Decatonics

I've drawn up a little diagram:

http://www.argonet.co.uk/users/mark.gould/images/C22_Decatonic.jpg

hopefully it explains itself.

Mark

In-Reply-To: <B8C4A26C.38C5%mark.gould@argonet.co.uk>
Mark Gould wrote:

> I've drawn up a little diagram:
>
> http://www.argonet.co.uk/users/mark.gould/images/C22_Decatonic.jpg
>
> hopefully it explains itself.

It doesn't follow either of your proposed methods for 7-limit diatonics
though. The two pentatonics don't intersect, which is a shame.

When looking for something else, I did happen to find a diatonic that
fits this pattern from 22-equal:

22
5 14
10 19
15 2
20 7
3 12
8 17
0

It's based on a 7:9:12 chord. I don't know if it fulfils all the right
criteria, but it does work with some simpler ones:

1) It's an octave-based MOS

2) Both intervals in the grid are a single step apart in the parent scale
(22-equal in the example)

3) It's a "diatonic" rather than "pentatonic", meaning the larger step
sizes are more common (9 steps of 2/22 and 4 of 1/22)

For (2) to work, you need an odd number of notes in the MOS and the
generator has to span an even number of diatonic steps. That's easy,
because you always have two generators to choose from, and one of them
will always be an even number of steps if there is an odd number to the
octave. But you also need the same generator to be an odd number of
chromatic steps, which won't always be the case. That is, it needs to be
an odd number of steps in the parent scale, which has to be the next step
down on the scale tree.

For (3) to work, I think the difference between the number of notes in the
parent and diatonic scales has to be more than half the number of notes in
the diatonic. In this case, 2*(22-13)>13.

I also found a 21 from 26 "pentatonic" which is a bit like Blackjack.

26
21 24
16 19
11 14
6 9
1 4
22 25
17 20
12 15
7 10
2 5
0

Call the defining chord 14:15:16

Graham

>
> It doesn't follow either of your proposed methods for 7-limit diatonics
> though. The two pentatonics don't intersect, which is a shame.
>
No, but I did say that there are many 3D shapes within a lattice, and I
didn't state that the two I chose for mention were the only ones.

In any case, the decatonic is the same as Paul's standard one, which was
what I was hoping people would notice.

> When looking for something else, I did happen to find a diatonic that
> fits this pattern from 22-equal:
>
> 22
> 5 14
> 10 19
> 15 2
> 20 7
> 3 12
> 8 17
> 0

This one I also found, but the scale 'fifth' is 17 steps wide and has a very
odd structure in terms of its keyboard"
b b b b b b b b b
W W W W W W W W W W W W W

I never gave this scale any more investigation.

> For (3) to work, I think the difference between the number of notes in the
> parent and diatonic scales has to be more than half the number of notes in
> the diatonic. In this case, 2*(22-13)>13.
>
> I also found a 21 from 26 "pentatonic" which is a bit like Blackjack.
>
> 26
> 21 24
> 16 19
> 11 14
> 6 9
> 1 4
> 22 25
> 17 20
> 12 15
> 7 10
> 2 5
> 0
>
This is inconsistent with my rule ii: it contains segments of 3 and more
adjacent PCs

0 1 2, 4 5 6 7, 9 10 11 12, etc

So it will show up as intervallically inchoerent (as defined by Balzano).

In-Reply-To: <B8C71E03.38EA%mark.gould@argonet.co.uk>
Mark Gould wrote:

> > It doesn't follow either of your proposed methods for 7-limit
> > diatonics
> > though. The two pentatonics don't intersect, which is a shame.
> >
> No, but I did say that there are many 3D shapes within a lattice, and I
> didn't state that the two I chose for mention were the only ones.

You didn't say very much at all.

> In any case, the decatonic is the same as Paul's standard one, which was
> what I was hoping people would notice.

The pentachordal one to be precise.

> This one I also found, but the scale 'fifth' is 17 steps wide and has a
> very
> odd structure in terms of its keyboard"
> b b b b b b b b b
> W W W W W W W W W W W W W

What's odd about it?

> I never gave this scale any more investigation.

Oh.

