AN INTRODUCTION TO THE MOMENTS OF SYMMETRY
By Kraig Grady (Cultural Liaison-North American Embassy of Anaphoria Island) 6/17/2007 all rights reserved.
(http://anaphoria.com/mos.PDF)
~Comments and Questions welcome~
Link to http://www.marcussatellite.com/mos.html for an interactive display of MOS patterns
[THIS IS THE CLIFF NOTE DEFINITION (added 6/2010)[revised 3a. 2/2011]
Definition of Moment of Symmetry
A Moment of Symmetry is a scale that consists of:
1. A generator (of any size, for example a 3/2 or a fifth in 12 equal temperament) which is repeatedly superimposed but reduced within the
2. Interval of Equivalence (of any size, for example most commonly an octave)
3. Where each scale degree or scale unit will be represented by no more than two sizes and two sizes only (Large = L and small = s)
[3a.] Wilson goes on and explores Bi-Level MOS patterns. That is where the units in an MOS are considered equivalent and one explores those subsets. His example are the Japanese pentatonics shown to him by Tanabe. One takes a 7 tone Pythagorean MOS and then all the five tone scales formed by the sucession of 3 units. It was this pattern that spawned the concept in Wilson's research.
A constant structure (commonly of a higher limit) is a scale where every occurrence of a ratio will always be the same number of steps or units. This term he added later than we find on the page 18.
THE BACKGROUND, THE QUESTION, THE APPROACH, AND THE RESULTS
Erv Wilson perceived two contrasting forces that influence what makes
up a scale, harmonic and melodic. Possibly along with this should be
added how instruments and their construction influence what type of
scales that are produced on them. This would require a separate study
in each individual case. Still Wilson’s ideas can and could be applied
to them.
Being that we have many cultures whose music is melodically based,
Wilson posed the question what makes people gravitate toward certain
types of scales and why we find so few of others. One of his approaches
was the concept of Moments of Symmetry. Not meant to be the final word
on the subject, it covers much territory not found in previous and
historical practices such as Tetrachordal scales, or those derived from
subharmonic scales. Often, though, it illustrates and overlaps these
constructions, often adding another dimension to them. More
importantly, it offers us a gateway into an almost infinite family of
new scales that the conditions have not been ripe to manifest. The
Moments of Symmetry appears to be a feature of scales that have been
known quite intuitively, both individually and culturally. His approach
is conservative even when the results are not as Wilson is careful to
remain on a firm and well-established foundation. Empirically the
results have been perceived as self-contained arrays of pitches that I
have yet to hear any objection to. While there are scales that fall
outside these boundaries, the real importance lies with the fact that
those which do appear to be are heard as scales without exception. On a
personal level I have received numerous comments from listeners about
how the scales I use hold together not knowing or being concerned that
they are moments of symmetries.
Others have found much of this similar to Rothenberg’s concept of
propriety which others, more mathematically inclined have found useful.
I include references at the end for the reader’s own comparison. They
do diverge most noticeably with Wilson’s illustration of bi-level
patterns, which will be discussed later. The importance of this should
not be underestimated for they represent what Wilson considers some of
our greatest scale resources. Needless to say, the 20th century concept
that any combination of pitches forms a scale is foreign to both of
their ideas of scales as the result of pattern making yet Wilson has
always been aware and a strong defender of how useful deviations have
or will be artistically viable driven by what the music maker is after.
Wilson sees the role of a theorist as someone that makes music makers
aware of the potential of given material or constructs which might or
might not be useful to the actual artist who makes music. One of the
features of Moments of Symmetry is the formation of a cycle from which
melodic material can be transposed up and down the scale and still
remains recognizable. This is such a common feature in music of all
kinds that it does not need to be defended much.
ENOUGH. WHAT ARE MOMENTS OF SYMMETRY?
A Moment of Symmetry is a scale where there are only two size steps.
