MOS = Myhill's Property = Self-similarity?

What does anybody make of this:

http://depts.washington.edu/pnm/CLAMPITT.pdf

?

In a message dated 10/15/99 2:42:48 AM,
From: Carl Lumma <clumma@nni.com>

What does anybody make of this:

http://depts.washington.edu/pnm/CLAMPITT.pdf

?>>

::scratches head furiously::: ::looks to & for John Chalmers:::
eh, John?

zHANg

In-Reply-To: <939976056.2247@onelist.com>
Carl Lumma, digest 354.9, wrote:

> What does anybody make of this:
>
> http://depts.washington.edu/pnm/CLAMPITT.pdf
>
> ?

The result that (at least for generated scales) Myhill's Property is the
same as an MOS or Well Formed scale is good. It means I don't have to
worry about what Myhill's Property is. It's good the mathematics was all
published, because then it can be checked, but I don't need to understand
it. I'm not sure what their concept of self similarity has to do with
mathematical self similarity, but it may still be of use to someone. The
rest of us can file it straight in the "same as MOS" basket.

The use of the inverse modulo calculation to convert between ways of
representing MOS scales was very useful for the calculations I've been
doing. It would have been nicer if they'd defined inverse modulo, for
those of us without fully stocked reference libraries in our bedrooms.
But I worked it out nonetheless.

The link between normal and neutral diatonics is meaningful, but then
there are only 3 types of 7-note MOS, so this isn't enormously surprising.
The neutral-third MOS isn't the only diatonic-like scale with those step
sizes, as I explain on this page:

http://www.cix.co.uk/~gbreed/7plus3.htm

The stuff on binary sequences may have some use, but I can't see any great
music coming out of it. My biggest problem with the paper is that it
seems to have so little connection with real music.

[Graham Breed:]
"I'm assuming the scale will be tuned to something like 24-, 31- or
38-equal. Plenty of other tunings are possible,"
(<http://www.cix.co.uk/~gbreed/7plus3.htm>)

Some of what Graham describes here is very similar to what I call
set-mapping (which I use as a compositional or structural tool, to aid
in the bridging of multiple tunings). In this case, I'll use it as a
way to define the equal divisions of the octave that would render a
scale of ten notes comprised of seven large step sizes and three small
step sizes (and more generally, where L=L s=s, and, L>s & s>0).

(7) 4 1 8 5 2 9 6 3 10
17 (14) 11 18 15 12 19 16 13 20
27 24 (21)(28) 25 22 29 26 23 30
37 34 31 38 (35) 32 39 36 33 40
47 44 41 48 45 (42)(49) 46 43 50
57 54 51 58 55 52 59 (56) 53 60
67 64 61 68 65 62 69 66 (63)(70)

By setting the horizontal cut-off point of this table at L=s+10, you
not only stave off an inexorable march to ~3e, but you also clearly
define the EDO's [equally divided octaves] where s=(-n), and where
s=0. And as the 7L 3s scale can be seen here as a linear mapping of an
interval >2/7 & <3/10 (or their inversions, i.e., >7/10 & <5/7), any
EDO that does not have an interval inside of this 1/70 space will map
as s=(-n), and will therefore be incapable of rendering a (functional)
scale (of 7L 3s). By the same token it's also clear to see that the
EDO's that only have an interval that is exactly equal to 2/7 or 3/10,
are also multiples of 7L where s=0, and as such, that they too are
also incapable of rendering the scale. All EDO's >7*10, e.g. L*(L+s),
will have an interval >2/7 & <3/10, and therefore all of them will
also create a 10-note linear sequence of 7L & 3s. However, this is
also an equally inexorable vertical march towards ~10e, and I
sometimes use this junction of L and L+s to set up a sort of rough
cut-off point... these would be the 7L 3s EDO's that occur within this
compass (34, 48, 51, 54, 62 & 68e are not included as they are
duplicate, i.e. reducible, mappings):

2 1 2 2 2 1 2 2 1 2
0 2 3 5 7 9 10 12 14 15 17
0 141 212 353 494 635 706 847 988 1059 1200

3 1 3 3 3 1 3 3 1 3
0 3 4 7 10 13 14 17 20 21 24
0 150 200 350 500 650 700 850 1000 1050 1200

3 2 3 3 3 2 3 3 2 3
0 3 5 8 11 14 16 19 22 24 27
0 133 222 356 489 622 711 844 978 1067 1200

4 1 4 4 4 1 4 4 1 4
0 4 5 9 13 17 18 22 26 27 31
0 155 194 348 503 658 697 852 1006 1045 1200

4 3 4 4 4 3 4 4 3 4
0 4 7 11 15 19 22 26 30 33 37
0 130 227 357 486 616 714 843 973 1070 1200