> This is inconsistent with my rule ii: it contains segments of 3 and more
> adjacent PCs
>
> 0 1 2, 4 5 6 7, 9 10 11 12, etc

Yes, that follows from it being a "pentatonic". You could always embed it
in 57- or even 47-equal to clear this criterion.

> So it will show up as intervallically inchoerent (as defined by
> Balzano).

Why is that important? There are some 5:6:7 "pentatonics" as well

9
5 8
1 4
0

13
7 11
14 5
8 12
2 6
0

but they become inconsistent if you try to remove this incoherence. The
classic pentatonic could be based on 6:7:8. The next I can find for 6:7:8
is 12/29 from 41 which is also a "pentatonic". 7:8:9 is your 11 from 41,
8:9:10 is Magic and 9:10:11 is a neutral third scale. 10:11:12 is
getting a bit tight, and 7:9:12 is the same complexity, but you didn't
like it.

Graham

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

> This is inconsistent with my rule ii: it contains segments of 3 and
more
> adjacent PCs
>
> 0 1 2, 4 5 6 7, 9 10 11 12, etc
>
> So it will show up as intervallically inchoerent (as defined by
Balzano).

this is what we call 'rothenberg improper'. but i don't think that's
a good reason to throw it out. the diatonic scale in pythagorean
tuning is rothenberg improper!

i sure hope my other responses to you show up, mark. but basically, i
wish there was some text accompanying your decatonic diagram. i have
no idea why you are using 4 (218¢) and 5 (273¢) as your 'generators',
for example.

In-Reply-To: <a7tcea+f5v4@eGroups.com>
Mark:
> > This is inconsistent with my rule ii: it contains segments of 3 and
> more
> > adjacent PCs
> >
> > 0 1 2, 4 5 6 7, 9 10 11 12, etc
> >
> > So it will show up as intervallically inchoerent (as defined by
> Balzano).

Paul:
> this is what we call 'rothenberg improper'. but i don't think that's
> a good reason to throw it out. the diatonic scale in pythagorean
> tuning is rothenberg improper!

It doesn't look like impropriety to me. The diatonic scale would only
fail in the degenerate case of 7-equal. The classic pentatonic would be
incoherent in 7-equal, because it's a "pentatonic".

> i sure hope my other responses to you show up, mark. but basically, i
> wish there was some text accompanying your decatonic diagram. i have
> no idea why you are using 4 (218�) and 5 (273�) as your 'generators',
> for example.

Yes, I'm looking forward to these. How come the only messages that come
through from you are ones where you say your messages aren't coming
through?

Graham

>>this is what we call 'rothenberg improper'. but i don't think that's
>>a good reason to throw it out. the diatonic scale in pythagorean
>>tuning is rothenberg improper!
>
>It doesn't look like impropriety to me. The diatonic scale would only
>fail in the degenerate case of 7-equal. The classic pentatonic would be
>incoherent in 7-equal, because it's a "pentatonic".

When I read Balzano's paper, incoherency was indeed equivalent to
propriety. I don't remember anything about "pentatonic" and
"diatonic" being a majority or minority of s vs. L, but maybe I'm
just repressing it cause it's such a poor idea.

-Carl

>this is what we call 'rothenberg improper'. but i don't think that's
>a good reason to throw it out. the diatonic scale in pythagorean
>tuning is rothenberg improper!

Paul, how long are you going to continue using this fallacious
application of Rothenberg? Who has been throwing out scales
for being improper in the way that the Pythagorean diatonic is
improper? Certainly not Rothenberg!

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >>this is what we call 'rothenberg improper'. but i don't think
that's
> >>a good reason to throw it out. the diatonic scale in pythagorean
> >>tuning is rothenberg improper!
> >
> >It doesn't look like impropriety to me. The diatonic scale would
only
> >fail in the degenerate case of 7-equal. The classic pentatonic
would be
> >incoherent in 7-equal, because it's a "pentatonic".
>
> When I read Balzano's paper, incoherency was indeed equivalent to
> propriety.

me too. so graham, what gives?

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >this is what we call 'rothenberg improper'. but i don't think
that's
> >a good reason to throw it out. the diatonic scale in pythagorean
> >tuning is rothenberg improper!
>
> Paul, how long are you going to continue using this fallacious
> application of Rothenberg? Who has been throwing out scales
> for being improper in the way that the Pythagorean diatonic is
> improper? Certainly not Rothenberg!

sorry -- just force of habit (acquired from john chalmers, i
believe). so who did introduce the terms 'proper' and 'improper' in
this context, if not rothenberg?

