CS

Really, Dave? No comment the CS alternative I fashioned at
your request? I'm surprised.

-Carl

--- In tuning-math@y..., carl@l... wrote:
> Really, Dave? No comment the CS alternative I fashioned at
> your request? I'm surprised.

Sorry Carl, but when I didn't know if this list was going to continue
...

So you're proposing a method that will give the degree of "CS-ness" of
a scale. The completely general one would probably work but is way too
complicated for the original purpose.

For MOS only, you say L/s but don't say how to use it. A MOS can fail
to be CS when L/s = 2 or when L/s = 3 etc, but not every MOS will do
so. On Wilson's definition, no irrational generator can result in a
non-CS MOS. But clearly some will be "nearly non-CS". Maybe this can
be refined, but I've lost interest I'm afraid.

Or maybe you mean to use purely the magnitude of L/s as a measure of
how improper a scale is (nothing to do with CS)?

The original purpose was in Grahams temperament-finder program, to
rank the temperaments. So we don't bother looking at complete junk we
use a Figure-of-Demerit that attempts to (crudely) include both
melodic and harmonic factors. Currently it is a normalised max-abs
error combined with the size of the smallest MOS containing a complete
otonality. Of course some of these MOS could be wildly improper. We
wondered whether we should use the smallest strictly proper MOS, but
this seemed too strong.

Paul suggested CS. I had a wrong idea about CS. Turns out it's pretty
meaningless for irrational-fraction-of-an-octave generators. I'll
forget it.

Carl maybe you're suggesting we use the size of the smallest
otonality-containing MOS that has L/s < 3 (or something), to make sure
we're not counting a MOS that's completely useless melodically?

Good idea.

-- Dave Keenan

> For MOS only, you say L/s but don't say how to use it.

An MOS is not CS iff L/s = 2. This is a corollary of a
formula by John Chalmers.

>A MOS can fail to be CS ... when L/s = 3

Example?

> The original purpose was in Grahams temperament-finder program,
> to rank the temperaments. So we don't bother looking at complete
> junk we use a Figure-of-Demerit that attempts to (crudely)
> include both melodic and harmonic factors.

I wasn't here for the beginning of the thread, so I don't know
what your after. I doubt CS (or anything like it) is what you're
looking for.

CS and propriety have no direct meaning for scales with more
than about 14 notes. They may simplify things as far as
transposing and chord patterns are concerned, but it makes no
sense to talk of melodically useful or not scales with 21 notes.
Any reasonably octave-periodic scale with 21 tones will be full
of both melodically useful and melodically useless scales, and
propriety hardly matters for chromatic ornamentation.

> Currently it is a normalised max-abs error combined with the
> size of the smallest MOS containing a complete otonality.

That sounds better, though I still don't know what you're
trying to find.

> Carl maybe you're suggesting we use the size of the smallest
> otonality-containing MOS that has L/s < 3 (or something), to
> make sure we're not counting a MOS that's completely useless
> melodically?

Not me.

> Good idea.

Maybe; I don't know.

-Carl

--- In tuning-math@y..., carl@l... wrote:
> Dave Keenan wrote:
> >A MOS can fail to be CS ... when L/s = 3
>
> Example?

Sorry I took so long to answer this. I remembered that I hadn't, while
lying in bed last night. ("Get a life" he says :-)

Example: Generator/Period = 6/19 (e.g. narrow major third/octave), 10
note improper MOS. This has steps of 3/19 and also has 3 consecutive
steps of 1/19.

> CS and propriety have no direct meaning for scales with more
> than about 14 notes. They may simplify things as far as
> transposing and chord patterns are concerned, but it makes no
> sense to talk of melodically useful or not scales with 21 notes.
> Any reasonably octave-periodic scale with 21 tones will be full
> of both melodically useful and melodically useless scales, and
> propriety hardly matters for chromatic ornamentation.

Good point.

> > Currently it is a normalised max-abs error combined with the
> > size of the smallest MOS containing a complete otonality.
>
> That sounds better, though I still don't know what you're
> trying to find.

Mainly just checking that the MIRACLE generator has no competitors,
without having to wade thru too many contenders. Weed most of them out
automatically.

> Sorry I took so long to answer this. I remembered that I hadn't,
> while lying in bed last night. ("Get a life" he says :-)

Not at all. I remember stuff like that all the time. I think I
was driving once when I remembered you hadn't replied. Sign of a
good memory, I'd say.