5 1 5 5 5 1 5 5 1 5
0 5 6 11 16 21 22 27 32 33 38
0 158 189 347 505 663 695 853 1011 1042 1200

5 2 5 5 5 2 5 5 2 5
0 5 7 12 17 22 24 29 34 36 41
0 146 205 351 498 644 702 849 995 1054 1200

5 3 5 5 5 3 5 5 3 5
0 5 8 13 18 23 26 31 36 39 44
0 136 218 355 491 627 709 845 982 1064 1200

6 1 6 6 6 1 6 6 1 6
0 6 7 13 19 25 26 32 38 39 45
0 160 187 347 507 667 693 853 1013 1040 1200

5 4 5 5 5 4 5 5 4 5
0 5 9 14 19 24 28 33 38 42 47
0 128 230 357 485 613 715 843 970 1072 1200

7 1 7 7 7 1 7 7 1 7
0 7 8 15 22 29 30 37 44 45 52
0 162 185 346 508 669 692 854 1015 1038 1200

7 2 7 7 7 2 7 7 2 7
0 7 9 16 23 30 32 39 46 48 55
0 153 196 349 502 655 698 851 1004 1047 1200

6 5 6 6 6 5 6 6 5 6
0 6 11 17 23 29 34 40 46 51 57
0 126 232 358 484 611 716 842 968 1074 1200

7 3 7 7 7 3 7 7 3 7
0 7 10 17 24 31 34 41 48 51 58
0 145 207 352 497 641 703 848 993 1055 1200

8 1 8 8 8 1 8 8 1 8
0 8 9 17 25 33 34 42 50 51 59
0 163 183 346 508 671 692 854 1017 1037 1200

7 4 7 7 7 4 7 7 4 7
0 7 11 18 25 32 36 43 50 54 61
0 138 216 354 492 630 708 846 984 1062 1200

7 5 7 7 7 5 7 7 5 7
0 7 12 19 26 33 38 45 52 57 64
0 131 225 356 488 619 713 844 975 1069 1200

8 3 8 8 8 3 8 8 3 8
0 8 11 19 27 35 38 46 54 57 65
0 148 203 351 498 646 702 849 997 1052 1200

9 1 9 9 9 1 9 9 1 9
0 9 10 19 28 37 38 47 56 57 66
0 164 182 345 509 673 691 855 1018 1036 1200

7 6 7 7 7 6 7 7 6 7
0 7 13 20 27 34 40 47 54 60 67
0 125 233 358 484 609 716 842 967 1075 1200

9 2 9 9 9 2 9 9 2 9
0 9 11 20 29 38 40 49 58 60 69
0 157 191 348 504 661 696 852 1009 1043 1200

Dan

BTW - Though no one else seems to of had a problem with it, I can't
seem to get that CLAMPITT.pdf to open.

In-Reply-To: <940062676.11571@onelist.com>
Dan Stearns, digest 356.4, wrote:

> [Graham Breed:]
> "I'm assuming the scale will be tuned to something like 24-, 31-
or
> 38-equal. Plenty of other tunings are possible,"
> (<http://www.cix.co.uk/~gbreed/7plus3.htm>)
>
> Some of what Graham describes here is very similar to what I call
> set-mapping (which I use as a compositional or structural tool, to
aid
> in the bridging of multiple tunings). In this case, I'll use it as
a
> way to define the equal divisions of the octave that would render
a
> scale of ten notes comprised of seven large step sizes and three
small
> step sizes (and more generally, where L=L s=s, and, L>s & s>0).

Right, you do that.

> By setting the horizontal cut-off point of this table at L=s+10,
you
> not only stave off an inexorable march to ~3e, but you also
clearly
> define the EDO's [equally divided octaves] where s=(-n), and where
> s=0.

The march to 3= isn't inexorable. You can use whatever scales you
like. Anyway, setting L=s+10 doesn't stop 3=. Set s=10000,
L=10010. Sounds like 3= to me.

Yes, you clearly define the EDOs, but no other scales.

I'd prefer saying 2/7 < L+s < 5/17, or even 11/38 < L+s < 7/24.

> And as the 7L 3s scale can be seen here as a linear mapping of an
> interval >2/7 & <3/10 (or their inversions, i.e., >7/10 & <5/7),

This is what the MOS or WF scales are about. Maybe you're talking
about the same thing.

> any
> EDO that does not have an interval inside of this 1/70 space will
map
> as s=(-n), and will therefore be incapable of rendering a
(functional)
> scale (of 7L 3s).

Yes, the generating interval has to be between 2/7 and 3/10 for a
7L+3s scale. That is, 2/7 <(=) L+s <(=) 3/10. What do 1/70 and
s=(-n) mean?

> By the same token it's also clear to see that the
> EDO's that only have an interval that is exactly equal to 2/7 or
3/10,
> are also multiples of 7L where s=0, and as such, that they too are
>also incapable of rendering the scale.

