83Z55H54B85B53H85H54B53H85H54B55Z83H54B55Z85Z53G84
13 535 229H
`6d 11h   864p`
`6d 11h   744p`
`6d 14h   174p`
Z55H54B83Z55H54B85B53H85H54B55Z83H54B55Z83H54B55Z85Z53G84
20 1,903 875H
`6d 22h   574p`
`6d 14h   179p`
`6d 22h 1,049p`
Z55H54B83Z55H54B85B53H85H54B55Z83H54B55Z83H55G54Z85Z53G84
61 4 305,483H
`6d 22h 1,054p`
`6d 22h 1,054p`
`6d 22h 1,069p`
Z55H54B83Z55H54B85B53H85H54B55Z83H54B55Z85Z53G54Z85Z53G84

The Qeviyot In Each Signature

The most significant finding of this analysis is that the Mahzor Qatan of type 1 = HebrewYear MOD 19 can have no fewer than 9 and no more than 13 different qeviyot.

More generally, 19 year periods measured from first years corresponding to Hebrew Year MOD 19 = either 12, 13, or 14 will contain all of the 14 existing qeviyot.

The Mahzor Qatan Type 1 Signatures
The Qeviyot in Each of the Signatures
StatisticsThe Qeviyot
SignatureFrequencyFirst Time 1 2 3 4 5 6 7 8 91011121314
26 1,605 1,464HZ55B53B55H54H55Z83B83G84H85
40 535 2,756HZ53Z55B53B55G54H54Z83Z85B85H83
56 4 114,780HZ53Z55B53B55G54H54Z85B85G84H85
48 4 35,873HZ53Z55B53B55G54H54Z85B85H83H85
29 535 1,654HZ53Z55B53B55G54H54Z85B85H83H85
16 833 476HZ53Z55B53B55H54Z83Z85B83G84H85
19 535 742HZ55B53B55G54H54Z83Z85B83B85H83
1 539 1HZ55B53B55G54H54Z83B83B85H83H85
51 4 75,051HZ53Z55B53B55G54H54H55Z83Z85B85H83
37 540 2,566HZ53Z55B53B55G54H54H55Z85B83G84H85
59 5 213,827HZ53Z55B53B55G54H54H55Z85B85G84H85
15 534 324HZ53Z55B53B55G54H54Z83Z85B83B85H85
54 5 88,541HZ53Z55B53B55G54H54Z83Z85B83B85H85
52 5 75,792HZ53Z55B53B55G54H54Z83Z85B85H83H85
18 829 666HZ53Z55B53B55G54H54Z83Z85B85H83H85
45 540 3,269HZ53Z55B53B55G54H54Z83B83B85G84H85
53 5 82,081HZ53Z55B53B55G54H54Z85B83B85G84H85
4 829 58HZ53Z55B53B55G54H54Z85B85G84H83H85
44 5 3,174HZ53Z55B53B55G54H54Z85B85G84H83H85
50 5 55,652HZ53Z55B53B55G54H54Z85B85G84H83H85
13 535 229HZ53Z55B53B55H54Z83Z85B83B85G84H85
47 5 23,124HZ53Z55B53B55H54Z83Z85B83B85G84H85
7 1,903 115HZ53Z55B55G54H54H55Z83B83G84H83H85
42 5 2,984HZ53Z55B55G54H54Z83Z85B83G84H83H85
36 535 2,490HZ53Z55B55G54H54Z83Z85B83G84H83H85
32 535 1,844HZ53Z55B55G54H54Z83B83B85G84H83H85
30 834 1,768HZ55B53B55G54H54H55Z83Z85B83B85H83
5 299 77HZ53Z55B53B55G54H54H55Z83Z85B83G84H85
11 1,065 191HZ53Z55B53B55G54H54H55Z83Z85B83G84H85
49 5 42,352HZ53Z55B53B55G54H54H55Z83Z85B85H83H85
31 1,065 1,787HZ53Z55B53B55G54H54H55Z83B83B85G84H85
17 833 552HZ53Z55B53B55G54H54H55Z85B83B85G84H85
34 298 2,319HZ53Z55B53B55G54H54H55Z85B83B85G84H85
27 299 1,483HZ53Z55B53B55G54H54Z83Z85B83B85G84H83
33 535 2,167HZ53Z55B53B55G54H54Z83Z85B83B85G84H85
61 4 305,483HZ53Z55B53B55G54H54Z83Z85B83B85G84H85
23 539 1,065HZ53Z55B53B55G54H54Z83Z85B83B85H83H85
8 829 134HZ53Z55B53B55G54H54Z83Z85B83B85H83H85
24 833 1,255HZ53Z55B53B55G54H54Z83Z85B83G84H83H85
58 5 167,448HZ53Z55B53B55G54H54Z83Z85B85G84H83H85
43 534 3,079HZ53Z55B53B55G54H54Z83Z85B85G84H83H85
39 535 2,680HZ53Z55B53B55G54H54Z83Z85B85G84H83H85
9 535 153HZ53Z55B53B55G54H54Z83B83B85G84H83H85
21 298 894HZ53Z55B53B55G54H54Z83B83B85G84H83H85
38 299 2,585HZ53Z55B53B55G54H54Z83B83B85G84H83H85
6 2,438 96HZ53Z55B53B55G54H54Z85B83B85G84H83H85
55 4 108,491HZ53Z55B53B55G54H54Z85B83B85G84H83H85
28 535 1,578HZ53Z55B53B55G54H54Z85B83B85G84H83H85
14 1,065 286HZ55B53B55G54H54H55Z83Z85B83B85G84H83
10 1,899 172HZ55B53B55G54H54H55Z83Z85B83B85H83H85
57 5 161,159HZ55B53B55G54H54H55Z83B83B85G84H83H85
25 1,899 1,388HZ55B53B55G54H54H55Z83B83B85G84H83H85
3 539 39HZ53Z55B53B55G54H54H55Z83Z85B83B85G84H83
22 833 932HZ53Z55B53B55G54H54H55Z83Z85B83B85G84H85
46 5 9,634HZ53Z55B53B55G54H54H55Z83Z85B83B85G84H85
20 1,903 875HZ53Z55B53B55G54H54H55Z83Z85B83B85G84H85
60 4 246,545HZ53Z55B53B55G54H54H55Z83Z85B83B85G84H85
41 295 2,908HZ53Z55B53B55G54H54H55Z83Z85B83B85G84H85
2 1,605 20HZ53Z55B53B55G54H54H55Z83Z85B83G84H83H85
12 1,903 210HZ53Z55B53B55G54H54H55Z83Z85B85G84H83H85
35 298 2,395HZ53Z55B53B55G54H54H55Z83Z85B85G84H83H85

