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