To Lisa Kalmar,- Dear Lisa, 1)Yes,since 1722 (J.S.Bach, who introduced the tempered scale for improving harmony)the humans are still listening imperfect harmony. For want of another system to find the solution to perfect chords and due to its small imperfections,the tempered scale still exists in every kind of instrument.The majority of audiences do not detect its tiny imperfection.If they have not heard better chords there is no reason to complain or to expect harmonious ones.--Some researchers confirmed the T. scale imperfections.For instance,W.T.Bartholomew "Acoustics of Music"wrote:"Equal temperament enables us to play equally well,or perhaps we should say equally badly,in all keys"--"Thus the "tuning" of a tempered instrument is in reality a process of controlled mistuning"----2)The T.scale is an arithmetical scale while the Piagui intonation is based on scientific parameters related with micro-consonance. There is only one T. scale.--Musical harmony sounded badly before J.S.Bach(1722)--I do not feel well knowing that thousands of piano teachers are playing imperfect chords and at the same time my book is trying to open the savage jungle and tell them how to tune the keyboard to achieve perfect harmony.Page 76 of my book gives the correct twelve tone frequencies (HZ) of the middle octave where note A keeps the international value of 440 HZ. Unfortunately, I could not attach to post (4) the Tempered and Piagui Chord Wave Peaks where you can SEE both kinds of harmony:Non-aesthetic and aesthetic responses respectively. These responses are only given in my web site and 24 of them in my book obviously. I am not allowed to give you my website address via The Piano Club D.F.-- However, if you get the approval of Jon,the administrator, I can send you the chord wave graphs by email.
I hope that the above explanations satisfy your expectancy.Merry Christmas to all of you.
Regards
Mario
Lima,December 13, 2005

I am aware that the tempered scale of Bach is imperfect. When I trained in the Kodaly method, we NEVER used the piano to teach a vocal piece. My teacher went on and on about how the piano is basically out of tune and to never play something on the piano and ask the children to sing it back. Instead, sing it to them. When we would sing "ti" for example, we would pitch it slightly higher than on the piano as we sing up, and sing it slightly lower coming down. Singing "between the tones" was very important. By singing a pitch slightly higher or lower than on a piano, our choir had the most ethereal sound and rave reviews. Our teacher said it had to do with the overtones, and like Lisa, my mind balks at going there. I presume orchestras do this also, as they are not tied down to the well-tempered scale of a keyboard instrument. Those of us who play the piano are so used to our slightly out of tune scale that I don't know if we would be interested in changing. Perhaps when Bach introduced well-tempered tuning, he was also met with skepticism.

I used to play with an orchestra. For those that don't know- orchestra's don't tune pitch with a piano. They tune to an oboe which is a different pitch all together since it is a woodwind. Conductors have 440 pitch tuners that take in the entire orchestra at once (or by section if they prefer) to tune to perfect 440 pitch of an A. If they are performing as a Baroque society they tune to lower 438 pitch range due to the tonality of the music and lowness of sound. If they use electronic instruments at all in their performances such as a professional synth or keyboard set up it is typically tuned and pitched to 441 since the frequency of sound is as bit higher on electric/electronically used instruments.

Well, I'm finding this all very interesting and thank you for explaining things in laman's terms.

To Susan,-- I agree with you, the pitch of note "ti" on a tempered piano sounds slightly lower so the correct pitch must be slightly higher. In fact, the pitch of the tempered "ti" in the middle octave is 493.8833 vibrations per second while the Piagui "ti" is : 495.--The pitch of the Piagui notes C#,D,E,F,G,Ab,Bb and B are slightly higher than the tempered ones while the pitch (HZ) of C,Eb,F# and A are the same in both systems.--I hope that you play a piano tuned to the Piagui scale very soon. If so,I know what intonation you will prefer.
C.Mario Pizarro--Lima,12-13-2005