Centre of Gravity of a Scale or Piece

I want to put forward the concept of the "Central of Gravity of a scale"
which may be though of as a sort of attractor which all the notes of the
scale hang off. This also gives very strong clues as to what non-scale
notes might be reasonable to occur in a piece. Actually a piece of
music will also have a "Central of Gravity of a Piece" which will
incorporate the various modulations within the piece.

To get the central ratio of a scale or piece, simply take the ratios of
frequencies and express them as prime factorisations and average the
indices of the primes. Here is an example for the JI major scale ...

Note C D E F G A B C
Ratio 24 27 30 32 36 40 45 48 Average
2^a 3 0 1 5 2 3 0 4 18/8 = 2.25
3^b 1 3 1 0 2 0 2 1 10/8 = 1.25
5^c 0 0 1 0 0 1 1 0 3/8 = 0.375
7^d 0 0 0 0 0 0 0 0 0/8 = 0.0

So we may say that the major scale when expressed as the above ratios is
centred on 2^2.25, 3^1.25, 5^0.375, 7^0 or 2^2 3^1 5^0 7^0 to the
nearest integer which gives C. However the exponent of 5 is almost
halfway to the next integer.

When considering the spread of notes that can be used about this central
value then the powers of 2 will have the greatest spread (+/-2 about )
while the powers of 3 will have a lesser spread (+/-1.5 say) and the
powers of 5 less still (+/-0.5 roughly) with no variation in 7 in this
particular scale. Actually, the power of 2 can vary much more because
several octaves of each note should also be included.

Therefore in the major scale, notes which have 5^0 will be slightly
favoured over 5^1 and other indices to 5 will be very unfavourable.
Likewise, 3^1 is favoured then 3^2 and then 3^0 and finally 3^3 with
others being less favourable (but happening in the order 3^-1, 3^4 etc).

Let's try another example, the blues scale mentioned recently, assuming
that I have the ratios correct ...

Note C Eb F F# G Bb B C
Ratio 24 28 32 35 36 42 45 48 Average
2^a 3 2 5 0 2 1 0 4 17/8 = 2.125
3^b 1 0 0 0 2 1 2 1 7/8 = 0.875
5^c 0 0 0 1 0 0 1 0 2/8 = 0.25
7^d 0 1 0 1 0 1 0 0 3/8 = 0.375

So this blues scale has a "centre" which is higher in the 7 index and
lower mainly in the 3 index than the major scale. That means that its
centre of gravity is in a different place in the multi-dimensional space
of all possible reasonable scales.

I am going to suggest that the "natural" spread of the indices of the
primes follows a rule something like this in proportion:

Prime 2 3 5 7 11 13 17 19 ...
Spread of Index 3.5 1.5 0.6 0.35 0.18 0.15 0.10 0.09 ...

Which can be calculated as in inverse proportion to p*log(p).

Just one more example. Suppose that a piece is played in the key of C
but modulates for half of the piece into G (actually we could add up all
the notes and find an exact weighted centre, but this will do as an
example). Then it will have a centre at the middle of C and G.
C is centred at 2^2.25, 3^1.25, 5^0.375, 7^0 and because G is 3/2 higher
than C it will be centred at 2^1.75, 3^1.75, 5^0.375, 7^0. That means
that our modulated piece is centred on 2^2.0, 3^1.5, 5^0.375, 7^0.

"So what?" you might ask. "What can this be used for?"

Well, I am suggesting that the strength of relationship of each note in
a scale or piece can be determined by considering how far it is from the
CoG (Centre of Gravity) of the scale or piece. This must be after
allowing for the relative spreads with respect to each prime because
powers of 2 are allowed to vary more than powers of 3 and much more than
powers of 5 and 7 and so on.

So let me take the centre at 2^2.0, 3^1.5, 5^0.3, 7^0 and allowing for
the spread factors above of 3.5, 1.5, 0.6 and 0.35 I will calculate the
relative importance of a whole set of notes. I will ignore the power of
2 altogether, but this is only to keep the problem manageable. In
actual fact the relative importance of the notes in a scale do vary with
octave, so that in C for example, F and C are more important in the low
octaves and C and G in the higher octaves.

