AN INTRODUCTION TO THE SCALES OF MT MERU

AND OTHER RECURRENT SEQUENCE SCALES

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

http://anaphoria.com/wilsonmeru.html

Like so many of Erv’s microtonal tuning developments, we find ourselves looking again at Pascal’s Triangle, the fountain that he continuely returned to. Here Erv refers to it by its ancient Indian name of Mt.Meru in reference to Meru Prastara, having become aware of its 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 great reverence.

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, Erv 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 Erv appears to be first to discover these other diagonals and
their
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 thought that someone there or India must have written about it. While the search still goes on for earlier investigations of these
diagonals, until then, we give him credit for his discovery. Since the release of these papers, others sources have appeared, but so far represent work after him. More
importantly for our concern, Wilson noticed that if treated as
harmonics, or subharmonics for that matter, these series produce scales
with in a unique and newprocess to any before.

Looking at the recurrent sequences generated by Mt. Meru. (http://www.anaphoria.com/meru.PDF).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. images are cropped from orininal to make viewing easier]

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). 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.

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 1.4655712318.

The next diagonal Hn=Hn-2+Hn-3, is a scale he likes to calls '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 contained 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 1, which is also We see
in the sequence 1/1, 1/2 , 2/3, 3/4 and 5/7 which is
basically you first scale of 7 notes of which our generating
interval or sequence will be 4 steps in this scale. The next would be a
10-tone scale in which the generator is 7 steps etc. ou might noticed that 23 is underlined. Wilson does this notation ot indicate it closeness to an ET.

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 grows into a plant the seed
disappears or is discarded, 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 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.

There is an extended mapping of EXTENDED DIAGONALS AND VARIOUS PRIMARY AND SECONDARY RECURRENT SERIES which extends the series

here http://anaphoria.com/secondaryrecurrents.html

~Comments and Questions welcome~