Term for horagramic division

In each ring of a horagram, the previous ring's large
interval (L) is divided into a new L and s (small
interval). As it does so, it also adopts the "s" from
the previous ring as either it's "s" or it's new "L."
(See http://www.anaphoria.com/wilson.html ) It seems to
me that there's something "proper" about a ring's "s"
becoming the next ring's "L," as opposed to adjacent
rings sharing an "s" size. The term "proper" already
has an important meaning in the world of scales so it
probably should not be called something like "proper
subdivision." For now, I'm calling it "sequential
subdivision." But that feels a little clumsy and
vague. Any suggestions?

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html

--

David Finnamore wrote,

>In each ring of a horagram, the previous ring's large
>interval (L) is divided into a new L and s (small
>interval). As it does so, it also adopts the "s" from
>the previous ring as either it's "s" or it's new "L."
>(See <http://www.anaphoria.com/wilson.html>
http://www.anaphoria.com/wilson.html ) It seems to
>me that there's something "proper" about a ring's "s"
>becoming the next ring's "L," as opposed to adjacent
>rings sharing an "s" size. The term "proper" already
>has an important meaning in the world of scales so it
>probably should not be called something like "proper
>subdivision." For now, I'm calling it "sequential
>subdivision." But that feels a little clumsy and
>vague. Any suggestions?

David, in every example of "s" from the previous ring becoming "L" in the
next ring, both rings have L:s in the golden ratio. In every example where
"s" remains "s", the inner ring (if not both) does not have L:s in the
golden ratio. You might want to verify this. Hence I suggest "golden
subdivision" as the term for what you're looking for.

-Paul

David and Paul!
I have a call into Erv to see what he calls it, which seems to be what should be the first step,
unless one is inclined to rip it out of his hands. Those layers where the s remains s (very inner
rings) are those points where the ratio has not converged to gold. You can also tell this in that
Erv moves the numbers to the right hand side when they do. The left hand side is when they haven't
converge yet. Often times the most interesting musically.

"Paul H. Erlich" wrote:

> David Finnamore wrote,
>
> For now, I'm calling it "sequential
> >subdivision." But that feels a little clumsy and
> >vague. Any suggestions?
>
> David, in every example of "s" from the previous ring becoming "L" in the
> next ring, both rings have L:s in the golden ratio. In every example where
> "s" remains "s", the inner ring (if not both) does not have L:s in the
> golden ratio. You might want to verify this. Hence I suggest "golden
> subdivision" as the term for what you're looking for.
>
> -Paul
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

David!
Two interval pattern sequence. from the source

"David J. Finnamore" wrote:

> Any suggestions?
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

Paul H. Erlich wrote:

> David, in every example of "s" from the previous ring becoming "L" in the
> next ring, both rings have L:s in the golden ratio.

> In every example where
> "s" remains "s", the inner ring (if not both) does not have L:s in the
> golden ratio. You might want to verify this. Hence I suggest "golden
> subdivision" as the term for what you're looking for.

It doesn't work out quite that way, unless you define phi so loosely that all rings are considered to
have a golden L:s. All the rings have divisions that relate to the golden ratio one way or another. If
phi is taken to be 0.618... [that is, (sqrt(5)-1)/2] then rings have L:s variously of 1+phi, 2+phi,
3+phi, 2-phi, etc. Some horagrams have rings with even more complicated relationships to phi. Not all
rings that take their "L" from the previous ring's "s" have L:s = 1+phi.

It appears that it always converges to 1+phi and stays there, at some point. This will be detailed for
each horagram when I put up my web site about it.

Kraig Grady wrote:

> I have a call into Erv to see what he calls it, which seems to be what should be the first step,

> unless one is inclined to rip it out of his hands.

Don't be silly! I'm not at all inclined to rip it out of his hands. At all. All I want to do is
discover their musical usefulness for myself, and share my discoveries with others. Naturally, then,
I'm very eager to learn any and all terminology that Erv uses in conjunction with his horagrams. If
he's willing to tell us, that would be wonderful. I got the idea he was reluctant to divulge any
further details, which is why I didn't ask you directly to ask him. I was hoping that someone out there
knew what he called it, or else knew what that sort of thing would customarily be called in mathematics.

Thank you for asking him for us, Kraig! While you're at it, do you think you could get him to give you
a glossary of horagram related terms? That way you won't have to bug him personally every time we have
a new question, right?