Furthermore it is formed bythe interaction of two intervals. One is an interval that is
superimposed over and over again called the ‘generator’ and another is
what is referred to as the ‘interval of equivalence’, most commonly,
but not limited to the octave. Whenever a superposition of ‘generator’
exceeds the ‘interval of equivalence’ it is transposed within its
compass. The most common example is the Pythagorean series where we
have a generator of a 3/2, a fifth, which is superimposed and placed
within the octave, the interval of equivalence. But each time we add a
note we do not arrive at a Moments of Symmetry scale as a result, only
when we have a certain number of tones do we find the symmetry in
question. The five-tone and seven-tone series are the most common
examples. These two have the defining property of a Moments of Symmetry
scale, that between successive steps we find two and only two size
steps. Wilson in his paper refers to these steps as large (L) and small
(s). To be an MOS though it must be true that every number of steps
will have two steps sizes,one large and one small size. Some readers will immediately
object
that we are dealing with a very narrow band of scales. Toward the end
of this paper I will show how Moments of Symmetry exert an influence on
scales Wilson calls ‘constant structures’, but I am getting ahead of
myself.
It is more that worth it by following Wilson in his showing how far we can go. So my goal here is to guide the
reader through this somewhat informal letter to John Chalmers.
THE LETTER. (http://anaphoria.com/mos.PDF)
Wilson’s first example of an MOS (an abbreviated form that Wilson finds
“unpoetic” but is widely used enough that one is forced to include it)
is found within 12-tone equal temperament. As stated, an MOS scale can
be recognized as having only two size steps. In the diagram on page 1,
the numbers along the top represent the 12 tone steps or units, and the
numbers below represent the number of ‘fourths’ or the 5 unit intervals
(e.g. unit equals steps as in 5 semitone steps in a fourth…) one has to
superimpose to find that position within 12. Wilson starts with 0
because at the starting point we have not yet proceeded up any fourths.
The pentatonic is formed by beginning at a starting point and going up a succession of fourths.
A word about the disjunction, which we find at the
end of the chain and atypical in size, will subtend (in between) the
same number of steps or units as the generator bringing us back to our
starting point, e.g…. Wilson points out that the disjunction functions
‘melodically’ within the context of the scale. Except in exceptional
cases, the disjunction interval will not exceed nor be smaller than
those intervals formed of larger or smaller number of steps. E.g.
In the middle of page two Wilson takes the 7 tone
MOS and looks at all the 5 tone subsets of it by treating the 7-tone
scale in the same fashion he treated the12-tone scale to create ‘binary
depth’. The upper set of numbers shows the chain of fourths. The lower
set, the seven pentatonics formed by taking two steps at a time. He
then octave reduces these in order to show the types ‘trichords’ that
are formed. A trichord being analogous to a tetrachord except we have
only two tones instead of three found before we reach each fourth.
On Page 3 Wilson shows the various trichords found in these
pentatonics.
On pages 4 and 5 he shows all the moments of symmetry using different generators.
On page 6 he again shows the idea of “binary depth” by taking 17 units at a time of the 41-tone scale which gives us a 17-tone MOS. He does something not uncommon but easy t ocause confusion here in that the generator goes up 12 places and down 4 notated as a minus sign. So be should realize that 10-2 should be read as two numbers one below the 6 and the other below the 7. His choice here is a musical one wanting to start at this location of the chain which becomes clearer when we compare it to the next two pages. Then he shows various 7-tone sub-MOS scales found taking 5 units at a time of this 17-tone scale.
Page 7-8 shows a Just Intonation interpretation of the MOS patterns on the previous page with the revised version first in order to make it easier to see it relation to the previous page.
At the top of page 9, Wilson illustrates how a generator that is 3/7ths of an octave and in the process of getting smaller passes through 12 -, 5 -, 13 -, and 8 ET . Wilson’s use of Greek letters was in keeping with Yasser, whose work was his starting point in investigating different tunings. Yasser’s work was well known at the time of his writing. . He recognizes that one could pick a scale formed at any point along this continuum, but the chart shows how at these points where it coincides with an ET it very nature can and does change. He has a chart illustrating the continuum between 0/1 and 1/2 here http://anaphoria.com/line.PDF
On page 9-10 Wilson uses annotation which I have not common elsewhere. He
explains on the bottom of page 12 the small “e” standing for equal, in
this 13 ET. The small numerical superscripts indicating how many units
he is counting to form the larger number of actual steps to the scale.
