Yahoo Tuning Groups Ultimate Backup tuning-math Jacks

Let me start by reviewing Farey sequences. The first row of the Farey
sequence is [0/1, 1/1] and the nth row is obtained by inserting the
fraction (p1+q1)/(p2+q2) between p1/q1 and p2/q2 if it is in reduced
form (that is, if gcd(p1+q1, p2+q2)=1) and if q1+q2 <= n. Hence the
second row is [0/1, 1/2, 1/1] and so forth.

Now suppose r = u/v is any rational number 0<r<=1, where u/v is in
reduced form. We can define a sequence s_i by setting s_0 = a_0/b_0
to be the Farey fraction adjacent to and below r in the vth row Farey
sequence and setting

s_{i+1} = a_{i+1}/b_{i+1} = (u+a_i)/(v+b_i),

so that s_i is the sequence of Farey fractions approaching r from
below.

Now let us define a function B for any positive integer n and any
rational number 0<r<1, and for any for any integer n>1 when r=1, by
setting

B(r, n) = s_n/s_{n-1}.

If p1/q1 and p2/q2 are two successive Farey fractions, then p2q1 -
p1q2 = 1, which implies p2q1/p1q2 is a superparticular ratio; hence B
takes values in superparticular ratios--that is in numbers of the
form n/(n-1). In exactly the same way, if r is a rational number
0<r<1 and n is a positive integer, we can define a function A(r, n)
of superparticular ratios obtained from Farey fractions approaching r
from above. Since the numerators and denominators of the sequences
s_i used to define A and B grow linearly, the numerators and
denominators of the values of A and B grow quadratically as a
function of n.

Some particular examples of this are

B(1, n) = n^2/(n^2-1) = n^2/(n+1)(n-1)

A(1/2, n) = (2n^2+n)/(2n^2+n-1) = n(2n+1)/(n+1)(2n-1)

B(1/2, n) = (2n^2+3n+1)/(2n^2+3n) = (n+1)(2n+1)/n(2n+3)

A(2/3, n) = (6n^2-n-1)/(6n^2-n-2) = (2n-1)(3n+1)/(2n+1)(3n-2)

B(2/3, n) = (6n^2+n-1)/(6n^2+n-2) = (2n+1)(3n-1)/(2n-1)(3n+2)

The numerators of B(1,n) are squares, and we get the sequence 4/3,
9/8, 16/15, 25/24, 36/35, ... . from it. The numerators of A(1/2, n),
when reduced to lowest terms, are triangular numbers for figurate
triangles with even sides; that is numbers 1/2 x (x+1) where x is
even. It gives us the sequence 3/2, 10/9, 21/20, 36/35, ... where we
might note that 36, which is both a square and a triangular number,
appears on both lists. The numerators of B(1/2, n) are triangular
numbers for triangles with odd size, giving us the sequence 6/5,
15/14, 28/27 and so forth.

These functions lead a multitude of multiplicative relationships,
including

B(1, n) = A(1/2, n)B(1/2, n-1)
A(1/2, n) = B(1, 2n)B(1, 2n+1)
B(1/2, n) = B(1, 2n+1)B(1, 2n+2)

Since this posting was inspired by one of Paul's which mentioned
Blackjack scales, I am going to propose the following definition:

A *jack* is a number which is the value of A(r, n) for n positive and
1/2 <= r < 1, or B(r, n) for n positive and 1/2 <= r <= 1.

(Since this is not being submitted to a peer-reviewed journal, no one
can stop me! Aint the Internet wonderful?)

It can happen that two jacks (n+1)/n and n/(n-1) are adjacent Farey
fractions. When this happens, the ratio of these two jacks is also a
jack, which "jumps" from one jack to another. This leads to the
following definition:

A *jumping jack* is the ratio of two jacks (n+1)/n and n/(n-1) which
are adjacent Farey fractions, so that the jumping jack is B(1, n) =
n^2/(n^2-1). Examples are 81/80 and 2401/2400 where the numerator is
the square of a square--ie, a fourth power--and 225/224, where the
numerator is the square of a triangular number.

