Re: Rational/Irrational Asymptote (RIA) [Correction: 32/31]

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:

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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