In the first example 3 units forms an 8-tone scale. The three units are
taken out, notated by the sign ‘)’. The next example might be easier to
read backward. One takes a generator (5) from 13 ET that when one has an 8-tone MOS the generator is now 3 units, then illustrates how this same generator will form a 5 tone sub-MOS scales whose generator is 2 units. Thus Wilson shows the possibility of taking
“binary depth” one-step further. Possibly one could refer this to
“trinary depth”, but Wilson does not introduce this term.
At the bottom of page 10 Wilson hints toward the Tanabe Cycle he puts forth on page 13. He does touch on one thing I have found quite useful in working with different scales, and I will expand upon it for a bit before continuing: He points out how the complementary set to the pentatonics and other MOS scales often form viable 7-tone scales. He has acknowledged this in other correspondence, and even earlier in this letter he points out a few of the complementary sets when illustrating the MOS scales found in 17. In a private communication to me, he lists 10 different 7-tone scales found in Xenharmonikon 9, The Marwa permutations, fig 1e. Page 3 (http://anaphoria.com/xen9mar.PDF). This set of permutations of 20 scales can be reduced to 11 different scales and their modes. Further he shows the complementary 5 tone scales. One of my own processes has been to take each of these 7-tone scales and to extract the cycle of pentatonics in the same way he illustrates here and in The Tanabe Cycle. These in turn produce some novel pentatonics, which in turn would generate other 7 tone scales. I have yet to extend this process out far enough to see where it breaks down for my own use. It is an area worth exploring. Page 10 actually corresponds to these 11 7-tone scale transposed.
Pages 12 shows a 13-tone keyboard that might be Wilson
having some fun. The joke being that if on retains the names
of fourths one finds that E and B are now higher than F and C.
Page 13 brings us to The Tanabe Cycle, which shows a
“historical” use of the MOS idea in Japanese music. It appears to be the fountainhead in which Wilson observed and noticed the
underlining pattern. So he is giving credit where he feels credit is
due.
Page 14 shows the variety of scales within the 7 tone scale by placing them on a single tonic. The letter names on the left shows where the scales could be found. It also shows the common tone modulations of the 7-tone scale in a
cycle of fourths, which results in 13 tones that one, is more likely to
hear as a 12-tone scale with a 'comma' inflection.
Pages 15 shows the 5 tone sub MOS of 7 tones scale and a possible 'Kornerup' type version of the scale. Kornerup envisioned the yasserian sequence as converging toward these relationships of large to small intervals.
Pages 16-17 clarified how every size steps will have the same properties ofonly two sizes (L and s)
FURTHER DEVELOPMENTS BY WILSON BEYOND THIS LETTER.
Many of Wilson’s later papers include various zigzag patterns that are
illustrations of the Moments-of-Symmetry of particular intervals, and
these can be the most bewildering of his illustrations for those who
have not had the benefit of personal communication. I don’t
understand exactly how this formula works and why. But this is
what you do:
Wilson refers these to 1/x patterns but some scientific calculators
have a x-1 button that means the same thing. First you start with the
log2 of your interval. If you don’t know how to figure out the log
based 2 of an interval, the formula is log (A/B)/log (2). To find
the cents, in case you don’t know, one multiplies the log2 of an
interval by 1200. Wilson personally prefers thinking of intervals in
terms of their log based 2 (as opposed to cents).
Let us take the 5/4 as an example to find what MOS scales it generates.
The log2 is .3219280949. Next we find the 1/x (or x-1) of this
interval, then subtract the number left of the decimal point before
repeating the 1/x again, always subtracting the numbers left of the
decimal point. In the case of 5/4, this gives us
In the case of 5/4 this gives us
3.106..
9.408..
2.446..
2.2405..
4.156
6.391..