> Those layers where the s remains s (very inner
> rings) are those points where the ratio has not converged to gold. You can also tell this in that
> Erv moves the numbers to the right hand side when they do. The left hand side is when they haven't
> converge yet.

There are two separate issues here. 1) Whether a ring gets its "s" size or its "L" size from the
previous ring. 2) What relationship the L:s ratio within a single ring bears to phi. The two are
independent of each other, although apparently they both always occur in the outer rings (the ones
beginning with the *second* ring numbered on the right in the horagrams drawn by Erv, seemingly).

Look at #5, for example. 1-2-3-4-5-9-14-19-33-52-85-137. For rings 5 and 9, "L" equals the previous
ring's "s" (but their L:s = 2-phi and 2+phi, respectively). But ring 14 shares the "s" with ring 9.
Rings 19 and on out are golden in both senses.

Horagram #8. 1-2-3-4-5-9-13-17-30-47-77-124. Ring 5 (3+phi) gets its "L" from the previous ring's
"s." 9 and 13 do not - even though 13 is golden in the sense of L:s = 1+phi. 17 on out appear to be
golden in both senses, at a glance.

It's not quite as pretty and clear cut as you might expect. But as you said, there's where you often
find the most musically interesting possibilities. It's easy enough to make a table column for each
ring that shows its L:s ratio in term of phi. What I need is a term simply for the property "L" = the
previous ring's "s" - regardless of the L:s of the ring in question.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

Kraig Grady wrote:

> Two interval pattern sequence. from the source

Argh. Are you sure? That seems more vague and clumsy than what I called it myself. In fact, it
appears to be merely synonymous with the term "MOS." Maybe for now I'll just stick with "L = previous
s."

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

David!
It is also referred to as the rabbit sequence where the s grows up to be an L and then has an
offspring l,s. This goes all the way back to Fibonacci in Liber Abaci.

the term inflation is also used. Steinhardt discussed much of this I am told

"David J. Finnamore" wrote:

> Kraig Grady wrote:
>
> > Two interval pattern sequence. from the source
>
> Argh. Are you sure? That seems more vague and clumsy than what I called it myself. In fact, it
> appears to be merely synonymous with the term "MOS." Maybe for now I'll just stick with "L = previous
> s."
>
> --
> David J. Finnamore
> Nashville, TN, USA
> http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
> --
>
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-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

"David J. Finnamore" wrote:

>
>
> Kraig Grady wrote:
>
> > I have a call into Erv to see what he calls it, which seems to be what should be the first step,
>
> > unless one is inclined to rip it out of his hands.
>
> Don't be silly! I'm not at all inclined to rip it out of his hands. At all. All I want to do is
> discover their musical usefulness for myself, and share my discoveries with others.

Don't be silly! This wasn't directed to you

> . I was hoping that someone out there
> knew what he called it, or else knew what that sort of thing would customarily be called in mathematics.

That is the basis of his use of terms. Often he is so engrossed in things it is to distracting from what he
is working on, but not always.

>
> > Those layers where the s remains s (very inner
> > rings) are those points where the ratio has not converged to gold. You can also tell this in that
> > Erv moves the numbers to the right hand side when they do. The left hand side is when they haven't
> > converge yet.
>
> There are two separate issues here. 1) Whether a ring gets its "s" size or its "L" size from the
> previous ring. 2) What relationship the L:s ratio within a single ring bears to phi. The two are
> independent of each other, although apparently they both always occur in the outer rings (the ones
> beginning with the *second* ring numbered on the right in the horagrams drawn by Erv, seemingly).
>
> Look at #5, for example. 1-2-3-4-5-9-14-19-33-52-85-137. For rings 5 and 9, "L" equals the previous
> ring's "s" (but their L:s = 2-phi and 2+phi, respectively). But ring 14 shares the "s" with ring 9.
> Rings 19 and on out are golden in both senses.

Yes you are right and i missed that in this context. Thanks for pointing this out.

>

> Horagram #8. 1-2-3-4-5-9-13-17-30-47-77-124. Ring 5 (3+phi) gets its "L" from the previous ring's
> "s." 9 and 13 do not - even though 13 is golden in the sense of L:s = 1+phi. 17 on out appear to be
> golden in both senses, at a glance.

correct again.

>
>
> It's not quite as pretty and clear cut as you might expect. But as you said, there's where you often
> find the most musically interesting possibilities. It's easy enough to make a table column for each
> ring that shows its L:s ratio in term of phi. What I need is a term simply for the property "L" = the
> previous ring's "s" - regardless of the L:s of the ring in question.

