72-tET bicycle chains

Around last May or so, when the "big" discussion of the Miracle scale
was taking place, there was also a discussion of 72-tET. Actually,
that might have even taken place in April...

Anyway, we came up with the idea of 6 "bicycle chains" in 72-tET and
there was an illustration of the factors that were used to multiply
one chain from the next.

I can't seem to find the post that illustrated this. Could somebody
either point it out to me or refresh my humble mind with what the 6
chains are multiplied by so they get so many just intervals.

Any help would be greatly appreciated...

Thanks!

__________ _________ ________
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> Around last May or so, when the "big" discussion of the Miracle
scale
> was taking place, there was also a discussion of 72-tET. Actually,
> that might have even taken place in April...
>
> Anyway, we came up with the idea of 6 "bicycle chains" in 72-tET
and
> there was an illustration of the factors that were used to multiply
> one chain from the next.
>
> I can't seem to find the post that illustrated this. Could
somebody
> either point it out to me or refresh my humble mind with what the 6
> chains are multiplied by so they get so many just intervals.
>
> Any help would be greatly appreciated...
>
> Thanks!

The central 12-tET chain (chosen arbitrarily) represents all the
ratios with prime numbers no higher than 3.

Then,

the 1/12-tone-lowered chain corresponds to multiplication by 5,
the 1/6-tone-lowered chain corresponds to multiplication by 7,
the 1/4-tone-altered chain corresponds to mult. or div. by 11,
the 1/6-tone-raised chain corresponds to division by 7,
the 1/12-tone-raised chain corresponds to division by 5.

Is that what you were looking for?

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

/tuning/topicId_29879.html#29886

>
> the 1/12-tone-lowered chain corresponds to multiplication by 5,
> the 1/6-tone-lowered chain corresponds to multiplication by 7,
> the 1/4-tone-altered chain corresponds to mult. or div. by 11,
> the 1/6-tone-raised chain corresponds to division by 7,
> the 1/12-tone-raised chain corresponds to division by 5.
>
> Is that what you were looking for?

Absolutely! Thanks, Paul! I know it's somewhere buried back in the
archives, but I couldn't find it yet....

Joseph

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

/tuning/topicId_29879.html#29886

>
> The central 12-tET chain (chosen arbitrarily) represents all the
> ratios with prime numbers no higher than 3.
>
> Then,
>
> the 1/12-tone-lowered chain corresponds to multiplication by 5,
> the 1/6-tone-lowered chain corresponds to multiplication by 7,
> the 1/4-tone-altered chain corresponds to mult. or div. by 11,
> the 1/6-tone-raised chain corresponds to division by 7,
> the 1/12-tone-raised chain corresponds to division by 5.
>
> Is that what you were looking for?

Paul... just because I'm a little dense, could you please multiply
out some ratios with this as an example, so I'm certain I see how it
works.

The multiplication and division by 5 obviously has something to do
with the syntonic comma and 5-limit sonorities, and the
multiplication and division by 7 seems to involve the septimal comman
and 7-limit sonorities, yes?? And quarter-tones are, I believe, near
the 11-limit... (??)

I'm getting this in a "general" way, since it seems clear that
lowering by 1/12 tone and using *that* chain with, let's say, the
fundamental C from the original chain will create a 5-limit or just
major third (or close) and the same process pertains with the minor
7th when a 1/6 tone chain lower is used...

However, I'm a little weak on the "details..."

How does that actually multiply out "in action" in a couple of
examples.

If you have time for this, I would *greatly* appreciate it!!!!!

Joseph

--- In tuning@y..., jpehrson@r... wrote:

> > Then,
> >
> > the 1/12-tone-lowered chain corresponds to multiplication by 5,
> > the 1/6-tone-lowered chain corresponds to multiplication by 7,
> > the 1/4-tone-altered chain corresponds to mult. or div. by 11,
> > the 1/6-tone-raised chain corresponds to division by 7,
> > the 1/12-tone-raised chain corresponds to division by 5.
> >
> > Is that what you were looking for?
>
>
> Paul... just because I'm a little dense, could you please multiply
> out some ratios with this as an example, so I'm certain I see how
it
> works.

Let's say C is in the central chain. Call it 1/1.

Now 5/4, since it requires multiplication by 5 (and no other primes
above 3 are involved), will be found in the 1/12-tone-lowered chain.

7/6, since it requires multiplication by 7 (and no other primes above
3 are involved), will be found in the 1/6-tone-lowered chain.

11/10, since it requires multiplication by 11, and division by 5 (and
no other primes above 3 are involved), will be found in the chain
1/12-tone higher than the 1/4-tone altered chain -- i.e., in the 1/6-
tone lowered chain.

99/70, since it requires multiplication by 11, and division by 5 and
by 7, will be (-1/4)+1/12+1/6 = 0 away from the original chain, that
is, back on the original chain!

And so on.

> The multiplication and division by 5 obviously has something to do
> with the syntonic comma and 5-limit sonorities,

Yup -- since 80/81 requires multiplication by 5 (and no other primes
above 3 are involved), it corresponds to a move to the 1/12-tone
lowered chain -- and 81/80 does the opposite.

> and the
> multiplication and division by 7 seems to involve the septimal >
comma
> and 7-limit sonorities, yes??

Yup -- since 63/64 requires multiplication by 7 (and no other primes
above 3 are involved), it corresponds to a move to the 1/6-tone
lowered chain -- and 64/63 does the opposite.