Now we use the whole numbers as a way of finding the moments of
symmetries by the process Erv calls “freshman sums” because it is the
‘wrong’ way to add fractions where you add the numerators together and
then the denominators together. We always start between 0/1 and 1/1 and
we zigzag our freshman summing starting at 1/1 and moving to the
zigging to the left however many steps we have as our first whole
number. Starting with 1/1, we add 0/1 and likewise with the answer
until we have moved 3 times finding ourselves at 1/4
0/1
1/1
1/4 1/3 1/2
Next we are going to zag to the right 9 times, each time toward a new
medient between 1/4 and 1/3, by adding 1/3 to each new answer until we
have proceeded 9 steps.
This gives us the following sequence
1/4
1/3
2/7 3/10 4/13 5/16 6/19 7/22 8/25 9/28 10/31
Next step we would zig to the left, then zag to the right as far as is useful to us.
Let me explain what this series is telling us. The denominator tells us
how many tones in the scale and the numerator how many units the
generator is in size. This gives us a chain of scales after the first
1, 2, 3 and 4 tone scales to a more viable 7-tone scale where a
generator is 2 units, to a 31-tone scale when it is now 10 units.
This lead to Wilson making what one might see as the mother of all
zigzag patterns he calls “The Scale Tree"
(http://anaphoria.com/sctree.PDF). Later it was found to have been
already discovered and called the “Stern-Brocot tree”. Since we are
working in music it seems to call it in the way it is functioning in
the musical world, but giving credit where it is due, first to those
who saw it as a mathematical abstract and the other seeing it as a
guide in which to place any scale with a generator of any size. It is
interesting that the way it came about historically was by people
involved in calculating gear ratios for clocks. So in a way it was
related to the division of time into arithmetical harmonic medients.
CONSTANT STRUCTURES [revised 17/1/2012]
Wilson has also noticed the influence of MOS patterns on scales of
higher limits (such as 5 and beyond) that cannot be explained as being
generated linearly by a single generator. This led to another term he
calls “Constant Structures”. These are defined simply as, “A tuning
system where each interval occurs always subtended by the same number
of steps. (That is all, no other restrictions). An example of this
would be say a 5 limit 12 tone tuning were the 5/4 would always be
subtended by 4 steps. While Constant structures are independent of MOS, they more often than not will be informed by the large/small patterns of MOS scales of the same number. It is for this reason that Wilson has referred to the MOS as being archetpal patterns that scales of more than one limit will fill accordingly. In this light, it is possible to look at Constant Structures also as a chain of a variable generator within the range a given limit imposes on the chain. These are easy to follow when we map such a scale to a generalized keyboard as seen in the examples below. With recurrent sequences, this becomes difficult in that if done as a numerical sequence no two intervals might ever repeat. In seeing the generator as possibly varied in size, we can still map such scales to a generalized keyboard, treating it as a representation of the golden/noble ratio these sequences converge on.
REFERENCES
A large collection of constant structures can be seen here just to cite a few.
http://anaphoria.com/xen3b.PDF
http://anaphoria.com/xen9mar.PDF
http://anaphoria.com/xen10pur.PDF
Carl Lumma made a PDF file of Rothenberg’s three main papers:
http://lumma.org/tuning/rothenberg/AModelForPatternPerception.pdf
The citations for these papers are:
Rothenberg, David.
"A Model for Pattern Perception with Musical Applications.
Part I: Pitch Structures as Order-Preserving Maps",
Mathematical Systems Theory vol. 11, 1978, pp. 199-234.
Rothenberg, David.
"A Model for Pattern Perception with Musical Applications
Part II: The Information Content of Pitch structures",
Mathematical Systems Theory vol. 11, 1978, pp. 353-372.
Rothenberg, David.
"A Model for Pattern Perception with Musical Applications
Part III: The Graph Embedding of Pitch Structures",
Mathematical Systems Theory vol. 12, 1978, pp. 73-101.
A special thanks to Adam Reese, Carl Lumma, and Terumi Narushima for help with article.
Any Errors though are mine