A counterexample to the hypothesis?

Let's see if I understand the hypothesis well enough to have found a
counterexample.

Suppose I decide to construct a scale with 15 notes to the octave in
7-limit harmony. I therefore turn my attention to the homomorphic map
[15, 24, 35, 42] which in spite of the way I have written it I may
consider to be a column vector. The kernel of this homomorphism is
generated by 28/27, 64/63 and 126/125, of which the latter two seem
to be good candidates for commatic unisons and the first for our
chromatic unison. Setting up a Graham-type matrix and inverting it, I
find 15 notes within an octave forming a scale:

1 - 21/20 - 10/9 - 9/8 - 6/5 - 5/4 - 4/3 - 7/5 -
10/7 - 3/2 - 8/5 - 5/3 - 16/9 - 9/5 - 40/21 - (2)

This scale has intervals of sizes 81/80, 50/49, 25/24, 21/20,
16/15 and 200/189. Since six intervals of widely differing sizes will
surely drive us nuts, we decide to temper out the two unison vectors,
namely 64/63 and 126/125. One simple way to do this is to use the 27-
et, which has both of them in its kernel (like the 15-et.) When we do
this, we find that 81/80 and 50/49 go to intervals of one scale step,
while the rest go to intervals of two scale steps. We now have the
pattern LLsLLLLsLLLLsLL, which has three repeating divisions of a
third of an octave. Isn't this a counterexample?

In-Reply-To: <9lvm8o+9v8r@eGroups.com>
Gene wrote:

> This scale has intervals of sizes 81/80, 50/49, 25/24, 21/20,
> 16/15 and 200/189. Since six intervals of widely differing sizes will
> surely drive us nuts, we decide to temper out the two unison vectors,
> namely 64/63 and 126/125. One simple way to do this is to use the 27-
> et, which has both of them in its kernel (like the 15-et.) When we do
> this, we find that 81/80 and 50/49 go to intervals of one scale step,
> while the rest go to intervals of two scale steps. We now have the
> pattern LLsLLLLsLLLLsLL, which has three repeating divisions of a
> third of an octave. Isn't this a counterexample?

No, it's an MOS with a period of a third of an octave for the purposes of
the hypothesis. The generator is the interval L.

Graham

--- In tuning-math@y..., genewardsmith@j... wrote:

> We now have the
> pattern LLsLLLLsLLLLsLL, which has three repeating divisions of a
> third of an octave. Isn't this a counterexample?

Why would this be a counterexample if LssssLssss wasn't?

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

> Why would this be a counterexample if LssssLssss wasn't?

I had somehow gotten the idea such a thing would be a counterexample.
I didn't look at 10 or 12 because they would have already been picked
over, so 15 looked like a good composite to try. I think by the way
that 15 notes out of a 27 scale would be a fine system to try for
anyone who likes sharp systems and who wants to wash the 12-et out of
their head for a while.

I have the idea that what we are talking about is some sort of
equidistribution property, but I can't seem to pin down the property.
Suppose I define a two-value scale structure to be a periodic
sequence of L and s values, containing both values. I could try the
following:

(1) Every contiguous set of n values has the same number of L's and
s's as every other such set.

The trouble with this is that nothing passes this condition.

(2) Contiguous sets of n values have on average the same number of
L's and s's.

The trouble with this is that everything passes this condition.

--- In tuning-math@y..., genewardsmith@j... wrote:
> I have the idea that what we are talking about is some sort of
> equidistribution property, but I can't seem to pin down the
property.

It has a rigorous mathematical definition which I find astonishing
because it is so simple and it doesn't directly say anything about
step sizes. It's essentially in
http://depts.washington.edu/pnm/CLAMPITT.pdf
and I've implemented it in an Excel spreadsheet at
http://dkeenan.com/Music/MyhillCalculator.xls

Take a generator g in cents (typically a fourth), an interval of
equivalence e in cents (typically an octave), and a whole number i
(typically 1, 2, 3). Then we define the period (or interval of
repetition) p = e/i.

Then a scale generated by iteratively adding that generator, modulo
that period (doing the same in all i periods), will be MOS iff the
number of notes per period n is the denominator of either a convergent
or a semiconvergent of g/p. Convergents and semiconvergents are
defined in relation to the continued-fraction expansion of p/g. See
Clampitt's paper above if you need the details.

The cardinality c of the scale is n * i. Musically, we are really only
interested in those with cardinalities between about 5 and 99.

Furthermore, if n (the number of notes per period) is a denominator of
a convergent (not a semiconvergent), the scale is not only MOS, but is
"strictly-proper", which we don't need to go into now, but
generally makes it musically more interesting.

-- Dave Keenan

OOps! I wrote:
> Convergents and semiconvergents are
> defined in relation to the continued-fraction expansion of p/g.

I meant "of g/p" (generator over period).

What makes the apparent relationship between MOS and
tempered periodicity blocks hard to fathom (for me at least) is that
with periodicity blocks we are dealing with the generator and interval
of equivalence as ratios of frequencies, not cents. Call them G and E
so that g = log(G) and e = log(E), where logs are to base 2^(1/1200).
So MOS is defined in terms of the convergents and semiconvergents of
log(G)/[log(E)/i].

-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> What makes the apparent relationship between MOS and
> tempered periodicity blocks hard to fathom ...

Well, tastes differ. I certainly hope your definition is correct,
because:

(1) It makes sense

(2) It certainly seems like a condition we would want a scale with
one generator in the octave to satisfy

(3) It explains why Paul's hypothesis is true.

For the last, note if for instance we want a q-step scale generated
by interating the p-th step, where p/q is approximately log_2(3/2),
so that fifths are represented by p steps in the q-division, then if
x is log_2 of our approximation for the fifth, we want p/q to be a
semiconvergent of x.