>> Paul, how long are you going to continue using this fallacious
>> application of Rothenberg? Who has been throwing out scales
>> for being improper in the way that the Pythagorean diatonic is
>> improper? Certainly not Rothenberg!
>
>sorry -- just force of habit (acquired from john chalmers, i
>believe). so who did introduce the terms 'proper' and 'improper'
>in this context, if not rothenberg?

I'm not sure. I've seen it around the list(s) on occasion, and
complained bitterly every time. :)

To recap: Rothenberg defines his "ideal measure" as one that works
on all the subsets of a scale. However, due mainly to computational
constraints, he uses stability, without a cutoff -- he lists all
the scales in 12-et and ranks them by stability. I've proposed a
varient of stability based on the actual log-frequency amount of
overlap, and this is available in Scala. Even for low stability
scales, Rothenberg does not throw them out -- he simply predicts
that a different kind of composition suits them better.

I don't endorse all of Rothenberg's model -- in fact, I only
understand a small part of it. I do agree that he tries to go too
far without considering harmony -- I suspect his derrivation of
the diatonic, pentatonic scale, etc, using only stability and
efficiency blows up in a tuning other than 12 where almost
everything is consonant. He may also have jumped to conclusions
with the ethno-musical data, though I've found that trying to
transpose themes over the modes of low-stability scales doesn't
seem to work.

-Carl

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

> >sorry -- just force of habit (acquired from john chalmers, i
> >believe). so who did introduce the terms 'proper' and 'improper'
> >in this context, if not rothenberg?
>
> I'm not sure. I've seen it around the list(s) on occasion, and
> complained bitterly every time. :)

i think we should ask john chalmers. look at his post of Tue Apr 19
09:04:34 1994 about 90% of the way down here, where he mentions
balzano coherence:

http://www.mills.edu/LIFE/CCM/ftp/tuning/list/archive/apr944

he didn't really get wilson's 'constant structures' right so maybe
he's mistaken on 'Rothenberg's "propriety"' as well?

>http://www.mills.edu/LIFE/CCM/ftp/tuning/list/archive/apr944
>
>he didn't really get wilson's 'constant structures' right so maybe
>he's mistaken on 'Rothenberg's "propriety"' as well?

Well, let me reply to his post:

> The problem as I saw it was that so many not very interesting
>looking scales were being generated that stronger criteria were
>necessary. Just ensuing that all the notes were parts of major or
>minor (or septimal) chords was not a fine enough filter. Melodic
>characteristics are also important.

So he has already an embarrassment of riches as far as he's
concerned, so erring on the side of fewer scales may not be
his worst fear.

> The criterion I've used the most is Rothenberg's "propriety,"
>which is equivalent essentially to Balzano's "coherence,"

So he agrees with us.

>and Wilson's "constant structure."

This isn't right. Both John and I independently made the mistake
that CS was equivalent to *strict* propriety in 1999, forgetting
that there are improper scales which are CS.

>A proper scale is one which holds together as a scale, rather than
>being perceived as a set of principal and ornamental tones.

This is a gloss, but seems okay. Rothenberg says that in practice,
especially without a drone, for scales with very low stability,
composition will tend to trace out proper subset(s) of the scale,
if they exist, and treat the propriety-breaking scale degrees as
ornamental. Playing with the harmonic series segment from 8-16,
I found that I tended to use 13 this way, and I later looked, and
indeed the scale was proper without it.

Yes, he should have said stability.

> A more important question is how are the scales going to be
>used. Are they collections of tones (gamuts, like the 12-tone set)
>or are they going to be projected as scales and are scalar passages
>going to be used thematically.

Ah-ha!

