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

http://anaphoria.com/wilsonmeru.html

Like so many of Erv’s microtonal 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 in reference to Meru Prastara, having become aware of its older history under this name. The other 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
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 1.4655712318.

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.

Thereis 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~