#### Mario Pizarro

##### New member

The smallest comma M= (32805/32768)=1.00112915039 and the new J and U ones I detected, define a progression of 612 relative frequencies from note Do = 1 up to 2Do=2.

The treatment is based on limited information from the ancient scales, as well as on data contained in the progression for detecting K and P semitone factors that rule the harmony of the Piagui scale and replace the Tempered T.

The combined work of K and P can set the needed twelve-tone frequencies for any octave. Their precise and suitable values determine that perfect fifths and perfect fourths link all tone frequencies of the piano keyboard in cycles. These remarkable results made possible the attainment of the best expressions of harmony.

The graphs given in the book show the quality of harmony and let the evaluation of all major and minor triads of both, Tempered and Piagui scales. It also deals with the application of the new scale to the piano, electronic organ, guitar and electronic tuner.

The equal-tempered octave is ruled by 1xTxTxTxTxTxTxTxTxTxTxTxT = 2 while

1 x K K P K K P K K P K K P = 2 corresponds to the Piagui octave. The values of semitone factors K and P are also deduced in the book.

Sufficient data are 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

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