> I have considerable doubt about JI in nature. Most sounds in
>nature are nonharmonic, if only because the oscillators and resonators
>are complex 3-D structures instead of ideal 1-D strings and air columns.
>It takes considerable effort to train the voice to make harmonic timbres,
>and the vocal quality is decidedly un-natural, whether in relation to
>speech or the usual singing voice. Most animal cries don't strike me
>as especially musical, except perhaps song birds. I suspect that our
>perception of JI as something special is an epiphenomenon of our
>auditory system. A system evolved to recognize the sounds of predators,
>human voices, and especially to decode speech, may find JI easy to
>process because of acoustic and/or neurological laws. Hence, we find
>it exceptional and special.

He's right. Many authors use the former (natural sounds) as a
justification for inharmonic/nonharmonic timbres/music, forgetting
the latter (biology).

> It may also be partly a learned phenomenon; JI may not be so
>easily perceived or appreciated in cultures whose musical instruments
>have mostly nonharmonic timbres (Indonesia, S.E. Asia). There is
>some speculation that in pre-columbian Mesoamerican music, melodic
>contour was more important than interval size. Perhaps we learn
>to perceive JI because it is part of our unnatural environment,
>even in its approximated tempered form. It's hard to escape harmonic
>timbres from muzak, radio, stereos, etc. But, this is speculation.

Well, I tend to think cultural conditioning is not significant here.
I don't buy the predator noises thing -- we have excellent spatial
stuff for that (not only timing between ears, but apparently also
the the direction sounds enter the outer ear -- somebody's even got
an algorithm to simulate this over headphones now). The key thing
I think is vowels and inflection in speech, which exist in all
cultures, and the biology here is so strong and exposed that we're
already beginning to discover it!

-Carl

Sorry to all of you who have already gotten this from the other
list. I've deleted it there and posted it here.

>>and Wilson's "constant structure."
>
>This isn't right. Both John and I independently made the mistake
>that CS was equivalent to *strict* propriety in 1999, forgetting
>that there are improper scales which are CS.

It's worth noting that CS can be interpreted in the same light
as propriety, though. Both can be viewed as measures of interval
recognizability -- linking acoustic intervals to scalar ones (ie
3:2 -> 5th). If you assume that the ear tracks acoustic intervals
by *relative* size, you get propriety/stablity. If you assume the
ear can recognize *particular* *absolute* intervals, you get CS
instead. Since I think the latter idea is complete nonsense (with
the possible exception of a few strong consonances), I think CS
should not be applied to the generalized diatonics problem. However,
you, Paul, have shown (and it seems that Wilson intuitively
understands) that CS is important with respect to periodicity
blocks. I can only remember one facet of this -- that all PBs with
unison vectors smaller than their smallest 2nd are CS (was there
ever a proof?).

-Carl

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

> > The criterion I've used the most is Rothenberg's "propriety,"
> >which is equivalent essentially to Balzano's "coherence,"
>
> So he agrees with us.

i thought you objected to calling it *rothenberg* propriety . . .
isn't that the whole point?

> > I have considerable doubt about JI in nature. Most sounds in
> >nature are nonharmonic, if only because the oscillators and
resonators
> >are complex 3-D structures instead of ideal 1-D strings and air
columns.
> >It takes considerable effort to train the voice to make harmonic
timbres,

did john really write this? john?

> >and the vocal quality is decidedly un-natural, whether in relation
to
> >speech or the usual singing voice.

this is completely backwards -- i wonder if john got this from the
nutty professor. harmonic sounds are the ones that sound most natural
and most vocal!

>>>The criterion I've used the most is Rothenberg's "propriety,"
>>>which is equivalent essentially to Balzano's "coherence,"
>>
>>So he agrees with us.
>
>i thought you objected to calling it *rothenberg* propriety . . .
>isn't that the whole point?

No, Rothenberg did coin the term propriety, and a scale is
either proper or not, and IIRC is is the same as Balzano
coherence. It's just that R. doesn't use it to eliminate
scales, and talks most often of stability.

I suppose I've been guilty here too. From now on, I'm saying
"Rothenberg stability".

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> >>>The criterion I've used the most is Rothenberg's "propriety,"
> >>>which is equivalent essentially to Balzano's "coherence,"
> >>
> >>So he agrees with us.
> >
> >i thought you objected to calling it *rothenberg* propriety . . .
> >isn't that the whole point?
>
> No, Rothenberg did coin the term propriety,

so then what's the problem?

