Reply to Petr

Petr,

The progression is composed by 612 relative frequencies with respect to Do = 1 and any of these frequencies is a DIRECT expression of a tone, tone relation or musical consonance; all the 21 tone relative frequencies of the Extended scale of Pythagoras as well as the 21 tone r. f. of the E. s. of Aristoxenus-Zarlino (1560), the Piagui tone relative frequencies, those Eb, F#, A of the e.t. scale, the Piagui K and P semitone factors, the ancient heptatonic scales of Pithagoras and Just Intonation together with their tone intervals like (256/243) = 1.05349794239, I repeat, all the mentioned scale parameters had to be given in decimal fractions, the same way as many books do, one of them is "Acoustics of Music" written by W. T. Bartholomew (1945).
Conversions to any other numeric system destroys the musical parameters content of the progression and destroys also the possibility of deducing the Piagui system since a group of six cells work as numbers to work out equations K(m) P(n) = 2 and (m + n) = 12. In the first equation, (m) and (n) are exponents.

In order to make it clear, I will copy for you two pages of my book. Next pages contain complex operations in the field of Superior Algebra. I suggest you to set aside the idea that decimal fractions can be replaced by another numeric system. With all my respect I say you forget it.
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Chapter IV

THE PIAGUI MUSICAL SCALE

IV.1 THE K AND P SEMITONE FACTORS

An inspection of the cells of the first segment of the progression shows that part of it is the Pythagorean semitone 256 / 243 as well as 16/15 and 10/9 of the Just Intonation Scale. So, some cells may not only be relative frequencies, but also semitone factors of two consecutive notes of a new musical scale.

The progression may contain two semitone factors that could replace the tempered T to establish ideal tone frequencies in any octave. This possibility would generate slight but important changes in harmony and tone intervals. Let us give the names K and P to the unknown semitone factors that would rule the new octave.

Both K and P should be progression cells, otherwise, their relative values with respect to C = 1 would give non-musical results.

It is presumed that the values of both factors to be deduced, when they lie in the progression, would be placed near the tempered T, due to the slight imperfection of this scale. Instead of the relation T12 = 2 of the tempered intonation, the equation Km Pn = 2 is the one that complies with the octave quadrature, when and only when (m + n) = 12, provided m and n are integer and positive numbers. Together, K and P should not be higher nor lower than T. If there is a solution, one of the factors, P, for instance, must be lower than T and K higher than the tempered factor to establish a new dodecatonic intonation. No cell equals T, it is a non-musical number.

Therefore, the following equations can be stated:

(A) K m P n = 2

(B) m + n = 12

The K and P sequence in the octave is a matter to be discussed should the equations be solved mathematically.

Equations (A) and (B) were supposedly considered in former times. Hundred of years ago, Ramos de Pareja, Zarlino, W. Holder, Delezenne and other researchers probably studied this set of equations with four unknown quantities. Obviously they were never worked out, since more mathematical information on musical elements was needed. Now, the required data work with the Natural Progression of Musical Cells. These equations will be solved in the following pages.

From cell No. 48 to No. 53, having values close to T, we see that Cells Nos. 48, 49 and 50 are lower than T, just as Cells Nos. 51, 52 and 53 are greater than this parameter. This group is indicated below.

CELL COMMA F(M,J,U) DECIMAL VALUES

48 M M30 J16 U2 1. 0558784008

49 J M30 J17 U2 1.05707299111

50 J M30 J18 U2 1.05826893295

51 M M31 J18 U2 1.05946387773

52 M M32 J18 U2 1.06066017178

53 M M33 J18 U2 1.06185781663

T = 21/12 = 1.05946309436

T is slightly lower than cell No. 51, therefore P may perhaps be Cell No. 50 or 49 or 48. Similarly, K may be No. 51 or 52 or 53, which are slightly greater than T.

If the mentioned equations were solved, then the importance and validity of cell progression as a link between science and music would be evident, provided K and P were progression cells. The next step would be defining the sequence of K and P within the octave in order to detect the probable theoretical frequencies of the twelve notes and proceed to further discussion regarding harmony.

Let Pa = (48) Ka = (51)

Pb = (49) Kb = (52)

Pc = (50) Kc = (53)

The next step is to work out equations (A) and (B) by using the combinations [(Ka)m(Pa)n], [(Ka)m(Pb)n], [(Ka)m(Pc)n], [(Kb)m(Pa)n], [(Kb)m(Pb)n], [(Kb)m(Pc)n], [(Kc)m(Pa)n], [(Kc)m(Pb)n], [(Kc)m(Pc)n], since probably one of them may solve the mentioned equations.

The four unknown quantities K, P, m and n may be determined by applying logarithms. The Neperian logarithm has been chosen, to work with a sufficient number of digits to reduce errors. By taking logarithms on both sides of equation (A) we find:

m (ln K) + n (ln P) = ln 2

Therefore, m = [ln 2 - n (ln P)] / ln K

Replacing the above value of m in equation (B), we get:

ln 2 - n (ln P) = 12 (ln K) - n (ln K)

Solving for n:

(C) n = (12 ln K - ln 2) / (ln K - ln P) ln 2 = 0.69314718056

K and P values have to be replaced in equation (C) by cells Nos. 51, 52, 53 and 48, 49, 50 termed Ka, Kb, Kc and Pa, Pb, Pc respectively. This procedure will set up nine auxiliary equations similar to equation (C) where Ki and Pi are parameters to yield nine values for n. When any of these results is an integer number, an important step in the analysis is accomplished.

