AN INTRODUCTION TO THE SCALES OF MT MERU
AND OTHER RECURRENT SEQUENCE SCALES
~Comments and Questions welcome~
This page is an introduction to the series of scales found in this section of the Wilson Archives
Like so many of Wilson’s microtonal tuning developments, we find Pascal’s Triangle in front of us. This is a figure he continuely found reappearing in his work in different contexts. Here it is referred to by its ancient Indian name of Mt. Meru in reference to Meru Prastara. Wilson had become aware of this triangle's older history in this area of the world before Pascal's rediscovery. Another reason for the reference is that the scales often bear an uncanny resemblance to scales we find in the Far East, Indonesia and even Africa, whose people hold the legendary mountains of this name in both locales in great reverence.
In began in discovering the Thomas M Green article (Mathematics Magazine Vol. 41 1968) that illustrated how the sums of the simplest diagonal of Mt Meru (Pascal’s Triangle) results in a the Fibonacci series. Wilson was prompted to investigate the sums of the other diagonals which seemed to go have gone unnoticed. These he found also generated other recurrent sequences of numbers, each with its own convergence like the former with an overall pattern of organization.
While Wilson appears to be first to discover these other diagonals and
recurrent sequences, he has stated that he finds this
hard to believe. Paul Beaver had pointed out to him years ago,
that the Chinese have been pouring over Pascal’s Triangle for thousands
of years and thus Wilson thought that someone there or India must have investigated these diagonals also. While the search still goes on for that type of historical evidence, until then, we give him credit for this discovery. While a few other have since noticed these they appeared after him, they remain absent of their musical appications that if treated as
harmonics, or subharmonics for that matter, produce scales
by a unique and new process to any before.
Looking at the recurrent sequences generated by the diagonal pathways of Meru #1 of Mt. Meru (http://www.anaphoria.com/meru.pdf) , we find that the series of numbers on the right is the sum of the numbers of the diagonal path but also in this case are also the sum of the two previous numbers.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,etc. being the result of : 1+1 =2, 1+2=3, 2+3=5, 3+5=8 etc. our classic Fibonacci series first pointed out by Leonardo of Pisa. Why this series is interesting is that the proportion of the two adjacent terms slowly begins to converge on a particular size. In this case one called the Golden Mean or Golden Proportion, which is 1.61803398875.
This series can be notated in a variety of ways. One is simply as A+B=C or as we see on our sheet toward the upper left Hn=Hn-2+Hn-1,(the standard mathematical formula which is Hn=Hn-x+Hn-y). Wilson drew diagonals are drawn through Mt Meru, in which the sums of the rows are indicated more often than not, to the right of the Meru. These sums in turn form patterns are known as recurrent sequences, which we will illustrate.
Now if we look at Meru #2 the sequence
We find the recurrent sequence requires 3 terms to start (which Erv calls the seed) in which we add the first and third together (Hn=Hn-3+Hn-1) or (A+C=D) as stated. In example starting with 1,1,1: 1+1=2, 1+2=3, 1+3=4, 2+4=6, 3+6=9 etc.
This gives us like before a series which converges on a particular proportion. This time 1.4655712318.
This next diagonal Hn=Hn-2+Hn-3, (or A + B = D) is a scale he called 'Meta-Slendro' for its uncanny sound compared to this family of Indonesian scales.
These charts are followed by illustrations of various zigzag patterns. These show the various Moments of Symmetries (MOS) each scale can produce. For those not familiar to these and the zigzag patterns, see http://anaphoria.com/wilsonintroMOS.html#zig. To recap them the bottom number is the total number of scale steps while the top being the number of scale units the generating interval is in relation to it. The math that this involves is explained in the above link. If we look at Meru 3, we see in the sequence 1/1, 1/2 , 1/3, 2/5 and 3/7 which is basically you first scale of 7 notes of which our generating interval or sequence will be 3 steps in this scale. The next would be a 12-tone scale in which the generator is 5 steps etc. Often Wilson will use a ! to indicate an MOS that is close to an ET, or underline but this is less common. These also should be understood as specific often isolated locations with the Scaletree if taken out to that many levels.