>Example: Generator/Period = 6/19 (e.g. narrow major third/octave),
>10 note improper MOS. This has steps of 3/19 and also has 3
>consecutive steps of 1/19.

I see the 3rd of 189 cents but not the 2nd. According to Scala
this is CS.

Please note that while my experience tells me Chalmers' formula
is correct, I do not have a proof. I've been thinking about one,
but haven't come up with anything yet. Probably came to a stop
sign at the key instant. :)

[Note: Myhill's property means there are only two kinds of 2nds,
and that they are distributed in a certain way, so that there will
be exactly two kinds of the other scalar intervals. Perhaps this
is where the 2 in Chalmers' formula comes from...]

-C.

--- In tuning-math@y..., carl@l... wrote:
> >Example: Generator/Period = 6/19 (e.g. narrow major third/octave),
> >10 note improper MOS. This has steps of 3/19 and also has 3
> >consecutive steps of 1/19.
>
> I see the 3rd of 189 cents but not the 2nd. According to Scala
> this is CS.

Oops. Sorry. You have to take it out to 13 notes before the 3/19 oct
seconds appear.

I wonder how much "slop" Scala allows in determining whether
products of intervals are equal for this purpose?

> Please note that while my experience tells me Chalmers' formula
> is correct, I do not have a proof. I've been thinking about one,
> but haven't come up with anything yet. Probably came to a stop
> sign at the key instant. :)

Hee hee.

> [Note: Myhill's property means there are only two kinds of 2nds,
> and that they are distributed in a certain way, so that there will
> be exactly two kinds of the other scalar intervals. Perhaps this
> is where the 2 in Chalmers' formula comes from...]

Clearly if L/s = n, an integer, and n consecutive "s"s appear in the
scale then it is not CS. That is the case with the 13 of 6/19 oct
generator above, where n = 3. But not all scales of L/s = 2 have 2
consecutive "s"s, so yes a proof looks tricky.

But I'm not much interested in a property that is defined so that it
dissapears with an inaudible, nay immeasurable!, change to the scale.

Regards,
-- Dave Keenan

>> I see the 3rd of 189 cents but not the 2nd. According to Scala
>> this is CS.
>
>Oops. Sorry. You have to take it out to 13 notes before the 3/19
>oct seconds appear.

Wow. "Show data" on this scale crashes my copy of Scala 1.7.
This reminds me of a bug I thought was fixed. Manuel?

"Show intervals" works, and the scale is clearly not CS.

Chalmers isn't wrong, it's my fault. If a scale is proper,
and isn't CS, then L:s = 2:1. But if a scale is improper,
L:s doesn't seem to apply.

> I wonder how much "slop" Scala allows in determining whether
> products of intervals are equal for this purpose?

My guess is none, within the precision setting. Manuel?

> But I'm not much interested in a property that is defined so
> that it dissapears with an inaudible, nay immeasurable!, change
> to the scale.

The property isn't designed to blindly measure scales in any
kind of search. It's just an idea that Wilson wants you to think
about.

In search of a measure, what did you think of my smooth-CS
suggestions?

-Carl

--- In tuning-math@y..., carl@l... wrote:
> In search of a measure, what did you think of my smooth-CS
> suggestions?

You mean those in
/tuning-math/message/118

The Consonant-CS thing makes sense, but for the more general case I
think you can leave out the computational complexity of Harmonic
Entropy and just say that a scale is "CS to a tolerance of X cents".
Which means that, so long as you don't identify (i.e. consider as
equal) interval sizes that differ by more than X cents (X cents or
more?), there is no interval size that can be arrived at by different
numbers of steps. So "CS to 0 cents" means not CS at all. I'd think CS
to 17 cents is somewhat borderline (one standard deviation). CS to 63
cents is obviously CS.

Blackjack (with 7/72 oct generator) is only CS to 17 cents.
It has
67c in 2 steps and 83c in 1 step,
183c in 4 steps and 200c in 3 steps,
300c in 6 steps and 317c in 5 steps,
417c in 8 steps and 433c in 7 steps,
533c in 10 steps and 550c in 9 steps,
etc.

So I don't think it's meaningful to say simply that Blackjack is CS.
It's borderline CS.

