Rational/Irrational Asymptote (RIA) Revisited

In subsequent personal posts on the meaning of friendship and
fair play, I am very happy to note that the RIA model served as a
catalyst for reconciliations. In the context of two (or more)
infinities, it seems to me that the subject of proximities is
unavoidable.

(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:

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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
32/31: (1000+968.8)/2 = 984.4……..32/31: _|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.

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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

Illustrations have their magic.

>
>Message: 9 > Date: Thu, 9 Jun 2005 11:26:02 -0400
> From: Cris Forster <76153.763@compuserve.com>
>Subject: Rational/Irrational Asymptote (RIA) Revisited
>
>In subsequent personal posts on the meaning of friendship and >fair play, I am very happy to note that the RIA model served as a >catalyst for reconciliations. In the context of two (or more) >infinities, it seems to me that the subject of proximities is >unavoidable.
>
> >
>
>
>
>
>
> >

--
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.

As builders of acoustic musical instruments, we both know that the
best tools are the simplest tools.

Cris

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> Illustrations have their magic.
>
> >
> >Message: 9
> > Date: Thu, 9 Jun 2005 11:26:02 -0400
> > From: Cris Forster <76153.763@c...>
> >Subject: Rational/Irrational Asymptote (RIA) Revisited
> >
> >In subsequent personal posts on the meaning of friendship and
> >fair play, I am very happy to note that the RIA model served as a
> >catalyst for reconciliations. In the context of two (or more)
> >infinities, it seems to me that the subject of proximities is
> >unavoidable.
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
>
> --
> 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