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http//www.pogus.com/21033.html
Narayanas Cows Music Tom Johnson Saxophones
Daniel Kientzy Realization Michel Waldschmidt
http//www.math.jussieu.fr/miw/
Update august 8, 2005
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Narayana was an Indian mathematician in the 14th.
century, who proposed the following problem A
cow produces one calf every year. Begining in
its fourth year, each calf produces one calf at
the begining of each year. How many cows are
there altogether after, for example, 17 years?
While you are working on that, let us give you a
musical demonstration.
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The first year there is only the original cow and
her first calf.
Year 1
Original Cow 1
Second generation 1
Total 2
long-short
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The second year there is the original cow and 2
calves.
Year 1 2
Original Cow 1 1
Second generation 1 2
Total 2 3
long -short -short
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The third year there is the original cow and 3
calves.
Year 1 2 3
Original Cow 1 1 1
Second generation 1 2 3
Total 2 3 4
long -short -short -short
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The fourth year the oldest calf becomes a mother,
and we begin a third generation of Naryanas cows.
Year 1 2 3 4
Original Cow 1 1 1 1
Second generation 1 2 3 4
Third generation 0 0 0 1
Total 2 3 4 6
long - short - short - short - long - short
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The fifth year we have another mother cow and 3
new calves.
Year 1 2 3 4 5
Original Cow 1 1 1 1 1
Second generation 1 2 3 4 5 1
Third generation 0 0 0 1 3 2
Total 2 3 4 6 9 3
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The sixth year we have 4 productive cows, 4 new
calves, and a total herd of 13.
Year 1 2 3 4 5 6
Original Cow 1 1 1 1 1 1
Second generation 1 2 3 4 5 6
Third generation 0 0 0 1 3 6
Total 2 3 4 6 9 13
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The seventh year brings the birth of the first
calf of the first calf of the first calf of
Naryanas original cow, and the fourth generation
begins.
Year 1 2 3 4 5 6 7
Original Cow 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7
Third generation 0 0 0 1 3 6 10
Fourth generation 0 0 0 0 0 0 1
Total 2 3 4 6 9 13 19
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In the eigth year the herd, which went from 1 to
2 to 3 to 4 to 6 to 9 to 13 to 19, now jumps to
28.
Year 1 2 3 4 5 6 7 8
Original Cow 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8
Third generation 0 0 0 1 3 6 10 15
Fourth generation 0 0 0 0 0 0 1 4
Total 2 3 4 6 9 13 19 28
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In the ninth year, 13 new calves are born. 1 is
the daughter of the original cow, 6 are
granddaughters and 6 are great-granddaughters.
Year 1 2 3 4 5 6 7 8 9
Original Cow 1 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8 9 1
Third generation 0 0 0 1 3 6 10 15 21 6
Fourth generation 0 0 0 0 0 0 1 4 10 6
Total 2 3 4 6 9 13 19 28 41 13
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In the tenth year, the herd of 41 grows to 60,
and the fifth generation begins on a new note.
Year 1 2 3 4 5 6 7 8 9 10
Original Cow 1 1 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8 9 10
Third generation 0 0 0 1 3 6 10 15 21 28
Fourth generation 0 0 0 0 0 0 1 4 10 20
Fifth generation 0 0 0 0 0 0 0 0 0 1
Total 2 3 4 6 9 13 19 28 41 60
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In the eleventh year an additional 28 calves are
produced by 28 mothers, and 32 other calves are
waiting to become mothers, making a total of 88.
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Year 1 2 3 4 5 6 7 8 9 10 11
Original Cow 1 1 1 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8 9 10 11
Third generation 0 0 0 1 3 6 10 15 21 28 36
Fourth generation 0 0 0 0 0 0 1 4 10 20 35
Fifth generation 0 0 0 0 0 0 0 0 0 1 5
Total 2 3 4 6 9 13 19 28 41 60 88
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In the twelfth year, Narayanas herd continues
its population explosion, going from 88 to 129,
and registrating an annual population growth of
46.59
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Year 1 2 3 4 5 6 7 8 9 10 11 12
Original Cow 1 1 1 1 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8 9 10 11 12
Third generation 0 0 0 1 3 6 10 15 21 28 36 45
Fourth generation 0 0 0 0 0 0 1 4 10 20 35 56
Fifth generation 0 0 0 0 0 0 0 0 0 1 5 15
Total 2 3 4 6 9 13 19 28 41 60 88 129
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Year 1 2 3 4 5 6 7 8 9 10 11 12
Original Cow 1 1 1 1 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8 9 10 11 12
Third generation 0 0 0 1 3 6 10 15 21 28 36 45
Fourth generation 0 0 0 0 0 0 1 4 10 20 35 56
Fifth generation 0 0 0 0 0 0 0 0 0 1 5 15
Year 2 3 4 6 9 13 19 28 41 60 88 129
Growth 50 33 50 50 44.4 46.1 47.3 46.42 46.34 46.66 46.59
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In the thirteenth year, the rate drops very
slightly to 46.51 as the sixth generation
begins.
