[Breed]

>5 is derived from 2 and 3. This means a 5 note MOS has 2 intervals of one

>size and 3 of another. Similarly a 7 note MOS has 5 intervals of one size

>and 2 of another and a 12 note MOS has 5 intervals of one size and 7 of

>another.

An important point for me was that nested MOS's are preserved; the 5-note

MOS has 2 and 3 note MOS's on every scale degree, and the 7-note MOS

contains 5 and 2 note MOS's on each of its scale degrees, and so on.

Daniel Wolf says it...

[Wolf]

>Over the whole collection of 12 tones, the melodic symmetry with the cycle

>of generating intervals is trivial, but the Moments of Symmetry in the

>subsets, which form closed cycles only with the addition of an atypical

>interval (e.g. wolfs, tritones and their analogs) to the cycle are the

>points of real interest. It is at this level that the non-trivial melodic

>symmetries take place.

C.

Can someone define/explain the term MOS ?

_________________________________________________________

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"skyler S." wrote:

> Can someone define/explain the term MOS ?

Take a look at Erv;s original paper found at

http://www.anaphoria.com/mos.html It remains the simplest explanation but

will try my best to clear up any points of unsureness!

-- Kraig Grady

North American Embassy of Anaphoria Island

www.anaphoria.com

>Beyond that, I don't think he's ever dealt with an MOS where the generating

>interval isn't taken to be a fourth. We may be on virgin territory!

? Wilson has played with more generators than I'd care to shake a stick at. He is especially fond of irrational ones, if you've heard _From on High_ or talked to Marcus about the tunings. He has these iterative formulas which converge on generators that give certain properties to the scale --- often difference tone properties, ala meta-meantone.

Wilson told me, with a grin a mile wide, that one of his favorite things in the world is varying the size of the generator and observing the points where the harmonic approximations rearrange on the linear series. He said that sampling this continuum at only the roots of two is often too crude to see the real action.

>Graham, In Wilson's term "Moment Of Symmetry", what symmetry is it

>referring to?

Kraig Grady replied...

>The resulting two interval patterns that a generator produces.

I take MOS to refer to those special sizes of chain, for a given link size and interval of equivalence (IE), at which every link subtends the same number of IE-equivalent joints.

I view this basic definition as the cause of the two-interval pattern, and its affect on the structures of the modes of the scale.

I think these properties are desirable because of the way they relate scale degrees to acoustic intervals.

Grady again...

>The scale tree is meant to be used both acoustically and/or logrithmically.

It is very interesting that you say that! I was just thinking the other day, that the Stern-Brocot tree (aka Farey tree, scale tree...) has got to be the most important structure in all of music theory. When applied to frequencies, it gives the best map of dyadic consonance there is (might as well define consonance for harmonic timbres). When applied to logarithms of frequencies, it gives the best lexicon of melodic resources there is (yet).

-C.

For those confused about MOS, here's something I've

emailed privately to a few people, which I learned

first-hand from Carl Lumma.

**********************************************

DISCLAIMER:

I EMPHASIZE THAT THIS IS ONLY MY VERSION OF IT.

Erv Wilson is adamant that his work should stand as

he wrote it and speak for itself. But I find that

this explanation helps make the whole thing much clearer.

Hope it helps the rest of you who are struggling with

this. Keep reading the Wilson Archives at

www.anaphoria.com

There may be stuff in this post that is just plain wrong

- corrections are appreciated.

************************************************

MOS stands for "Moment of Symmetry" (a really

lovely title), and was discovered in the

1960s by Erv Wilson.

http://www.anaphoria.com/mos.html

is the letter describing MOS.

The first thing to take note of is that

Wilson always uses his theories and

diagrams to depict ideas of complete

generality. He will sometimes use specific

ratios in his beautiful mandala-like

lattice diagrams, but it is always

assumed that they can representent any

kind of set of anything.

The idea behind MOS is that it links

together two different ways we listen

to or perceive music:

1) the sonance of the interval (i.e.,

its harmonicity).

2) how many scale degrees the intervals

subtend (Wilson's word, I think;

it kind of means "passes over"

or "is divided into").

Harmonic listening is bound to force one

to think in terms of ratios, while scalar

listening encourages thinking in terms

of "steps" (unequal or equal).

