UR1136/KH0152 - Pendant Pendant Sums

Drawings:

Overview	Detail View
All Khipus' Ascher Sum Fieldmarks	Right-Handed XRay
All Pendant Pendant Sum Relationships	Left-Handed XRay
This Khipu's Fieldmark Catalog	Right-Handed Sum Detail
This Khipu's Drawing	Left-Handed Sum Detail

Right Handed Sums: # Sums = 43, Max # Summands = 27, (Min, Mean, Max) Sum Values = (16, 56, 155)
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Left Handed Sums: # Sums = 40, Max # Summands = 21, (Min, Mean, Max) Sum Values = (17, 58, 306)
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Right Handed Sum Detail: - Click on column name to sort

#	Sum Cord	Sum Cord Value	# Summands	Summands
1	p1_{1, 1} : 140_W	140	4	p37: 100_W + p38: 5_W + p39: 16_W:BB + p40: 19_W:BB
2	p6_{1, 6} : 64_W	64	3	p30: 10_W:G + p31: 24_W:BB + p32: 30_W
3	p8_{1, 8} : 68_W	68	4	p29: 4_W + p30: 10_W:G + p31: 24_W:BB + p32: 30_W
4	p9_{1, 9} : 48_W	48	2	p34: 14_W + p35: 34_W
5	p10_{2, 1} : 146_W	146	4	p35: 34_W + p36: 7_W + p37: 100_W + p38: 5_W
6	p14_{2, 5} : 60_W	60	3	p31: 24_W:BB + p32: 30_W + p33: 6_W
7	p15_{2, 6} : 50_W	50	3	p32: 30_W + p33: 6_W + p34: 14_W
8	p16_{2, 7} : 81_W	81	6	p101: 13_B + p102: 11_B + p103: 12_B + p104: 20_B + p105: 14_B + p106: 11_B
9	p17_{2, 8} : 39_W	39	2	p41: 19_W + p42: 20_W
10	p18_{2, 9} : 35_W	35	2	p39: 16_W:BB + p40: 19_W:BB
11	p19_{3, 1} : 147_W	147	5	p36: 7_W + p37: 100_W + p38: 5_W + p39: 16_W:BB + p40: 19_W:BB
12	p21_{3, 3} : 46_W:G	46	3	p65: 8_W + p66: 20_W:BB + p67: 18_W:BB
13	p23_{3, 5} : 38_W	38	3	p29: 4_W + p30: 10_W:G + p31: 24_W:BB
14	p25_{3, 7} : 52_W	52	2	p59: 26_W + p60: 26_W
15	p26_{3, 8} : 36_W	36	2	p32: 30_W + p33: 6_W
16	p28_{4, 1} : 95_W	95	4	p58: 11_W:BB + p59: 26_W + p60: 26_W + p61: 32_W
17	p31_{4, 4} : 24_W:BB	24	2	p74: 15_W + p75: 9_W:BB
18	p32_{4, 5} : 30_W	30	2	p48: 18_W:BB + p49: 12_W:BB
19	p35_{4, 8} : 34_W	34	3	p74: 15_W + p75: 9_W:BB + p76: 10_W:BB
20	p37_{5, 1} : 100_W	100	3	p119: 46_B + p120: 34_B + p121: 20_B