Huh? If the generating interval is 3/10, then L=s. These boundary
scales depend on how you define the limits. I don't think it's
important. 7= and 10= are both aurally indistinguishable from
scales that are in the category, so the limit isn't musically
important.

> BTW - Though no one else seems to of had a problem with it, I
can't
> seem to get that CLAMPITT.pdf to open.

Can you normally open pdf files?

[Graham Breed:]
> The march to 3= isn't inexorable. You can use whatever scales you
like.

If s=3 it is... (BTW, it would be entirely dependent on what you set s
to.)

>Anyway, setting L=s+10 doesn't stop 3=.

Yes it does.

>Set s=10000, L=10010. Sounds like 3= to me.

Vertically your only going to get to ~10e... If L=10010 & s=10000,
then you get 0, 10010, 20020, 30020, 40030, 50040, 60040, 70050,
80060, 90060, 100070, at 0 120 240
360 480 600 720 840 960 1080 1200.

> Yes, you clearly define the EDOs, but no other scales.

Well, I'm not really sure of your point, as that was all I was trying
to show.

>This is what the MOS or WF scales are about. Maybe you're talking
about the same thing.

Yes, and I think that I go about it from a slightly different angle,
and because I was doing this before I ever heard the terms MOS & WF
scale there could be some overlaps that aren't always readily apparent
to me. But any n (where n is any number representing the amount of
notes in a scale), taken by any two (non-reducible) L & s's, will also
clearly define the body of EDO's of that linear mapping.

>What do 1/70 and s=(-n) mean?

If you want two step sizes (i.e., seven large and three small), then
the linear interval must fall inside a 1/70th of an octave space
between the 3/10 and the 2/7. s=(-n), just means that the small step
size is a negative number, and therefore these EDO's can't be used to
create the 7L 3s scale... I think it is good to know what doesn't
work, and why, and again in this format, this is all clear at a
glance... and I guess the real point of my post, *was* an attempt
(using the 7+3) to show how clearly all this can be defined at a
glance (and applied to any two step size scale).

Dan

>BTW - Though no one else seems to of had a problem with it, I can't
>seem to get that CLAMPITT.pdf to open.

What pdf reader are you using? It has a nav pane, which although being
completely useless, may not open with earlier versions of Acrobat?

-C.

I guess what I was getting at here- does anybody buy the self-similarity
part? Does that fact that MOS's have a like proportion of semitone steps
in all interval classes have anything to do with anything? Can anybody
hear it in Gregorian chant, as the authors claim (page 66)? What about for
scales where semitones aren't so rare (say 3 + 4 scales)? What about
tunings where the size of the semitone approaches the size of the whole
tone (such as the diatonic scale in 47tET)?

In any case, as Graham pointed out, the article is cool for showing
Myhill's property equivalent to MOS.

-C.

In-Reply-To: <940148307.26295@onelist.com>
Dan Stearns (digest 357.5) wrote:

> >Anyway, setting L=s+10 doesn't stop 3=.
>
> Yes it does.

Sorry, of course it does. But there's still 10.

> > Yes, you clearly define the EDOs, but no other scales.
>
> Well, I'm not really sure of your point, as that was all I was
trying
> to show.

A theory is judged by how much it explains for its size. A theory
that only covers EDOs is inferior to a theory that explains other
scales as well, without getting more complex.

> >But any n (where n is any number representing the amount of
> notes in a scale), taken by any two (non-reducible) L & s's, will
also
> clearly define the body of EDO's of that linear mapping.

Sure does. You can also convert between L&s and the generator of
the scale in steps. Hence getting the m in m/n. I take L and s to
be the sizes of the intervals in logarithmic units (cents). I see
no reason to constrain them to be integers. Do you think otherwise?

> >What do 1/70 and s=(-n) mean?
>
> If you want two step sizes (i.e., seven large and three small),
then
> the linear interval must fall inside a 1/70th of an octave space
> between the 3/10 and the 2/7.

Gotcha.

> s=(-n), just means that the small step
> size is a negative number,

s<0, then.

> and therefore these EDO's can't be used to
> create the 7L 3s scale... I think it is good to know what doesn't
> work, and why, and again in this format, this is all clear at a
> glance... and I guess the real point of my post, *was* an attempt
> (using the 7+3) to show how clearly all this can be defined at a
> glance (and applied to any two step size scale).

Yes, it can be clearly defined at a glance, so no need for all this
verbiage. I find the scale tree the clearest way of showing it. I
think it should be taken for granted that the two step sizes be
positive. Otherwise, I worked out a shorthand way of specifying
things like "L should be larger than s" and "L should be no more
than twice the size of s" but maybe it's easier to write the
inequalities. So, we have a 7L+3s scale with 0<s<L<2s.

