The roots

The roots of authentic musical elements, that is, the smallest distinguishable interval called schisma whose symbol is M and the new commas J and U ones I detected, define the Progression of Musical Cells, an able set of 624 relative frequencies from note Do = 1 up to (9/8)6. Twelve frequencies before, works the cell Nº 612 which corresponds to the octave 2 Do = 2. A theoretical infinite number of octaves can be coupled to the first one. 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.

The new comma U exhibits peculiar properties. It is remarkable that the product of four classical frequencies formed by a comma, a semitone and two tone frequencies of ancient scales and multiply this product by the new comma U = 1,0012136965065., the square root of 3 is obtained. In fact, by taking the following musical parameters:

ZC = Zarlino comma = (81/80) = 1.0125

ZS = Zarlino semitone = (16/15) = 1.06666666..

D = Pythagorean and Just Intonation note D = (9/8) = 1.125

EAZ = Note E of the Aristoxenus-Zarlino scale = (5/4) = 1.25

Let us determine the product of the following functions of ancient musical values:

[ (ZC)2 (ZS) D2 EAZ ] = 1.02515625 x 1.066666.x 1.265625 x 1.25 = 1.72995117186

By multiplying the product obtained by the new comma U = 1,0012136965065., we get:

1.72995117186 x 1,0012136965065 = 1.73205080755 = 31/2 = Square root of 3.

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The connection between progression cells and useful numbers follows:

Since cell Nº 484 equals to M298 J168 U18, therefore:

(M298 J168 U18) U = (M298 J168 U18) (1,0012136965065) = (M298 J168 U19) = (M299 J168 U18) (U/M) =

(Cell number 485) (U/M) = 1.73205080755 = 31/2 = Square root of 3.

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Similarly:

2(Cell Nº 98) M1/2= 2 (1.11740330854) M1/2= 2 (1.11740330854)(1.00056441591) =2.2360679775 = 51/2

Another peculiar result is that values 21/4, 21/2, 23/4, which are the equal tempered frequencies of notes Eb, F#, and A, are progression cells whose values are accurately defined by the schisma M and the commas J and U.

The progression also followed the ancient Pythagoras and Aristoxenus heptatonic scales and their limited information for detecting K and P semitone factors. When using 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 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 equal tempered T and rule the harmony of the new musical system. Their precise and suitable values determine that perfect fifths and perfect fourths link most tone frequencies on the piano keyboard in cycles. These unexpected and remarkable results made possible the attainment of the best expressions of harmony.

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 equal tempered sinusoidal components yields non-aesthetic chord wave peaks, except for one diminished chord.

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.

MARIO PIZARRO

Lima, July 18, 2008

piagui@...

Mario,

> 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.

Music does not have a problem. Maybe some tunings do, but the best
music transcends this. Balinese and Javanese gamelans don't have a
problem. Georgian liturgical singing does not have a problem. North
and South Indian musics do not have a problem.

I'm glad you've found a tuning that *you* find can improve certain
kinds of music, but please stop with the idea that Music itself has a
problem. Because, in fact, music is more than tuning.

Mario wrote:

> ZC = Zarlino comma = (81/80) = 1.0125

> ZS = Zarlino semitone = (16/15) = 1.06666666..

> D = Pythagorean and Just Intonation note D = (9/8) = 1.125

> EAZ = Note E of the Aristoxenus-Zarlino scale = (5/4) = 1.25

#1. It's not good to say that "5/4 is E" and "9/8 is D" and so on. Why can't you just say "major second" or "major third"? If you want to label 5/4 as E, then you have to also specify the pitch of the 1/1 as C, which is what you probably meant but didn't say.

#2. Why are you still writing the factors as decimal numbers? It would be much better understandable for most of us if you said "5/4 = ~386.314 cents" instead of "5/4 = 1.25". Decimal numbers are not of much use here.

#3. I still don't understand what is the interval that you call "the new comma U".

Petr