Wilson's MOS

Kraig,

I was too quick in my point, but I didn't mean to ascribe to Wilson the
point that the fifth is special as a generator. This is an idea that
interests me particularly because it accounts for a large portion of the
world's scales if it is in fact the case. Even improper scales such as the
pelog type scale you mention are usually derived from fifth cycles (ie are
subsets of a diatonic or some such scale). I say all this allowing some
intonational margin in what I would call a fifth. Accually, my original
idea may be wrong because, as I have been looking closer at this point I
find that slightly flat fifths work better as generators than more
acoustically correct fifths. Wilson's attractive method of showing how the
various intervals generate scales pointed out to me that the fact that
certain intervals generate good scales in Rothenberg's sense after fewer
repetitions than others depends in a regular way upon the rough amount of
the octave which that interval takes up, and that the best always seem to
be those in the vicinity of the fifth.

You said:

>Wilson also explain very basic bedrock
>scales that are omitted in Rothenbergs case such as the pentatonics formed
out of the
>diatonics. That it includes such scales as f a b c e. That these exist
historically cannot be
>overlooked. Another problem with the Rothenberg (out of just being not
understandable to the
>average musician) is those scales which he thinks are of importance like
the whole tone (and
>the other symmetrical scales shows) have had limited historical use. His
aim is close but
>misses.

I don't agree that Rothenberg overlooks historical impromper scales. In
fact, he has a lot to say about them, in particular about the Indonesian
Pelog scale and ragas based on improper scales. I don't think his system
should be interpreted a la Balzano as placing a value judgement on scales,
where proper scales are judged as better. The model is explainatory; it
says that proper scales imply a certain style of music for which improper
scales would be inappropriate, and it explains certain stylistic features
of musics using improper scales. The fact that a body of music exists
(albeit small in perspective) in which composers grappled with the problems
of symmetrical scales is interesting in the applicability of his model to
the style which those composers developped. The implication is definitely
not to favor those scales as musical material.
I agree with you, though, that R.'s work unfortunately isn't
understandable to the average musician. I feel this is more a matter of
presentation: the average musician hasn't read the principia mathematica.
However, the concepts involved, I believe, are intimately and uniquely
understood by musicians.

jason

>This is an idea that interests me particularly because it accounts for a
>large portion of the world's scales if it is in fact the case.

I think we can explain this by the highly-consonant nature of the 3/2,
the fact that it occurs early in the harmonic series, etc.

>I don't agree that Rothenberg overlooks historical impromper scales. In
>fact, he has a lot to say about them, in particular about the Indonesian
>Pelog scale and ragas based on improper scales. I don't think his system
>should be interpreted a la Balzano as placing a value judgement on scales,
>where proper scales are judged as better. The model is explainatory; it
>says that proper scales imply a certain style of music for which improper
>scales would be inappropriate, and it explains certain stylistic features
>of musics using improper scales. The fact that a body of music exists
>(albeit small in perspective) in which composers grappled with the problems
>of symmetrical scales is interesting in the applicability of his model to
>the style which those composers developped. The implication is definitely
>not to favor those scales as musical material.

Well written.

-Carl

Paul,

You may be right here. I don't know exactly how much the elaboration of
"better" in my previous post restricts the range of the generator, or
whether there is some theoretically best fifth:

>All kinds of fifths can work well as generators for various purposes (see
>Dave Keenan's work for more on this). What do you mean by "better"?
>Certainly for 5-limit diatonicism, you're correct -- but would you not admit
>other possibilities?

Prime limits definitely aren't a consideration. Diatonicism (as in III =
4V) may be a consideration, but the point is that diatonicism should be a
result and not a premise. I think I explained the meaning of "better" in
my previous post. It makes assumptions about the musical purposes that the
scale must fulfill. Any musical system can be undermined by a shift in
aesthetic goals. The inclusiveness of my assumptions is up for debate.

>Aha -- I think I see what you're getting at. I think you're mistaken. Try,
>for example, a generator which is 2^phi or 741.64 cents. It produces more
>evenly-spaced scales more often than any other generator (in a certain
>mathematical sense). This will get into a discussion of noble numbers, etc.