> and a scale is
> either proper or not, and IIRC is is the same as Balzano
> coherence. It's just that R. doesn't use it to eliminate
> scales,

so what? i don't get what you were yelling at me about! i was just
saying exactly the same thing you're saying here -- that it's the
same as balzano coherence. balzano (and some of his followers) *do*
use it to eliminate scales, and *that's* what i was responding to.

>>It's just that R. doesn't use it to eliminate scales,
>
>so what? i don't get what you were yelling at me about! i was just
>saying exactly the same thing you're saying here -- that it's the
>same as balzano coherence. balzano (and some of his followers) *do*
>use it to eliminate scales, and *that's* what i was responding to.

Ok, sorry. I did ask...

>>Who has been throwing out scales for being improper in the way
>>that the Pythagorean diatonic is improper?

...and in the past, you've used this argument to say that
'propriety doesn't matter', which is true but incomplete,
since it assumes the strictest possible definition for the
term "propriety".

-Carl

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

> ...and in the past, you've used this argument to say that
> 'propriety doesn't matter', which is true but incomplete,
> since it assumes the strictest possible definition for the
> term "propriety".

point well taken. so i'm agreeing with you that balzano shouldn't be
so strict about 'coherence' (though of course there are far worse
problems with balzano's theory).

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

> This isn't right. Both John and I independently made the mistake
> that CS was equivalent to *strict* propriety in 1999, forgetting
> that there are improper scales which are CS.

Are either of these equivalent to the epimorphic property? Are there implications either way?

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:
> I've drawn up a little diagram:
>
> http://www.argonet.co.uk/users/mark.gould/images/C22_Decatonic.jpg
>
> hopefully it explains itself.
>
> Mark

that's the pentachordal decatonic scale -- hopefully you're also
aware of the symmetrical decatonic i proposed. each of the two
decatonics can be seen as a pair of interlaced 3/2-generated
pentatonics -- in the symmetrical case the separation is 600 cents
instead of 109 cents.

(note that there is no 'equal' in the title of my paper).

it seems you are choosing a mode without a 4/3 over the tonic --
nothing inherently wrong with this choice, but i wonder what is
motivating it. most likely we have different views about which
properties of the diatonic scale are appropriate to keep in the
process of generalization -- it would be fun to flesh this out.

>> This isn't right. Both John and I independently made the mistake
>> that CS was equivalent to *strict* propriety in 1999, forgetting
>> that there are improper scales which are CS.
>
>Are either of these equivalent to the epimorphic property? Are there
>implications either way?

I was hoping you could tell me. I've read the defs. for epimorphic
and val in monz's dictionary, but I had problems with val.

I thought you knew what all this stuff was. Maybe an example.
The diatonic scale:

in 1/3-comma meantone is strictly proper and CS
in 12-et is proper but not strictly so, not CS
in Pythagorean tuning is improper and CS

In general: SP -> CS, but no other relation exists.

Manuel, I think there's a bug in Scala 1.8 (are you still
maintaining it?),

Pythag
7, 2/1, 0, 3/2, 0,
normalize
sort
show data -> Rothenberg stability = 1.000000 = 1

-Carl

Carl:
> > When I read Balzano's paper, incoherency was indeed equivalent to
> > propriety.

Paul:
> me too. so graham, what gives?

I don't know about Balzano, but what Mark describes here:

> > This is inconsistent with my rule ii: it contains segments of 3 and
> more
> > adjacent PCs
> >
> > 0 1 2, 4 5 6 7, 9 10 11 12, etc
> >
> > So it will show up as intervallically inchoerent (as defined by
> Balzano).

is not propriety. The fifth-generated pentatonic in 7-equal is 0 1 3 4 5
7. That has 3, 4 and 5 as adjacent pitch classes. But it's proper

0 1 3 4 5 7
1 2 1 1 2
3 3 2 3 3
4 4 4 4 5
5 6 5 6 6
7 7 7 7 7

Perhaps "intervallic coherence" is different to plain old "coherence"?
Although that still doesn't cover the second rule (i) at the bottom of
page 94 of Mark's paper. *cough*

Graham

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

> I thought you knew what all this stuff was.

Ha!

Maybe an example.
> The diatonic scale:
>
> in 1/3-comma meantone is strictly proper and CS
> in 12-et is proper but not strictly so, not CS
> in Pythagorean tuning is improper and CS

All of the above are epimorphic.