The difference between < and <= is musically irrelevant, so assume
an implied = sign everywhere.

This TD exchange between myself and Graham Breed:

> [Graham:]
> > The march to 3= isn't inexorable. You can use whatever scales you
> like.

> [me]
> If s=3 it is... (BTW, it would be entirely dependent on what you set
s
> to.)

>
> [Graham:]
> >Anyway, setting L=s+10 doesn't stop 3=.

> [me]
> Yes it does.
>

was (unfortunately) based on a typo in my previous post that I didn't
pickup on until just looking at it now... 3e, was supposed to be 7e
(and therefore when I wrote "BTW, it would be entirely dependent on
what you set s to," it should have read: it would be entirely
dependent on what you set L to.

Another (perhaps clearer) way to look at this (as opposed to a
horizontal cut-off point), would be as a diagonal cut-off created by
the multiples of L... can anybody offer a nice rule of thumb method
for setting both the vertical and a diagonal cut-off points which
would clearly illuminate only the EDOs that 'sufficiently differ' from
either L+s (equal) or L (equal)?

Dan

Carl wrote,

>Does that fact that MOS's have a like proportion of semitone steps
>in all interval classes have anything to do with anything?

Can you explain this claim more clearly?

In-Reply-To: <940235417.24422@onelist.com>
Carl Lumma, digest 358.4, wrote:

> I guess what I was getting at here- does anybody buy the self-similarity
> part? Does that fact that MOS's have a like proportion of semitone
> steps
> in all interval classes have anything to do with anything? Can anybody
> hear it in Gregorian chant, as the authors claim (page 66)? What about
> for
> scales where semitones aren't so rare (say 3 + 4 scales)? What about
> tunings where the size of the semitone approaches the size of the whole
> tone (such as the diatonic scale in 47tET)?

I don't think "self-similarity" is the right word for it. But whatever,
it might have some meaning in some musics. The fact that a scale covering
an octave will have proportionately the same number of semitones as the
same scale covering a fourth is useful to know. But you'd expect this to
happen for an MOS, because the rarer steps size is evenly distributed.
Inventing a term and then declaring it to be redundant is a bit odd.

>>Does that fact that MOS's have a like proportion of semitone steps
>>in all interval classes have anything to do with anything?
>
>Can you explain this claim more clearly?

I assumed reading of the paper. Somebody said they had problems getting
it. Were you one of them?

In the diatonic scale, in the span of any 5th, if you had to walk the span
a 2nd at a time, it is always the the case that 1/5 of the 2nds are s, and
4/5 of the seconds are L. These ratios are the best approximations to 2/7
and 5/7 with denominator five. 2/7 and 5/7 are the ratios of s and L steps
in any octave span, respectively. And the authors show that this is the
case for all melodic intervals in the diatonic scale (and in any WFS/MOS),
and propose that diatonic music takes advantage of this sort of scaling
symmetry.

I think it's bunk. There are two ways to test it I could think of. First,
you can listen to Gregorian chant and see if you can hear it. The authors
imply (on pg. 66) that it is easy to hear in chant because it's one of the
only tricks being turned, in a manner of speaking.

Another way would be to ask if this sort of thing would be at all
discernable in scales where the semitone steps are less rare (like 3 + 4
scales), or where they differ in size by far less (like the diatonic scale
in 47tET), and if not to play music in such scales and see if they still
work melodically.

-C.

I noticed that a host of old disembodied posts have suddenly
materialized just in time for Halloween...

[me:]
> > s=(-n), just means that the small step
> > size is a negative number,
[Graham Breed:]
> s<0, then.

Well I find that being able to see in a glance the EDOs that don't
work in a particular mapping (where s=(-n)), to also be a useful
convenience... but as you point out, not a musically significant
mapping.

[Graham:]
> Yes, it can be clearly defined at a glance, so no need for all this
verbiage.

Yes... verbiage... Rereading most any post I've ever done does cause
(even) me to cringe in horror... but no matter how many times I say to
myself: "self, you've really got to tone it down a bit," I just seem
to come right back and embarrass myself all over again... oh well, so
it goes I guess...

[Graham:]
>I find the scale tree the clearest way of showing it.

By the scale tree, in this context (i.e., a 7L+3s linear mapping of an
interval inside a 1/70th of an octave space between a 3/10 and a 2/7),
I would take that to mean (something along the lines of):

3/10 2/7
\ /
5/17
/ \
7/24 8/27
/ \ / \
9/31 10/34 11/37
/ \ / \ / \
11/38 12/41 13/44 14/47
/ \ / \ / \ / \
13/45 14/48 15/51 16/54 17/57

(etc.)

or:

3/10 2/7
5/17
8/27 7/24
11/37 13/44 12/41 9/31

(etc.)

Please let me know if you mean something different.

(thanks,)
Dan