Finally, we can have a jack which is defined as a jack in more than
one way. To weed out uninteresting cases, let us consider only A(p/q,
n) and B(p/q, n) where n>=q. We then may define a *high jack* as a
jack which is a jack in two different ways, with the condition on n
above for both of the ways. Since 36 is both the 8th triangular and
6th square, it defines the high jack 36/35.

When we focus on one particular type of jumping jack or high jack, we
can determine all jacks of that type by solving a Pell's equation,
which is a Diophantine equation (an equation for which we seek
solutions in the integers) of the form x^2 - dy^2 = N, where d and N
are fixed positive integers and d is square-free. We then seek all
solutions in pairs x, y of integers. In this way we can determine a
formula (and a recurrence relationship) for all square numbers which
are one less than a triangular number, and so forth.

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> The q1+q2 rule leads not to a Farey sequence, but to what we've
> called a "Mann" sequence. For the Farey sequence, the rule is
>
> reduced form and q1 <= n and q2 <=n.
>
> See Hardy and Wright, for example.

I just saw Hardy and Wright, and they say F_5, the fifth Farey
sequence, is just what I said it would be. I defined it interatively,
as for instance Niven and Zuckerman do, and Hardy and Wright define
it directly, but either way it comes out the same. This is a
completely standard definition in elementary number theory, but I'm
afraid I don't know what a Mann sequence is--from the way you refer
to it, it seems it is not a standard definition.

> Anyhow, the rest looks very interesting . . . what post was it
> inspired by . . . and how does it relate to tuning?

It was inspired by something you posted saying the Blackjack was
derived from 36/35 (a high jack) and 81/80, 225/224, and 2401/2400
(jumping jacks.) This suggests how to find something similar; I
thought I would write up an explanation for why certain
superparticular intervals keep popping up in music theory.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > The q1+q2 rule leads not to a Farey sequence, but to what we've
> > called a "Mann" sequence. For the Farey sequence, the rule is
> >
> > reduced form and q1 <= n and q2 <=n.
> >
> > See Hardy and Wright, for example.
>
> I just saw Hardy and Wright, and they say F_5, the fifth Farey
> sequence, is just what I said it would be.

Hmm . . . isn't it

0 1 1 1 2 1 3 2 3 4 1
-, -, -, -, -, -, -, -, -, -, - ?
1 5 4 3 5 2 5 3 4 5 1

This is commonly known as the "Farey series of order 5" -- I thought
I remembered seeing it in Hardy and Wright this way -- is it
a "series" vs. "sequence" thing?

Anyhow, it seems that the "determinant" of two consecutive fractions
being equal to one is a property of many differently-defined series --
for example, a product limit on numerator times denominator, or a
layer of the Stern-Brocot tree . . .

> I defined it interatively,
> as for instance Niven and Zuckerman do, and Hardy and Wright define
> it directly, but either way it comes out the same.

Maybe I misunderstood your iterative definition -- does my rule
correspond to Hardy and Wright's direct derivation?

> This is a
> completely standard definition in elementary number theory, but I'm
> afraid I don't know what a Mann sequence is--from the way you refer
> to it, it seems it is not a standard definition.

No, Mann wrote a book called _Analytic Study of Harmonic Intervals_
or something like that.
>
> > Anyhow, the rest looks very interesting . . . what post was it
> > inspired by . . . and how does it relate to tuning?
>
> It was inspired by something you posted saying the Blackjack was
> derived from 36/35 (a high jack) and 81/80, 225/224, and 2401/2400
> (jumping jacks.)

81/80 shouldn't be there.

> This suggests how to find something similar; I
> thought I would write up an explanation for why certain
> superparticular intervals keep popping up in music theory.

I'll have to look at it more closely . . . I posted my own
explanation for that recently, in a discussion with Monz . . .