Yes, I was wrong to imply that approximations to the ratio 3/2 had
anything to do with the propriety of the generated scales, but, of course,
2^phi does fall into the vinicity of the 5th in common sense terms (the
interval of the fifth note of a diatonic scale), although its far on the
sharp side, and doesn't generate scales resembling the diatonic. I'm
interested in looking up some of the things you cite: where can I find
David Keenan's work and the mathematical definition of 2^phi?

jason

--- In tuning@egroups.com, Jason_Yust <jason_yust@b...> wrote:
> Paul,
>
> You may be right here. I don't know exactly how much the
elaboration of
> "better" in my previous post restricts the range of the generator,
or
> whether there is some theoretically best fifth:
>
> >All kinds of fifths can work well as generators for various
purposes (see
> >Dave Keenan's work for more on this). What do you mean by "better"?
> >Certainly for 5-limit diatonicism, you're correct -- but would you
not admit
> >other possibilities?
>
> Prime limits definitely aren't a consideration.

No, I meant an odd-limit of 5.

> Diatonicism (as in III =
> 4V) may be a consideration,

That's what I meant by 5-limit diatonicism -- as opposed, for
example,
to the 3-limit diatonicism of the Gothic era.

but the point is that diatonicism should be a
> result and not a premise. I think I explained the meaning of
"better" in
> my previous post. It makes assumptions about the musical purposes
that the
> scale must fulfill. Any musical system can be undermined by a
shift in
> aesthetic goals. The inclusiveness of my assumptions is up for
debate.
>
> >Aha -- I think I see what you're getting at. I think you're
mistaken. Try,
> >for example, a generator which is 2^phi or 741.64 cents. It
produces more
> >evenly-spaced scales more often than any other generator (in a
certain
> >mathematical sense). This will get into a discussion of noble
numbers, etc.
>
> Yes, I was wrong to imply that approximations to the ratio 3/2 had
> anything to do with the propriety of the generated scales, but, of
course,
> 2^phi does fall into the vinicity of the 5th in common sense terms
(the
> interval of the fifth note of a diatonic scale), although its far
on the
> sharp side, and doesn't generate scales resembling the diatonic.

Right -- again, its proper scales are 1, 2, 3, 5, 8, 13, 21, 34, 55,
89, 144, etc. But wasn't diatonicism supposed to be a result, and not
an assumption?

Anyway, the generator for the scales in Yasser's series is Kornerup's
golden fifth of 696.21 cents, which is defined by its property that
the
ratio of the logarithmic (i.e., cents) sizes of certain pairs of
intervals is phi. These pairs include the minor third to perfect
fourth, the major second to minor third, the minor second to major
second, and the chromatic semitone to minor second. We can calculate
the size of the golden fifth exactly using the fact that two perfect
fourths plus a major second equals an octave:

2*x + x*phi^2 = 1 octave

x=1/(2+phi^2) octave

which is approximately .419821 octave, or 503.786 cents. So the
golden
fifth is 696.214 cents.

> I'm
> interested in looking up some of the things you cite: where can I
find
> David Keenan's work and the mathematical definition of 2^phi?

Dave Keenan has some web pages up at http://www.uq.net.au/~zzdkeena/
Music/index.html, but the bulk of his work on this subject is only
available in the archives of this list. Phi is (sqrt(5)-1)/2, so you
can either think of the generator as phi octaves, or 1200*phi cents,
or
as a frequency ratio of 2^phi.

To further clarify:

Phi = 1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1...... = 0.618...

In general, any real number can be expressed (uniquely, with some
minor
proviso that I forget) as a continued fraction of the form

a+1/(b+1/(c+1/(d+1/(e+1/(f+1/(g+1/(h+1/......

where a, b, c, etc., are whole numbers. In our case, a is zero, since
we're interested in fractions of an octave.

