AN INTRODUCTION TO THE MOMENTS OF SYMMETRY
By Kraig Grady (Cultural Liaison-North American Embassy of Anaphoria Island) 6/17/2007 [revised 10/2012] all rights reserved.
~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)
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 also includes Secondary or Sub-moment MOS patterns. Here a smaller MOS found embedded within a larger MOS is rotated freelyby treating the step sizes in the larger MOS as if they are equivalent. His example is the Japanese pentatonics shown to him by Tanabe. Starting with the 7 tone Pythagorean MOS, the 5 note MOS [small small large small large pattern] is rotated through the 7 possible formations possible . This is illustrated in The Tanabe Cycle. He often referred to this as what directed to his concept.
A constant structure (commonly of a multi-limit tuning) is a scale where every occurrence of a ratio will always be the same number of steps or units. This term he added later to differentiate from the strict MOS formed of a consistent generator size.
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. While he understood possibly the influence of timbre, he discarded as not effecting his line of pursuit
Being that we have many cultures whose music is melodically based, Wilson posed a question. What makes people gravitate toward certain types of scales like a 5 or 7 tone ones and why we find so few of others say that are six or eight. His concept of Moments of Symmetry satisfied many of these examples not included in other scale building practices such as Tetrachordal scales, or those derived from subharmonic scales. It can though overlaps these to a large degree 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 have been felt quite intuitively, both by individually and culturally even in higher limit tunings. If his approach is conservative in being careful to remain on a firm and well-established foundations, the results are not.
While there are scales that fall
outside these boundaries, and Erv explores some of these elsewhere, the real importance lies with the fact that
those which do, are heard as conheasive scales that hold together. 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.
While possibly more mathematically defined, some have found Rothenberg’s concept of propriety more useful for that reason. I include references at the end for the reader’s own comparison. They do diverge most noticeably with Wilson’s illustration of Secondary MOS patterns discussed later. The importance of this should not be underestimated for they represent what Wilson considers often the more interesting scale formations. Erv was not convinced of Propriety in that he was able to show Chalmers an improper scale that showed no ambiguity in the fourth being different than the fifth or third which according to his theory should have been there.
Wilson sees the role of a theorist as someone that makes music makers
aware of the potentials of given material or constructs which might be useful to the actual artist who makes music. It is for this reason he prefers to present his ideas on a level intelligable to that audience. His work, if any crititism can be said to be too brief for this form, but it was always intended to be accompany verbal descriptions.
WHAT ARE MOMENTS OF SYMMETRY?
The cliff note version above covers it but will explain more fully. A Moment of Symmetry is a scale where there are only two size steps. It is formed by the interaction of two intervals. The first is what is reffered as the 'generator' being an interval that is superimposed over and overin a chain with the second referred to as the ‘interval of equivalence’, most commonly ( but not limited to) the octave. Whenever a superposition of a ‘generator’ exceeds this ‘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.
It is more that worth it by following Wilson in his showing how far and where he takes this concept. 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 we succumb since it so widely used) is shown within 12-tone equal temperament. 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 o cause 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 of only 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
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
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
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 4/4/2012]
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 archetypal 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.
The concept of a Constant Structure is in many ways superior to identifying these structures as having 'multiple generators' or of different "ranks".
First it can be observed historically that the constant structures we have as examples, they have been thought of as variations of MOS scales like the Pythagorean. Constant structures preserves this conception and it is important to note that they also have a tendency to imitate with subtle variations the Large and small patterns of MOS scales which act as their prototype.
Second in observing the occurrence of the new intervals resulted from adding other limits, these new intervals do not result in other chains of repeated intervals, and when they do they rarely form an cycle which makes the term 'generator' most unprohibited.
There are examples of multiple generators where complete chains are repeated. Such as Margo Schulter's use of two Pythagorean chains a seven limit apart. In these instances the term is appropriate
A large collection of constant structures can be seen here just to cite a few.
Carl Lumma made a PDF file of Rothenberg’s three main papers:
A special thanks to Adam Reese, Carl Lumma, and Terumi Narushima for help with article.
Any Errors though are mine
~Comments and Questions welcome~