Centre of Gravity at 3^1.5 5^0.4 7^0.0

Prime Distance Weighted Distance Total Actual Note in
Index from Centre from Centre Weights Ratio Scale
3 5 7 3 5 7 3 5 7

0 0 0 1.5 0.4 0.0 1.0 0.7 0.0 1.7 32 F
0 0 1 1.5 0.4 1.0 1.0 0.7 2.8 4.5 28 Eb7
0 1 0 1.5 0.6 0.0 1.0 1.0 0.0 2.0 40 A
0 1 1 1.5 0.6 1.0 1.0 1.0 2.8 4.8 35 F#
1 0 0 0.5 0.4 0.0 0.3 0.7 0.0 1.0 24 C
1 0 1 0.5 0.4 1.0 0.3 0.7 2.8 3.8 42 Bb7
1 1 0 0.5 0.6 0.0 0.3 1.0 0.0 1.3 30 E
1 1 1 0.5 0.6 1.0 0.3 1.0 2.8 4.1 26.25 C#?
2 0 0 0.5 0.4 0.0 0.3 0.7 0.0 1.0 36 G
2 0 1 0.5 0.4 1.0 0.3 0.7 2.8 3.8 31.5 F-
2 1 0 0.5 0.6 0.0 0.3 1.0 0.0 1.3 45 B
2 1 1 0.5 0.6 1.0 0.3 1.0 2.8 4.1 39.375 A-
3 0 0 1.5 0.4 0.0 1.0 0.7 0.0 1.7 27 D
3 0 1 1.5 0.4 1.0 1.0 0.7 2.8 4.5 47.25 C-
3 1 0 1.5 0.6 0.0 1.0 1.0 0.0 2.0 33.75 F#
3 1 1 1.5 0.6 1.0 1.0 1.0 2.8 4.8 29.93 E*?
-1 0 0 2.5 0.4 0.0 1.7 0.7 0.0 2.4 42.67 Bb
-1 1 0 2.5 0.6 0.0 1.7 1.0 0.0 2.7 26.67 D-
4 0 0 2.5 0.4 0.0 1.7 0.7 0.0 2.4 40.5 A-
4 1 0 2.5 0.6 0.0 1.7 1.0 0.0 2.7 25.31 C#?

Now comes the result. Let us put the notes in order by the minimum
total weights which is the measure of how important each note is in this
particular piece (C modulated to G) ...

Weights 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 ...
C E F A A-
G B D F# Bb
Most important <-----------------------------> Least important

So the result is that C and G are central with A and F# much less so.
We might expect that F# is less important as it is in the key of G but
not C, however the same argument goes for F (the other way) whereas it
is actually A that is the other least important note.

Although I put a whole set of notes with 7 in the ratio in the table
these were found to be bad fits and so are not likely to occur at all.

It is interesting to note that there are two A notes, one at 40 and one
at 40.5 (labelled A-) which is the old 80/81 dilemma. In this case it
can be seen that both are of moderate importance in the scale and could
reasonably occur and so any fixed tuning instrument cannot deal with
this. It is desirable that both notes can be played.

It is possible to explore various other scales by fiddling around with
the centre of gravity. If the centre is moved along the 5 axis then a
minor scale results for example. Note that a movement by a whole unit
takes us back to the same place again (it just changes all the ratios by
a constant factor) and so for each prime there is a single unit of space
that can be explored. I will write a simple program do demonstrate some
examples by allowing each of these "prime knobs" to be twiddled and see
what scales result and which are the important notes. If anyone has
some sample scales or even better sample note frequencies from a piece
of music then it is possible to calculate the centre of gravity "CoG"
for that scale or piece.

If ratios like 11 and 13 (or even 7 for that matter) are to occur in
some music then the centre needs to be somewhere near 0.5 on the 11 or
13 axis to begin with because the weighting otherwise will not make any
notes with that ratio come near to those without that ratio. I will
explore this on another occasions after seeing whether people understand
what I am on about and what they think of it.

This scheme can accommodate a very rich variety of scales and probably
already incorporates all the common scales around the world as well as
many that have never been tried but which might be quite interesting.

