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Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
moment of symmetry, MOS
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.
[from Kraig Grady]
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.
[from Paul Erlich, Yahoo tuning-math group
message 661
(Tuesday, August 07, 2001 10:18 AM)
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: monz@juno.com
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:
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.)
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:
which means we've completed the cycle
if we're in 12-Eq (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.
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:
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:
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
updated:
To: Dan Stearns
Date: Sat, 6 Mar 1999 13:50:33 -0500
Subject: MOS
> 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?
1) the sonance of the interval (i.e.,
its harmonicity).
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,
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.
#degrees size of "5th" in cents
17 706
29 703
41 702.4
#degrees size of "5th" in cents
12 700
19 695
31 697
2002.10.11 -- email explanation to Dan Stearns added
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