I like the rabbit sequence more and more. The rabbit grows up sometimes slower but once an adult, produces
offspring. Is this true? a s will become a s or L but an L will always result in a L,s

> --
> David J. Finnamore
> Nashville, TN, USA
> http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
> --

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

Kraig Grady wrote:

> I like the rabbit sequence more and more. The rabbit grows up sometimes slower but once an adult, produces
> offspring. Is this true? a s will become a s or L but an L will always result in a L,s

Yes, that's the way it works, and it's a good, simple, concise summary statement. I like the term "rabbit
sequence," too - it's fun! Thanks a ton!

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

I wrote:

> All the rings have divisions that relate to the golden ratio one way or another. If
> phi is taken to be 0.618... [that is, (sqrt(5)-1)/2] then rings have L:s variously of 1+phi, 2+phi,
> 3+phi, 2-phi, etc. Some horagrams have rings with even more complicated relationships to phi.

Couldn't remember which at the time, nor exactly what. Here's one: horagram #21 [(8phi+3)/(21phi+8)], ring
5 has L=288.44 cents, s=167.34 cents (pattern sLsLL). That yields an L:s of 1.7236068... (using higher
precision values). That's equal to - are you ready for this? - 1+(1/(2-phi))! What's that all about?

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

David wrote,

>That's equal to - are you ready for this? - 1+(1/(2-phi))! >What's that
all about?

This is related to the continued fraction representation of noble numbers
and of phi (something you should read up on -- see Manfred Schroeder's
_Fractals, Chaos, Power Laws_ for example.

Any ratio can be written uniquely as a continued fraction:

a + 1/(b + 1/(c + 1/(d + . . . )))

where a, b, c, d are whole numbers. Now

Phi = 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/ . . . ))))

So any noble number can be written as

a + 1/(b + 1/(c + 1/(d + . . . /(z + phi)) . . . )

since all their continued fractions representations end in all 1's.

For example, the ratio you posted,
(8phi+3)/(21phi+8),
can be written as
1/(2 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + phi))))).

Every time you make a "rabbit hop" move in expanding though the rings of the
horagrams, the ratio L:s drops off the first piece of its continued fraction
representation.

So the first ring, obviously, has s:L equal to

1/(2 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + phi))))).

The next rabbit hop move takes you to

1/(1 + 1/(1 + 1/(1 + 1/(2 + phi)))).

and the next takes you to

1/(1 + 1/(1 + 1/(2 + phi))).

Which you might observe is equal to the 1+(1/(2-phi)) value you posted.

Paul Erlich wrote:

> David wrote,
>
> >That's equal to - are you ready for this? - 1+(1/(2-phi))! >What's that
> all about?
>
> This is related to the continued fraction representation of noble numbers
> and of phi (something you should read up on -- see Manfred Schroeder's
> _Fractals, Chaos, Power Laws_ for example.

Etc.

Paul, that was so very helpful! Would you mind if I quoted some of your post in the introductory
page of my Horagram website? I'll have to try to squeeze Schroeder's book further up my Must Read
list.

Thanks!

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

Thanks Paul had missed that one!

"Paul H. Erlich" wrote:

> This is related to the continued fraction representation of noble numbers
> and of phi (something you should read up on -- see Manfred Schroeder's
> _Fractals, Chaos, Power Laws_ for example.
>
> Any ratio can be written uniquely as a continued fraction:
>
> a + 1/(b + 1/(c + 1/(d + . . . )))
>
> where a, b, c, d are whole numbers. Now
>
> Phi = 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/ . . . ))))
>
>
> So any noble number can be written as
>
> a + 1/(b + 1/(c + 1/(d + . . . /(z + phi)) . . . )
>
> since all their continued fractions representations end in all 1's.
>
> For example, the ratio you posted,
> (8phi+3)/(21phi+8),
> can be written as
> 1/(2 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + phi))))).
>
> Every time you make a "rabbit hop" move in expanding though the rings of the
> horagrams, the ratio L:s drops off the first piece of its continued fraction
> representation.
>
> So the first ring, obviously, has s:L equal to
>
> 1/(2 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + phi))))).
>
> The next rabbit hop move takes you to
>
> 1/(1 + 1/(1 + 1/(1 + 1/(2 + phi)))).
>
> and the next takes you to
>
> 1/(1 + 1/(1 + 1/(2 + phi))).
>
> Which you might observe is equal to the 1+(1/(2-phi)) value you posted.

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm