More advanced information

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 = Consonant comma.

The scale proposals I sent you before as well as all twelve tone scales I know,
contain narrow fiths and the Pythagorean comma. In all cases common numbers
give the value of this especial comma.

Recently, I derived a scale that presents interesting features which might be signs
of the searched scale. The scale works with six narrow fifths; each one is
obviously complemented by a factor to give 1.5. One of its interesting features
concerns to the six complementary factors which are not common numbers;
instead, the square of M and the square of J factors 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.

..........................................................................................----------------------------------------

.....................................................................................Pythagorean comma = (M^8)(J^4)

It is also remarkable that the twelve tone relative frequencies comply with the

following ratios:

(D/C#) = (E/Eb) = (F#/F) = (G/F#) = (A/Ab) = (B/Bb) = (9/8)^(1/2)

C# =(Eb/D)= (F/E)= (Ab/G)= (Bb/A)= (2C/B)= [(9/8)^(1/2)]/[(Schisma)^2]

Tomorrow I will post the scale and I hope you will study the proposal.

Thanks

Mario Pizarro

piagui@...

Lima, February 2, 2009