In the diatonic scale, all "5th"s are fixed

to the perception of a 3/2 (what Wilson calls

the "3-function"), except for the last one

(the "tritone").

The "3rd"s are fixed to the "5-function",

but not as rigidly as the "5th"s are to 3/2:

the "3rd"s may be either "major" (5/4)

or "minor"(6/5).

His concept of MOS gives us a neat diatonic

way of relating the two modes of listening.

Now on to the details . . .

-------------------

MOS assumes octave-equivalency, and is

based on a linear mapping of notes, for

example, a Pythagorean JI system (open),

or 12-(or other-)Eq (closed). It has

2 basic intervals: one called a "generator"

and another which acts as octave-reducer.

The generator is an interval which is

cycled thru at more-or-less the same size,

to create all the different notes in the

system (for instance, a "5th"; we need not

specify the tuning). The octave-reducer

is typically 2/1, as in most music theories.

I'm deliberately going to fudge the

discrepancy between JI and ET in this

description - trust me, it will make

things easier. Start by imagining a circle . . .

(You should draw it yourself on paper as

I describe it - that helps a lot to

understanding it.)

<I'll wait, go get your pencil . . .>

OK, start by drawing a circle to represent

the octave, 1/1 at the top (12 o'clock).

We'll use approximate clock positions just

to keep things simple. Just put a tick

mark on the circumference of the circle

and label it for each note as we progress.

The first "5th" takes us to 3/2, 7 o'clock

the second "5th" to 9/8, 2 o'clock

the third "5th" to 27/16, 9 o'clock

the fourth "5th" to 81/64, 4 o'clock . . .

Hell, let's switch to prime-factor notation,

and give them the typical musical letter-names

too. So, that's:

C 3^ 0 at 12:00

G 3^ 1 at 7:00

D 3^ 2 at 2:00

A 3^ 3 at 9:00

Fb/E 3^ 4 at 4:00

Cb/B 3^ 5 at 11:00

Gb/F# 3^ 6 at 6:00

Db/C# 3^ 7 at 1:00

Ab/G# 3^ 8 at 8:00

Eb/D# 3^ 9 at 3:00

Bb/A# 3^10 at 10:00

F /E# 3^11 at 5:00

C /B# 3^12 at 12:00,

which means we've completed the cycle

if we're in 12-Eq (or its Pythagorean

near-miss). (But this process can be

carried out much further, and Wilson

did . . .)

Now, here's my definition of Moment

of Symmetry (MOS): it's when every

"link" in the chain *subtends the same

number of steps*, even if the last link

is not exactly the same size as the others.

I can only illustrate by way of our circle.

The first example doesn't have to be drawn -

it's trivial. It would be a chain of 2 links,

going from C 3^0 (12:00) to G 3^1 (7:00) and

back again. The only notes in the system

are C and G. Obviously 2 is a MOS, because

each link subtends 1 "step". The 1st step

is a "5th", (to 7:00) and the 2nd step is a

"4th", bringing us back to the origin C (12:00)

or "octave".

If you're clever, you should have the idea

already. If not, like me, go ahead and

draw each example as I describe it.

(Draw that last one if you need to.)

Remember that "step" only refers to the

*number of steps in, and the specific

steps derived from, THAT division*.

We'll have to draw many circles now, one

for each division, to discover which ones

are a MOS and which are not. Do that.

The division into 3 goes like this:

1st link C 3^0 (12:00) to G 3^1 (7:00).

2nd link G 3^1 ( 7:00) to D 3^2 (2:00).

3rd link D 3^2 ( 2:00) back to C 3^0 (12:00).

Thus our system is made up of C, D, and G.

The 1st link subtends 2 steps: C-D and D-G.

The 2nd link subtends 2 steps: G-C and C-D.

The 3rd link subtends 2 steps: D-G and G-C.

Therefore 3 is also a MOS.

Division into 4:

1st link C 3^0 (12:00) to G 3^1 (7:00).

2nd link G 3^1 ( 7:00) to D 3^2 (2:00).

3rd link D 3^2 ( 2:00) to A 3^3 (9:00).

4th link A 3^3 ( 9:00) back to C 3^0 (12:00).

Our system is made up of C, D, G, and A.

1st link subtends 2 steps: C-D and D-G.

2nd link subtends 3 steps: G-A, A-C, and C-D.

