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
There is no figure more than Pascal’s Triangle that mysteriously reappeared to Wilson in developing his various tuning developments. It was Paul Beaver who had pointed out to him years ago, that the Chinese have been pouring over Pascal’s Triangle for thousands of years. Here Wilson is showing preference to its ancient Indian name, Mt. Meru, in reference to Meru Prastara along with its uncanny resemblance to scales found in the Far East, Indonesia and even Africa. These are also the places where the people also hold the legendary mountains of this name in great reverence.
The discovery of this family of scales began via Lou Harrison, who brought to his attention the Thomas M Green article (Mathematics Magazine Vol. 41 1968), that illustrated how the sum of the simplest diagonal of Mt Meru (Pascal’s Triangle) results in the Fibonacci series and that the sums of the other diagonals likewise generated other recurrent sequences, each with its own convergence.
Wilson appears to be first to investigate these other diagonals and
their recurrent sequences musically, he stated that he found 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. Regardless of their history, musical appications when treated as
harmonics, or subharmonics produce scales
via a unique and new process.
Meru #1 of Mt. Meru (http://www.anaphoria.com/meru.pdf) , coresponds with Thomas M Green's diagonal. The series of numbers on the right is the sum of the numbers of the diagonal path but 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 through Mt Meru, in which the sums of the rows are indicated to the side of the Meru Triangle. These sums in turn also form 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 he would latter call Meta-Pelog.
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 like the one above. It shows the various Moments of Symmetries (MOS) produced by the convergent ratio. For those not familiar to these and the zigzag patterns, see this here. 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. This works for both the converged and using the numeric series also if one starts ones series not at the beginning but when it starts to converge. This will be explained later. It also as a guide for placing the scales on a generalized keyboard. In the above diagram, 3/7 refers to a 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 ! or show the degree of error to the side to indicate an MOS that is close to an ET, or underline but this is less common. These zig-zag patterns should be understood as specific locations upon the Scaletree taken out to further levels illustrated there.
It is worth highlighting what properties make these scales 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 a self contained acoustical and
perceivable structure. The proportional triad is found in the sum
triplet by 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 the two lower are numbers are raised an octave, we have 18:21:24 all
separated by 3, hence an equal beating triad. This 3 is found in our recurrent sequence. Further up we
have 86+114 = 200 which for the sake of simplicity place the top number an
octave lower to get 86-100-114, which reduces to 43:50:57 with 7 as a sucessive different tone and also a member of our series. Further
properties of the sum triplet to the proportional triad was developed by Wilson in
a later papers.
The choice is where do we start our scale to presents us with a musical and artistic choice as opposed to strictly mathematical results. The beginning seed more often falls short of giving us a real cue as to where we will end up when it is converged. Wilson thought of it like when a seed grows into a plant the seed disappears or is discarded in the sprout or shoot, and so it is when the case we form scales using these numerical 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 self known. 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. A good guide would be if the harmonics fit well into the same keyboard pattern as the convergd generator. Even within a single series seeded by the same numbers, different people can 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. Others might have quite different and maybe even opposing experiences so it really wide open.
Jacques Dudon independently and concurrently worked and developed many of these series. He used a method of finding what he calls the P series. This can be found by multiplying the converged generator by the set of ascending whole numbers at least up to the number of seed numbers needed and rounding out the results.
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. It should be understood that each 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. Wilson explored seeding Mt. Meru itself with outher numbers besides the classic ones elsewhere.
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~