In general the 3 axis is that of modulating keys (adding and subtracting
sharps and flats), the 5 axis is the major-minor variation and the 7
axis is the blues variation. Who knows what the 11 and 13 axis are?
Actually there is enormous variation in just the 3, 5 and 7 axes.
If anyone has an easily programmable keyboard I can suggest a few sample
tunings that will produce potentially interesting and totally new
scales.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

Ray Tomes introduced this concept, ooh, over a day and a half ago now:

> I want to put forward the concept of the "Central of Gravity
> of a scale"
> which may be though of as a sort of attractor which all the
> notes of the
> scale hang off. This also gives very strong clues as to what
> non-scale
> notes might be reasonable to occur in a piece. Actually a piece of
> music will also have a "Central of Gravity of a Piece" which will
> incorporate the various modulations within the piece.

I think this is a useful concept in some situations. I notice from these
Euler translations that have appeared on the web (sorry, don't have the
URLs, look back a few digests) that he seemed to have the same idea. But
that's by the by. The primary use, that I see, is in adaptive tuning.

To clarify, I'll start by cutting the adaptive tuning process into two
steps:

1) Find all the possible interpretations for each note

2) Decide on the best tuning for each note

This is really stating the obvious, but I feel it has to be done. Ray
outlined a similar process more recently:

>Computers are so fast now that it can get the answer easily fast enough
>by considering each unique position of the slide and looking at all the
>notes and doing a weighted score on each fit to get the best answer,
>possibly incorporating information about the previous centre of gravity
>also to resolve ambiguous cases. No problem!

In this case, the two steps are not completely independent, however. I'll
pretend they are nonetheless, and concentrate on step 2.

The thing is, in the most general case, the tuning of each chord depends on
the tuning of each other chord. So, the number of calculations required to
tune each note is proportional to the number of notes it has to be tuned to.
Therefore, the number of calculations required to tune all the notes in a
piece is proportional to the square of the number of notes. Assuming you
think they're all important, anyway. With a lot of notes, this calculation
could take a noticeable time. If you're tuning outside of real-time, it can
get even more complicated. Retuning one note will alter a load of notes you
already thought were tuned right. So, some simplification may be useful.

An answer is only to consider the most recently played notes. That's fine,
so long as you don't mind wandering tonics. Otherwise, this is where the
centre of gravity comes in. Ray again:

> To get the central ratio of a scale or piece, simply take the
> ratios of
> frequencies and express them as prime factorisations and average the
> indices of the primes. Here is an example for the JI major scale ...
>
> Note C D E F G A B C
> Ratio 24 27 30 32 36 40 45 48 Average
> 2^a 3 0 1 5 2 3 0 4 18/8 = 2.25
> 3^b 1 3 1 0 2 0 2 1 10/8 = 1.25
> 5^c 0 0 1 0 0 1 1 0 3/8 = 0.375
> 7^d 0 0 0 0 0 0 0 0 0/8 = 0.0

So, for 7-prime-limit JI, the centre of gravity can be expressed by four
numbers. My idea is this: get your adaptive tuning algorithm to tune
relative to the centre of gravity, rather than all existing notes. Then,
the time required to tune each note does not depend on the number of notes
it's being tuned relative to. So, tuning a whole piece is an order N rather
than order N-squared process.

> I am going to suggest that the "natural" spread of the indices of the
> primes follows a rule something like this in proportion:
>
> Prime 2 3 5 7 11 13 17 19 ...
> Spread of Index 3.5 1.5 0.6 0.35 0.18 0.15 0.10 0.09 ...
>
> Which can be calculated as in inverse proportion to p*log(p).

This is a good rule of thumb, but I think we can do better. The spread of
each index can be proportional to the variance or standard deviation of the
indices already chosen. A standard statistics text (or scientific
calculator's instruction manual) should tell you how to calculate both the
mean and variance/standard deviation from these quantities:

The total number of items (n)
The sum of those items (s)
The sum of the squares (q)

Obviously, the mean is s/n. I should have looked up the variance, but I
think it's something like this:

q/n - (s/n)^2

And the standard deviation is the square root of the variance.

So, you can work through the piece, update n, s and q for each index, and
tune each note accordingly. One detail is that n needn't be an integer.
So, you can use a weighting where more recent notes are more important than
distant ones.

There'll need to be some guesses as to the beginning state. You can assume
something like C-major with index weights as Ray gives above. And use a
small enough weighting that the notes that are played become much more
important than this initial state.

