AN INTRODUCTION TO THE MOMENTS OF SYMMETRY -WILSON'S THEORIES FOR GENERATING MICROTONAL SCALES OF ANY SIZE INTERVAL
By Kraig Grady (Cultural Liaison-North American Embassy of Anaphoria Island) 6/17/2007 [revised 10/2014] all rights reserved.
~Comments and Questions welcome~
Let it be known that these archives have no relation to the Xenharmonic Alliance or any of its offshoots. While terms found herein have been borrowed, used, or even undermined by others, these archives are essential for fostering the vision of Wilson's original work and thereby deflecting misinterpretations.
Link to Wilsonic app which includes interactive MOS exploration --- Link to Stiltner's MOS generator for equal divisions
a bit of Wilson's humor
Definition of Moment of Symmetry CLIFF NOTE DEFINITION
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
2. An Interval of Equivalence commonly called a period (of any size, for example most commonly an octave).
3. A Moment of Symmetry is formed where each scale degree or scale unit size will be represented by no more than two sizes and two sizes only (Large = L and small = s).
4.The relative number of L and s intervals is coprime, i.e. they share no common factors other than 1.
5. The numerator (generator) and denominator (period) representing MOS are also co-prime.
[3a.] Wilson expands the process to create Secondary or Sub-moment MOS patterns. Following the example of cycles of Japanese pentatonics derived from the 7 tone Pythagorean chain (an MOS scale) shown to him by Tanabe, he generalizes the principle to include all MOS scales containing a smaller MOS embedded within it. While keeping the units of scale steps constant, one rotates the smaller MOS set around the parent MOS until the cycle is complete. The Tanabe Cycle is illustrated on page 13 of the document. He often referred to this secondary level as what directed him to his concept of MOS patterns
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 later used to differentiate from the strict MOS formed of a consistent generator size. A constant structure can be thought of as a 'MOS' with a variable generator size accommodating the similar size intervals found when using tunings with more than one harmonic limit.
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 the influence of timbre, he felt it had little influence in the actual making of a chain that started and stopped at certain points.
Being that we have many cultures whose music is melodically more than harmonically based, Wilson posed a question.
"What makes people gravitate toward certain types of scales like a 5 or 7 tone one and why do we find so few of others, say that are six or eight."
Moments of Symmetry satisfied many of these examples as an influence even if not strictly applied. On one level it is a very simple idea that gaps in a scale would be filled with similar or like intervals. Hence it is often done intuitively. The Moments of Symmetry appear to have been felt quite intuitively, both individually and culturally even in higher limit tunings. Both the scales of Partch and Novaro exhibit this intuitive application. More importantly, it offers us a gateway into an almost infinite family of new scales that the conditions have not been ripe to manifest.
Moments of Symmetry do not to attempt to explain all scales. The enharmonic and other tetrachordal scales are good examples of those that don't fit. What is important about them is that they always sound like cohesive scales. 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 Symmetry.
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 intelligible to that audience. His work, if any critctism can be said to be too brief for this form, but it was always intended to accompany verbal descriptions.
WHAT ARE MOMENTS OF SYMMETRY?
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 referred to as the 'generator', an interval that is superimposed over and over in a chain. The second is referred to as the ‘interval of equivalence’ , or period, 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 reduced
within the octave, the interval of equivalence. Each time we add a
note we do not arrive at a Moment of Symmetry scale as a result, only
when we have two and only two size
steps. Wilson in his paper refers to these steps as large (L) and small
(s). The five-tone and seven-tone series are the most common
examples of those that occur with the 3/2. To be an MOS though it must be true that every number of steps or units of the scale
will also only have two step sizes, one large and one small size.
THE LETTER. (http://anaphoria.com/mos.pdf)
The goal here is to guide the reader through this somewhat informal letter to John Chalmers. Wilson’s first example of an MOS (an abbreviated form that Wilson finds “unpoetic” but we succumb since it is so widely used) is shown within a 12-tone scale. 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.
After the 2 and 3 note scales the pentatonic is formed by going up a succession of 4 fourths above our beginning note.
At this point we return to our beginning note in what is called the disjunction. The disjunction is found at the end of the chain and a typical in size, but will be subtended (in between) by the same number of steps or units. This brings use cyclically back to our starting point.. Wilson states that the disjunction functions ‘melodically’ within the context of the scale even if of a different size. Except in exceptional cases, the disjunction interval will not exceed nor be smaller than those intervals formed of larger or smaller number of steps.
In the middle of Page 2 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 the 12-tone scale to create what he calls ‘binary depth’. After mapping the chain of fourths under the 12 units, he provides, 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 within the fourth.