Now, if memory serves, if |x - p/q| < 1/q^2 then p/q is a
semiconvergent for x, and the converse is "almost" true. The details
don't much matter, unless you wanted to write this up for
publication, because it is clear that if x is anything in a certain
interval around p/q, then p/q will be a semiconvergent for it. If p/q
is a good enough approximation for log_2(3/2), say one that satisfies
|log_2(3/2) - p/q| < 1/(2*q^2), then anything close to log_2(3/2)
will be close enough to p/q so that p/q must be a semiconvergent for
it. Since the approximation x ought to be better than that given by
p/q, we would expect to find (and will find, if we wrote this up with
the details worked out as if for publication) that it gives a MOS.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > What makes the apparent relationship between MOS and
> > tempered periodicity blocks hard to fathom ...
>
> Well, tastes differ. I certainly hope your definition is correct,

It's almost correct. There can be an interval of repetition which is
1/P of the interval of equivalence, with P an integer.

> because:
>
> (1) It makes sense
>
> (2) It certainly seems like a condition we would want a scale with
> one generator in the octave to satisfy
>
> (3) It explains why Paul's hypothesis is true.
>
> For the last, note if for instance we want a q-step scale generated
> by interating the p-th step, where p/q is approximately log_2(3/2),
> so that fifths are represented by p steps in the q-division, then
if
> x is log_2 of our approximation for the fifth, we want p/q to be a
> semiconvergent of x.
>
> Now, if memory serves, if |x - p/q| < 1/q^2 then p/q is a
> semiconvergent for x, and the converse is "almost" true. The
details
> don't much matter, unless you wanted to write this up for
> publication, because it is clear that if x is anything in a certain
> interval around p/q, then p/q will be a semiconvergent for it. If
p/q
> is a good enough approximation for log_2(3/2), say one that
satisfies
> |log_2(3/2) - p/q| < 1/(2*q^2), then anything close to log_2(3/2)
> will be close enough to p/q so that p/q must be a semiconvergent
for
> it. Since the approximation x ought to be better than that given by
> p/q, we would expect to find (and will find, if we wrote this up
with
> the details worked out as if for publication) that it gives a MOS.

Lost me there. Did you see my "proof" of the hypothesis?

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

> Lost me there. Did you see my "proof" of the hypothesis?

No--where is it hiding?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Lost me there. Did you see my "proof" of the hypothesis?
>
> No--where is it hiding?

Gene, I'm getting the feeling that somehow you're missing a good
number of my posts. Like the chromatic vs. commatic unison vector
deal . . . I explained that quite a number of times before you seemed
to acknowledge it.

Anyway, as I tried to tell you before, my sketch of a proof is in
message #591.

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

> Gene, I'm getting the feeling that somehow you're missing a good
> number of my posts. Like the chromatic vs. commatic unison vector
> deal . . . I explained that quite a number of times before you
seemed
> to acknowledge it.

Sorry about that. I did read that message before, but that was in an
attempt to understand definitions--what is a MOS, hyper-MOS, PB,
unison vector and so forth. The definitions are gradually becoming
clear, so these things began to make sense to me.

You were discussing the two-scale-step condition, which made it sound
to me as if the whole business was defined in terms of large and
small steps, without reference to tuning. I like the semiconvergents
at this point!

--- In tuning-math@y..., genewardsmith@j... wrote:

> You were discussing the two-scale-step condition, which made it
sound
> to me as if the whole business was defined in terms of large and
> small steps, without reference to tuning. I like the
semiconvergents
> at this point!

In my proof, I actually get at _two different_ (but mathematically
equivalent) definitions of MOS. One concerns not a "two-scale-step"
condition, as you say, but rather a "Myhill" condition, which says
that every generic interval, not just steps but _any_ interval, aside
from the interval of repetition, will come in exactly two step sizes.

But the other definition of MOS I was trying to get at in my proof
is, I believe, the "semiconvergent" one. I'm talking about vectors
which have length M/N in the direction of the period boundaries. This
is clearly very closely related (much more so than the Myhill stuff)
to the continued fraction approximation process. Do you see that?

Anyway, if you're beginning to think the hypothesis is true, then for
any set of n-1 (or n-2, if we're counting your way) commatic unison
vectors, you should be able to find the generator of the resulting
linear temperament. Do you see a direct mathematical way of
determining what this generator is?

I wrote,
>
> In my proof, I actually get at _two different_ (but mathematically
> equivalent) definitions of MOS. One concerns not a "two-scale-step"
> condition, as you say, but rather a "Myhill" condition, which says
> that every generic interval, not just steps but _any_ interval,
aside
> from the interval of repetition, will come in exactly two step
sizes.

Ack! You got me doing it! I meant _two sizes_ at the end there,
scratch the word "step"!

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

One concerns not a "two-scale-step"
> > condition, as you say, but rather a "Myhill" condition, which
says
> > that every generic interval, not just steps but _any_ interval,
> aside
> > from the interval of repetition, will come in exactly two step
> sizes.

> Ack! You got me doing it! I meant _two sizes_ at the end there,
> scratch the word "step"!

Let's see if I can phrase this purely in terms of steps, without
reference to sizes:

Let S be a sequence consisting of steps of sizes L and s, which is
periodic with period P. Every set of N contiguous steps will consist
of p L's and q s's, such that p+q = N. Then the Myhill' condition
says that exactly two counts of L's, p1 and p2, occur for every N.
(Hence of course only two counts of s's, so that we have p1+q1 = N
and p2+q2 = N.)

Is the Myhill' condition the same as the Myhill condition?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> One concerns not a "two-scale-step"
> > > condition, as you say, but rather a "Myhill" condition, which
> says
> > > that every generic interval, not just steps but _any_ interval,
> > aside
> > > from the interval of repetition, will come in exactly two step
> > sizes.
>
> > Ack! You got me doing it! I meant _two sizes_ at the end there,
> > scratch the word "step"!
>
> Let's see if I can phrase this purely in terms of steps, without
> reference to sizes:
>
> Let S be a sequence consisting of steps of sizes L and s, which is
> periodic with period P. Every set of N contiguous steps will consist
> of p L's and q s's, such that p+q = N. Then the Myhill' condition
> says that exactly two counts of L's, p1 and p2, occur for every N.
> (Hence of course only two counts of s's, so that we have p1+q1 = N
> and p2+q2 = N.)

