Presentation of musical relationships

Monzo wrote:
>On a Monzo lattice:

> 27 D
> /
> 45 B /
> / '-._ /
> / 36 G
> / /
> 30 E /
> / '-._ /
> / 24 C
> / /
> 40 A /
> '-._ /
> 32 F

I have to say that I incredibly admire how much you get into some of
your lattices (not specifically the above one) and how beautiful they
look. I am much lazier and prefer to just do the above as:

A E B
F C G D

but you are most welcome to reproduce everything in 3D.

>[Tomes]
>> To determine the frequencies for G multiply all the above
>> by 3/2 and the only note that actually shifts is A which shifts
>> by 81/80 ratio which I assume is the 9/8 vs 10/9 discussion.
>> I say that this note should shift by that ratio.

>Following your math to the letter:

> 40.5 A
> /
> 67.5 F# /
> / '-._ /
> / 54 D
> / /
> 45 B /
> / '-._ /
> / 36 G
> / /
> 60 E /
> '-._ /
> 48 C

>First of all, you surely don't mean to have decimal points
>in your *ratios* (not that it's going to affect the final
>result, but still...).

You are right that it is messy and also that it makes no difference.
I have a tendency to reduce all notes to the 24 to 48 range of ratios
and accept things like 40.5 knowing what it means. Of course that is
only personal taste and almost certainly not someone else's taste.
Please excuse my bad habits.

>Since you're assuming 'octave'-equivalence anyway (i.e.,
>ignoring powers of 2), I find it much simpler to use the
>prime-factor notation:

Absolutely. However it is useful to present things in several different
ways because they highlight different aspects of what is going on. The
diagrams are great but are a lot of work and can get overcomplicated
when we start doing all primes up to 13. In that case the prime-factor
notation is the only way to go.

However I also do make a case in a recent post for non-equivalence of
octaves.

>A comparison of these two tables makes it very easy
>to see which notes have changed and which haven't:

>'C major': 'G major':

>fac. 3 5 fac. 3 5
>----------- ------------
>B | 2 1 | B | 2 1 |
>A | 0 1 | A | 4 0 |
>G | 2 0 | G | 2 0 |
> F# | 3 1 |
>F | 0 0 |
>E | 1 1 | E | 1 1 |
>D | 3 0 | D | 3 0 |
>C | 1 0 | C | 1 0 |

>There are two notes which 'shift' here:

>The obvious metamorphosis (which you didn't even mention)
>is the shift of F ( 0 0 ) to F# ( 3 1 ).

Yeah, I asumed that everyone knew about that ;^)

>The subtle one, which you did point out, is the shift of
>A ( 0 1 ) to A ( 4 0 ).

And another notation for that is the 81/80 shift which is what I always
think of this as. Hey, I know things have proper music names but I am
not really a musician (should I really mention this when I am trying to
revolutionise music theory? answer: might as well, as it is obvious
anyway).

As regards the answer to "what's the problem?" I would just say that I
answered already a couple of very similar ones and so hopefully these
will give the flavour of what I believe. In general I would say that
the individual chords should be consonant and that there will be some
place where something gives and a note does the 80/81 thing between
successive chords (or held notes). If the music is made tight enough so
that this doesn't happen then the composer successfully negotiated a way
to move the scale by a 81/80 ratio and all power to him for that.
Nobody cares if you move it by 9/8 intentionally so why shouldn't you
move by 81/80 if your heart is really set on it.

Sideline: If music really follows the subtle cycles in nature (as I
believe, and as supported by some composers saying they just listen)
then such things will not really happen in natural music. Or, if they
do they will likely return but not keep wandering off as (81/80)^n.

>I've frequently created JI chord progressions that looked
>good on paper, and I thought would sound great, and the
>individual chords *did* sound great. But put them all together,
>and... yuk!

This is where I think my post on the big field of music comes into its
own. I have used the matrix space rather than the prime notation
(although I did that previously just to allow several ways to look at
it) because then it is very clear what sort of distances are involved in
any given leap. It is my opinion that using this big field (or the
prime factorisations which is the same thing) and understanding that
changing the index of higher primes is much more severe than changing
the index of lower primes allows the correct decision to be made in all
cases. Of course clever Charlie's will find borderline cases. These
will have their own special attractions.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
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