New and perfect musical scale--The PIAGUI SCALE

Mario Pizarro

New member
To Frederik Magle,
My best regards Mr. Magle, my name is Mario Pizarro. After twelve years of
research and good results, let me offer to you the tone frequency ratios and details of the PIAGUI middle octave. From Lima, PERU, the best wishes to you and to your company.
I first got involved with musical scales after reading W. T. Bartholomew’s “Acoustics of Music”, Joaquin Zamacois’, “Teoría de la Música”, M. Caillaud’s, “Notions D’Acoustique”, Julian Carrillo’s “Sonido 13” and other fascinating works. Authors usually start analyzing the theme by studying the works of Pythagoras and Aristoxenus, Zarlino, William Holder, Ramos de Pareja, Delezenne and other well known researchers on the improvement of musical scales.
All of us pay well-deserved homage to Pythagoras for his heptatonic scale set down about 550 BC. Nearly 250 years later, Aristoxenus conceived a second heptatonic scale, also known as the scale of physicists and geometricians. The work of these great researchersenabled the incipient audience of that era to listen to melodies that had only seven pitches per octave. Then, at the beginning of the second millenium, sharps and flats were added to the heptatonic scales extending them to twenty-one pitches per octave, based on the Pythagoras and Aristoxenus scales. At about 1560, Zarlino set out the “commas”, such as (81/80) = 1.0125 and Dr. Delezenne deduced the chromatic semitone 135/128. In 1985, the smallest comma known was used: 1.00112915 = (32805 / 32768) = M, obtained by dividing the Delezenne and Pythagoras semitones, that is (135/128) ÷ (256/243), which played a fundamental role in the establishment of an eminently good scale. Considering the transcendental contributions of Zarlino and Delezenne, they must also be included in our homage to the masters.
In 1957, I had the opportunity to talk to one of the greatest guitar players of our time, Andrés Segovia, in Lima during his concert tour period. I asked him if he was satisfied with the quality of guitar chords. He wanted to know why I asked this question. I told him that I had detected discords when playing the guitar, and that I could perceive a slight incoherence in the tone frequencies. He replied that he did not approve such chords either, but that it was an old problem and he hoped to see it solved some day.
Twenty-five years later, I decided to study the harmony problem. I found interesting information in books from the National Library, but solving the problem seemed to require about as much work as finding the Philosophers’ stone. In 1482 Ramos de Pareja proposed a dodecaphonic scale, named “Tempered”, as a practical solution to imperfect chords. The sole semitone factor T of the Tempered scale set the same frequency relation between any two consecutive notes within the octave. J. S. Bach introduced the Tempered scale in 1722.
From 1994 to 1997, I wrote to manufacturers of musical instruments abroad, trying to convince them of the advantages of using a piano tuner ruled by the new scale. I also sent them some computer-plotted graphs for comparing Tempered and “Piagui” chords, a term used for the new intonation. Some of them replied, congratulating me on my endeavor to resolve the harmony problem, but that was all. At that time, I realized that the design of electronic organ keyboards and electronic tuners interrupted the use of all the hardware in their designs and products. Most manufacturers were already using software.
After long years of research, I finally decided to set down my findings and started writing this book in June 2002. I apologize for including so many mathematical reasoning. I had to avail myself this opportunity to provide the full and complete details on the subject.
A well known analyst once stated, somewhat pessimistically: “Musical scale is not one, not natural or even founded necessarily on laws of constitution of musical sound, but very diverse, very artificial and very capricious”.
Chapter I of this book contain opinions and criticisms of the Tempered scale by notable analysts who, apparently, are resigned to listening forever to discords. I hope that the research and analyses described herein will contribute to perfecting musical expression.
I have set down here new concepts of the scientific basis of a new musical scale and its application in the manufacture of musical instruments. Universities, musical students and analysts will find interesting information regarding micro-consonance and harmony, but it is desirable that the reader have some familiarity with basic concepts of music, moderate competence in mathematics and an elementary knowledge of physics.
The roots of authentic musical elements, that is, the smallest comma M and the new J and U ones I detected, define the Natural Progression of Musical Cells, an able set of 624 relative frequencies from note Do = 1 up to (9/8)6. Since several features of this progression acknowledge it as a scientific source of natural consonance, it deserves to be included in the acoustics field of physics.
A concise explanation of the ancient Pythagoras and Aristoxenus heptatonic scales was made, emphasizing intervals between eachtwo consecutive notes. The treatment is based on limited information from the ancient scales, as well as on data contained in the Natural Progression of Musical Cells for detecting K and P semitone factors.
When used properly, the combined work of K and P can set the needed twelve-tone frequencies for any octave, these being the most suitable for yielding aesthetic complex waves when the chord tone frequencies are computer-plotted and added to show perfect harmony. As most countries now use 440 cycles per second for note La, it is one of the pitches of the new middle octave. The K and P semitone factors replace the tempered T and rule the harmony of the new musical system. Their precise and suitable values determine that perfect fifths and perfect fourths link all tone frequencies of the piano keyboard in cycles. These unexpected and remarkable results made possible the attainment of the best expressions of harmony.
Mathematical analyses and the proposal presented here were made to resolve the problem of the slight discordance produced by the Tempered scale. Some audiences who are able to distinguish discords detect small harmony imperfections.
Chapter VIII analyzes the harmony of the Tempered and the Piagui scale. Chord waves, as well as chord wave peaks, are plotted to compare them and decide on the qualities of harmony. An examination of Piagui chord wave peaks detected aesthetic displays. Compared with the new chords, where frequency ratios are exclusively K and P functions, the sum of tempered sinusoidal components yields non-aesthetic chord wave peaks, except for one diminished chord. These are the origin of the discords that humans have endured since 1722.
Chapter V deals with the application of the new scale to the piano, electronic organ, guitar and electronic tuner. Sufficient data is given to permit manufacturers of musical instruments to introduce the new sound in harmony on the world market.
On October 4th, 2003, the first string of the available Piagui guitar was tuned to obtain a 440 Hz corresponding to the note La. Then, by audible comparisons, the twelve reference tones of the middle octave on a grand piano were tuned and the tones extended throughout the keyboard. When the pianist Luis E. Colmenares-Perales played the first of the classic works listened to that evening, he said enthusiastically:"Not only the harmony but everything is clearly more pleasant than what is heard on a piano tuned to the Tempered scale".
Major performers on the piano, electronic organ, guitar, cello and other instruments will be familiar with the new harmony produced by the new tone frequencies of C#, D, E, F, G, Ab, Bb and B.
The Piagui scale can be distinguished from the Tempered scale when musical chords are maintained for the few tenths of a second required by the brain to classify the harmony. Adagios, nocturnes, serenades and many, but not all, classical and selected pieces of music show the difference in the new musical system. However, Chopin’s Fantasia Impromptu and Rimsky-Korsakov’s Flight of the Bumblebee do not demonstrate this difference.
The Piagui Musical Scale is not an invention; it is a discovery based on years of research on micro-consonance to resolve a problem that has existed since music was born.
Mankind has sought and achieved, over millenia, the perfection of almost all things that are linked to him. Musical harmony, with which we have lived for many centuries, was achieved and had only a small degree of imperfection left to solve. From now on, we can say that this problem no longer exists.