Seems correct.
>
> Is the Myhill' condition the same as the Myhill condition?

Almost. The difference is that there may be an interval of
repetition, which is exactly a factor of the number of steps per
interval of equivalence, for which there may be only one count of
L's. For example, for the scale LssssLssss, if N=5, then p1 always
equals 1 and p2 always equals 4.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> > --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > > What makes the apparent relationship between MOS and
> > > tempered periodicity blocks hard to fathom ...
> >
> > Well, tastes differ. I certainly hope your definition is correct,
>
> It's almost correct. There can be an interval of repetition which is
> 1/P of the interval of equivalence, with P an integer.

Huh? I included that. I used "i" where you've used "P" and be careful
because I used "P" to mean something else, i.e. the interval of
repetition (for which "period" is a shorter word) as a frequency
ratio p = log(P). And i is not just any integer, it's a whole-number
(not zero, not negative).

-- Dave Keenan

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > What makes the apparent relationship between MOS and
> > tempered periodicity blocks hard to fathom ...

It's not so hard as I first thought. In MOS we are dealing with the
log of the generator over the log of the period. In PB's we are
dealing with vectors of logs (except they are all logs to different
prime bases).

> Well, tastes differ. I certainly hope your definition is correct,
> because:
...
> (3) It explains why Paul's hypothesis is true.
>
> For the last, ...

Gene, you seem to be only proving (or making plausible) that
cardinality of MOS = denominator of convergent or semiconvergent.
While it's nice to have you confirm it, I believe Carey and Clampitt
proved this in 1989 and we have accepted it.

This is not Paul's hypothesis (or should it be called conjecture, I
thought hypotheses were for science, not math?).

-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> > --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > > What makes the apparent relationship between MOS and
> > > tempered periodicity blocks hard to fathom ...
>
> It's not so hard as I first thought. In MOS we are dealing with the
> log of the generator over the log of the period. In PB's we are
> dealing with vectors of logs (except they are all logs to different
> prime bases).

They can be anything. The just intervals forming the basis for the
lattice are irrelevant for the hypothesis. They can be any intervals
you want, as long as they're linearly independent.

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

> In my proof, I actually get at _two different_ (but mathematically
> equivalent) definitions of MOS. One concerns not a "two-scale-step"
> condition, as you say, but rather a "Myhill" condition, which says
> that every generic interval, not just steps but _any_ interval,
aside
> from the interval of repetition, will come in exactly two step
sizes.

Dave told me I was barking up the wrong tree with the semiconvergent
business, though it seems to me it should imply Myhill's property.
Did what I said then strike you as using a wrong definition?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > In my proof, I actually get at _two different_ (but mathematically
> > equivalent) definitions of MOS. One concerns not a
"two-scale-step"
> > condition, as you say, but rather a "Myhill" condition, which says
> > that every generic interval, not just steps but _any_ interval,
> aside
> > from the interval of repetition, will come in exactly two step
> sizes.
>
> Dave told me I was barking up the wrong tree with the semiconvergent
> business, though it seems to me it should imply Myhill's property.

Yes it certainly does. We already knew that.

> Did what I said then strike you as using a wrong definition?

No. The definition is fine.

But the conjecture/hypothesis we're trying to prove is

tempered-PB = semiconvergent

This is where we could really use your help Gene.

-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> No. The definition is fine.
>
> But the conjecture/hypothesis we're trying to prove is
>
> tempered-PB = semiconvergent
>
> This is where we could really use your help Gene.

I need to see how "tempered-PB" leads to a tuning. For instance, what
does and does not count as a mean-tone tuning?

If we take 25/24 and 81/80 as our canonical example, leading to the 7-
note scales related to the h_7(2^a*3^b*5^c) = 7*a+11*b+16*c 7-et
homomorphism, we can temper out 81/80 (commatic unison) and not 25/24.
We then arrive at the mean-tone systems in the 5-limit. We can
generate our scale with any number of scale steps relatively prime to
7; for instance 4 steps representing a fifth. If we tune the fifth by
setting it to x cents, then x/1200 is log_2 of the approximation to
the fifth are using. This will have 4/7 (4 scale steps out of 7) as a
semiconvergent if |x/1200 - 4/7| < 1/49, but if it is very far out of
this range we are in trouble. In other words, in terms of cents we
want x to be in the interval |x - 685.7 ...| < 171.4...; this
includes a wide range of tunings for a fifth, but not all tunings
whatever.

--- In tuning-math@y..., genewardsmith@j... wrote:

> this range we are in trouble. In other words, in terms of cents we
> want x to be in the interval |x - 685.7 ...| < 171.4...; this
> includes a wide range of tunings for a fifth, but not all tunings
> whatever.

This should be |x - 685.7| < 24.5, sorry.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > No. The definition is fine.
> >
> > But the conjecture/hypothesis we're trying to prove is
> >
> > tempered-PB = semiconvergent
> >
> > This is where we could really use your help Gene.
>
> I need to see how "tempered-PB" leads to a tuning. For instance,
what
> does and does not count as a mean-tone tuning?
>
> If we take 25/24 and 81/80 as our canonical example, leading to the
7-
> note scales related to the h_7(2^a*3^b*5^c) = 7*a+11*b+16*c 7-et
> homomorphism, we can temper out 81/80 (commatic unison) and not
25/24.
> We then arrive at the mean-tone systems in the 5-limit. We can
> generate our scale with any number of scale steps relatively prime
to
> 7; for instance 4 steps representing a fifth. If we tune the fifth
by
> setting it to x cents, then x/1200 is log_2 of the approximation to
> the fifth are using. This will have 4/7 (4 scale steps out of 7) as
a
> semiconvergent if |x/1200 - 4/7| < 1/49, but if it is very far out
of
> this range we are in trouble. In other words, in terms of cents we
> want x to be in the interval |x - 685.7 ...| < 171.4...; this
> includes a wide range of tunings for a fifth, but not all tunings
> whatever.

What if we first stick with the case where the chromatic unison
vector is unchanged in size -- so in this case, 2/7-comma meantone.

