Yahoo Tuning Groups Ultimate Backup tuning Bigger Field for Notes

The object of this post is to clarify the (dare I say) "natural" field
of notes and relationships as I see it. This field is recognised in
Indian music (see http://www.kcbbs.gen.nz/users/rtomes/in_music.gif).

In this table each horizontal displacement to the right represents a
3/2 ratio. Each displacement to the upper right equals a 5/4 ratio.
Therefore to the lower right is 6/5. The notation is designed to be
totally unambiguous and the table goes far enough that we shouldn't
normally fall off the edges if we start around the natural notes.

A- E- B- F#- C#- G#- D#- A# E# B#
F- C- G- D- A E B F# C# G# D#
Ab Eb Bb F C G D A+ E+ B+
Fb Cb Gb Db Ab+ Eb+ Bb+ F+ C+ G+ D+

Any shape like this grouping ...
A E B
F C G D
... can be picked out anywhere and will have the standard JI tuning
ratios for a major scale of ...

Note C D E F G A B
Ratio 1 9/8 5/4 4/3 3/2 5/3 15/8

The additional notations have the following meanings

Notation # b + - ^ v
Meaning 135/128 128/135 81/80 80/81 64/63 63/64

Notation #- b+ bv #^
Meaning 25/24 24/25 14/15 15/14 etc.

So for example G#- is (3/2)*(135/128)*(80/81) = 25/16
and Bbv is (15/8)*(128/135)*63/64 = 7/4

A separate table with the v notes can be made which is like the above
but displaced by a factor of 7/4 from it. Actually the table is 4
dimensional because it has the 4 axes for the primes 2, 3, 5 and 7.
I am going to ignore 11 and 13 although clearly additional dimensions
can be included for these.

In our tables or diagrams C represents all the "C"s, C1, C2, C3 etc. and
although we call this an identity it isn't really. The reason is that
after going up about 4 octaves in the key of C, the tonic will actually
want to shift to G, or if movig down 4 octaves to F. This may be an
idea of mine and not generally accepted, but we may note that monster
5th chords like F-C-G-D are much more acceptable than the equivalent
4ths, D-G-C-F because the former is like building a split level house
with the lie of the land and the latter is like trying to do it the
other way. I would be interested in other people's comments on this
because it is not something that I have ever seen anyone else mention.
A separate post about this "Key drift up the scale" will be made.

The key point about this great big playing field is that at any one time
we can only light up a small part of it with our spotlight. If we try
to play A- Db and E+ together then that is musically nonsense. The zone
that we nomally light up (although this is subject to some slight
variation) is about 7 ratios of 2, 3 ratios of 3 and 1 ratio of 5.

The centre of the spotlight does not have to be on a note. It can be
somewhere between notes. For example the centre of the key of C major
is normally between C, E and G but a little nearer to C perhaps. If we
make a minor key then it moves into an inverted triangle rather than an
upright triangle. However the spotlight can gradually move around by
fractions of the cell spacing during a piece. If it moves by a whole
space horizontally then we modulate by adding or subtracting a sharp or
flat. But sometimes it will just flirt with doing this which means that
it moves maybe 1/2 or 3/4 of a cell horizontally. Of course the
individual notes and chords may jump around a within the spotlight and
the spotlight just moves slowly so as to keep the average position of
the dancing in the centre.

It is clear from the above table that when we play the chord denoted as
D F A that we are actually playing D- F A and when we play G B D we are
playing the real D. This is another way of saying that compared to C
being 1, in D F A the D is 10/9 while in G B D it is 9/8. In this case
it is quite clear that confusion is created by calling two different
notes by the same name. Even if we are in the key of C which contains
the note A, it is quite possible that we may play a lot of notes around
G B D and then an A might really be an A+ because it is much nearer to D
than A is. It would be obvious if an F# was played that the A was A+
because we would know that we had modulated into the key of G.

