a thought on ratio notation

Often we end up using large ratios which if one is unfamiliar can stop
the flow of one thought. I have often thought much could be gained by a
simplier system , possibly used in conjunction with the more common one
that better ilustrated the actual function of interval. This would be
done by breaking it down accoerding to it factors and could be
considered an exstention of notation used in functional harmony where
secondary dominants are notated according to how they are used, such as
v / v etc.
An example of the is in a simple case as 28/27 could be notated as
7/3 x 9. such a notation bring to the front what one needs and wants to
know right to the front.
A more complex example would be Gene's 4375/4374 planar.
which would be notated as 5x5x5x7/3x3x3x3x3x3x3 which you end up having
to figure out anyway (i don't know how to do superscripts. likewise less
confusing than the matrix notation.

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

hi Kraig,

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:

> Often we end up using large ratios which if one is
> unfamiliar can stop the flow of one thought.

*exactly* why prime-factor ("monzo") notation is great!

> I have often thought much could be gained by a
> simplier system , possibly used in conjunction with the
> more common one that better ilustrated the actual function
> of interval. This would be done by breaking it down
> accoerding to it factors and could be considered an
> exstention of notation used in functional harmony where
> secondary dominants are notated according to how they are
> used, such as v / v etc.
> An example of the is in a simple case as 28/27 could
> be notated as 7/3 x 9.

not correct: your notation for it should be 7/3 / 9 .
7/3 x 9 gives the ratio 63/48.

the [3 5 7]-monzo of 28/27 is [-3 0 1].

another way of doing it, combining the "ratio fraction"
style with prime-factoring, is 7 / 3^3 .

> such a notation bring to the front
> what one needs and wants to know right to the front.

i agree completely.

> A more complex example would be Gene's 4375/4374 planar.
> which would be notated as 5x5x5x7/3x3x3x3x3x3x3 which you
> end up having to figure out anyway (i don't know how to do
> superscripts. likewise less confusing than the matrix notation.

that's also incorrect -- you left out one "5".

its [3 5 7]-monzo is [-7 4 1].

in the combined "ratio fraction and prime-factor" style
it's (5^4 * 7) / 3^7 .

my software uses both of these notations, along with regular
ratios.

-monz

Well, related to your enquire is the plain decomposition of the ratio
into powers of prime factors. Monzo uses the vector notation, which
is a n-uple containing the powers of the first prime factors (that
conform a single ratio).

In any case, your 28/27 ratio should be expressed as:

2^2 x 3^-3 x 7

...which is an octave-equivalence of the factor decomposition
menctioned in your post.

Regarding the core of your mail, the functional aspect of interval, I
must say that the concept cannot be applied without a precise
definition of the context involved. For instance, in 12-eq
temperament, something like v/v is perfectly understood; however, no
ratio including the factor 7 can be properly stated within that
system.

My last words: I believe it is indeed impossible to produce
superscripts here.

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> Often we end up using large ratios which if one is unfamiliar can
stop
> the flow of one thought. I have often thought much could be gained
by a
> simplier system , possibly used in conjunction with the more common
one
> that better ilustrated the actual function of interval. This would
be
> done by breaking it down accoerding to it factors and could be
> considered an exstention of notation used in functional harmony
where
> secondary dominants are notated according to how they are used,
such as
> v / v etc.
> An example of the is in a simple case as 28/27 could be
notated as
> 7/3 x 9. such a notation bring to the front what one needs and
wants to
> know right to the front.
> A more complex example would be Gene's 4375/4374 planar.
> which would be notated as 5x5x5x7/3x3x3x3x3x3x3 which you end up
having
> to figure out anyway (i don't know how to do superscripts. likewise
less
> confusing than the matrix notation.
>
> -- -Kraig Grady
> North American Embassy of Anaphoria Island
> http://www.anaphoria.com
> The Wandering Medicine Show
> KXLU 88.9 FM WED 8-9PM PST

At the moment, I am working on a list of important intervals which
presents intervals in different ways, including the ratio fraction
and the prime factor notation. The list is published in html, so I
can user superscripts. It also cantains decompositions of the
intervals in smaller ones, in different ways.
Whes the first draft of the list is ready, I put it somewhere on a
site, en I'll advertise for it here on the list.
Prime factor notation indeed helps a lot in understanding how an
interval in constructed, it is also the base of any calculations I
make wirh intervals, inluding the decompositions.
The risk is that you automatically come to linear algebra, and thus a
banishment to tuning-math ...

Bart Pauwels

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> Often we end up using large ratios which if one is unfamiliar can
stop
> the flow of one thought. I have often thought much could be gained
by a
> simplier system , possibly used in conjunction with the more common
one
> that better ilustrated the actual function of interval. This would
be
> done by breaking it down accoerding to it factors and could be
> considered an exstention of notation used in functional harmony
where
> secondary dominants are notated according to how they are used,
such as
> v / v etc.
> An example of the is in a simple case as 28/27 could be
notated as
> 7/3 x 9. such a notation bring to the front what one needs and
wants to
> know right to the front.
> A more complex example would be Gene's 4375/4374 planar.
> which would be notated as 5x5x5x7/3x3x3x3x3x3x3 which you end up
having
> to figure out anyway (i don't know how to do superscripts. likewise
less
> confusing than the matrix notation.
>
> -- -Kraig Grady
> North American Embassy of Anaphoria Island
> http://www.anaphoria.com
> The Wandering Medicine Show
> KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> Often we end up using large ratios which if one is unfamiliar can
stop
> the flow of one thought. I have often thought much could be gained
by a
> simplier system , possibly used in conjunction with the more common
one
> that better ilustrated the actual function of interval. This would
be
> done by breaking it down accoerding to it factors and could be
> considered an exstention of notation used in functional harmony
where
> secondary dominants are notated according to how they are used,
such as
> v / v etc.
> An example of the is in a simple case as 28/27 could be
notated as
> 7/3 x 9. such a notation bring to the front what one needs and
wants to
> know right to the front.
> A more complex example would be Gene's 4375/4374 planar.
> which would be notated as 5x5x5x7/3x3x3x3x3x3x3 which you end up
having
> to figure out anyway (i don't know how to do superscripts. likewise
less
> confusing than the matrix notation.
>
> -- -Kraig Grady
> North American Embassy of Anaphoria Island

Complex combinations of prime factors become rather unwieldy in a
performance notation, and the intervals involved can be described as
anything but consonant. In the high-precision version of sagittal JI
we would treat this combination by approximating it to a simpler
interval and dispensing with the very small schisma (what we call
a "schismina") by which it differs. Taking C as 1/1, this particular
combination of prime factors would be notated as either Ab/| (a
pythagorean A-flat raised by a 5-comma, 80:81) or G#.! (a pythagorean
G-sharp lowered by a 5-schisma, 32768:32805). With either spelling a
schismina of ~0.4 cents is dispensed with.

In medium precision sagittal JI (in which the 5-schisma is ignored),
the preferred spelling is still Ab/| (with ~0.4 cents error) and the
alternate spelling simply becomes G#, with the error increasing to
~1.6 cents.

In the case of 28/27, we can notate that exactly in both medium and
high-precision sagittal as Db!) (D-flat lowered by a 7-comma,
63:64). The alternate spelling in high-precision JI is also exact,
as C.(|\, but in medium-precision JI this becomes C(|\, with the 5-
schisma being ignored (for an error of ~1.954c).

These are all symbols that could be read in real-time *performance*.

--George

--- In tuning@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:

> These are all symbols that could be read in real-time
*performance*.
>
> --George

I've not followed the whole thread here, but I wonder how to
interpret this rather diplomatic claim. The saggital symbols
'could' be read in real-time performace, and if 'real-time' means
reading at sight, or 'sight-reading', we must bear in mind that
sight-reading is a difficult skill already for many trained
performers reading traditional notation, but certainly there are
many who excel at this skill.

Perhaps all that is said here is that it is conceivable that
someone 'could' sight read music in this notation. This is
plausible enough. Given enough time, concentrated effort, and
support, just about anything along these lines is possible.

That this system actually 'can' work in a real-time situation, with
the ideal being at 'sight-reading' level will take quite a lot of effort
to demonstrate. Someone has to learn to play an instrument
(violin? trombone? voice?) according to the system. At least one
composer must learn to write music using the system. For the
experiment to work, the performer and the composer cannot be
the same person. The performer may then sight read new
music, and his or her performance may be judged by the
composer. This process could be accomplished in a few years,
but until something like that happens, I suspect most people will
agree that claims to performance practicality, no matter how
noncommital, remain unconvincing.

By no means am I offering a discouraging word. By all means,
please carry out the experiment.

Best regards,
Aaron Hunt

--- In tuning@yahoogroups.com, "pitchcolor" <Pitchcolor@a...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor"
> <gdsecor@y...> wrote:
> > These are all symbols that could be read in real-time
> *performance*.
>
> I've not followed the whole thread here, but I wonder how to
> interpret this rather diplomatic claim.
...
> By all means,
> please carry out the experiment.

Hi Aaron,

I believe you have interpreted it correctly. We would dearly love to
see such experiments carried out. We cannot do them ourselves. We are
just two guys working on this stuff at home in whatever spare time we
can scrape together. We are not tenured musical academics, or
composers whose microtonal work is regularly performed by others.

We had hoped that Joseph Pehrson would help us with this experiment,
but he has understandably declined until we make some improvements.

We are working on an improvement and would like to hear from anyone
else willing to help us with such experiments.

Are you offering to help, Aaron?

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:

> Often we end up using large ratios which if one is unfamiliar can
stop
> the flow of one thought. I have often thought much could be gained
by a
> simplier system , possibly used in conjunction with the more common
one
> that better ilustrated the actual function of interval. This would
be
> done by breaking it down accoerding to it factors

kraig, every time i did this you complained that you couldn't "crack
the code". but see below . . .

> and could be
> considered an exstention of notation used in functional harmony
where
> secondary dominants are notated according to how they are used,
such as
> v / v etc.
> An example of the is in a simple case as 28/27 could be
notated as
> 7/3 x 9. such a notation bring to the front what one needs and
wants to
> know right to the front.
> A more complex example would be Gene's 4375/4374 planar.
> which would be notated as 5x5x5x7/3x3x3x3x3x3x3 which you end up
having
> to figure out anyway (i don't know how to do superscripts. likewise
less
> confusing than the matrix notation.

it's almost identical, though. if we were to write 3^-7 * 5^3 * 7,
wouldn't that give exactly the same information but even more
concisely? because you don't even have to count the 3s, you're told
immediately that there are 7 of them . . .

I wrote:
"We would dearly love to see such experiments carried out."

Its interesting to consider what the "control" should be in such an
experiment. Which _other_ universal microtonal notation should it be
compared against. ;-)

And of course it would need to be tested on many different instruments
by many different people using many different tunings. In fact no one
has the time or money to conduct such a controlled experiment.

It will just have to be done in an anecdotal manner by whoever is
brave enough to try something that is completely untested, and comes
with no warranty.

--- In tuning@yahoogroups.com, "Dave Keenan"
<d.keenan@b...> wrote:
> I wrote:
> "We would dearly love to see such experiments carried out."
... of course it would need to be tested on many different
instruments
> by many different people using many different tunings. In fact
no one
> has the time or money to conduct such a controlled
experiment.
----

Ideally many performers would be involved, but an initial
experiment need only involve one performer and one composer,
and this could be carried out within the time span of just a few
years. I believe that there are in fact people interested in carrying
out research such as this. Put together a clear, concise
instructional book for the system and hand the project over to the
right PhD student. It's a great dissertation topic.

regards,
Aaron Hunt

i agree with everything Aaron says here.

-monz

--- In tuning@yahoogroups.com, "pitchcolor" <Pitchcolor@a...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan"
> <d.keenan@b...> wrote:
> > I wrote:
> > "We would dearly love to see such experiments carried out."
> ... of course it would need to be tested on many different
> instruments
> > by many different people using many different tunings. In fact
> no one
> > has the time or money to conduct such a controlled
> experiment.
> ----
>
> Ideally many performers would be involved, but an initial
> experiment need only involve one performer and one composer,
> and this could be carried out within the time span of just a few
> years. I believe that there are in fact people interested in
carrying
> out research such as this. Put together a clear, concise
> instructional book for the system and hand the project over to the
> right PhD student. It's a great dissertation topic.
>
> regards,
> Aaron Hunt