> And quarter-tones are, I believe, near
> the 11-limit... (??)

33/32 is sometimes called the "undecimal comma".

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

/tuning/topicId_29879.html#29902

> --- In tuning@y..., jpehrson@r... wrote:
>
> > > Then,
> > >
> > > the 1/12-tone-lowered chain corresponds to multiplication by 5,
> > > the 1/6-tone-lowered chain corresponds to multiplication by 7,
> > > the 1/4-tone-altered chain corresponds to mult. or div. by 11,
> > > the 1/6-tone-raised chain corresponds to division by 7,
> > > the 1/12-tone-raised chain corresponds to division by 5.
> > >
>
> Let's say C is in the central chain. Call it 1/1.
>
> Now 5/4, since it requires multiplication by 5 (and no other primes
> above 3 are involved), will be found in the 1/12-tone-lowered chain.
>
> 7/6, since it requires multiplication by 7 (and no other primes
above 3 are involved), will be found in the 1/6-tone-lowered chain.
>

Thanks, Paul... This post was *exactly* what I was looking for...

However, I have one probably "silly" question that I can't
immediately see.

Why is it when you multiply the ratios by integers do the chains get
*lower??* The obvious thought would be that the chains would get
*higher...* I must be missing something obvious....

Thanks!

Joseph

--- In tuning@y..., jpehrson@r... wrote:

> Why is it when you multiply the ratios by integers do the chains
get
> *lower??* The obvious thought would be that the chains would get
> *higher...* I must be missing something obvious....
>
> Thanks!
>
> Joseph

That's just the way it works out. Be glad that they're all in the
same direction -- for most bicycles other than 72-tET, it wouldn't
work out that way!

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

/tuning/topicId_29879.html#29918

> --- In tuning@y..., jpehrson@r... wrote:
>
> > Why is it when you multiply the ratios by integers do the chains
> get
> > *lower??* The obvious thought would be that the chains would get
> > *higher...* I must be missing something obvious....
> >
> > Thanks!
> >
> > Joseph
>
> That's just the way it works out. Be glad that they're all in the
> same direction -- for most bicycles other than 72-tET, it wouldn't
> work out that way!

Well... now naturally you've piqued my curiosity...

Why should this work out like this for 72 and not for the other kinds
of bikes??

Joseph

--- In tuning@y..., jpehrson@r... wrote:

> Well... now naturally you've piqued my curiosity...
>
> Why should this work out like this for 72 and not for the other
kinds
> of bikes??
>
> Joseph

Well, it just works out that way. It's "luck". Or, it's because 12-
tET is about 1/12-tone sharp for the 5, 1/6-tone sharp for the 7, and
1/4-tone sharp for the 11. Why? Well, it just works out that way.
It's "luck".

Let's consider 217-tET, which consists of seven 31-tET chains.

Multiplying by 3 corresponds to moving up 1 chain.
Multiplying by 5 corresponds to staying on the same chain.
Multiplying by 7 corresponds to staying on the same chain.
Multiplying by 11 corresponds to moving up 2 chains.
Multiplying by 13 corresponds to moving down 2 chains.
Multiplying by 17 corresponds to moving down 2 chains.
Multiplying by 19 corresponds to moving down 2 chains.

Now that's different, isn't it? Certainly not as easy to remember as

3 -> down zero
5 -> down one
7 -> down two
11 -> down three

More examples:

58-tET as two 29-tET chains:
3 -> stay on same chain
5 -> move to other chain
7 -> move to other chain
11 -> move to other chain
13 -> move to other chain

68-tET as four 17-tET chains:
3 -> stay on same chain
5 -> move two chains
7 -> move down one chain

and good ol'
72-tET as six 12-tET chains:
3 -> stay on same chain
5 -> move down one chain
7 -> move down two chains
11 -> move three chains
13 -> move up two chains
17 -> stay on same chain

In-Reply-To: <9s7n3c+b90a@eGroups.com>
There's also 188 as two chains of 94

3 -> stay on same chain
5 -> move to other chain
7 -> stay on same chain
11 -> move to other chain
13 -> stay on same chain
17 -> move to other chain
19 -> move to other chain

Graham

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

/tuning/topicId_29879.html#29924

> --- In tuning@y..., jpehrson@r... wrote:
>
> > Well... now naturally you've piqued my curiosity...
> >
> > Why should this work out like this for 72 and not for the other
> kinds
> > of bikes??
> >
> > Joseph
>
> Well, it just works out that way. It's "luck". Or, it's because 12-
> tET is about 1/12-tone sharp for the 5, 1/6-tone sharp for the 7,
and
> 1/4-tone sharp for the 11. Why? Well, it just works out that way.
> It's "luck".
>
> Let's consider 217-tET, which consists of seven 31-tET chains.
>
> Multiplying by 3 corresponds to moving up 1 chain.
> Multiplying by 5 corresponds to staying on the same chain.
> Multiplying by 7 corresponds to staying on the same chain.
> Multiplying by 11 corresponds to moving up 2 chains.
> Multiplying by 13 corresponds to moving down 2 chains.
> Multiplying by 17 corresponds to moving down 2 chains.
> Multiplying by 19 corresponds to moving down 2 chains.
>
> Now that's different, isn't it? Certainly not as easy to remember as
>
> 3 -> down zero
> 5 -> down one
> 7 -> down two
> 11 -> down three

Of course... well that's quite something that 72-tET works out that
way...

JP