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moment of symmetry, MOS

[Kraig Grady]

A term coined by Erv Wilson in 1975. The process of producing a scale of melodic integrity by the superposition of a single interval (generator). Those points where there are only 2 different size intervals are called "moments of symmetry". This cycle has the property that any occurrence of an interval will always be subtended by the same number of steps.

When a harmonic system is used instead of a single interval and all the melodic gaps are filled the scale is referred to as a "constant structure".

It has been found useful to utilize deeper levels within the MOS. The best example of this Bifocal MOS are the pentatonics taken from the Diatonic. Here from a Parent MOS (Diatonic) another smaller MOS (Pentatonics) set can be found which, although it will not have the property of every interval being subtended by the same number of steps, the generator interval (fourth or fifth) will be.

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[Paul Erlich, Yahoo tuning-math, message 661 (Tuesday, August 07, 2001 10:18 AM)]

An MOS is a linear tuning (since there is a single generator along with an associated interval of repetition), carried out to some number of notes such that there are only two step sizes.

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[Joe Monzo, private email to Dan Stearns (Sat, 6 Mar 1999 13:50:33 -0500)]

[Below is an explanation of MOS which i wrote in an email a few years ago, when i was still having a hard time understanding it myself.]

[From Dan:]

I'm unfamiliar with the acronym MOS... (I've seen it used apparently in relation with n-tET's that create a [seven note] scale consisting of [five] same size diatonic major seconds, and [two] same size diatonic minor seconds...) Could you give me an explanation of it?

OK, you asked for it.

You're describing a specific MOS property - (at least I think you are - I'm still struggling with this stuff myself). It's MUCH MUCH more general than that. MOS stands for "Moment of Symmetry" (a really lovely title), and was discovered in the 1960s by Erv Wilson. I've only just gotten a good understanding of it myself (thanks to in-person discussions with [Paul] Erlich and [Carl] Lumma). Sit back, this is a long one.

I begin with this MAJOR caveat - Erv Wilson doesn't like other people to speak for him. He INSISTS on letting his own work stand entirely on its own. So the best thing you can do is visit the Wilson Archives on Kraig Grady's website (Anaphoria). http://www.anaphoria.com/mos.PDF is the letter describing MOS. I'll admit it didn't mean anything to mean when I read it - I need to go back and do so again now.

So, I'll tell you what I know about it, but, remember, it's ONLY MY VERSION.


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-3rds" (5/4) or "minor-3rds" (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-)EDO (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) [i.e., 12 Semitones]. 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 [i.e., 7 Semitones] 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 [assume "C" = 1/1 = n0]. 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-edo (or its pythagorean near-miss [because the 13th note in the chain introduces the pythagorean comma]). (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: (1, 2, 3,) in parentheses, the rest of the MOS numbers without parentheses and separated by commas.

For positive mapping -- that is, a system that has "5th"s that are 700 cents or wider -- you get a MOS at (1, 2, 3,) 5, 7, 12, 17, 29, ... . 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 of 700 cents or narrower -- we get a MOS at (1, 2, 3,) 5, 7, 12, 19, 31, ... . These include the following ETs:

#degrees  size of "5th" in cents
  12        700
  19        695
  31        697

Thus, the boundary between positive and negative systems is that which separates the progression of MOS beyond 12, the positive systems giving MOS of 17, 29 ... and the negative systems giving MOS of 19, 31, ... .

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".

Wilson 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

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