3rd link subtends 2 steps: D-G and G-A.

4th link subtends 1 step: A-C.

The steps sizes are not all the same,

so 4 is *not* a MOS.

Division into 5:

1st link C 3^0 (12:00) to G 3^1 (7:00).

2nd link G 3^1 ( 7:00) to D 3^2 (2:00).

3rd link D 3^2 ( 2:00) to A 3^3 (9:00).

4th link A 3^3 ( 9:00) to E 3^4 (4:00).

5th link E 3^4 ( 4:00) back to C 3^0 (12:00).

Our system is made up of C, D, E, G, and A.

1st link subtends 3 steps: C-D, D-E, and E-G.

2nd link subtends 3 steps: G-A, A-C, and C-D.

3rd link subtends 3 steps: D-E, E-G, and G-A.

4th link subtends 3 steps: A-C, C-D, and D-E.

5 is a MOS.

I'll skip the rest - you can draw them yourself.

6 is *not* a MOS,

7 is a MOS,

8, 9, 10, and 11 are *not* a MOS,

12 is a MOS.

Scales below 5 notes are considered

insignificant, so I will use Wilson's notation.

For positive mapping, that is, a system that

has "5th"s that are 3/2s [= 702.0 cents] or

wider, you get a MOS at

(1, 2, 3,) 5, 7, 12, 17, 29, and 41.

These include the following ETs:

#degrees size of "5th" in cents

17 706

29 703

41 702.4

For negative mapping, a system with "5th"s

narrower than a 3/2, we get a MOS at

(1, 2, 3,) 5, 7, 12, 19, and 31.

These include the following ETs:

#degrees size of "5th" in cents

12 700

19 695

31 697

Scales with more than 41 are not considered

necessary (I can't remember why now -

ask Lumma).

That's MOS.

------------------

A little further on in your study of Wilson,

you'll find that in his article "On the

Development of Intonational Systems by Extended

Linear Mapping" he recommends taking advantage

of the fact that the 12-tone scale allows

us to perceive either positive or negative

mapping, to switch from the current negative

mapping to one which is positive and

"acoustically advantageous".

He only discusses the cultural imprinting

of the dual mapping on our consciousness in

relation to the 12-tone scale, but it seems

to me that dual mapping would already be

ingrained from the previous historical use

of both the 5-tone and 7-tone scales, both

of which are MOS in both mappings. (I'm

not sure tho - I'd have to explore it a lot

more. Maybe you can see it.)

- Monzo

Joseph L. Monzo Philadelphia monz@juno.com

http://www.ixpres.com/interval/monzo/homepage.html

|"...I had broken thru the lattice barrier..."|

| - Erv Wilson |

--------------------------------------------------

___________________________________________________________________

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Joe Monzo wrote:

> The first "5th" takes us to 3/2, 7 o'clock

> the second "5th" to 9/8, 2 o'clock

> the third "5th" to 27/16, 9 o'clock

> the fourth "5th" to 81/64, 4 o'clock . . .

>

> Hell, let's switch to prime-factor notation,

> and give them the typical musical letter-names

> too. So, that's:

Why not keep it simple and stick with ratios?

> --

* D a v i d B e a r d s l e y

* xouoxno@virtulink.com

*

* J u x t a p o s i t i o n N e t R a d i o

* M E L A v i r t u a l d r e a m house monitor

*

* http://www.virtulink.com/immp/lookhere.htm

Joe Monzo wrote,

>For positive mapping, that is, a system that

>has "5th"s that are 3/2s [= 702.0 cents] or

>wider, you get a MOS at

>(1, 2, 3,) 5, 7, 12, 17, 29, and 41.

>These include the following ETs:

>#degrees size of "5th" in cents

> 17 706

> 29 703

> 41 702.4

>For negative mapping, a system with "5th"s

>narrower than a 3/2, we get a MOS at

>(1, 2, 3,) 5, 7, 12, 19, and 31.

>These include the following ETs:

>#degrees size of "5th" in cents

> 12 700

> 19 695

> 31 697

Correction: a positive system has fifths wider than 700 cents, and a

negative system has fifths narrower than 700 cents.

