Dear Bill,

Some of the 31 flavors,

Starting with the most tasty first.

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**31 ****= { ^{6}_{2}}**
is the 6th Sterling Number.

(which counts the number of ways to partition a set of n things into k non empty sets.

i.e. there are 31 ways to group 6 things 2 ways.

e.g. the 4^{th} sterling
number{^{4}_{2}}
= 7 :

there are seven partitions of four things two ways:

(1 2 3) (4) (1 2) (2 3)

(1 2 4) (3) (1 3) (2 4)

(1 3 4) (2) (1 4) (2 3)

(2 3 4) (1).

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There are three interesting concepts you might consider: partitions, cycles and the modulus as we talked about in Bockenhiemer Depot.

The **Modulus** is important because it is at work in all the numerical
operations you showed me. The modulus is related to a number system base (e.g.
base 10 decimal numbers, base 2 binary numbers). It is also related to division
for integers (no fractions allowed). MOD is like the remainder in division.
After all the whole cycles of the modulus value n have been removed from k
the remainder is the answer is “k MOD n”.

1 MOD 3 = 1 2 MOD 3 = 2 3 MOD 3 = 0 4 MOD 3 = 1

5 MOD 3 = 2 6 MOD 3 = 0 7 MOD 3 =1 …

16 MOD 12 = 4

11 MOD 9 = 2 etc

In the horizontal numerological additions you were studying, you always reduced numbers to the range 0..9 so there was an unconscious MOD 10 being invoked. It was additive characteristics of 10 that you were exploring.

Mod 9: if you ignore 0 as a number. I.e. if you eliminate 0 in your horizontal
additions you are looking at base 9 numbers (musicians and dancers count 1..9.
unlike mathematicians who see 8 as the 9^{th} number. So is
31 = 1..31 or 0..31 I.e. is the number of interest 31 or 32?

What would be revealing
is to perform the same studies with __different__ bases, keeping in mind
different numbers as the reference. This Base/Modulus is the significant grouping
element at work. However taking 31 as 3*10 + 1 is not the most interesting
case because 10 is not the most musically interesting number. When you look
at other bases things get more exciting.

31= 43 base 7 7+7+7+7 + 3

**31= 37 base 8
8+8+8 + 7 ( or 3+3+2 3+3+2 3+3+2 3+2+2)**

31= 111 base 5
25 + 5 + 1 =1*5^{2} + 1*5^{1}
+ 1

31= 1011 base 3
27+3+1 i.e. 1*3^{3} + 0*3^{2}
+ 1*3^{1} + 1

31= 51 base 6 6+6+6 + 6+6 +1

31= 34 base 9 9+9+9 +4

31= 1F base 16 16+15 or 4+4+4+4 + 4+4+4 +3

**31= 27 base 12
12+12+7 or 4+4+4 + 4+4+4 + 4+3.**

Notice also the different ways of making the sum.

31 = 12+12 +3+4 or 4+4+4 +4+4+4 +4+3 or 8+8+8+7

These are the **Partition****s**** of 31**.

The **modulus**
groups groupings for you, organizes the partitions by interesting numbers
eg. by 12’s or 8’s or 6’s…

We can generate interesting cyclic patterns with in our base via the recursion x + k mod n. I.e. we start with some number add a constant k and wrap it with in some modulus n. Wrap can be imagined as the numbers which exceed the modulus cyclically wrap around the possibilities like the octave on the piano so that c3 is still c.

so **n+5 mod 16**
generates the pattern

0 5 10 15 4 9 14 3 8 13 2 7 12 1 6 11 0…

like this wrapping numbers which go beyond the modulus limit:

0+5 =5 mod16= 5

5+5 =10 mod16= 10

10+5 =15 mod16=15

1+15 =20 mod 16=4

4+5 =9

9+5 =14=14

14+5 =19mod16=3

… and so on.

It generates a set of numbers that seem a bit random in their occurrence. We could have started with any number and we would simply have jumped into the cycle at another point but arrived at the same series.

Consider n+5 mod 12 which generates

0 5 10
3 8 1 6 11 4 9 2 7 0… interesting because this time the whole set of
numbers between 0 and the modulus is revealed unlike with 5 above. This is
also one of the older such series since is the famous musical cycle of 5^{th}s
(7 semitones) where by you visit all the tonal centers by jumps of a 5^{th}

**Cycles** are more than subsets or partitions
because they include the order of the elements.

The set [A B C] has two cycles ABC and ACB:

Think of a triangle A

C B

you can make a cycle from A to B to C (clockwise) or A to C to B (anticlockwise),

the starting point in not important here. The sequence. ABCABCABC is not ACBACBACB

**The set
[A B] has only one l cycle **because there is no difference between
going from AB or BA in the cycle ABABABAB, its all the same ‘direction’
with only two elements. With four it gets even more interesting because you
have more dimensions or degrees of freedom. You can go ABCD or ACBD or ADBC
or ADCBOr BACD or BADC…

but CDAB or BDCA or BCAD or CBAD are just phase shifted versions of the first few. This concept of cycles of permutations is musically significant be cause in a cyclic repetitive form we also do not hear the extracts abcd or dcab as anything but parts of the repetitions abcdabcdabcd ………

**31 is interesting**
because it is the end of a cycle of ** 32** things 0..31

and perhaps it is the relation between 31 and the more symmetrical and super familiar 32 = 16 16 = 8 8 8 8 = 4444 4444

**Interesting because
31 is prime**, meaning indivisible by any other integer except 1.

31 is, in binary (base 2), the number 11111.

I.e. each one marks a power added 16+8+4+2+1

(The 5^{th}
power of 2 minus 1 = 31 = (2*2*2*2*2)-1

This makes 31 the
5^{th} Mersenne number and the 3^{rd} Mersenne Prime.

**31 is interesting
because**

31 =
1 + 5 + 5^{2} = 1 + 2 + 4 + 8 + 16,

i.e. 31 is one of
only two known numbers that can be written __two__ __ways__ as the sum
of successive powers, the other is 8191.

31 is the numbers of moves required to transport 5 disks in the infamous “Towers of Hanoi” game (moving the golden disks on the three spindles)

Fibonnacci figured 31 to be the maximum number of different measurementsthat could be made using a one-pan balance system using 5 weights of 1,2,4,8,16!

1145.036, the 31st Harmonic expressed in cents, ie the ratio 31/16, or 31 times the fundimental normalized back into the octave by dividing by 16)