21	p46_{6, 1} : 91_W	91	7	p101: 13_B + p102: 11_B + p103: 12_B + p104: 20_B + p105: 14_B + p106: 11_B + p107: 10_B
22	p50_{6, 5} : 25_W	25	2	p105: 14_B + p106: 11_B
23	p51_{6, 6} : 19_W	19	2	p75: 9_W:BB + p76: 10_W:BB
24	p52_{6, 7} : 23_W	23	2	p102: 11_B + p103: 12_B
25	p55_{7, 1} : 115_W	115	27	p78: 19_W + p79: 13_W + p80: 14_W + p81: 13_W + p82: 0_W + p83: 0_W + p84: 0_W:BB + p85: 0_W:BB + p86: 0_W + p87: 0_W + p88: 0_W + p89: 0_W + p90: 0_W + p91: 0_B + p92: 0_B + p93: 0_B + p94: 0_B + p95: 0_B + p96: 0_B + p97: 0_B + p98: 0_B + p99: 0_B + p100: 0_B + p101: 13_B + p102: 11_B + p103: 12_B + p104: 20_B
26	p57_{7, 3} : 27_W:BB	27	2	p79: 13_W + p80: 14_W
27	p60_{7, 6} : 26_W	26	21	p81: 13_W + p82: 0_W + p83: 0_W + p84: 0_W:BB + p85: 0_W:BB + p86: 0_W + p87: 0_W + p88: 0_W + p89: 0_W + p90: 0_W + p91: 0_B + p92: 0_B + p93: 0_B + p94: 0_B + p95: 0_B + p96: 0_B + p97: 0_B + p98: 0_B + p99: 0_B + p100: 0_B + p101: 13_B
28	p62_{7, 8} : 36_W	36	3	p101: 13_B + p102: 11_B + p103: 12_B
29	p63_{7, 9} : 35_W	35	3	p105: 14_B + p106: 11_B + p107: 10_B
30	p70_{8, 7} : 32_W	32	2	p78: 19_W + p79: 13_W
31	p73_{9, 1} : 89_W	89	3	p108: 17_B + p109: 47_B + p110: 25_B
32	p77_{9, 5} : 21_W	21	2	p106: 11_B + p107: 10_B
33	p108_{13, 8} : 17_B	17	2	p147: 7_G + p148: 10_G
34	p109_{13, 9} : 47_B	47	2	p132: 23_B + p133: 24_B
35	p111_{14, 2} : 31_B	31	3	p137: 15_G + p138: 6_G + p139: 10_G
36	p114_{14, 5} : 21_B	21	2	p137: 15_G + p138: 6_G
37	p118_{14, 9} : 62_B	62	4	p150: 10_G + p151: 32_B + p152: 12_B + p153: 8_G
38	p121_{15, 3} : 20_B	20	2	p146: 13_G + p147: 7_G
39	p122_{15, 4} : 16_B	16	2	p138: 6_G + p139: 10_G
40	p127_{15, 9} : 92_B	92	6	p145: 34_G + p146: 13_G + p147: 7_G + p148: 10_G + p149: 18_G + p150: 10_G
41	p133_{16, 6} : 24_B	24	2	p143: 15_B + p144: 9_G
42	p136_{16, 9} : 60_B	60	4	p141: 10_G + p142: 26_B + p143: 15_B + p144: 9_G
43	p172_{20, 9} : 155_G	155	3	p174: 67_G + p175: 52_G + p176: 36_G