Paul,

OK, looks like our discussion on this topic is beginning to heave its last

breaths. I apologize if I've been (or continue to be) long winded. If

anyone is still interested I'll post my equations for calculating the

efficiency of any MOS, which are almost finished (in the case where the

scale is strictly proper, the efficiency depends only on m, the number of

tones in the scale, and for the merely proper case, there's a difficult

equation depending on m and also n, the number of tones in the system.)

We're almost down to barefaced opinions now, so I'll just clarify a couple

of my own tenative opinions.

Paul wrote:

>So you're saying that _all_ scales are perceptually forced into the

>lowest equal temperament which gives a good fit? That kind of assumption

>may sit well with theorists like Balzano and Clough-Carey-Clampitt,

>who've "explained" a lot of things on these sorts of bases, but I don't

>buy it, not one bit -- there's always a better, more acoustical

>explanation, with fewer arbitrary assumptions.

I think acoustical explainations definitely have their place. I believe

that sensory consonance, which has a clear physical basis which has been

studied extensively by Plomp et al, plays a role in harmony. But sound

acoustical explainations of music, at present, are mostly limited to

sensory consonance. Models which explain premises of music theory on a

pysiological/neurological basis, such as Terhardt's, are interesting but

speculative. There's much more to music theory and tuning theory than

sensory consonance unless the variety of meanings of consonance which James

Tenney outlines in his book are all ultimately reducible to sensory

consonance, a shaky theory at best.

You wrote:

>Equal temperament is a contrivance that offers many great benefits, but I

>don't believe it is of perceptual origin (though in the _particular_ case

>of the Pythagorean scale the fit to 12 is so good that it is hard to

>avoid, even perceptually).

I agree, if you mean, equal tunings considered as sets of irrational

frequency ratios aren't perceptually real. I have meant to emphasize

(although I have been less than clear) that I'm not looking at ET's

considered as such, but at divisions of the 8ve into some number of

categories. Equal temperment is nothing more than a keyboard or lute

tuning designed to produce 12 equally (sensory) consonant major and minor

triads.

I said:

>>Yes, and this fact gives us a useful method of finding the generator of

>>the scale at any step of the process. But here's where I cash in on the

>>observation above that each choice of a previous number of places to go

>>back to find the number to add to the last, given the weak criterion,

>>gives a unique l:s in the following scale: to explain the value of the

>>strong criteria. If the choice is always 1, then l:s is always 2:1.

And you said:

>??? Actually, it approaches 1:phi, unless you're dealing with a

Yasser->type situation where you use an equal temperament which is chosen

by >using the next term in the sequence (i.e., using an additional "choice

of >1").

This point actually requires a much more extensive explaination than I

gave. Let's just look at one alternative, for simplicity: going back 2

steps rather than one. Say I've started a sequence (2), 3, 5, 8, 13. At

the last step of the sequence I have a system of 13 notes and 13 interval

categories, and if I exhaust that system, I have a scale 1111111111111 (in

terms of interval categories). I also have the 8 note scale 21221212. The

next choice is a choice in extending the number of categories, and

according to our limitations, I can extend it to 21 or 18. With the former

choice, My 8-note scale becomes 32332323, which implies than my 13 note

scale becomes 2122121221221. In the latter case my 8-note scale becomes

31331313, and the 13-note scale 2112121121121. Of course, l:s is always

going to reduce to 2:1, I should have been contrasting the proportions of

the 8-note scales, 3:2 and 3:1. These proportions translate into the 2nd

level of structure in the 13-t scale. The situation becomes more

complicated as the next system becomes smaller: I need to study this

problem further. Suffice it to say that the scales become less stable as

the next system becomes smaller, which I think is evident.

Let me make one more point which I think I have not made clear yet. We're

becoming confused as to whether I'm talking about scales tuned in some ET,

or according to the reiteration of some irrational generator. Actually I'm

talking about neither. Let's look at it this way: the scale forms I'm

proposing expressed as interval categories could include an infinite number

of scales expressed in terms of cents or freq ratio (I'm changing the

terminology here). A number of non-equal intervals of these scales will

fall into the same interval category given some translation. My

supposition, then, is that the interval categories are more perceptually

relevant in many respects than the exact freq ratios. To a some extent

this _has_ to be true because we can only detect a certain degree of

precision in freq.