John Watt

looks like notes to me

Mario Pizzaro! What a historic and scientific posting! I read it all. What are you trying to do? I think I understand. You are beginning to see tempered tuning for what it is, and starting to see scientific tuning.

I'm totally ignoring notes, scales and chords. They can be anything, and are a byproduct of tuning.

As far as I have seen, scientific tuning started with Leo Fender and his invention of two-way adjustable individual bridges for each string. For the first time in strung history, you could position a string and tune the harmonics scientifically. I haven't seen an acoustic guitar built with this tuning ability. You might think an electronic instrument with notes activated by key contact would be scientifically tuned. But it's not tuned, it's built. Tuning involves temperature, atmosphere, humidity and heat. A lot of rock bands, the biggest entertainment business in their day, had to tune to synthesizers because they would change day to day or when they got hot under the lights. That's not even tempered or scientific tuning, but it sounds the same for most people. It doesn't matter if you are talking amplifier effects onstage or digital and computer recording, if you tune scientifically everything is more precise, occupies it's space cleanly, and permits finer tuning permutations.

Instead of boggling your mind, and the minds of innocent professionals, with copyous amounts of technical trivia, you should get a computer symphony program where you can input music you love and use editing to clean up each note, removing the tempered quality. That wouldn't take thirteen years as you have so far, but it would be a tangible listening experience that would demonstrate, firstly for you, the difference between tempered and scientific. But be careful! Your system might reduce the overall sound envelope and leave you with a completed, but tonally thin work that would impress no-one. If you don't want to do that, you could start with violins, making adjustable metal bridges so they can tune scientifically. Once you've got all the violins, the other instruments will follow. The paradox here is that when musicians aren't tempered, playing tempered tuning pieces, the overtones and harmonics won't work as much, and if the original composer was here to listen, he would add more notes to compensate, or further his expression. Since dead composers aren't available, interest in such performances would be negligible. However, you could patent and copywrite your detempered music as your own, just like film colourists own their product and don't pay royalties. You could rerelease anything with acoustic instruments as your own product, perhaps a payday with more potential than retuning the world as we know it.

When Leo Fender further invented the Stratocaster, he added a tremolo unit to help imitate the action of steel guitars, which loosened or tightened strings with pedal action. He wasn't thinking about loud, loud amplifiers with effects, so scientific and penetrating, so reactive to the environment, that this tremolo would be used more for slight detuning to stabilize and control sounds. I hope you can understand how being in tune goes beyond tuning when you deal with live musical situations. If this continues to frustrate you, maybe you can try to deal with this question along the way. When you are singing or playing a song, how do you know where you are? And to further this concept, when you are spontaneously improvising, how do you know which note or chord you will play next?
Answers to these and other mean tempered tunings can be found here, now, on Magle International Forums.
Thread properties: It took one Watt to type this.
This is a fascinating subject. As a person who has a lifelong interest in music and music theory I too have ideas about this.

I think some of the most basic assumptions about music as a science are simply wrong. And the errors need correction.

For example (and this is perhaps relevant) one of the first things taught at music school is that a given note of fixed pitch produces a series of harmonics. Now, that, of course, is true. Nobody doubts it. Standing next to a bell one hears lots of different harmonics. But then we are told (often in the same breath) by the same teachers that the number of harmonics produced by such a single vibrating note are 'infinite'. And here is the problem.

It seems quite obvious to me that the number of tones created by a single tone cannot possibly be infinite. How can a thing which begins be 'infinite' ? It cannot. So the number of harmonics MUST be finite. And yet the dogma remains of an 'infinite' number of harmonics coming from a single note.

So what ? Well, if the number of harmonics coming from a single note are FINITE in number we might ask - 'How do you explain the huge number of harmonics' ? The answer is of course that these other harmonics are produced from the interaction of these basic harmonics, later in the process. But not instantly.

And therefore, contrary to the textbooks, there may be, in fact, only a small number of actual harmonics, which, in turn, create others. But not instantly.

And, as every colour in painting can be produced from a small number of prime colours in combination so too every musical note can be produced from only a very small number of musical frequencies in combination.

The future of music may be far more simple and profound than we think. Musicianship of the future may consist of musicians together creating harmonics. Of learning to create them as part of the music making process. It may be the new skill of the future for composers and musicians to learn these things as they really are.

It seems to me that the interval of the octave is both the simplest and most profound interval in all of music.
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John Watt

Sometimes with music you can think beyond the infinite to the eternal. Those vibrations translate into energies beyond our hearing, or that of any other mammal. And your description of an octave is correct, being what defines our human hearing. If we think of human hearing as say a five octave range, imagine five octaves as just a linear measurement on a vibrating thread of hearing potential. What sounds we hear make us as invidividuals, on a primal scale, more than words.


New member
Sorry to necropost, but something that would really help the adoption of the scale is a YouTube video with a recording of the same song in both Tempered and Piagui. Many of us who do not know how to tune our pianos would love to hear what the Piagui scale sounds like, but we can't because there aren't any good comparison samples available on the internet.
Also, it would be awesome if you would post the Piagui frequencies (440hz, 495hz, etc.) online so that people can tune their pianos to the scale.
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John Watt

Thank you, sweyn78! While I detracted from the overall concept,
what you're saying is the only actual way to hear and assess this.

And.... more than that, I like your term "necropost". I'm gonna get into that some more.
Sometimes I type after myself, seeing my name as a last poster and thinking it's not adding anything.
Why I'm so shy and worried about broadband online just must be part of my Watt electrical affinity.
Just not all good.