Talking

To Brad Lehman,

Your analyses and comments contain the needed explanations. I see that you know more than I about the Piagui scales and your opinions show that you are a sharp - sighted man for detecting the basic and hidden features of the new system.

Mike Battaglia; wrote that "the only intervals in the Piagui scale that will be distinguishable from equal temperament will be the ones involving the 696 cent flat perfect fifths, and those will be distinguishable in that they will be MORE out of tune than equal temperament, and not less."Mike perfectly knows that the inclusion of cents is unavoidable in all kind of scales to comply with twelve tones per octave. In the variant Piagui II, eight tones have only plus or minus 2 cents and four tones work with 0 cents, practically, here the out of tuning does not exist. The other two variants show eight tones with 0 cents; other eight with +2 and +4 for Piagui I and Piagui III with ?2 and ? 4.

Mike takes again the reference of the failed equal tempered scale where ALL the tones are flat, it was criticized by its own adopter J. S. Bach and he also states that some of the Piagui flat fifths will be MORE out of tune than equal temperament and not less.

I apologize for using in the following lines some references that have been exposed in my book "The Piagui Musical Scale: Perfecting Harmony" to discuss this point.

Since the Piagui chords are mainly based on K and P semitone factors, one of its characteristics is the exact tuning of tone frequencies derived from products and quotients of K and P, so I will take the figures involved in the flat fifths to demonstrate that these frequencies are perfectly tuned:

(1,49492696 / 1,25707872) = 1,189207115 = Perfect Minor Third = K(2) P

(1,49492696) = K(4) P(3) = G (Piagui III)

(1,25707872) = K(2) P(2) = E (Piagui III)

(1,5 / 1,49492696) = (K(5) P(2) / K(4) P(3)) = (K / P)

The above relations conclude that the frequencies and quotients that define the needed flat fifths that are criticized by Mike are not common numbers, otherwise the triad chord wave peaks would not have been aesthetic responses.

Should you like to get some graphs of Piagui triad responses let me know it for sending you the graphs you want. They are photos of the harmony.

Regards

MARIO PIZARRO

Lima, July 8, 2008

> Mike perfectly knows that the inclusion of cents is unavoidable in all kind
> of scales to comply with twelve tones per octave. In the variant Piagui II,
> eight tones have only plus or minus 2 cents and four tones work with 0
> cents, practically, here the out of tuning does not exist. The other two
> variants show eight tones with 0 cents; other eight with +2 and +4 for
> Piagui I and Piagui III with －2 and － 4.

Yes.

> Mike takes again the reference of the failed equal tempered scale where ALL
> the tones are flat, it was criticized by its own adopter J. S. Bach and he
> also states that some of the Piagui flat fifths will be MORE out of tune
> than equal temperament and not less.

What?

> The above relations conclude that the frequencies and quotients that define
> the needed flat fifths that are criticized by Mike are not common numbers,
> otherwise the triad chord wave peaks would not have been aesthetic
> responses.

You mean they would have been.

-Mike