Also, I don't know if this discussion has been seen as heated, but let me

say that at no point in it have I felt heated. I value challenges to my

opinions as well as mathematical corrections as I'm sure very few final

words have yet been spoken.

jason

Paul Erlich wrote:

> Did Erv every specifically mention any MOS scales that repeat at

> the half-octave?

I've posted about this more than once in the past. Years ago, Erv pointed out

several MOS tetrachords to me, so he must have recognized that the MOS idea is

extendable, in principle, to any interval.

Dave Keenan wrote,

<<MOS is fundamentally a melodic property. A scale can be a MOS and

contain no good approximations to any consonance. If the generator

doesn't need to have any particular relationship to consonance, why

should the period. Kraig Grady, Graham Breed, Dan Stearns and I all

now agree that your symmetric decatonic is MOS. If you want to be more

specific you can say it is MOS at the half-octave.>>

I've been saying this for years on this list -- see the many posts in

the archives about fractional periodicity and how all two-term, single

generator scales relate to adjacent fractions. I don't seem to recall

anyone (at least not Paul or Carl who I do remember responding back

then) agreeing with any of it...

The other point that I've been saying right along with it was that all

M-out-of-N maximally even scales are always "bivalent" (or some other

suitable term). And that this might be a better type of approach to

these ideas, thereby not trying to tie them too tightly to Erv

Wilson's MOS, which seemed problematic for people.

The idea as I saw it being that the concept can be more generally tied

to series -- i.e., Fibonacci, Tribonacci, etc. So this seemed like a

potentially productive way to go if generalization was the goal.

I also put a lot of effort into many obviously related

L-out-of-M-out-of-N "trivalence" posts. And if there was ever a thread

that I personally was hoping for some group type of participatory

collaboration and feedback on, that was it! With the exception of

Robert Walker, I don't think anyone said boo... oh well, so it goes!

Some things catch fire and others simply don't.

--Dan Stearns

--- In tuning@y..., "Daniel Wolf" <djwolf1@m...> wrote:

> Paul Erlich wrote:

>

> > Did Erv every specifically mention any MOS scales that repeat at

> > the half-octave?

>

> I've posted about this more than once in the past. Years ago, Erv

pointed out

> several MOS tetrachords to me, so he must have recognized that the

MOS idea is

> extendable, in principle, to any interval.

You didn't answer the question at all. I guess the answer is

probably "no, to the best of Daniel Wolf's knowledge".

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> Dave Keenan wrote,

>

> <<MOS is fundamentally a melodic property. A scale can be a MOS and

> contain no good approximations to any consonance. If the generator

> doesn't need to have any particular relationship to consonance, why

> should the period. Kraig Grady, Graham Breed, Dan Stearns and I all

> now agree that your symmetric decatonic is MOS. If you want to be

more

> specific you can say it is MOS at the half-octave.>>

>

> I've been saying this for years on this list -- see the many posts

in

> the archives about fractional periodicity and how all two-term,

single

> generator scales relate to adjacent fractions. I don't seem to

recall

> anyone (at least not Paul or Carl who I do remember responding back

> then) agreeing with any of it...

Well, Carl and I were going by the small amount of Wilson that we had

seen. Apparantly we were wrong . . . and until this week Kraig Grady

never replied all the times I said LssssLssss was not an MOS. So

forgive us.

Paul!

Pardon that i missed this . It is not what one would think of as an MOS unless you really look

closely. in a sense then all symmetrical scales are MOS. The thing is you are kinda cheating in

the sense in that you originally conceive of something as an octave based scale and because it is

symmetrical at the half octave you can all it an MOS. But not on the level you originally

constructed your scale. technically an MOS but not one is "spirit".

Paul Erlich wrote:

> Well, Carl and I were going by the small amount of Wilson that we had

> seen. Apparantly we were wrong . . . and until this week Kraig Grady

> never replied all the times I said LssssLssss was not an MOS. So

> forgive us.

>

>

-- Kraig Grady

North American Embassy of Anaphoria island

http://www.anaphoria.com

The Wandering Medicine Show

Wed. 8-9 KXLU 88.9 fm

[Dave Keenan wrote...]

>>>MOS is fundamentally a melodic property. A scale can be a MOS and

>>>contain no good approximations to any consonance. If the generator

>>>doesn't need to have any particular relationship to consonance, why

>>>should the period. Kraig Grady, Graham Breed, Dan Stearns and I all

>>>now agree that your symmetric decatonic is MOS. If you want to be

>>>more specific you can say it is MOS at the half-octave.