> Well, I am suggesting that the strength of relationship of
> each note in
> a scale or piece can be determined by considering how far it
> is from the
> CoG (Centre of Gravity) of the scale or piece. This must be after
> allowing for the relative spreads with respect to each prime because
> powers of 2 are allowed to vary more than powers of 3 and
> much more than
> powers of 5 and 7 and so on.

We seem to be in agreement, at least as this relates to adaptive tuning.

> This scheme can accommodate a very rich variety of scales and probably
> already incorporates all the common scales around the world as well as
> many that have never been tried but which might be quite interesting.

I don't think this is true at all. As well as the whole idea of just
approximations being redundant for a lot of musics, there are plenty of
cases where the centre of gravity will shift during the piece.

I should have written - Centre of Gravity of a Chord/Passage/Piece

Graham Breed <g.breed@tpg.co.uk> wrote:
>I think this is a useful concept in some situations.
>The primary use, that I see, is in adaptive tuning.

Yes. A secondary use is in understanding the harmonic structure of a
piece. I was really amazed at some of the things that I learned from
this. Please remind me some time to go into this in detail.

Another is for finding some variations to the normal scales to explore.
More about this below.

...
>Therefore, the number of calculations required to tune all the notes in a
>piece is proportional to the square of the number of notes.
> ... If you're tuning outside of real-time, it can
>get even more complicated.

My intention was to do it in real time (so that it can be put as
software into keyboards as a feature) and so I have concentrated on
that. Of course there are some situations where a bit of knowledge
about what is going to happen next what make things a lot easier and so
the possibility of off-line use should be allowed or.

Generally I don't think that there is much benefit in looking at all the
notes at once, although this would allow solving the possible problem of
tonic drift. I would have a parameter for the range of recent notes to
look at which would have a low of one chord, a medium of about a bar and
a high of say 4 bars. This is the parameter that controls how slowly
the "spotlight" moves compared to the "dancer".

>So, for 7-prime-limit JI, the centre of gravity can be expressed by four
>numbers. My idea is this: get your adaptive tuning algorithm to tune
>relative to the centre of gravity, rather than all existing notes. Then,
>the time required to tune each note does not depend on the number of notes
>it's being tuned relative to. So, tuning a whole piece is an order N rather
>than order N-squared process.

Thanks for clarifying that which I didn't. That is exactly my thinking.
The centre of gravity is 4 numbers which each chord updates as an
exponential moving average with the parameter mentioned above, e.g.

newprime7index = oldprime7index*(1-param) + presentprime7index*param
(from present chord)

where param is 0.5 for a rapidly following setting, 0.25 for a medium
and 0.05 for a slow following setting.

>> Prime 2 3 5 7 11 13 17 19 ...
>> Spread of Index 3.5 1.5 0.6 0.35 0.18 0.15 0.10 0.09 ...

Actually I think that these values may drop off slightly too harshly.

>This is a good rule of thumb, but I think we can do better. The spread of
>each index can be proportional to the variance or standard deviation of the
>indices already chosen.

I think that this is a good improvement for making a parameter for how
big the "spotlight" is - i.e, what is the reasonable range of jumps.
However I would not use adaption for the relativities of the different
primes as I think that this is rather fixed and some of the variances
would head to zero through lack of use and therefore cause problems when
that prime suddenly turned up.

>So, you can use a weighting where more recent notes are more important than
>distant ones.

Exactly, that is what the exponential average above does. It makes the
recent notes very important, the ones a little before that somewhat
important and what came long ago of almost no importance.

>There'll need to be some guesses as to the beginning state. You can assume
>something like C-major with index weights as Ray gives above. And use a
>small enough weighting that the notes that are played become much more
>important than this initial state.

Absolutely.

In my final comment about possible new scales I was actually changing
subject a bit without making that clear. The interesting thing about 4D
space (or 6D with 11 and 13) is that there is a lot of it. Although
most music drifts around quite a lot on the 2 and 3 axes it doesn't seem
to on the 5 and 7 axes and these axes seem to have just a couple of
notches rather than the full range getting used. That is why I think
there is some room for inventiveness. However this is actually in the
composing side rather than the tuning side.

Graham, thanks for the good ideas.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm