Rational/Irrational Asymptote (RIA)

I prefer to think of a tuning on my canon as a swash of colors on a
canvas. No one has ever successfully analyzed or described a
painting by extracting and systematically classifying the colors into
"warm" and "cold" categories. Warm needs Cold. Cold needs
Warm. Rational needs Irrational. Irrational needs Rational.

(Please note that in the table below, the symbol ( _| ) represents
the Square Root Symbol.)

The following quote is from my manuscript Musical Mathematics:
A Practice in the Mathematics of Tuning Instruments and
Analyzing Scales, Chapter 10:

******************************

Table 10

Arithmetic Mean Geometric Mean

2/1: (1000+500)/2 = 750 2/1: _|1000*500 = 707.1
4/3: (1000+750)/2 = 875 4/3: _|1000*750 = 866.0
8/7: (1000+875)/2 = 937.5 8/7: _|1000*875 = 935.4
16/15: (1000+937.5)/2 = 968.8 16/15: _|1000*937.5 = 968.2
32/21: (1000+968.8)/2 = 984.4 32/21: _|1000*968.8 = 984.3

(64/63, etc.)

According to these figures, the arithmetic mean of the "octave"
expressed as length ratio 2/1 requires a bridge at 750.0 mm,
whereas the geometric mean requires a bridge at 707.1 mm.
Table 10 shows that for a sequentially decreasing progression of
length ratios, the bridge locations of these two different means
begin to converge ! For example, the arithmetic mean of length
ratio 8/7 requires a bridge at 937.5 mm, and the geometric mean,
a bridge at 935.4 mm; here the difference is 2.1 mm. Next, the
arithmetic mean of length ratio 16/15 requires a bridge at 968.8
mm, and the geometric mean, a bridge at 968.2 mm; here the
difference is only .6 mm. However, because the arithmetic means
represent rational numbers, and the geometric means represent
irrational numbers, these two sets of numbers can never be
identical. Mathematicians refer to values or lines that converge
but never meet or intersect as asymptotes.

******************************

Just/Rational Intonation, Tempered/Irrational Intonation, I don't
think it really matters what you call it. All experienced composers
know that the aural experience of a Rational/Irrational Asymptote
(RIA) accurately describes how far or how near the intervals of a
given tuning are to just/rational or tempered/irrational values.

Cris Forster, Music Director
www.Chrysalis-Foundation.org

http://www.Chrysalis-Foundation.org/musical_mathematics.htm

yes Apollo and Dionysis were brothers and Greeks understood how they related and were not antagonistic toward each other.
( these cultures still has not raised to this understanding)

if one looks at the leaf of the laurel, Apollo's plant , one will notice that the veins are not perfectly symetrical at all. something he learn from his brother.

Both series fills infinity, hence indistinguishable in the end.
How they get there is another matter

Message: 16 Date: Tue, 7 Jun 2005 23:32:57 -0400
From: Cris Forster <76153.763@compuserve.com>
Subject: Rational/Irrational Asymptote (RIA)

I prefer to think of a tuning on my canon as a swash of colors on a canvas. No one has ever successfully analyzed or described a painting by extracting and systematically classifying the colors into "warm" and "cold" categories. Warm needs Cold. Cold needs Warm. Rational needs Irrational. Irrational needs Rational.

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Hi Kraig. My intention was not to overstate the obvious.
Allow me to rephrase: Are your mental processes and aural
perceptions always precisely aware of the "infinite" distinctions
between "Apollo" with "Dionysus"?

Cris

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> yes Apollo and Dionysis were brothers and Greeks understood how
they related and were not antagonistic toward each other.
> ( these cultures still has not raised to this understanding)
>
> if one looks at the leaf of the laurel, Apollo's plant , one will
notice that the veins are not perfectly symetrical at all. something
he learn from his brother.
>
> Both series fills infinity, hence indistinguishable in the end.
> How they get there is another matter
>
>
>
>
> Message: 16
> Date: Tue, 7 Jun 2005 23:32:57 -0400
> From: Cris Forster <76153.763@c...>
> Subject: Rational/Irrational Asymptote (RIA)
>
> I prefer to think of a tuning on my canon as a swash of colors on
a
> canvas. No one has ever successfully analyzed or described a
> painting by extracting and systematically classifying the colors
into
> "warm" and "cold" categories. Warm needs Cold. Cold needs
> Warm. Rational needs Irrational. Irrational needs Rational.
>
>
>
>
>
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