The Qeviyot Missing From Each Signature

Since a type 1 Mahzor Qatan can have missing between 1 to 5 qeviyot it is interesting to note that the qeviyot Z55, B55, and H54 are always present in all the type 1 Mahzorim Qatanim.

The Mahzor Qatan Type 1 Signatures
The Qeviyot Missing in Each of the Signatures
StatisticsThe MissingQeviyot
SignatureFrequencyFirst Time 1 2 3 4 5
26 1,605 1,464HZ53G54Z85B85H83
40 535 2,756HH55B83G84H85
56 4 114,780HH55Z83B83H83
48 4 35,873HH55Z83B83G84
29 535 1,654HH55Z83B83G84
16 833 476HG54H55B85H83
19 535 742HZ53H55G84H85
1 539 1HZ53H55Z85G84
51 4 75,051HB83G84H85
37 540 2,566HZ83B85H83
59 5 213,827HZ83B83H83
15 534 324HH55G84H83
54 5 88,541HH55G84H83
52 5 75,792HH55B83G84
18 829 666HH55B83G84
45 540 3,269HH55Z85H83
53 5 82,081HH55Z83H83
4 829 58HH55Z83B83
44 5 3,174HH55Z83B83
50 5 55,652HH55Z83B83
13 535 229HG54H55H83
47 5 23,124HG54H55H83
7 1,903 115HB53Z85B85
42 5 2,984HB53H55B85
36 535 2,490HB53H55B85
32 535 1,844HB53H55Z85
30 834 1,768HZ53G84H85
5 299 77HB85H83
11 1,065 191HB85H83
49 5 42,352HB83G84
31 1,065 1,787HZ85H83
17 833 552HZ83H83
34 298 2,319HZ83H83
27 299 1,483HH55H85
33 535 2,167HH55H83
61 4 305,483HH55H83
23 539 1,065HH55G84
8 829 134HH55G84
24 833 1,255HH55B85
58 5 167,448HH55B83
43 534 3,079HH55B83
39 535 2,680HH55B83
9 535 153HH55Z85
21 298 894HH55Z85
38 299 2,585HH55Z85
6 2,438 96HH55Z83
55 4 108,491HH55Z83
28 535 1,578HH55Z83
14 1,065 286HZ53H85
10 1,899 172HZ53G84
57 5 161,159HZ53Z85
25 1,899 1,388HZ53Z85
3 539 39HH85
22 833 932HH83
46 5 9,634HH83
20 1,903 875HH83
60 4 246,545HH83
41 295 2,908HH83
2 1,605 20HB85
12 1,903 210HB83
35 298 2,395HB83

The Surviving Qeviyot

The above tables demonstrate that 19 year cycles contain neither the same number nor the same combinations of qeviyot. They also demonstrate that even though there are 14 qeviyot possible the type 1 Mahzor Qatan only can have from 9 to 13 different qeviyot.

An analysis over all of the 19 different types of Mahzorim Qatanim reveals the following statistics.

Number of Qeviyot By Mahzor Qatan Type
Mahzor Qatan
Types
Minimum
Number
Maximum
Number
7, 8, 18, 19912
1, 2, 3, 4, 5, 11, 15, 16, 17913
12, 13, 14914
6, 9, 101013

Mahzor Qatan types 12, 13, and 14 are the only kinds of 19 year cycles that can possibly contain all of the 14 qeviyot within their 19 year spans. None of the type 1 cycles have all of the 14 qeviyot. This is the cycle on which we base the Hebrew year count and which leads to the leap year distribution known as GUChADZT.

Looking at the qeviyot occurring in all of the 19 year spans in the full Hebrew calendar cycle, it is possible to notice that between 1 and 3 will always appear within the signatures for any Mahzor Qatan type. These qeviyot may be described as the Surviving Qeviyot. They are Z55, B55, and H54. In particular, the qeviah H54 is found in any and all periods of 19 Hebrew years.

The Surviving Qeviyot By Mahzor Qatan Type
Mahzor Qatan TypesSurviving Qeviyot
3, 4, 11, 12, 14, 15  H54
13Z55 H54
1, 2, 5, 6, 7, 8, 9, 10, 16, 17, 18, 19Z55B55H54

```First  Paged 19 Oct 1997