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

> What if we first stick with the case where the chromatic unison
> vector is unchanged in size -- so in this case, 2/7-comma meantone.

That would work, but it seems to me you are relying on the fact that
the 7-et is relatively good. Suppose we take instead 512/375 as our
chromatic unison vector, and 10/9 as our commatic unison vector. We
then get the [7, 12, 17] system. If we temper out the 10/9 and keep
the 512/375 value just, we want to solve the linear system

a = 1200
a-2*b+c = 0
9*a-b-3*c = 1200 * log_2(512/375) = 512/375 expressed in cents.

Solving this gives us

a = 1200
b = 1980.12820
c = 2760.25604

The value for b is 77 cents flat from the 1200*(12/7) mark, well over
the allowed amount; we would not have a MOS.

Whoops . . . I meant "at least one step size is larger than the
chromatic unison vector" . . .

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > What if we first stick with the case where the chromatic unison
> > vector is unchanged in size -- so in this case, 2/7-comma
meantone.
>
> That would work, but it seems to me you are relying on the fact
that
> the 7-et is relatively good. Suppose we take instead 512/375 as our
> chromatic unison vector,

Whoa -- that's 539 cents!

> and 10/9 as our commatic unison vector. We
> then get the [7, 12, 17] system. If we temper out the 10/9 and keep
> the 512/375 value just, we want to solve the linear system
>
> a = 1200
> a-2*b+c = 0
> 9*a-b-3*c = 1200 * log_2(512/375) = 512/375 expressed in cents.
>
> Solving this gives us
>
> a = 1200
> b = 1980.12820
> c = 2760.25604
>
> The value for b is 77 cents flat from the 1200*(12/7) mark, well
over
> the allowed amount; we would not have a MOS.

Carl brought this sort of thing up a while back (actually, he showed
that some PBs are not CS, also using very large unison vectors). I
replied that there needs to be some notion of "good" PBs. What's the
weakest such condition we can come up with? If at least one step size
is smaller than the chromatic unison vector . . . will that work?

>/.../ What's the weakest such condition we can come up with? If
>at least one step size is smaller than the chromatic unison
>vector . . . will that work?

Well, in this non-CS example from way-back,

|4 -1|
|0 2| = 8
1/1 135/128 9/8 5/4 45/32 3/2 27/16 15/8

The smallest 2nd is 135:128, and the chromatic UV is 32:25.

>Whoops . . . I meant "at least one step size is larger than the
>chromatic unison vector" . . .

The largest 2nd is 9:8, so this condition is not met, though
still not 100% on why the 81:80 can't be chromatic (isn't this
just a matter of choice?). IIRC, we agreed way back that all
steps being larger than all vectors would give CS, so this is
indeed weaker.

That's non-CS for you. Not exactly clear on what property
Gene has found.

-Carl

--- In tuning-math@y..., "Carl" <carl@l...> wrote:
> >/.../ What's the weakest such condition we can come up with? If
> >at least one step size is smaller than the chromatic unison
> >vector . . . will that work?
>
> Well, in this non-CS example from way-back,
>
> |4 -1|
> |0 2| = 8
> 1/1 135/128 9/8 5/4 45/32 3/2 27/16 15/8
>
> The smallest 2nd is 135:128, and the chromatic UV is 32:25.
>
> >Whoops . . . I meant "at least one step size is larger than the
> >chromatic unison vector" . . .
>
> The largest 2nd is 9:8, so this condition is not met,

Cool . . . so my proposal looks good so far.

> though
> still not 100% on why the 81:80 can't be chromatic (isn't this
> just a matter of choice?).

You haven't tempered any unison vectors out yet, so at this point,
there's really no difference between commatic and chromatic. If you
called the 81:80 chromatic and 25:16 commatic, you'd have to temper
out the 25:16 first before looking at the step sizes.

[I wrote...]
>That's non-CS for you. Not exactly clear on what property
>Gene has found.

Sorry for the 3rd-person, Gene!

[Gene wrote...]
>The value for b is 77 cents flat from the 1200*(12/7) mark, well
>over the allowed amount; we would not have a MOS.

So it's MOS, then!

-Carl

>> The largest 2nd is 9:8, so this condition is not met,
>
> Cool . . . so my proposal looks good so far.

Well, to me the key thing about the Hypothesis is that L-s
is the chromatic vector. So if this proposal gives MOS,
then the hypothesis is true. But it seems there are scales
with more than two sizes of 2nd that meet the proposal
condition...

>>though still not 100% on why the 81:80 can't be chromatic (isn't
>>this just a matter of choice?).
>
>You haven't tempered any unison vectors out yet, so at this point,
>there's really no difference between commatic and chromatic. If you
>called the 81:80 chromatic and 25:16 commatic, you'd have to temper
>out the 25:16 first before looking at the step sizes.

Oh, right. It blows my mind that this could work, or that anybody
could think of it.

-Carl

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

> Carl brought this sort of thing up a while back (actually, he
showed
> that some PBs are not CS, also using very large unison vectors). I
> replied that there needs to be some notion of "good" PBs. What's
the
> weakest such condition we can come up with? If at least one step
size
> is smaller than the chromatic unison vector . . . will that work?

I don't know what the weakest condition is, and I haven't thought
about the one you propose above. I'm not sure what the point of it is-
-a step of the 7-et is larger than 25/24, after all.

However, the point of what I was sketching out before was that it is
certainly possible to come up with conditions which make sense and
suffice to produce a MOS.

What's CS?

--- In tuning-math@y..., genewardsmith@j... wrote:
> What's CS?

The property that every interval in a scale appears in only
one interval class. For example, 3:2 appears only as a 5th
in the diatonic scale... but in 12-tET, the tritone appears
as both a 4th and a 5th, so the diatonic scale in 12-tET is
non-CS.

CS is our acronym for Constant Structures. ...You don't
want to know. :)

Actually, terms like MOS and CS come to us from Erv Wilson,
who admits he has trouble naming the things he thinks about,
and suggests we call them whatever we like. But, names seem
to stick.