To see the points where you get an MOS scale (which is often a proper
scale) from a given generator, you truncate the expansion at each
instance of a "+", and see what fraction you get.

i.e:

1/1 = 1
1/(1+1/1) = 1/2
1/(1+1/(1+1/1)) = 1/(1+1/2) = 1/(3/2) = 2/3
1/(1+1/(1+1/(1+1/1))) = 1/(1+2/3) = 1/(5/3) = 3/5
1/(1+1/(1+1/(1+1/(1+1/1)))) = 1/(1+3/5) = 1/(8/5) = 5/8
1/(1+1/(1+1/(1+1/(1+1/(1+1/1))))) = 1/(1+5/8) = 1/(13/8) = 8/13
1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1)))))) = 1/(1+8/13) = 1/(21/13) = 13/
21

Since these fractions tell you approximately what fraction of an
octave
the generator is, the denominators tell you how many times you have
to
stack the generator upon itself before an additional stacking takes
you
very close to your starting point, i.e., how many notes are in the
MOS
scale. So the phi-part-of-an-octave generator gives you MOS scales
with
1, 2, 3, 5, 8, 13, and 21 notes.

The golden fourth is also a proportion of an octave whose continued
fraction expansion ends in all 1s, i.e., a noble number. Let's call
it
K for Kornerup. It is

K = 1/(2+1/(2+1/(1+1/(1+1/(1+1/(1+1/(1+1...... = 0.4198...

Its convergents are

1/2 = 1/2
1/(2+1/2) = 1/(5/2) = 2/5
1/(2+1/(2+1/1)) = 1/(2+1/3) = 1/(7/3) = 3/7
1/(2+1/(2+1/(1+1/1))) = 1/(2+1/(2+1/2)) = 1/(2+2/5) = 1/(12/5) = 5/12

and (leaving out the calculations) the next ones are 8/19, 13/31, 21/
50, 34/81, etc., giving the Yasser/Kornerup sequence of scales: 2, 5,
7, 12, 19, 31, 50, 81.

I was wrong when I said, a few minutes ago, that this series was not
shown on the scale tree. Look at http://www.anaphoria.com/ST07.html.
Here we see that the noble number that we derived as the size of the
golden fourth in octaves, 1/(2+phi^2) = .41982112717, is the
generator
of the series of MOS scales that zigzags in the tree around the
dashed
line that originates between 2/5 and 3/7.

Note that K can also be written as 1/(2+1/(2+phi)).

Now, in case you thought that good generators like these had to be in
the vicinity of the fifth or fourth, let's look at another noble
number.

1/(1+1/(2+1/(1+1/(1+1/(1+1/(1+1/(1+1...... = 1/(1+1/(2+phi)) =
.7236067977

This generator is 1200*(.7236067977) = 868.33 cents, clearly a
"sixth"
and not a "fifth". Calculating the convergents of this scale:

1/1 = 1
1/(1+1/2) = 1/(3/2) = 2/3
1/(1+1/(2+1/1)) = 1/(1+1/3) = 1/(4/3) = 3/4

Continuing the calculation, or following the scale tree, the next
convergents are 5/7, 8/11, 13/18, 21/29, 34/47, 55/76, 89/123,
indicating MOSs with 1, 3, 4, 7, 11, 18, 29, 47, 76, 123. I believe
all
these scales are proper -- you may want to check, though.

The question as to which generator is "best" in your sense is a
difficult one, Jason, since all noble numbers share the property that
the ratio of the size of the denominator of one convergent to the
size
of the denominator of the next convergent rapidly approaches phi as
you
go farther and farther out into the expansion. So with enough notes,
any noble number (of which there are an infinite number in any finite
range) will eventually get you a sequence of MOS scales where the
number of notes increases by a factor of 1/phi [= phi+1] from one
scale
to the next. Also, the new notes do the best possible job of avoiding
existing notes, as the steps in one scale are divided into two
smaller
steps, whose sizes are in the proportion 1:phi [= phi:(1-phi)] to one
another.

However, you may be more interested in the first few MOS scales you
get
than in the behavior of MOS scales out toward infinity. Observe that
the last property I mentioned, dividing existing steps in the
proportion 1:phi, starts happening at the point in the process of
taking convergents where the rest of the continued fraction expansion
contains all 1s. Since that happens right away with the "2^phi" or
phi-
of-an-octave generator, it would seem that this generator best
captures
the property you're after. I know that you are interested in
propriety;
I leave it as an exercise for you to catalogue the first few proper
scales that each of these generators generates, and let me know how
they compare. My bets are on the the phi-of-an-octave generator.

I'm afraid this generator, and the scales it produces, do not agree
with what we observe in scales around the world. In fact I'm unaware
of
any music that uses this generator and any of the Fibonacci series of
MOSs that it produces.