The Piagui Musical Scale
Perfecting Harmony
by
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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 each two 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
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
When Tempered and Piagui chord progressions are compared by playing two pianos tuned to these scales, not only does the new scale reveal its flawless chords but also demonstrates how the new intonation reharmonizes the chord progressions. These achievements will be appreciated as soon as a piano tuned to the Piagui scale is available. There are, of course, tho
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About the Book
The
Tempered scale proposed in 1482 as a practical solution to discords was only
introduced and applied 240 years later by J. S. Bach. Since then, this scale
has ruled the tone frequencies in all variety of chords. Due to its simple conception, small imperfections
in harmony are unavoidable.
Now
a new musical scale is proposed, and this book details the new concepts and
features and their application in the manufacture of musical instruments, to
introduce the new sounds in harmony to the world market.
The
Natural Set of forty-seven elements was the beginning of the research. The M comma, the smallest consonance that can
be distinguished by the ear, together with J and U, allowed the attainment of
the Natural Progression of Musical Cells, while its 624 elements led to the
discovery of K and P semitone factors to establish the Piagui
octave. The proper sequence of eight K and four P replace the twelve T factors
of the Tempered intonation.
The
origins of K and P are the ten tone frequencies found in the Pythagoras and Aristoxenus heptatonic scales.
Piagui
and Tempered chord wave peaks of basic twenty-four triads are drawn by computer
to demonstrate the true concords and discords respectively.
About the Author
C. Mario Pizarro’s serious interest in music developed in his university years. He graduated in 1957 as an electrical engineer from the