AN INTRODUCTION TO THE SCALES OF MT MERU
AND OTHER RECURRENT SEQUENCE SCALES

    Like so many of Erv’s tuning developments, we once again find ourselves looking at Pascal’s Triangle, that fountain that he continues to return to. In this case, Erv refers to it by its ancient name, Mt.Meru, having become aware of it around the time of discovering these scales. The other reason for the reference is that the scales often bear an uncanny resemblance to scales we find in the Far East and Indonesia, whose people hold this legendary mountain in great reverence.

    It was discovered by Thomas M Green (Mathematics Magazine Vol. 41 1968) that if one adds up the sums of the simplest diagonal of Mt Meru (Pascal’s Triangle), one produces the Fibonacci series. This prompted Erv Wilson to investigate the sums of the other diagonals which likewise generated other recurrent sequences of numbers, each with its own consistent pattern. While Erv appears to be first to look at these other diagonals and their recurrent sequences he has stated that he finds this hard to believe knowing what Paul Beaver had said to him years ago, That the Chinese have been pouring over Pascal’s Triangle for thousands of years. The search still goes on for earlier investigations of these diagonals, until then, we give him credit for his insight. More importantly for our concern, Wilson noticed that if treated as harmonics, or subharmonics for that matter, these series produce scales with interesting properties and possibilities. They also represent a unique and new form of scale building.
 
Perhaps we should take a moment to look at the some of the recurrent sequences generated by Mt. Meru.  (http://www.anaphoria.com/meruone.PDF)-. As stated, different 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 that are called recurrent sequences, which we will illustrate.  In Meru #1, we find that in the series of numbers, each number is the result of the two previous numbers as we mentioned above.

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 Hn=Hn-2+Hn-1, derived from a standard mathematical formula which is
Hn=Hn-x+Hn-y. Now if we look at Meru #2 the sequence 1,1,1,2,3,4,6,9,13,19,28 etc
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 1.46557123188 .
The next being Hn=Hn-2+Hn-3, a scale he likes to call Meta Slendro for its uncanny sound compared to this family of Indonesian scales.    
    It is always important to ask why one would use these scales and the answer lies in its unique acoustical properties. Each of these scales has the property of forming a series of equal beating or what is called proportional triads that in turn generates difference tones that occur in the scale or its seed. Thus these scales are constantly reinforcing themselves selves, making self contain acoustical and perceivable units. This proportional triad is found by taking 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 7+9 =16. If we raise the two lower ones an octave, we have 14,16,18 all separated by 2, hence an equal beating triad. If we look further up we see that 86+114 +200 and for the sake of simplicity lower the top an octave we get 86-100-114, which gives use 14, the octave of 7 which occurs as a tone in the scale series already. The properties of the sum triplet to the proportional triad is developed in a later paper.

In quite a few of these charts there follows sheets illustrating various zigzag patterns. These show the various Moments of Symmetries each scale produces. For those not familiar to these, one should look at http://anaphoria.com/wilsonintroMOS.html . Breifly the bottom number being the total number of scale steps while the top being the number of scale units the generating interval is inrelation to it. The math that this involves is explained in the above link. If we look at Meru 3, which is also called and referred to as Meta-Slendro, We see in the sequence which after 1/1, 1/2 and 1/3 we have 2/5 which is basically you first real scale of 5 notes of which our generating interval or sequence will be 2 steps in this scale. The next would be a 7-tone scale in which the generator is 4 steps etc.

At this point though we must step outside of mathematics because it has taken us as far as it can and thank it for what it has provided us. Now we confronted with musical and artistic choices as opposed to mathematical choices.

    Just as when a seed becomes a plant the seed disappears or shell discarded, so it is when we form scales using these sequences. 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. This point is not easy to define, although if we look at the Moment of Symmetry patterns, we will find often the seed will not fit into this pattern as easily. Starting past this point we are given choices, which depend on ones preferences and ones musical sensibilities. 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.

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 7 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. In effect a different ‘Slendro’ on each of 7 steps. The other extreme 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 seed that Erv is quite fond of is one that starts 10,13,17,23,30,40,53,70,93, etc. The real importance of this method it allows a method 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 family 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.