i do ot assume anything obvious about two infinite series and how they
interact

--- In tuning@yahoogroups.com, "c_ml_forster" <76153.763@c...> wrote:
> Hi Kraig. My intention was not to overstate the obvious.
> Allow me to rephrase: Are your mental processes and aural
> perceptions always precisely aware of the "infinite" distinctions
> between "Apollo" with "Dionysus"?
>
> Cris
>
>
>
>
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> >
> > yes Apollo and Dionysis were brothers and Greeks understood how
> they related and were not antagonistic toward each other.
> > ( these cultures still has not raised to this understanding)
> >
> > if one looks at the leaf of the laurel, Apollo's plant , one will
> notice that the veins are not perfectly symetrical at all. something
> he learn from his brother.
> >
> > Both series fills infinity, hence indistinguishable in the end.
> > How they get there is another matter
> >
> >
> >
> >
> > Message: 16
> > Date: Tue, 7 Jun 2005 23:32:57 -0400
> > From: Cris Forster <76153.763@c...>
> > Subject: Rational/Irrational Asymptote (RIA)
> >
> > I prefer to think of a tuning on my canon as a swash of colors on
> a
> > canvas. No one has ever successfully analyzed or described a
> > painting by extracting and systematically classifying the colors
> into
> > "warm" and "cold" categories. Warm needs Cold. Cold needs
> > Warm. Rational needs Irrational. Irrational needs Rational.
> >
> >
> >
> >
> >
> >
> > --
> > Kraig Grady
> > North American Embassy of Anaphoria Island <http://anaphoria.com/>
> > The Wandering Medicine Show
> > KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Then it's a mystery worthy of a Greek tragedy.

--- In tuning@yahoogroups.com, "banaphshu" <kraiggrady@a...> wrote:
> i do ot assume anything obvious about two infinite series and how
they
> interact
>
>
> --- In tuning@yahoogroups.com, "c_ml_forster" <76153.763@c...>
wrote:
> > Hi Kraig. My intention was not to overstate the obvious.
> > Allow me to rephrase: Are your mental processes and aural
> > perceptions always precisely aware of the "infinite"
distinctions
> > between "Apollo" with "Dionysus"?
> >
> > Cris
> >
> >
> >
> >
> >
> > --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...>
wrote:
> > >
> > > yes Apollo and Dionysis were brothers and Greeks understood
how
> > they related and were not antagonistic toward each other.
> > > ( these cultures still has not raised to this understanding)
> > >
> > > if one looks at the leaf of the laurel, Apollo's plant , one
will
> > notice that the veins are not perfectly symetrical at all.
something
> > he learn from his brother.
> > >
> > > Both series fills infinity, hence indistinguishable in the end.
> > > How they get there is another matter
> > >
> > >
> > >
> > >
> > > Message: 16
> > > Date: Tue, 7 Jun 2005 23:32:57 -0400
> > > From: Cris Forster <76153.763@c...>
> > > Subject: Rational/Irrational Asymptote (RIA)
> > >
> > > I prefer to think of a tuning on my canon as a swash of colors
on
> > a
> > > canvas. No one has ever successfully analyzed or described a
> > > painting by extracting and systematically classifying the
colors
> > into
> > > "warm" and "cold" categories. Warm needs Cold. Cold needs
> > > Warm. Rational needs Irrational. Irrational needs Rational.
> > >
> > >
> > >
> > >
> > >
> > >
> > > --
> > > Kraig Grady
> > > North American Embassy of Anaphoria Island
<http://anaphoria.com/>
> > > The Wandering Medicine Show
> > > KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los
Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> yes Apollo and Dionysis were brothers and Greeks understood how they
related and were not antagonistic toward each other.
> ( these cultures still has not raised to this understanding)

Oh yes! What a marvelous analogy. That's exactly what we have here.
Appolonian (rational) and
Dionysian (sensory) definitions of JI.

-- Dave Keenan