It is important to highlight what properties these scales have make possible that is unique. Harmonically they have the property of forming a series of proportional (or equal
beating triads) that in turn
generates difference tones
that occur in the scale or its seed. Thus these scales are constantly
reinforcing themselves, making self contained acoustical and
perceivable units. This proportional triad is in the sum
triplet and placing the top tone in between the two numbers we used to
generate it. For instance in this latter sequence (Meru #3) we have 9+12 = 21. If we raise the two lower ones an octave, we have 18-21-24 all
separated by 3, hence an equal beating triad. Further up we
see have 86+114 = 200 which for the sake of simplicity place the top number an
octave lower to get 86-100-114, which gives use 14, the octave of 7 which occurs also as a tone in the scale series already. Further
properties of the sum triplet to the proportional triad is developed by Wilson in
a later paper.
Now we confronted with musical and artistic choices as opposed to strictly mathematical choices. The choice is where do we start our scale to use. Depending on how many steps our recurrent sequence spans, it is only after so many steps will we truly feel the individual quality of the sequence beginning to make its will known. Looking at the scales that the converging ratio forms in the zig zag patterns we might find that the early numbers will not fit into the pattern easily. While we could easily wait till the series converges to the point where the differences between successive numbers are small, often the most musically interesting parts of these series lies before this area, when the sequence is in the ball park so to speak and has not completely narrowed to a small fluctuations. So even with a single series seeded by the same numbers, different people can and will choose different starting places. I would recommend starting low and proceeding upward. More often than not, I end up settling a bit higher than where I start. Although I imagine others might have quite different and maybe even opposing experiences. It really is wide open.
Wilson thought of it like when a seed grows into a plant the seed disappears or is discarded in the sporut or shoot, and so it is when the case we form scales using these numerical sequences.
Let us take our last example Meru #3 Meta Slendro as an example of applying the above. It is this scale that I have most investigated, building an orchestra of instruments as well of composing numerous pieces and shadow plays around. I began my tuning on 9 and proceed up to 200, which produces 12 different pitches and the type of variety I sought to enable each 5 tone pentatonic to give me a different intervallic variation. Some instruments now take the scale to 17 tones with 816 being the highest harmonics. In effect a different ‘Slendro’ occurs on each of steps. The other possibility would be to take the interval the infinite series converges on which Erv includes on each page. In this case 1.32471795725…. which one would want to convert to Log based 2 (sometimes included). The advantage to this method is that the subharmonic and harmonic versions are the same and the basic triad produce by the scale will be usable in both forms, the disadvantage being a lack in variation in scale shape. Traditionally people seem to prefer unequal size steps in their scales to equal ones.
The numerical seed also offers a myriad of variations as one can seed these recurrent series how one wishes. Another meru 3 seed that Wilson was quite fond of is starts with 10,13,17,23,30,40,53,70,93, etc. One can also seed the triangle with different numbers which can be found here.The real importance of this method it allows for each individual to make their own Slendro, in much the same way a different village might do so in Indonesia. This makes it difficult to put down in stone, or in Scala, the most definitive version of the scale that each produces even though the latter contains these classic ones. It should be understood that these are only representatives of a whole family of scales. Each slant likewise should be understood as a genus of scales, open to such seeding.
Also there are various recurrent sequences not found on the tree but that come about due to specific problems or situations. Such are Meta Meantone and Meta Mavila, named after the Chopi Village that has a tuning from which this is derived discussed in a latter paper.
There is an extended mapping of EXTENDED DIAGONALS AND VARIOUS PRIMARY AND SECONDARY RECURRENT SERIES which extends the series
~Comments and Questions welcome~