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To the tuning members,

Most of you are not acquainted with the M, J, U micro-intervals which determine the 612 musical cells of a non uniform geometric progression that starts on Do = 1 and ends on 2Do = 2. The origins of these three elements are explained in my book.

M = (32805/32768) = 1.00112915039 = Schisma.
J = [(33554432 x 2^1/4)/39858075] = 1.0011313711

The scale proposals I sent you since last October as well as all twelve tone scales I know work with narrow fifths which comply with the Pythagorean comma insertion and in all cases common numbers give the partial factors associated with the narrow fifth to get this especial comma.

Recently, I derived a scale that presents interesting features which might be signs that it is the searched scale. The scale works with six narrow fifths; each one is obviously complemented by a factor to give 1.5. The main feature is that each of the six complementary factors are not common numbers; instead, the square of M and the square of J produce the Pythagorean comma: (M^2)(M^2)(M^2)(M^2)(J^2)(J^2) =
(M^8)(J^4) = Pythagorean comma = 1.013643264.....

Regarding the six narrow fifths, we have:

(2D/G) = 1.49661827761 and (1.5/1.49661827761) = 1.00225957576 = M^2.

(2E/A) = 1.49661163802 and (1.5/1.49661163802) = 1,00226402222 = J^2.

(2F#/B) = 1.49661827761 and (1.5/1.49661827761) = 1.00225957576 = M^2.

(2C#/F#)=1.49661827761 and (1.5/1.49661827761) = 1.00225957576 = M^2.

(2Eb/Ab)= 1.49661163802 and (1.5/1.49661163802) = 1,00226402222 = J^2.

(2F/Bb) = 1.49661827761 and (1.5/1.49661827761) = 1.00225957576 = M^2.

I will post the scale in about three days; let me ask you to study my proposal.

Thanks

Mario Pizarro

piagui@...

Lima, February 2, 2009