[Dan Stearns replied...]

>>I've been saying this for years on this list -- see the many posts

>>in the archives about fractional periodicity and how all two-term,

>>single generator scales relate to adjacent fractions. I don't seem to

>>recall anyone (at least not Paul or Carl who I do remember responding

>>back then) agreeing with any of it...

[Paul Erlich wrote...]

>Well, Carl and I were going by the small amount of Wilson that we had

>seen. Apparantly we were wrong . . . and until this week Kraig Grady

>never replied all the times I said LssssLssss was not an MOS. So

>forgive us.

In October of last year I wrote,

"

... how a non-octave MOS can be viewed as a superposition of multiple,

octave-based MOSs. Take the wholetone scale in 12-tET (0 2 4 6 8 10 12).

It doesn't have Myhill's property, so it isn't an MOS. But it can be

described as the superposition of (0 2 4 6 8 12) and (0 4 6 8 10 12),

both of which have Myhill's property. Now, call 1300 cents the interval

of equivalence. The set (0 2 4 6 8 10 13) _is_ a single MOS here. Did

I make a mistake?

In short, a set is MOS if and only if it has Myhill's property at the

interval of equivalence specified; scales can be MOS at a given IE but not

at another.

"

In August of 1999 I offered the following definition for MOS,

"

A pythagorean-type scale (the generator need not be the 3:2, and the

interval of equivalence need not be the 2/1) is MOS iff the generating

interval occurs at only one scale degree in all modes of the scale (i.e.

3:2 is always a 5th in the diatonic scale).

"

Defining MOS with a generalized IE was also what John Chalmers did in

1997 when I asked him to explain MOS to me.

-Carl

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:

> Paul!

> Pardon that i missed this . It is not what one would think of as an MOS unless you really look

> closely. in a sense then all symmetrical scales are MOS. The thing is you are kinda cheating in

> the sense in that you originally conceive of something as an octave based scale and because

it is

> symmetrical at the half octave you can all it an MOS. But not on the level you originally

> constructed your scale. technically an MOS but not one is "spirit".

I accept the charges and also put forth my favorite non-MOS, LsssLsssss.

Nice to see you back, Carl! Check out the tuning-math list. Anyway, I've always seen a

distinction between "Interval of Equivalence" and "Interval of Repetition". The symmetrical

decatonic scale, and Messaien's modes of limited transposition, to me, have an "Interval of

Repetition" that is a fraction of the "Interval of Equivalence". So that's why I never thought they

were MOS.

Clarification: These modes are always presented in the context of octave-equivalence. The

pitches are given within one 2:1 span, and that's it -- the rest are considered repetiions.

Musically, in no way are any of the pitches within the 2:1 span considered _equivalent_ to one

another -- although many of them may pull _equally strongly_ with the feeling of a tonal center.

LsssLsssss is an MOS, or was this a typo :)

Paul Erlich wrote:

> I accept the charges and also put forth my favorite non-MOS, LsssLsssss.

-- Kraig Grady

North American Embassy of Anaphoria island

http://www.anaphoria.com

The Wandering Medicine Show

Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:

> LsssLsssss is an MOS, or was this a typo :)

No, this was not a typo. OK, Kraig, how is it an MOS???

Carl: Thank you very much for the reference to the paper-- I hadn't seen it somehow. I do remember a chart by John Cuciureanu that delimited the various types of scale, but I don't have it handy either. I've downloaded and will read it over as I think having a standardized nomenclature is an excellent advance.

I agree that the octatonic can be considered an MOS either in 3-TET or 1/3 of an octave, but it seems to me that it is the BROKEN symmetry of MOS scales sui generis that gives them their distinctive scale-like properties. On the other hand, I have come across scales such as two copies of the black key pentatonic compressed into a 24-tet octave (as 2 3 2 2 3 2 3 2 2 3), but in another context.

I don't recall Erv or for that matter Joel Mandelbaum, who came upon the same concept in 19-tet (as Quasi-Equal-Interval Symmetric or QEIS), claiming that things like the whole tone scale, the string-of-pearls (octatonic) or the other of Messiaen's "Scales of Limited Transposition" to be MOS.

--John