The following tabulation of ln Ki , ln Pi , and 12 ln Ki gives the values that should be replaced in equation (C). Term Ki represents any of the values Ka, Kb, Kc. Likewise Pi represents Pa or Pb or Pc.

K i ln K i 12 ln K i

Ka = 1.05946387773 0.05776300444 0.69315605339

Kb = 1.06066017178 0.05889151782 0.70669821393

Kc = 1.06185781663 0.06002003121 0.72024037452

ln Pa = ln 1.0558784008 = 0.05437302789

ln Pb = ln 1.0570729911 = 0.05550375948

ln Pc = ln 1.05826893295 = 0.05663449107

By replacing the known values in (C), nine solutions for n are obtained:

(1) n1 = (12 ln Ka - ln 2) / (ln Ka - ln Pa) = 0.00261733963

(2) n2 = (12 ln Ka - ln 2) / (ln Ka - ln Pb) = 0.00392729436

(3) n3 = (12 ln Ka - ln 2) / (ln Ka - ln Pc) = 0.00786230826

(4) n4 = (12 ln Kb - ln 2) / (ln Kb - ln Pa) = 2.99901814321

(5) n5 = (12 ln Kb - ln 2) / (ln Kb - ln Pb) = 4

(6) n6 = (12 ln Kb - ln 2) / (ln Kb - ln Pc) = 6.0039311807

(7) n7 = (12 ln Kc - ln 2) / (ln Kc - ln Pa) = 4.79780024071

(8) n8 = (12 ln Kc - ln 2) / (ln Kc - ln Pb) = 5.99901766761

(9) n9 = (12 ln Kc - ln 2) / (ln Kc - ln Pc) = 8.00262079304

The integer number n5 = 4 leads to important conclusions.

If n5 = 4, then m = (12 - 4) = 8, so that the new octave is obtained by four Pb semitone factors and eight Kb. Auxiliary equation (5) shows that Kb and Pb together with m = 8 and n = 4 solve equation (A).

The semitones K and P of a new musical octave have been sought and worked out:

K = Kb = (Cell No. 52) = (9/8)1/2 = 1.06066017178

P = Pb = (Cell No. 49) = (8/9) ( 21/4 ) = 1.0570729911

K8 P4 = [ (9/8)1/2 ]8 [ (8/9)( 21/4 ) ]4 = 2

Contrasted with T = 21/12 = 1.05946309436, the K and P values show slight discrepancies.

These remarkable results confirm that the Natural Progression of Musical Cells establishes the scientific base of the art of music. However, until it is proved that K and P resolve the harmony problem, they cannot be accepted as providing the desired solution. Their validity will depend on chord evaluations.

Whereas Cells Nos. 52 and 49 could be the semitone factors of the best musical scale, the tone relative frequencies with respect to note C will depend on their sequence within the octave. The sequences need to be analyzed, since tone frequencies and chords will depend on the K and P arrangements within the octave.

IV. 2 THE K AND P SEQUENCE IN THE NEW OCTAVE

The eight K and four P factors comply with proper distribution within the octave if the same factor sequence is maintained from note C to B. When an arrangement of twelve semitone factors is attempted, the idea of four identical groups arises, since there are four P factors working within the octave.

Elements of each group: K, K, P.

Number of groups = 4

Semitone arrangements: KKP, KPK and PKK.

Following are the three types of Piagui octaves:

I) KKP KKP KKP KKP = 2

II) KPK KPK KPK KPK = 2

III) PKK PKK PKK PKK = 2

Four groups of semitone factors comprised by KKP, KPK and PKK rule the relations of tone frequencies in the new octaves of Piagui I, Piagui II and Piagui III scales respectively. However, the new octaves work simultaneously. In fact, the set KKPKKPKKPKKPKKP shows that Piagui II starts from the second K and Piagui III from the first P.

I was not speaking about common fractions, I was speaking about logarithmic conversions. If you want to divide the octave into 612 steps, then one step is ~1.9607843 cents, which makes, when multiplied by 612, 1200 cents, which is the octave. Similarly as the 612-equal scale divides the 12-equal semitone into exactly 51 steps, so do cents divide it into exactly 100 steps. You can read more here: http://tonalsoft.com/enc/c/cent.aspx
> < The relative frequencies of notes D, E, F, G, .....with respect to Do of the Just Intonation Scale: 9/8, 5/4, 4/3, 3/2 ..... (Most of this fractions are also used in the Pythagoras scale as you know) are normally used when necessary.

> < Since there are various "major second" or "major third" depending on the scale, some times I have to precise their values.

Calling an interval by a note name does not specify it any further than calling it by an interval name.

> < To deduce and discern about the Piagui system the only numeric system that can be used is the fractional decimal; the cents system cannot be applied.

Why not?

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I hope that now you understand WHY I must use decimal fractions and no other numeric system.

Thanks

MARIO PIZARRO

piagui@...