Left Handed Sum Detail: - Click on column name to sort

#	Sum Cord	Sum Cord Value	# Summands	Summands
1	p21_{3, 3} : 46_W:G	46	2	p12: 36_W:G + p13: 10_W:BB
2	p42_{5, 6} : 20_W	20	2	p33: 6_W + p34: 14_W
3	p46_{6, 1} : 91_W	91	5	p32: 30_W + p33: 6_W + p34: 14_W + p35: 34_W + p36: 7_W
4	p55_{7, 1} : 115_W	115	6	p31: 24_W:BB + p32: 30_W + p33: 6_W + p34: 14_W + p35: 34_W + p36: 7_W
5	p62_{7, 8} : 36_W	36	2	p32: 30_W + p33: 6_W
6	p63_{7, 9} : 35_W	35	3	p47: 5_W + p48: 18_W:BB + p49: 12_W:BB
7	p64_{8, 1} : 105_W	105	2	p37: 100_W + p38: 5_W
8	p73_{9, 1} : 89_W	89	4	p50: 25_W + p51: 19_W + p52: 23_W + p53: 22_W
9	p77_{9, 5} : 21_W	21	2	p38: 5_W + p39: 16_W:BB
10	p108_{13, 8} : 17_B	17	2	p44: 11_W + p45: 6_W
11	p109_{13, 9} : 47_B	47	2	p71: 33_W + p72: 14_W
12	p110_{14, 1} : 25_B	25	2	p105: 14_B + p106: 11_B
13	p111_{14, 2} : 31_B	31	2	p76: 10_W:BB + p77: 21_W
14	p114_{14, 5} : 21_B	21	2	p106: 11_B + p107: 10_B
15	p118_{14, 9} : 62_B	62	2	p22: 24_W:BB + p23: 38_W
16	p119_{15, 1} : 46_B	46	3	p103: 12_B + p104: 20_B + p105: 14_B
17	p120_{15, 2} : 34_B	34	2	p115: 18_B + p116: 16_B
18	p124_{15, 6} : 24_B	24	2	p101: 13_B + p102: 11_B
19	p127_{15, 9} : 92_B	92	4	p110: 25_B + p111: 31_B + p112: 22_B + p113: 14_B
20	p129_{16, 2} : 23_B	23	2	p102: 11_B + p103: 12_B
21	p142_{17, 6} : 26_B	26	21	p81: 13_W + p82: 0_W + p83: 0_W + p84: 0_W:BB + p85: 0_W:BB + p86: 0_W + p87: 0_W + p88: 0_W + p89: 0_W + p90: 0_W + p91: 0_B + p92: 0_B + p93: 0_B + p94: 0_B + p95: 0_B + p96: 0_B + p97: 0_B + p98: 0_B + p99: 0_B + p100: 0_B + p101: 13_B
22	p151_{18, 6} : 32_B	32	3	p139: 10_G + p140: 12_G + p141: 10_G
23	p154_{18, 9} : 27_G	27	2	p107: 10_B + p108: 17_B
24	p155_{19, 1} : 31_G	31	3	p137: 15_G + p138: 6_G + p139: 10_G
25	p156_{19, 2} : 42_G	42	2	p150: 10_G + p151: 32_B
26	p157_{19, 3} : 50_G	50	3	p142: 26_B + p143: 15_B + p144: 9_G
27	p159_{19, 5} : 23_G	23	2	p134: 11_B + p135: 12_B
28	p161_{19, 7} : 45_B	45	4	p147: 7_G + p148: 10_G + p149: 18_G + p150: 10_G
29	p163_{19, 9} : 180_G	180	8	p149: 18_G + p150: 10_G + p151: 32_B + p152: 12_B + p153: 8_G + p154: 27_G + p155: 31_G + p156: 42_G
30	p164_{20, 1} : 40_G	40	3	p115: 18_B + p116: 16_B + p117: 6_B
31	p166_{20, 3} : 47_G	47	3	p152: 12_B + p153: 8_G + p154: 27_G
32	p170_{20, 7} : 39_B	39	2	p114: 21_B + p115: 18_B
33	p172_{20, 9} : 155_G	155	9	p147: 7_G + p148: 10_G + p149: 18_G + p150: 10_G + p151: 32_B + p152: 12_B + p153: 8_G + p154: 27_G + p155: 31_G
34	p173_{21, 1} : 60_G	60	2	p168: 27_G + p169: 33_B
35	p174_{21, 2} : 67_G	67	2	p167: 40_G + p168: 27_G
36	p175_{21, 3} : 52_G	52	3	p151: 32_B + p152: 12_B + p153: 8_G
37	p176_{21, 4} : 36_G	36	2	p141: 10_G + p142: 26_B
38	p177_{21, 5} : 24_G	24	2	p143: 15_B + p144: 9_G
39	p178_{21, 6} : 72_B	72	2	p169: 33_B + p170: 39_B
40	p181_{21, 9} : 306_G	306	6	p167: 40_G + p168: 27_G + p169: 33_B + p170: 39_B + p171: 12_G + p172: 155_G

Khipu Notes:

Ascher Databook Notes:

The khipu is attached to a carved wooden bar./li>
Marker M1 is a cord of length 49.0 cm. tied around the main cord.
Subsidiary 180s1s1 is tied around pendant 180 as well as around subsidiary 180s1.
UR1136 and UR1140 were acquired by the Museum in 1904. The provenance is given as Nasca. For a comparison of them, see #10 below.
Photographs of UR1136 are in Ascher (1975) and Schmidt, Max, 1929, Kunst und Kultur von Peru, Impropyläen-Verlagzu, Berlin.
The marker and the way in which the bar is threaded, separate the khipu into 2 parts each containing 10 groups of 9 pendants each.

