Features of the eight schismas scale

To the tuning members,

A new scale proposal.

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.

If we call M and J to the corresponding Schisma and comma that operate in the six

narrow fifths detailed below, they have the following values:

M = (32805/32768) = 1.00112915039 = 1.95372 cents = Schisma.

J = [(33554432 x 2^1/4)/39858075] = 1.0011313711 = 1.957561 cents =

Consonant comma.

The new scale tone frequencies follows:

C = 1........... = 0 Cents

C# = 1.05826893294 .. = 98.04756.. = About -2 cents from ETT.

D = 1.12246370821.. = 200.00256 = About 0 cents from ETT.

Eb = 1.18786320091.. = 298.04243.. = About -2 cents from ETT.

E = 1.25991918672 .. = 399.997439 = About 0 cents from ETT.

F = 1.33333333333.. = 498.045 = About -2 cents from ETT.

F# = 1.41421356237.. = 600......= 0 cents from ETT.

G = 1.5 ......... = 701.955... = About +2 cents from ETT.

Ab = 1.58740339942.. = 800.00256 = About 0 cents from ETT.

A = 1.68369556231.. = 901.95756.. = About +2 cents from ETT.

Bb = 1.78179480135... = 999.99743 = About 0 cents from ETT.

B = 1.88987878008.. = 1101.95244 = About +2 cents from ETT.

2C = 2........... = 1200..... = 0 cents from ETT.

It is 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]

All twelve tone scales operate with narrow fifths and the Pythagorean comma which are

given by common numbers or cents.

The eight schismas scale 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.....

The first six lines of the following tabulation correspond to the six narrow fifths.

We can see that they are distanced from 1.5 by M^2 or by J^2.

(2D/G) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(2E/A) = 1.496611638 and (1.5/1.496611638)=1,002264022= J^2.

(2F#/B) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(2C#/F#)=1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(2Eb/Ab)= 1.496611638 and (1.5/1.496611638)=1,002264022= J^2.

(2F/Bb) = 1.496618277 and (1.5/1.496618277)=1.002259575= M^2.

(G/C) = 1.5 Pure fifth

(A/D) = 1.5 Pure fifth

(B/E) = 1.5 Pure fifth

(Ab/C#) = 1.5 Pure fifth

(Bb/Eb) = 1.5 Pure fifth

(2C/F) = 1.5 Pure fifth

For the first time, all narrow fifths are distanced from 1.5 by the

squared schisma (M^2) and by the comma (J^2) which

are musical parameters. These features really confirm the musical

validity of the eight schismas scale. Its numerical proximity to

the equal tempered scale is understandable since the e-tempered

set is not a badly system.

More information regarding the six narrow fifths:

1,00225957576 = M^2 = 3,90744157266 Cents

1,00226402222 = J^2 = 3,91512206388 Cents

1,00225957576 = M^2 = 3,90744157266 Cents

1,00225957576 = M^2 = 3,90744157266 Cents

1.00226402222 = J^2 = 3,91512206388 Cents

1,00225957576 = M^2 = 3,90744157266 Cents

Total=(M^8)(J^4)=Pythagorean comma = 23,4600104 Cents

Thanks

Mario Pizarro

piagui@ec-red.com

Lima, February 02, 2009