-Carl

--- In tuning-math@y..., "Carl" <carl@l...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> > What's CS?
>
> The property that every interval in a scale appears in only
> one interval class. For example, 3:2 appears only as a 5th
> in the diatonic scale... but in 12-tET, the tritone appears
> as both a 4th and a 5th, so the diatonic scale in 12-tET is
> non-CS.

It seems to me that in a 12-et, a tritone would always be 6 steps.
Can you clarify?

In 12-tET, six steps can be used "enharmonically" to represent either
an augmented fourth or a diminished fifth. If we think in the key of
C, this implies that the sixth step above C can be used as either an
F# or a Gb.

In Just Intonation, these pitches form intervals with C that are not
equal. In 12-tET, the irrational approximation of both intervals (sq
root of 2) lies between F# and Gb. On the other hand, in Just
Intonation F# is 45/32 of the frequency of C(1.40625*Fc)and Gb is
36/25 (1.44)of C. So it becomes clear that F# is lower than Gb, and
the 12-tET interval of 1.414... is an irrational approximation in
between them.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Carl" <carl@l...> wrote:
> > --- In tuning-math@y..., genewardsmith@j... wrote:
> > > What's CS?
> >
> > The property that every interval in a scale appears in only
> > one interval class. For example, 3:2 appears only as a 5th
> > in the diatonic scale... but in 12-tET, the tritone appears
> > as both a 4th and a 5th, so the diatonic scale in 12-tET is
> > non-CS.
>
> It seems to me that in a 12-et, a tritone would always be 6 steps.
> Can you clarify?

--- In tuning-math@y..., BobWendell@t... wrote:

> In Just Intonation, these pitches form intervals with C that are
not
> equal.

Your definition said every interval appears in only one class--
however, these are two different intervals. Naturally, they will be
represented as different in many systems, and will be the same only
in those systems that has their ratio in the kernel.

>>>What's CS?
>>
>>The property that every interval in a scale appears in only
>>one interval class. For example, 3:2 appears only as a 5th
>>in the diatonic scale... but in 12-tET, the tritone appears
>>as both a 4th and a 5th, so the diatonic scale in 12-tET is
>>non-CS.
>
>It seems to me that in a 12-et, a tritone would always be 6 steps.
>Can you clarify?

Scale steps. What recent threads have called "steps" are
actually 2nds. Then, there's 3rds (major and minor), etc.

Here's what Rothenberg calls a "interval matrix" for the
diatonic scale,

1sts 2nds 3rds 4ths 5ths 6ths 7ths 8ths
ionian 0 2 4 5 7 9 11 12
dorian 0 2 3 5 7 9 10 12
phrygian 0 1 3 5 7 8 10 12
lydian 0 2 4 [6] 7 9 11 12
mixolydian 0 2 4 5 7 9 10 12
aeolian 0 2 3 5 7 8 10 12
locrian 0 1 3 5 [6] 8 10 12

The tritone is what R. calls an "ambiguous interval".
CS is equivalent to no ambiguous intervals.

In meantone, the augmented 4th is smaller than the
diminished 5th, so the meantone diatonic is CS.

In Pythagorean tuning, the aug. 4th is larger than
the dim. 5th, so it is also CS. But Rothenberg would
call this scale "improper", since its scale steps
overlap in interval space.

When certain other conditions are met, R. claims
proper scales make possible a compositional style in
which melodies may be transposed across the modes
of a scale without loosing their identity.

-Carl

>>In Just Intonation, these pitches form intervals with C that are
>>not equal.
>
>Your definition said every interval appears in only one class--
>however, these are two different intervals. Naturally, they will be
>represented as different in many systems, and will be the same only
>in those systems that has their ratio in the kernel.

Just for the record, that was Bob, not me.

"Interval class" is just a bad way to say "scale step". "Every
interval" is just a bad way to say "every acoustic interval".
Does that help? See also my previous message in this thread.

Gene, was it ever decided if a kernel is equivalent to a set
of unison vectors, as we use them?

-Carl

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Carl brought this sort of thing up a while back (actually, he
> showed
> > that some PBs are not CS, also using very large unison vectors).
I
> > replied that there needs to be some notion of "good" PBs. What's
> the
> > weakest such condition we can come up with? If at least one step
> size
> > is smaller than the chromatic unison vector . . . will that work?
>
> I don't know what the weakest condition is, and I haven't thought
> about the one you propose above. I'm not sure what the point of it
is-
> -a step of the 7-et is larger than 25/24, after all.

You must have missed the message where I corrected this -- I
meant "at least one step size is larger than the chromatic unison
vector".
>
> However, the point of what I was sketching out before was that it
is
> certainly possible to come up with conditions which make sense and
> suffice to produce a MOS.
>
> What's CS?

Every specific interval size is always subtended by the same number
of steps. Seems to be synonymous with "good" PBs in the untempered JI
case.

--- In tuning-math@y..., BobWendell@t... wrote:
> In 12-tET, six steps can be used "enharmonically" to represent
either
> an augmented fourth or a diminished fifth. If we think in the key
of
> C, this implies that the sixth step above C can be used as either
an
> F# or a Gb.
>
> In Just Intonation, these pitches form intervals with C that are
not
> equal. In 12-tET, the irrational approximation of both intervals
(sq
> root of 2) lies between F# and Gb. On the other hand, in Just
> Intonation F# is 45/32 of the frequency of C(1.40625*Fc)and Gb is
> 36/25 (1.44)of C.

In Just Intonation there are actually several possible ratios for F#,
as well as for Gb. So for these purposes, a well-defined tuning such
as Pythagorean or meantone would have been better.

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> Gene, was it ever decided if a kernel is equivalent to a set
> of unison vectors, as we use them?

Here are some questions:

(1) Is the unison a unison vector?

(2) If q is a unison vector, is q^2 a unison vector?

(3) Are products of unison vectors unique--that is, if we have unison
vectors {v1, ... vn} and v1^e1 * ... * vn^en = q, are the exponents
ei determined?