We may see also that chord sequences such as I-vi-ii-V-I are actually
designed to be movements from C E G to C- E- G-. This is only seen as a
problem because we don't have a name for C- that is distinct from C.
When someone writes a piece that deliberately modulates from C to C#
then we don't care that the frequency of C# is different to C.
Therefore if someone does succeed in taking us from C to C- then that is
clearly what they did do although in my opinion it needs a slightly
stronger sequence than the above to do this (that is a bit more
dilly-dallying along the way).

I will repeat again that the shape of the spotlight is not circular but
that the following number of steps are about equal in each of the axes.

Prime 2 3 5 7 11 13 17 19 ...
Spread of Index 5.8 2.4 1.0 0.6 0.3 0.25 0.17 0.15 ...

Quite clearly then we cannot reasonably have 3 different powers of the
higher primes (7 up) in a single scale or it will have too much tension.
Even to have two different powers of the higher primes (11 up) in a
scale requires that the spotlight be accurately placed halfway between
two different planes so that both are equally valid. In the case of 7
then we can choose places where either two ratios of 7 are quite natural
or where they only one is (i.e no 7 ratios present).

If we think in terms of this big field, which is just an extension of
all the diagrams people post, then we have a clearer idea of what
happens during a piece of music, or how the composer has decided to just
explore a certain part of the big field.

Tradition has it that there is a different quality to major and minor
which is the location in the 5 axis. The Blues have a different quality
due to choosing a location between layers in the 7 axis. It is possible
to explore locations between the 11 and 13 axes as well or even the 17
and 19 axes. Even the 3 axis can lead to different qualities of scale
and we can find a region of notes that is half way between C and G if we
want. This is a lot less radical however because most music floats the
spotlight around in this axis relatively a lot more because of the wider
compass of the spotlight in this direction.

-- 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

I just realised that if I spaced things differently like this ...

A- E- B- F#- C#- G#- D#- A# E# B#

F- C- G- D- A E B F# C# G# D#

Ab Eb Bb F C G D A+ E+ B+

Fb Cb Gb Db Ab+ Eb+ Bb+ F+ C+ G+ D+

... then it is possible to use a circular spotlight and we get the
correct relative distance between notes in terms of their musical
harmonic relationship to each other. We can also put little marks where
the centre of chords are and show the flow and consider the
possibilities ...

A- E- B- F#- C#- G#- D#- A# E# B#

6 5

F- C- G- D- A E B F# C# G# D#

9 4 2
8 3 1 7

Ab Eb Bb F C G D A+ E+ B+

So for example the sequence I-vi-IV-ii-V-I would follow the numbers
starting from 1 in the above as the location of the dance. The dilemma
is whether to go 1 2 3 4 5 6 and so end up on a different tonic or to go
1 2 3 4 7 1 and end up at the start. The latter is my preference in
this particular case because although the leap from 4 to 7 is possibly
greater (but certainly not by much) than from 4 to 5 the spotlight is
still centred somewhere between 1 2 3 and 4 and so 7 is still in the
spotlight and 5 is not.

That is why I say that if a bit more dilly-dallying were done the answer
would be different. Playing the sequence I-vi-IV-ii-VIIb-v-V-I would
achieve the path 1 2 3 4 8 9 5 6 for example. In this case the composer
has made it fairly clear that he wants to end on up in C- not in C and
so who am I to disagree.

Sure, ET gets back to the tonic by making lots of small errors to hide
the shift. However we can actually hide it in JI also by using a
gradual drag back to the tonic in such cases and have even smaller
errors in JI than in ET but I personally don't think that we should
because I think that the shift has actually occurred. If a piece of
music repeated such a shift and gradually drifted then I am of the
opinion that it is either bad music or intention.

If music is performed from traditional scores then I would expect
results as indicated above. If someone wants something different then
they should annotate for it. In marginal cases make it clear what the
intention is. No problem!?

Bigger Field for Notes

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