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Year 1 2 3 4 5 6 7 8 9 10 11 12 13
Original Cow 1 1 1 1 1 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8 9 10 11 12 13
Third generation 0 0 0 1 3 6 10 15 21 28 36 45 55
Fourth generation 0 0 0 0 0 0 1 4 10 20 35 56 84
Fifth generation 0 0 0 0 0 0 0 0 0 1 5 15 35
Sixth generation 0 0 0 0 0 0 0 0 0 0 0 0 1
Total 2 3 4 6 9 13 19 28 41 60 88 129 189
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Year 4 5 6 7 8 9 10 11 12 13
Original Cow 1 1 1 1 1 1 1 1 1 1
Second generation 4 5 6 7 8 9 10 11 12 13
Third generation 1 3 6 10 15 21 28 36 45 55
Fourth generation 0 0 0 1 4 10 20 35 56 84
Fifth generation 0 0 0 0 0 0 1 5 15 35
Sixth generation 0 0 0 0 0 0 0 0 0 1
Total 6 9 13 19 28 41 60 88 129 189
Growth 50 50 44.4 46.1 47.3 46.42 46.34 46.66 46.59 46.51
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The fourteenth year brings 88 new calves,
advancing the herd from 189 to 277.
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Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Original Cow 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Third generation 0 0 0 1 3 6 10 15 21 28 36 45 55 66
Fourth generation 0 0 0 0 0 0 1 4 10 20 35 56 84 120
Fifth generation 0 0 0 0 0 0 0 0 0 1 5 15 35 70
Sixth generation 0 0 0 0 0 0 0 0 0 0 0 0 1 6
Total 2 3 4 6 9 13 19 28 41 60 88 129 189 277
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Narayana's Cows, inspired by an Indian
mathematician of the 14th century, and playable
on any combination of instruments, is written on
three staves the complete melody, the reduced
bass melody, and the drone. The present
multi-track saxophone version is probably as
rich and energetic as any of the large ensemble
versions. The melody is played by three
overdubbed sopranino saxophones in unison, the
bass line is played by three baritones, and the
drone is played by three altos.
http//www.pogus.com/21033.html
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Tom Johnson, born in Colorado in 1939, received
degrees from Yale University, and studied
composition privately with Morton Feldman.
After 15 years in New York, he moved to Paris,
where he has lived since 1983. He works with
simple forms, limited scales, and generally
reduced materials, but proceeds more logically
than most minimalists, often using formulas,
permutations, and predictable sequences.
http//www.tom.johnson.org
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Daniel Kientzy is an international avant-garde
artist, dedicated to contemporary music,
exploiting the potential of all seven saxophones
and the musical power of the electronic as
well. Prior to his 25 years dedicated to this
work, he participated professionally in almost
all genres of western music, playing instruments
as diverse as bass guitar, double bass, viola da
gamba, recorder, bagpipes, and crumhorn.
novamusica_at_infonie.fr
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By the fifteenth year, the herd numbers 406.
This includes the original cow, 15 daughters,
78 granddaughters, 165 great-granddaughters,
126 great-great-granddaughters, and 21
great-great-great-granddaughters.
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Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Original Cow 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Third generation 0 0 0 1 3 6 10 15 21 28 36 45 55 66 78
Fourth generation 0 0 0 0 0 0 1 4 10 20 35 56 84 120 165
Fifth generation 0 0 0 0 0 0 0 0 0 1 5 15 35 70 126
Sixth generation 0 0 0 0 0 0 0 0 0 0 0 0 1 6 21
Total 2 3 4 6 9 13 19 28 41 60 88 129 189 277 406
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Narayana's Cows and Delayed Morphismsby
Jean-Paul Allouche and Tom Johnson
http//kalvos.org/johness1.html
Six or eight years ago one of us (T.J.) found a
German edition of a little book on the history
of mathematics by a Ukranian scholar named
Andrej Grigorewitsch Konforowitsch. The book
was full of curious information, but I was
particularly struck by the following, which
Konforowitsch attributed to Narayana, an Indian
mathematician in the 14th century
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A cow produces one calf every year. Beginning in
its fourth year, each calf produces one calf at
the beginning of each year. How many cows and
calves are there altogether after 20 years? In
working this out, T.J. came to know a unique
numerical sequence, and a year or so later I
found a way to translate this into a composition
called Narayana's Cows. It begins with the
original cow and her first calf long-short.
The second year she has another calf
long-short-short .
http//kalvos.org/johness1.html
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The third year long-short-short-short. Then in
the fourth year, the first calf also becomes a
mother and the herd grows from four to six
long-short-short-short-long-short. The music
continues like this, though it doesn't go all
the way to the 20th year, because by the 17th
year there are already more than 800 cows and
calves and 15 minutes of music.
http//kalvos.org/johness1.html
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Many things can be said about the mathematics of
Narayana's cows, about different ways to
translate them into music, about the point at
which the calves begin to outnumber the cows,
about the rate of population increase, the limit
which this rate approaches, and so on. The
essence of the problem, however, is simply the
sequence resulting as the years go by 1, 2,
3, 4, 6, 9, 13...
http//kalvos.org/johness1.html
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1, 2, 3, 4, 6, 9, 13...