>>Gene, was it ever decided if a kernel is equivalent to a set
>>of unison vectors, as we use them?
>
> Here are some questions:
>
> (1) Is the unison a unison vector?

No.

> (2) If q is a unison vector, is q^2 a unison vector?

No, but it does point to a unison.

> (3) Are products of unison vectors unique--that is, if we have
> unison vectors {v1, ... vn} and v1^e1 * ... * vn^en = q, are the
> exponents ei determined?

I don't know.

-Carl

Sorry Bob, I did not see your message until after I had posted
mine (we both answered in the same way). Paul also makes a good
point -- you're assuming the 'classical' 12-tone, 5-limit scale
here (such as Ellis' "duodene"), which is common practice in
many music theory text books, but often leads to trouble here,
where we take nothing for granted when it comes to JI!

-Carl

> In Just Intonation, these pitches form intervals with C that are
> not equal. In 12-tET, the irrational approximation of both
> intervals (sq root of 2) lies between F# and Gb. On the other hand,
> in Just Intonation F# is 45/32 of the frequency of C(1.40625*Fc)and
> Gb is 36/25 (1.44)of C. So it becomes clear that F# is lower than
> Gb, and the 12-tET interval of 1.414... is an irrational
> approximation in between them.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
>
> > Gene, was it ever decided if a kernel is equivalent to a set
> > of unison vectors, as we use them?
>
> Here are some questions:
>
> (1) Is the unison a unison vector?

Yes.
>
> (2) If q is a unison vector, is q^2 a unison vector?

Yes.

Don't know about (3).

"Unison vector" sometimes means any element of the kernel, but
sometimes "the set of unison vectors of G" means "the generators of
the kernel for G" . . . and in the case of chromatic unison vectors,
we're pointing to an _altered_ equivalence, not a true equivalence.

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

Paul's answers were yes, yes, don't know and Carl's answers were no,
no, don't know. I've remarked that I don't know if a unison vector is
an element of the kernel or a generator of the kernel, and apparently
that has not been decided. If your answer to (1) is "no", then your
answer to (3) should probably be "yes", since otherwise a product of
unison vectors will equal 1. Let's assume Paul's answer to (3) is
also yes, then we have two types of definition:

(1) Carl type: Unison vectors are defined to be generators of the
kernel of some homomorphism.

(2) Paul type: Unison vectors are defined to be members of the kernel
of some homomorphism.

To pin this down further, here is another question:

(4) If we are considering octaves to be equivalent, is 2 a unison
vector?

>>> Gene, was it ever decided if a kernel is equivalent to a set
>>> of unison vectors, as we use them?
>>
>> Here are some questions:
>>
>> (1) Is the unison a unison vector?
>
> Yes.

Really? Why would it be? And how do you define 'unison vector',
then?

>> (2) If q is a unison vector, is q^2 a unison vector?
>
> Yes.

I think I get a different PB if I use 5:4 instead of 25:16...

> "Unison vector" sometimes means any element of the kernel, but
> sometimes "the set of unison vectors of G" means "the generators
> of the kernel for G" . . .

Whew.

-Carl

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
>
> Paul's answers were yes, yes, don't know and Carl's answers were
no,
> no, don't know. I've remarked that I don't know if a unison vector
is
> an element of the kernel or a generator of the kernel, and
apparently
> that has not been decided. If your answer to (1) is "no", then your
> answer to (3) should probably be "yes", since otherwise a product
of
> unison vectors will equal 1. Let's assume Paul's answer to (3) is
> also yes, then we have two types of definition:
>
> (1) Carl type: Unison vectors are defined to be generators of the
> kernel of some homomorphism.
>
> (2) Paul type: Unison vectors are defined to be members of the
kernel
> of some homomorphism.

Gene, did you read the rest of my message???

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

> "Unison vector" sometimes means any element of the kernel, but
> sometimes "the set of unison vectors of G" means "the generators of
> the kernel for G" . . . and in the case of chromatic unison
vectors,
> we're pointing to an _altered_ equivalence, not a true equivalence.

It seems to me we should decide which way it's going to be. As for
your last point, the chromatic unison vector is in the kernel of one
homomorphism but not of another.

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> >>> Gene, was it ever decided if a kernel is equivalent to a set
> >>> of unison vectors, as we use them?
> >>
> >> Here are some questions:
> >>
> >> (1) Is the unison a unison vector?
> >
> > Yes.
>
> Really? Why would it be?

It's mapped to a unison.> >> (2) If q is a unison vector, is q^2 a
unison vector?
> >
> > Yes.
>
> I think I get a different PB if I use 5:4 instead of 25:16...

Just because it's also a unison vector, doesn't mean the resulting PB
is the same!
>
> > "Unison vector" sometimes means any element of the kernel, but
> > sometimes "the set of unison vectors of G" means "the generators
> > of the kernel for G" . . .
>
> Whew.

I hoped Gene would understand this, but apparently he skipped over
this and came to the same conclusion independently (based on your
answer and the first part of mine).

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > "Unison vector" sometimes means any element of the kernel, but
> > sometimes "the set of unison vectors of G" means "the generators
of
> > the kernel for G" . . . and in the case of chromatic unison
> vectors,
> > we're pointing to an _altered_ equivalence, not a true
equivalence.
>
> It seems to me we should decide which way it's going to be.

Too late -- it's already been used both ways. Why don't we just drop
the "unison vector" terminology on this list and use "kernel"
terminology instead, as I suggested before?

>>"Unison vector" sometimes means any element of the kernel, but
>>sometimes "the set of unison vectors of G" means "the generators
>>of the kernel for G" . . .

Ah, so my answers were based on the latter, and yours on the
former?

>>>>(2) If q is a unison vector, is q^2 a unison vector?
>>>>
>>>Yes.
>>
>>I think I get a different PB if I use 5:4 instead of 25:16...
>
>Just because it's also a unison vector, doesn't mean the resulting
>PB is the same!

And your reasoning here an example of the former?

>>>> (1) Is the unison a unison vector?
>>>
>>> Yes.
>>
>> Really? Why would it be?
>
> It's mapped to a unison.

