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.
INTRODUCTION
I first got involved with musical scales after reading W. T. Bartholomews Acoustics of Music, Joaquin Zamacois, Teoría de la Música, M. Caillauds, Notions DAcoustique, Julian Carrillos 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, Chopins Fantasia Impromptu and Rimsky-Korsakovs 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.
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.
INTRODUCTION
I first got involved with musical scales after reading W. T. Bartholomews Acoustics of Music, Joaquin Zamacois, Teoría de la Música, M. Caillauds, Notions DAcoustique, Julian Carrillos 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, Chopins Fantasia Impromptu and Rimsky-Korsakovs 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.