Note: The pendant order on the listing proceeds from one end of the bar to the other, goes around the end, and continues on the other side. However, similar color and number patterns are found in both parts if the pendants are read instead from one end of the bar to the other, and then beginning at the original end on the other side of the bar. We will, therefore, refer to the pendants and groups on side 2 as if following the second scenario. Part 1 is pendants 1--->90 and references to the j^th pendant in the i^th group correspond to pendant 9 (i-1) +j. Part 2 is pendants 180--->91 and references to the j^th pendant in the i^th group correspond to pendant 190-9i-j.

Each of the groups in part 1 has the same color pattern: W, W, BB:W or G:W, BB: W, W, W, W, W, W. The first 5 groups in part 2 have a similar pattern involving different colors. Each group is G, G, B, B, G, G, G, G, G. The last 5 groups in part 2 consist of all B colored pendants.
Subsidiaries are attached to pendants in 2 distinctly different positions. One position is about 1.0 cm. down from the main cord and the other position is at about 13.0 cm. Part 1 has only subsidiaries in the lower position and, where there is only 1, it is always color B. (Where there are 2 subsidiaries, the second is G.) Part 2 has subsidiaries in both positions. As in part 1, the lower position always is B. The upper is BB with one exception. The exception is the subsidiary described in #3 which itself has a subsidiary of BB.
Parts 1 and 2 have several numerical similarities.
1. In each part, all pendant values in the 10th group are 0.
2. In all non-zero groups, the first pendant in the group has the maximum value.
3. In each part, the sum of the values in the 4th and 5th positions of the 10 groups is the same. That is:
  \[ \sum\limits_{i=1}^{10} (P_{i4}+P_{i5}) = 433 \]
4. With a discrepancy of 1 in part 1, in each part, the sum of the values in the 4th group equals the sum of the values in the 5th group. That is:
  \[ \sum\limits_{j=1}^{9} P_{4j} = \sum\limits_{j=1}^{9} P_{5j} \]
  Also, in the second part only, this sum equals half the sum of the values in the 7th group. Thus:
  \[ \sum\limits_{j=1}^{9} P_{4j} = \sum\limits_{j=1}^{9} P_{5j} = \frac{1}{2} \sum\limits_{j=1}^{9} P_{7j}\]
  so that
  \[ \sum\limits_{j=1}^{9} (P_{4j} + P_{5j}) = \sum\limits_{j=1}^{9} P_{7j}\]
5. Excluding the first pendant in each group, the sum of the values in the 6th group is the same for each part.
  \[ \sum\limits_{j=2}^{9} P_{6j} = 142 \]
By color, part 2 is separated into 2 subparts of 5 groups each. There are a couple of regularities within the subparts.
1. In the second subpart (groups 6-10), the sum of the values in the 2nd position of the groups equals the sum of the values in the 3rd positions. That is:
  
  \[ \sum\limits_{i=6}^{10} P_{i2} = \sum\limits_{i=6}^{10} P_{i3} \]
2. The sum of the values in group 2, added to the sum of the values in the corresponding group in the next subpart (hence, group 7), equals the sum of the values in group 3 and its corresponding group. Namely:
  \[ \sum\limits_{j=1}^{9} (P_{2j}+P_{7j}) = \sum\limits_{j=1}^{9} (P_{3j}+P_{8j}) \]
Comparison of UR1136 and UR1140:
1. Each khipu is attached to a carved wooden bar. Each is 180 pendants. The manner of attachment to the bar separates each into 2 parts of 10 groups of 9 pendants.
2. Both khipus are made of cords of the same basic colors: B, BB, and G.
3. The khipus can be oriented so that they have a common color pattern. The pattern separates the 9 pendants in a group into 2, 2, and 5 pendants with the first 2 and last 5 being the same color. In 5 of the groups of UR1136, the remaining 2 pendants are also the same color. But, in most of the other groups, those 2 pendants are the same color as each other but different from the rest of the group.
4. When oriented so that they show the same color pattern, in all groups on both khipus, the value on the first pendant is the maximum value in the group.