I think of scales in terms of the properties epimorphic,
convex, and connected.

Carl wrote:
>Manuel, I think there's a bug in Scala 1.8 (are you still
>maintaining it?),

>Pythag
>7, 2/1, 0, 3/2, 0,
>normalize
>sort
>show data -> Rothenberg stability = 1.000000 = 1

You're correct, it was solved in the 2.x version.
Still maintaining it, I should make an update soon.

Manuel

>> in 1/3-comma meantone is strictly proper and CS
>> in 12-et is proper but not strictly so, not CS
>> in Pythagorean tuning is improper and CS
>
>All of the above are epimorphic.
>
>I think of scales in terms of the properties epimorphic,
>convex, and connected.

If I'm right about the terms convex and connected, they
only apply to periodicity blocks. What about non-just
scales? Rothenberg claims some of the stuff of melody
doesn't have anything to do with harmony.

-Carl

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

> If I'm right about the terms convex and connected, they
> only apply to periodicity blocks.

They apply to JI scales, block or not, and to tempered scales with a specified defining mapping of primes.

>> If I'm right about the terms convex and connected, they
>> only apply to periodicity blocks.
>
>They apply to JI scales, block or not, and to tempered scales with
>a specified defining mapping of primes.

Howabout harmonics 8-16. I'm guessing it's 15-limit connected,
but is it convex?

Is the entry for val in monz's tuning dictionary the only version
there is?

I searched the archives for "val definition" and yahoo returned what
looks like an or search. I tried "val and definition" and got the
same results. A search for "val" returned about 1 out of 3 messages.

-Carl

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

> Is the entry for val in monz's tuning dictionary the only version
> there is?

There's one in my contribution to the paper project, which I haven't finished.

In-Reply-To: <B8C71E03.38EA%mark.gould@argonet.co.uk>
Mark Gould wrote (27th March):

> This is inconsistent with my rule ii: it contains segments of 3 and more
> adjacent PCs
>
> 0 1 2, 4 5 6 7, 9 10 11 12, etc
>
> So it will show up as intervallically inchoerent (as defined by
> Balzano).

>From the discussion that followed, it emerged that Balzano's intervallic
coherence is the same as Rothenberg's propriety. The scale in question is

0 1 2 4 5 6 7 9 10 11 12 14 15 16 17 19 20 21 22 24 25 26

It's a bit tedious to work out the propriety by hand, but I've written a
script to do it now. The magic table is

1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1
2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2
4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 3
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 5
6 6 7 6 6 6 7 6 6 6 7 6 6 6 7 6 6 6 6 6 6
7 8 8 7 7 8 8 7 7 8 8 7 7 8 8 7 7 7 8 7 7
9 9 9 8 9 9 9 8 9 9 9 8 9 9 9 8 8 9 9 8 8
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 9 10 10 10 9 10
11 11 12 11 11 11 12 11 11 11 12 11 11 11 11 11 11 11 11 11 11
12 13 13 12 12 13 13 12 12 13 13 12 12 12 13 12 12 12 13 12 12
14 14 14 13 14 14 14 13 14 14 14 13 13 14 14 13 13 14 14 13 13
15 15 15 15 15 15 15 15 15 15 15 14 15 15 15 14 15 15 15 14 15
16 16 17 16 16 16 17 16 16 16 16 16 16 16 16 16 16 16 16 16 16
17 18 18 17 17 18 18 17 17 17 18 17 17 17 18 17 17 17 18 17 17
19 19 19 18 19 19 19 18 18 19 19 18 18 19 19 18 18 19 19 18 18
20 20 20 20 20 20 20 19 20 20 20 19 20 20 20 19 20 20 20 19 20
21 21 22 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21
22 23 23 22 22 22 23 22 22 22 23 22 22 22 23 22 22 22 23 22 22
24 24 24 23 23 24 24 23 23 24 24 23 23 24 24 23 23 24 24 23 23
25 25 25 24 25 25 25 24 25 25 25 24 25 25 25 24 25 25 25 24 25

Which looks proper to me. So this rule ii remains a spectacularly bad way
of predicting propriety.

Graham