Sounds like a tautology to me.

-Carl

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> "Interval class" is just a bad way to say "scale step".

Eek! No it isn't. I remember you had this problem before.

A scale step is the distance between two _consecutive_ scale degrees.
For example, in western diatonic scales we have whole-tone steps and
half-tone steps. We don't have minor third steps or perfect fifth
steps.

> "Every
> interval" is just a bad way to say "every acoustic interval".
> Does that help?

I don't think it helps. "Acoustic" simply means "relating to sound".
Everything we deal with here is acoustic.

I don't think that any the above terminology is very good, when trying
to define CS or propriety for newbies.

Instead of your "interval class" I use "number of scale steps", and
instead of your "interval" I use "size (in cents)". In these
definitions I use "interval" to mean "the distance between a specific
pair of notes of the scale".

Now the definitions:

A scale is proper if all intervals spanning the same number of scale
steps, have a range of sizes (in cents) that does not overlap but may
meet, the range of sizes for any other number of scale steps.

A scale is strictly-proper if all intervals spanning the same number
of scale steps, have a range of sizes (in cents) that is disjoint from
(does not meet or overlap), the range of sizes for any other number of
scale steps.

Examples.

1. Improper
Number of steps in interval 4
1 2 3 <---------->
Ranges <--------> <--------> <---------->
| | | | | | | |
0 100 200 300 400 500 600 700 etc.
Interval size (cents)

2. Proper
Number of steps in interval
1 2 3 4
Ranges <--------> <--------> <--------x-------->
| | | | | | | |
0 100 200 300 400 500 600 700 etc.
Interval size (cents)

3. Strictly proper
Number of steps in interval
1 2 3 4
Ranges <--------> <--------> <-------> <------->
| | | | | | | |
0 100 200 300 400 500 600 700 etc.
Interval size (cents)

The following is supposedly Erv Wilson's definition of CS, as conveyed
by Kraig Grady.

A scale is CS if all intervals of the same size (in cents), span the
same number of scale steps.

CS is supposed to be a useful property for a scale to have, but notice
that, by this definition, any random scale that has
no-two-intervals-the-same-size is trivially CS, even if it has two
intervals that differ by only 0.00001 cent spanning different numbers
of scale steps!

A more meaningful definition for CS would be of the form:

A scale is CS if all intervals in the same range of sizes (in cents),
(with all ranges defined so as to be disjoint), span the same number
of scale steps.

Notice that this is almost equivalent to strict-propriety, written
conversely. However a scale which is not strictly proper (i.e. it has
number-of-step ranges that meet or overlap) might be able to have
these ranges split into sub-ranges in such a way thay they no longer
overlap and it is thereby CS.

4. CS?
Number of steps in interval 4 4
1 2 3 <-> 3 <--->
Ranges <--------> <--------> <---> <->
| | | | | | | |
0 100 200 300 400 500 600 700 etc.
Interval size (cents)

But clearly, this division into non-overlapping sub-ranges must be
musically meaningful and in particular the sub-ranges must not be
allowed to be too narrow, or else we are back to the trivial case
where every subrange can consist of a single size.

Various ways of defining allowable ranges for CS, have been proposed,
but none universally agreed upon. I ask their authors to explain what
they are, should they be so inclined.

-- Dave Keenan

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> What recent threads have called "steps" are
> actually 2nds.

Yes. "Seconds" are the _only_ things that are called steps. Lest ye
doubt, please see p63 of
http://depts.washington.edu/pnm/CLAMPITT.pdf
"A step interval is an interval whose two boundary pitches are
adjacent pitches of a scale."

-- Dave Keenan

>> "Interval class" is just a bad way to say "scale step".
>
> Eek! No it isn't. I remember you had this problem before.
>
> A scale step is the distance between two _consecutive_ scale
> degrees. For example, in western diatonic scales we have
> whole-tone steps and half-tone steps. We don't have minor third
> steps or perfect fifth steps.

You're right. I usually use "scale interval" here.

I think my past problem was "scale degrees", which of course
are pitches, not intervals.

>> "Every interval" is just a bad way to say "every acoustic
>> interval". Does that help?
>
> I don't think it helps. "Acoustic" simply means "relating to
> sound". Everything we deal with here is acoustic.

Not really. Propriety has only to do with the relative sizes
of a scale's intervals, not with the actual sizes -- in fact,
Rothenberg discards the interval matrix in favor of the rank-
order matrix very early on. Now, "acoustic" may not be the
best way to get this across, I'll agree. How would you say it?

> //propriety stuff//

Great job!

> A more meaningful definition for CS would be of the form:
>
> A scale is CS if all intervals in the same range of sizes (in
> cents), (with all ranges defined so as to be disjoint), span
> the same number of scale steps.

Now you've got me confused.

> Notice that this is almost equivalent to strict-propriety, written
> conversely. However a scale which is not strictly proper (i.e. it
> has number-of-step ranges that meet or overlap) might be able to
> have these ranges split into sub-ranges in such a way thay they no
> longer overlap and it is thereby CS.
>
> 4. CS?
> Number of steps in interval 4 4
> 1 2 3 <-> 3 <--->
> Ranges <--------> <--------> <---> <->
> | | | | | | | |
> 0 100 200 300 400 500 600 700
>
> Interval size (cents)
>
> But clearly, this division into non-overlapping sub-ranges must be
> musically meaningful and in particular the sub-ranges must not be
> allowed to be too narrow, or else we are back to the trivial case
> where every subrange can consist of a single size.

There are many scales which have overlapping intervals subtending
the same number of scale steps, which also lack ambiguous intervals.
Therefore they are CS but not proper. These cases don't represent
a failure of either concept, though. Propriety is based on the
idea that listeners order scale intervals by size. CS is based on
the idea that listeners can recognize particular intervals.

Actually, both properties can result in what I'd call "convenience
items", such as transpositional coherence (cough!), and handy
things for PBs. I think these are primarily what motivated Wilson
with the concept. But on perceptual grounds, I'd say CS doesn't
have much of a leg to stand on (Paul's consonance-only CS may be
another matter), but if it does, it would still have the leg for
improper CS's.

