1 x (Q R Q Q Ro Q) (Q R Q Q Ro Q) (Q R Q Q Ro Q) (Q R Q Q Ro Q)...= 2, where
Q= (M M J J M M M M J J M M M M J J M M)
R= (J J M M U U M M J J M M M M J J)
Ro = (J J M M M M J J M M U U M M J J) (Inverse sequence of R).
The successive products of the first 612 commas give 2 Do = 2.
M= (32805/32768)=1.00112915039 (The smallest comma)
J=1.0011313711..
U=1.0012136965..
It was guessed that some progression frequencies may not only be tone frequencies of a new musical scale, but also semitone factors of two consecutive notes of this scale. The names K and P were given to the Piagui semitone factors that rule the new octave. Their values are placed near the Tempered T= 1.059463094.., due to the slight imperfection of the equal-tempered scale.
The equation (K x K x K...m times) (P x P x P....n times) = 2, (First equation), is the one that complies with the octave quadrature, when and only when (m + n) = 12, (Second equation), provided m and n are integer and positive numbers. P must be lower than T and K higher than the tempered factor.
The two equations given above, contain four unknown quantities: K, P, m, n. The additional data required to work out the equations were deduced by using the information contained in the progression, as detailed in my book:: “The Piagui Musical Scale: Perfecting Harmony”.
The graphs shown in the book show and evaluate the quality of harmony of all major and minor triads of both, Tempered and Piagui scales. The book also deals with the application of the new intonation to the electronic organ, guitar and electronic tuner.
The Piagui octave was finally worked out:
K x K x P x K x K x P x K x K x P x K x K x P = 2 , where m = 8 and n = 4
Sufficient data is given for tuning the piano to the top quality musical scale.
The Piagui scale is not an invention; it is a discovery that resolves the slight discordance problem of the equal-tempered scale.
C. Mario Pizarro
[email]piagui@ec-red.com[/email]
http://www.piaguimusicalscale.com