With a 3/31 oct generator it is not CS at all (but it's proper with
L/s = 2).
With a 4/41 oct generator it is improper (L/s = 3) but at its maximum
CS-ness, having a 29c (one step of 41-EDO) tolerance.
But these extremes are no fun harmnonically.

-- Dave Keenan

Carl wrote:
>Wow. "Show data" on this scale crashes my copy of Scala 1.7.
>This reminds me of a bug I thought was fixed. Manuel?

I haven't replaced 1.7, but it is (should) be fixed in 2.0.

>> I wonder how much "slop" Scala allows in determining whether
>> products of intervals are equal for this purpose?

>My guess is none, within the precision setting. Manuel?

Virtually nothing, something in the range of 1/1000 of a cent,
although the internal precision is higher.

Manuel

>> In search of a measure, what did you think of my smooth-CS
>> suggestions?
>
> You mean those in
> /tuning-math/message/118

Yep.

> The Consonant-CS thing makes sense, but for the more general case
> I think you can leave out the computational complexity of Harmonic
> Entropy and just say that a scale is "CS to a tolerance of X
> cents".

Well, I suppose it would be the same difference as rating an ET
by max cents error, or something like Erlich's accuracy.

> Which means that, so long as you don't identify (i.e. consider as
> equal) interval sizes that differ by more than X cents (X cents or
> more?), there is no interval size that can be arrived at by
> different numbers of steps. So "CS to 0 cents" means not CS at
> all. I'd think CS to 17 cents is somewhat borderline (one standard
> deviation). CS to 63 cents is obviously CS.

Of what is 17 cents one standard deviation?

My feeling is that this ought to be expressed as a fraction of the
scale's smallest 2nd.

-Carl

--- In tuning-math@y..., carl@l... wrote:
> > The Consonant-CS thing makes sense, but for the more general case
> > I think you can leave out the computational complexity of Harmonic
> > Entropy and just say that a scale is "CS to a tolerance of X
> > cents".
>
> Well, I suppose it would be the same difference as rating an ET
> by max cents error, or something like Erlich's accuracy.

Not really. In the case of a CS tolerance, more cents is better. i.e.
it can tolerate more confusion between interval sizes and still be
perceived as CS.

> > Which means that, so long as you don't identify (i.e. consider as
> > equal) interval sizes that differ by more than X cents (X cents or
> > more?), there is no interval size that can be arrived at by
> > different numbers of steps. So "CS to 0 cents" means not CS at
> > all. I'd think CS to 17 cents is somewhat borderline (one standard
> > deviation). CS to 63 cents is obviously CS.
>
> Of what is 17 cents one standard deviation?

The probability of the average listener perceiving as the same, two
pitches differing by some amount. It's from that dude's experiment
that Paul quotes for HE. 1% change in frequency ~= 17 cents.

> My feeling is that this ought to be expressed as a fraction of the
> scale's smallest 2nd.

i.e. ... of the scale's smallest step. Yes, that would be useful to
know, as well as the absolute tolerance. But as a figure of merit in
comparing improper MOS, the absolute tolerance seems more important to
me. Can you explain further. Maybe with an example.

Regards,
-- Dave Keenan

>>> The Consonant-CS thing makes sense, but for the more general
>>> case I think you can leave out the computational complexity of
>>> Harmonic Entropy and just say that a scale is "CS to a tolerance
>>> of X cents".
> >
> > Well, I suppose it would be the same difference as rating an ET
> > by max cents error, or something like Erlich's accuracy.
>
> Not really. In the case of a CS tolerance, more cents is better.

Yes, but a scale with many (bad errors, near collisions) may have
the same (max error, min tolerance) as a scale with only one (error,
near collision).

> The probability of the average listener perceiving as the same, two
> pitches differing by some amount. It's from that dude's experiment
> that Paul quotes for HE. 1% change in frequency ~= 17 cents.

My goodness! I don't think we want Goldstein's 1% bit involved here.

> i.e. ... of the scale's smallest step. Yes, that would be useful to
> know, as well as the absolute tolerance. But as a figure of merit
> in comparing improper MOS, the absolute tolerance seems more
> important to me. Can you explain further. Maybe with an example.

Well, I'd argue that a near-collision of 10 cents would not be as
important in an otherwise-even pentatonic as it would in a decatonic.

I'll note here that non-CS may be desirable. As I once said,
'ambiguous intervals [collisions] are the common tones of melodic
modulation'.

-Carl