Like the Fibonacci sequence, each number is
calculated by adding earlier numbers, but
instead of adding the two previous numbers, as
one does for the Fibonacci series, one adds the
previous number in the sequence plus the number
two places before that Sn Sn-1
Sn-3. The last number above is 13 (9 4) and
the next must be 19 (13 6).
http//kalvos.org/johness1.html
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In the sixteenth year, we have 1 new daughter,
13 new granddaughters, 55 new
great-granddaughters, 84 new great-great-granddau
ghters, 35 new great-great-great-granddaughters
and the very first great-great-great-great-grandd
aughter.
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Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Original Cow 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Third generation 0 0 0 1 3 6 10 15 21 28 36 45 55 66 78 91
Fourth generation 0 0 0 0 0 0 1 4 10 20 35 56 84 120 165 220
Fifth generation 0 0 0 0 0 0 0 0 0 1 5 15 35 70 126 210
Sixth generation 0 0 0 0 0 0 0 0 0 0 0 0 1 6 21 56
Seventh generation 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Total 2 3 4 6 9 13 19 28 41 60 88 129 189 277 406 595
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Quiz What is the ultimate annual population
growth of this herd?
The first correct answer wins a CD of Narayanas
cow
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Let Sn be the total number of cows at the end of
the year n
S0 1,
Sn Sn-1 Sn-3 .
S1 2,
S2 3,
S3 4,
1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129,
189, 277, 406, 595,
S4 6,
S5 9,
S6 13,
S7 19,
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Let Un satisfy Un Un-1 Un-3
and let aU0 , bU1 , cU2 .
Then U3 ac,
Recall Sn S0 1, S1 2, S2 3, S3 4, S4
6,
U4 abc,
U5 ab2c,
Hence Un Sn-5 a Sn-6 b Sn-4 c.
U6 2ab3c,
U7 3a2b4c,
U8 4a3b6c,
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Let x be such that Un xn satisfies Un Un-1
Un-3 .
Then x3 x2 1.
Positive solution Narayanas constant ?
1.465571231876
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Now we arrive at the seventeenth and final year
of the problem. Most of you have probably already
calculated what the population has to be now, but
if you havent, or if you want to check your
work, you can always just count the notes as we
play.
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PV Numbers A PV number is a positive algebraic
integer ? greater than 1, all of whose conjugate
elements have absolute value less than 1

• ? is a root of an equation
• ?n an-1 ?n-1 a1 ? a0 0 with integers ai
• and all other roots of that equation are complex
• numbers in the disc zlt1.

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PV numbers
Pisot (1938) and Vijayaraghavan (1941)
independently studied these numbers, and
Salem (1943) proposed calling such values
Pisot-Vijayaraghavan numbers.
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The smallest PV number is given by the positive
root 1.324717957... of x3-x-10. This
number was identified as the smallest known by
Salem (1944), and proved to be the smallest
possible by Siegel (1944).
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The ultimate annual population growth of this
herd is 46.55
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a1.3134, ? 1.46557
For n 4, Sn is the nearest integer to a ?n.
n 4, 5, 6, 7,
8, 9, a ?n 6.05,
8.88, 13.01, 19.07, 27.94, 40,96, Sn
6, 9, 13, 19,
28, 41,
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Further examples
Fix k 1, let un un-1un-k .
For k 1 geometric sequence un 2un-1 1,
2, 4, 8, 16, 32, 64, 128, un 2n
A mouse produces one new mouse every year.
Begining in its second year each mouse produces
one mouse at the begining of each year. Annual
rate of growth 100 .
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Further examples
Fix k 1, let un un-1un-k .
For k 2, un un-1un-2, Fibonacci rabbits A
rabbit produces one new rabbit every year.
Begining in its third year each rabbit produces
one rabbit at the begining of each year
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Further examples
Fix k 1, let un un-1un-k .
For k 3, un un-1un-3 , Narayanas cows 1, 2,
3, 4, 6, 9, 13, 19, 28, 41, 60,
Annual rate of growth 46,55
k sequence rate growth
1 Exponential 1, 2, 4, 8, 16, 32, 100
2 Fibonacci 1, 2, 3, 5, 8, 13, 21, 61,80
3 Narayana 1, 2, 3, 4, 6, 9, 13, 19, 46,55
4 un un-1un-4 1, 2, 3, 4, 5, 7, 10, 14, 38,02
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Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Original Cow 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Second generation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Third generation 0 0 0 1 3 6 10 15 21 28 36 45 55 66 78 91 105
Fourth generation 0 0 0 0 0 0 1 4 10 20 35 56 84 120 165 220 286
Fifth generation 0 0 0 0 0 0 0 0 0 1 5 15 35 70 126 210 330
Sixth generation 0 0 0 0 0 0 0 0 0 0 0 0 1 6 21 56 126
Seventh generation 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7
Total 2 3 4 6 9 13 19 28 41 60 88 129 189 277 406 595 872
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http//www.pogus.com/21033.html
Narayanas Cows Music Tom Johnson Saxophones
Daniel Kientzy Realization Michel Waldschmidt
http//www.math.jussieu.fr/miw/