On Page 3 Wilson shows the various character of trichords found in these pentatonics.
On Page 4 and 5 shows all the Moments of Symmetry of a 17 tone scale using different generators. He points out complementary sets as he found them musically useful.
On Page 6 the "depth" is extended even further. First taking 17 units at a time of the 41-tone scale he forms a 17-tone MOS (the 10-2 should be read as two numbers one below the 6 and the other below the 7). Then he takes 5 units of this 17 to form a 7 tone scale. Then he goes through all of the 7-tone sub-MOS scales found taking 5 units at a time of this 17-tone scale in the same way he went around the pentatonics in the 7 tone scale.
Page 7 and 8 shows a Just Intonation interpretation of the MOS patterns on page 6 with the revised version first repeating the numbers at the top while the next page shows how it articulate Helmholtz's 1/8 skisma temperament.
At the top of Page 9, Wilson illustrates a sequence in which a generator that is 3/7ths of an octave as it get progressively smaller will form MOS scales of process of getting smaller passes through 12 ET, 5ET, 13 ET, and finally 8 ET . Wilson at this point used Greek letters in keeping with Yasser, who was both well known and Wilson's starting point in investigating different tunings. Wilson recognized that one could pick a scale formed of a generator at any point along this continuum. 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 and 10 Wilson uses annotation unique to this paper. He explains on the bottom of page 12 the small “e” standing for equal, in this 13 ET. The small numerical superscripts indicating the generator is size of units with the large number the actual steps to the scale. In the first example an 8-tone scale made of a 3 unit generator [ the sign ‘ ) ' means 'taken out of' in this case 13]. The next example might be easier to read backward. One makes an 8-tone MOS out of a generator of 3 units of 13ET as before, then one then forms a 5 tone scale the generator is now 2 units (5) from 13 ET that when one has an , then illustrates how this same generator will form a 5 tone sub-MOS scales whose generator is 2 units of the 8 tone scale. Thus Wilson shows the possibility of having what one could refer to “trinary depth”, but Wilson does not introduce this term. How far can one go with this process? This is up to ones taste and sense of scale in the material one is using.
At the bottom of Page 10 Wilson returns to the pentatonic found on page 13 as the Tanabe Cycle. On Page 11 he lists without explanation 11 different 7-tone scales found later in Xenharmonikon 9, The Marwa permutations, fig 1e. Page 3 (http://anaphoria.com/xen9mar.pdf). Something the reader is free to pursuit there.
Page 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. This appears to be the fountainhead in which Wilson observed what Tanabe had shown him 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.
Page 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.
Page 16 and 17 clarify how every size steps will have the same properties of only two sizes (L and s)
FURTHER DEVELOPMENTS BY WILSON BEYOND THIS LETTER.
Zigzag or 1/x patterns
These Zigzag or 1/x patterns can be the most bewildering of his illustrations for those who have not had the benefit understanding their significance. Many of Wilson’s later papers include these various zigzag patterns which are illustrations of the Moments-of-Symmetry of a particular generator. He learned these from Larry Hanson.
I only know how to use the formula, not it background or logic, but this is what you do:
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).
Scientific calculators will have a 1/x button as in Wilson's case or some will have a x-1 button that means the same thing.
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
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 mediant 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
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.
Next step we would zig to the left, then zag to the right as far as is useful to us. Here is a further chart which contains the data of the next set of zigzags. Due to the limits of most hand held calculators, going past 10 or 12 steps is questionable as the calculator will round off and one can end up computing artefacts.
The series and branches of freshman sums lead to Wilson making what one might see as the mother of all
zigzag patterns he called “The Scale Tree"
(http://anaphoria.com/sctree.pdf). In the mid 90's it was discovered to already found and called the “Stern-Brocot tree”. Wilson nevertheless deserve credit for being the first one to see the it application to music both in terms of MOS scales but also in regard to keybaord mappings and horograms all subjects discussed elsewhere in the archives. He does one thing not found in the stern Brocot tree and that is to calculate the noble numbers that the different zigzags generate from the first 12 layers of the tree. 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 mediants.
Definition of a Constant Structure CLIFF NOTE DEFINITION
“A tuning system where each interval occurs always subtended by the same number of steps. (That is all, no other restrictions)"
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 gravitate to 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. These keyboards can be understood as two dimensional model of constant structures in that the same interval will always be laid out on the keyboard in the same way. With recurrent sequences, like the fibonacci series, 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, or cycles which makes the term 'generator' awkward.
There are examples of multiple generators where complete chains are repeated and are appropriate. Margo Schulter's use of two Pythagorean chains a seven limit apart is a good example of this.
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~