-Carl

Guys, CS is useful in the context in which it was intended -- strict
JI scales that are connected in the lattice. Forget about random
scales or ETs. Now, it seems that if a PB, in untempered from, is CS,
then tempering all but one of the unison vectors will lead to an MOS.
I'm hereby betting that this is true. Can we prove it?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Guys, CS is useful in the context in which it was intended -- strict
> JI scales that are connected in the lattice. Forget about random
> scales or ETs.

The silly definition of CS works fine for subsets of ETs-with-less-
than-100-notes. In fact it works best for these. Rational scales have
the same problem as random scales.

Consider a connected rational scale that has, for example, a 5:7
spanning 8 steps while a 343:480 (still 7-prime-limit and only 0.7c
different) spans 9 steps. That would be CS according to the silly
definition. But everyone would hear the 343:480 as a 5:7 and so this
definition is useless.

I agree with Carl, that a useful definition of CS would recognise some
intervals as the same, even if they differ by a few cents. Paul, I
understand your consonant-CS satifies this requirement.

My current thinking is still that CS needs to be qualified with a cent
value, as in "this scale has a CS-level of 33 cents" which means "this
scale is CS provided you don't conflate any intervals differing by
more than 33 cents". Bigger numbers are better.

In that case, the above hypothetical rational scale only has a
CS-level of 0.7 cents, which everyone should realise is useless.

CS-level is meaningful for any kind of scale and doesn't require us to
agree on which intervals are consonant, or what deviations are still
recognisable as the same interval, or on deviations from what.

To determine the CS-level of a scale:

1. Consider the set of pairs (interval-size-in-cents,
number-of-scale-steps) over all the scale's intervals. List these
pairs in order of interval-size-in-cents. (If two or more have the
same size, their order relative to each other doesn't matter, and we
know immediately that the scale has a CS-level of zero.)

2. Partition this list so that each sub-list is as large as possible
with all its members having the same number-of-scale-steps.

3. For each pair of adjacent sub-lists, consider the difference in
size between the intervals on either side of the partitioning boundary
between them. The minimum of all such differences is the CS-level.

-- Dave Keenan

>>Guys, CS is useful in the context in which it was intended --
>>strict JI scales that are connected in the lattice. Forget
>>about random scales or ETs.
>
>The silly definition of CS works fine for subsets of ETs-with-less-
>than-100-notes. In fact it works best for these. Rational scales
>have the same problem as random scales.

There are really two issues here:

(1) How does CS fit, if at all, into a model of melodic perception?
(2) What happens to a PB when it is CS?

Items belonging to (2) I referred to in my last message as
"convenience items". I did mention that Wilson probably had
these in mind when he coined "CS".

It is for issue (1) that my criticisms of the measure, and yours
Dave, are important. And (1) is also the category which Paul's
consonance-CS addresses (or so it seems to me).

>Consider a connected rational scale that has, for example, a 5:7
>spanning 8 steps while a 343:480 (still 7-prime-limit and only 0.7c
>different) spans 9 steps. That would be CS according to the silly
>definition. But everyone would hear the 343:480 as a 5:7 and so
>this definition is useless.

For purposes of issue (2), PBs containing a step 2401:2400 would
be odd -- and unless all their uv's were smaller still, they
wouldn't be CS by Wilson's definition anyway. In short, I think
Wilson's definition is splendid for (2).

>I agree with Carl, that a useful definition of CS would recognise
>some intervals as the same, even if they differ by a few cents.
>Paul, I understand your consonant-CS satifies this requirement.

As I understood it, Paul's consonance-CS (as I've called it) is
the same as Wilson's, except it only cares about ambiguous
intervals if they are consonances in the scale. So this doesn't
address near-collisions.

Once again, my interp. of a possible application of CS for (1) is
that it speeds scale interval tracking if listeners can recognize
exact intervals.

I don't think they can, with the possible exception of strong
consonances. For melodic intervals in general, I agree with
Rothenberg that listeners simply order melodic intervals by size.

But you're right Dave -- for either Rothenberg or CS(1), near
collisions are a problem! R. addresses this somewhat by replacing
propriety with stability. I've addressed it for propriety by
creating what is called in Scala, "Lumma stability" and "Lumma
impropriety".

> My current thinking is still that CS needs to be qualified with
> a cent value, as in "this scale has a CS-level of 33 cents" which
> means "this scale is CS provided you don't conflate any intervals
> differing by more than 33 cents". Bigger numbers are better.

That's a good idea.

-Carl

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > Guys, CS is useful in the context in which it was intended --
strict
> > JI scales that are connected in the lattice. Forget about random
> > scales or ETs.
>
> The silly definition of CS works fine for subsets of ETs-with-less-
> than-100-notes. In fact it works best for these. Rational scales
have
> the same problem as random scales.
>
> Consider a connected rational scale that has, for example, a 5:7
> spanning 8 steps while a 343:480 (still 7-prime-limit and only 0.7c
> different) spans 9 steps. That would be CS according to the silly
> definition. But everyone would hear the 343:480 as a 5:7 and so
this
> definition is useless.

It may become useful in the context of proving the hypothesis. In the
case above, 2400:2401 would not be a commatic unison vector in the
scale and thus would probably be enlarged when the commatic unison
vectors are tempered out. I'd prefer to think of it as an
abstract, "group-theoretical" property of the set of ratios at this
stage, rather than a perceptual characterization of the scale.

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> >Consider a connected rational scale that has, for example, a 5:7
> >spanning 8 steps while a 343:480 (still 7-prime-limit and only
0.7c
> >different) spans 9 steps. That would be CS according to the silly
> >definition. But everyone would hear the 343:480 as a 5:7 and so
> >this definition is useless.
>
> For purposes of issue (2), PBs containing a step 2401:2400 would
> be odd -- and unless all their uv's were smaller still, they
> wouldn't be CS by Wilson's definition anyway. In short, I think
> Wilson's definition is splendid for (2).

Great minds think alike, Carl. ;^)