Fw: Re:Offline comments MOS

Kraig Grady provided some very insightful comments to me
privately, in response to the posting I sent about MOS
[TD 276.1].

In response to some of Kraig's questions to me, I can only
reiterate that what's in my post was basically what I understood
of Carl Lumma's explanation of it to me.

Kraig has given me permission to forward this to the List.
Further feedback is most welcome. Carl? Daniel Wolf?

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

--------- Forwarded message ----------
From: Kraig Grady <kraiggrady@anaphoria.com>
To: Joe Monzo <monz@juno.com>
Date: Mon, 09 Aug 1999 18:01:32 -0700
Subject: Re:Offline comments MOS
Message-ID: <37AF79E4.97E9ECB8@anaphoria.com>
References: <19990809.125829.-847969.63.monz@juno.com>

Joe Monzo wrote:

> From: Joe Monzo <monz@juno.com>
>
> 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.

He has had problems of misquotes and misrepresentations so he
just wants the originals to be handy to those interest. Outside
of this he hopes his work will be of use to all. Including other
tyheorist! Comment as you will!

> 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:

I think (in my opinion) that Wilson is only concerned with
melodic scale archetypes. MOS as an Archetype is another music
shaping dynamic in contrast to harmonic. He has stated that any
tuning systems is a compromise between the two.

>
>
> 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").

I'm curious where you got this one!

>
>
> 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,

In a true MOS all the intervals are the same size. Otherwise
he refers to them as constant structures.

>
> 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

He calls the last link, the disjunction

>
> 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).

I know Wilson does't think so!

>
>
> 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
>
>

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

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