CORRECTIONS

CORRECTIONS IN THE PRECEEDING MESSAGE:
PLEASE DELETE THE PHRASE
"Nature decided two ways of making good scales, only two."

The two ways I meant are:
-- By determining the octave tones as normally do the groop members.
-- By determining the two semitone factors like K and P (uncommon way).
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In "regarding the fifths", please replace "pure" by "sharp" and delete
"pure = 702 cents (1,5)"
-------------------------------------------------------------------------------------------

Good news,

About two months ago I gave to the tuning members the details of the three Piagui scales. Piagui II presents a more uniform solution I think. The tone arrangement of this layout is given here again:

Note---Semitone factor--Relative Freq./C -----Frequency (Hz)----Cents/ Equal T.
C-------------------------1--------------------------261.6255----------0
C#-----K-----------------1.06066017 =K----------277.4958----------+2
D-------P-----------------1.12119522 =KP---------293.3333---------- -2
Eb------K-----------------1.18920711 =K(2)P-----311.1269----------- 0
E-------K-----------------1.26134462 =K(3)P------330--------------- +2
F-------P-----------------1.33333333 =K(3)P(2)--348.8341---------- -2
F#------K----------------1.41421356 =K(4)P(2)--369.9944---------- 0
G-------K-----------------1.5----------=K(5)P(2)---392.4383--------- +2
Ab------P-----------------1.58561808-=K(5)P(3)---414.836---------- -2
A-------K-----------------1.68179283-=K(6)P(3)---440--------------- 0
Bb------K-----------------1.78381067-=K(7)P(3)---466.6905---------+2
B-------P------------------1.88561808-=K(7)P(4)---493.3259--------- -2
2C------K-----------------2-------------=K(8)P(4)---523.2511--------- 0

The numbers given within parenthesis are exponents.

K = [9/8](1/2) =Square root of (9/8) = 1.06066017178...

P = [8/9] x 2(1/4) = 1.05707299111...

Some members of the tuning group declared that this layout is about the equal tempered scale due to the low values of tone deviation cents; it may be so. I would say that when comparing the Piagui tones with those of the equal tempered scale, they can be distinguished except those of 0 cents. The matter is that a new group of three Piagui scales has been discovered at the end of August, just a few days ago; one of them will be given in this message. Its +3,91 and -3,91 tone deviation cents with respect to the corresponding equal tempered tones are the suitable ones. The new group have been achieved thank to the parameters given in my book: "The Piagui Musical Scale: Perfecting Harmony".

Below I copied a few lines of the Progression of Musical Cells where we can see the semitone factors K and P that correspond to cells #52 and #49 respectively.

Between semitone factors K and P works the interval MMJ. The same interval links the cells #52 and #55 =1.064259525555... I availed this second and last opportunity to develope a second group of Piagui scales IV, V and VI by using the semitone factors P =1.05707299111... and W = #55 =1.064259525555...

CELL...............................RELATIVE FREQUENCY

No......COMMA................F(M,J,U) DECIMAL VALUE

43 J M26J15U2 1.04993498456918

44 J M26J16U2 1.05112285067072

45 M M27J16U2 1.05230972644815

46 M (256/243) = M28J16U2 1.05349794238683 *

47 M (135/128) = M29J16U2 1.0546875

48 M M30J16U2 1.05587840080261

49 J (C#)III = M30J17U2 1.05707299111353 = P

50 J M30J18U2 1.05826893294940

51 M M31J18U2 1.05946387772843

52 M (C#)I-II = M32J18U2 1.06066017177982 = K

53 M M33J18U2 1.06185781662711

54 M M34J18U2 1.06305681379554

55 J M34J19U2 1.06425952555548------------------------- W

56 J M34J20U2 1.06546359802875

57 M (16/15) = M35J20U2 1.066666666... ** ZS

58 M M36J20U2 1.06787109375 *

Therefore, the Piagui scale Nº IV is detailed as follows:

Note--Semitone factor--Relative Freq./C --------Frequency (Hz)----Cents/ Equal Tempered
C-------------------------1-----------------------------261.6255-----------0
C#----xP-----------------1.05707299111=P----------276.5572---------- -3.91
D------xW----------------1.125 =PW------------------294.3287---------- +3.91
Eb-----xP-----------------1.189207115 =P(2)W------311.1269----------- 0
E------xP-----------------1.25707872211 =P(3)W-----328.8838---------- -3.91
F------xW----------------1.33785800438 =P(3)W(2)--350.0177---------- +3.91
F#----xP-----------------1.41421356237=P(4)W(2)---369.9944----------- 0
G-----xP------------------1.49492696045 =P(5)W(2)--391.1111--------- -3.91
Ab----xW-----------------1.59099025767=P(5)W(3)---416.2436---------- +3.91
A-----xP------------------1.68179283051=P(6)W(3)---440----------------- 0
Bb----xP------------------1.77777777777=P(7)W(3)---465.112----------- -3.91
B-----xW-----------------1.89201693432=P(7)W(4)---495--------------- +3.91
2C----xP------------------2----------------=P(8)W(4)---523.2511----------- 0

The numbers given within parenthesis are exponents.

Regarding the fifths:

C G D A E B F# C# G# Eb Bb F
narrow sharp narrow narrow sharp narrow narrow sharp narrow narrow sharp

narrow = 696,09 cents (1,49492696)
sharp = 707,8 cents (1,50509025)

Regarding the major thirds:

C Major ----- 1,257078722
C# Major --1,265625
D Major ----- 1,257078722
E bemol ----- 1,257078722
E ----------1.265625
F ------------ 1,257078722
F# ---------- 1,257078722
G ---------1,265625
Ab ---------- 1,257078722
A ----------- 1,257078722
Bb ------- 1,265625
B ----------- 1,257078722

Lima, September 3, 2008

MARIO PIZARRO

piagui@...