The validity of K and P Piagui factors

Dear friends,

Today I found that Piagui semitone factors K = (9/8)^(1/2) = 1.06060017178.... and P = (8/9)^[2^(1/4)] = 1.05707299111... that defined the three Piagui scale variants are also working in the classical twelve tone sets derived from the ancients greek seven tone scales.

In fact, one of the well known scales is formed by the following tone frequencies and semitone factors:

C = 1
C# = (3/4) [2^(1/2)] = 1.06066017178....(9/8)^(1/2)
D = (9/8)
Eb = (32/27) = 1.185185185....
E = (8/9) [(2)^(1/2)] = 1.25707872.....
F = (4/3)
F# = (2)^(1/2) = 1.4142......
G = (3/2)
Ab = (9/8) [(2)^(1/2)] = 1.59099025.....
A = (27/16) = 1.6875
Bb = (16/9) = 1.77777.....
B = (4/3) [(2)^(1/2)]
2C = 2

The semitone factors of this scale are given as folows:

(C# / C) = 1.0606601717... = Piagui K
(D / C#) = 1.0606601717...= Piagui K
(Eb / D) = (256 /243) = 1.053497942.....= R
(E / Eb) = 1.0606601717...= Piagui K
(F / E) = 1.0606601717...= Piagui K
(F# / F) = 1.0606601717...= Piagui K
(G / F#) = 1.0606601717...= Piagui K
(Ab / G) = 1.0606601717...= Piagui K
(A / Ab) = 1.0606601717...= Piagui K
(Bb / A) = (256 /243) = 1.053497942.....= R
(B / Bb) = 1.0606601717...= Piagui K
(2C / B) = 1.0606601717...= Piagui K

Semitone factor R works in the Pythagoras scale while Piagui scales do not use this factor; both K and P were calculated. The scale given above complies with the following relation:

[K^(10) R^(2) = 2

Since the Piagui scale variants work with K and P semitone factors, a similar relation is given below:

[K^(8) P^(4) = 2

The point is that (K^2) (R^2) = (P^4), hence:

[K^(10) R^(2) = [K^(8) P^(4) = 2

Factors K and P were calculated in my book "The Piagui Musical Scale: Perfecting Harmony", so what explained above proves that both K and P semitone factors are valid and useful musical constants.

Since a few weeks ago, about 11 files containing interesting information on the Piagui system were placed.
I suggest you to take a look at them and see how the harmony of major and minor equal tempered and Piagui triads are shown as if they were photographed.

Thanks

Mario Pizarro

piagui@...

Lima, August 04, 2009