Yahoo Tuning Groups Ultimate Backup tuning What on earth .. ?

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> > Something new from Peru -
> >
> > http://www.geocities.com/piaguiscale/page.html
> >
> > The 'motivation' is a division of the octave into 612 (unequal)
> > parts.
> Not to be confused with 612-EDO:
> http://www.xs4all.nl/~huygensf/doc/measures.html
> "A similar useful unit is 1/612 part of an octave, or one step of
> 612-tone equal temperament, because this temperament has extremely
> accurate approximations of fifth and thirds, and because 612 is
> divisible by 12. So one step is also very close to 1/12 of a
> Pythagorean comma and the schisma."
> Already Isaac Newton considered 612-EDO.
> Lit: New Grove (2001) Vol.17 p.815
> /tuning/topicId_71935.html#72166
>
> >If anyone can understand what he's doing you have more
> > patience than me...
> >
> Appearently he's a member in:
> http://www.magle.dk/music-forums/members/mario-pizarro.html
> There you can ask him about the details in that particular forum:
> http://www.magle.dk/music-forums/1380-tempered-piagui-chord-
wave.html#post9204
> about the details of his book:
> http://www.authorhouse.com/BookStore/ItemDetail~bookid~21407.aspx
> > -------------------------------------------------
>
> >
> > M = [(3^8 x 5) / 2^15] = [(32805 / 32768)] =
> = ~1.00112915039062...
> that's the common usual:
> http://en.wikipedia.org/wiki/Schisma
> >
> > J = [(2^25x2^1/4)/(3^13x5^2)] = [(33554432x2^1/4)/39858075]=
> = ~1.001131371103...
> >
> > U = [(2^12x 5^2 x 3^1/2) / 3^11] = [(102400 x 3^1/2)/177147] =
> = ~1.0012136965066...
> >
> Probably -i guess- he uses the other units J and U
> in a similar way alike already Kirnberger
> his so called "atom" ,
> as defined in the last entry at:
> http://209.85.129.104/search?
q=cache:kt7EBmv0fZAJ:lumma.org/music/theory/tctmo/5-
limit_linear_temp.xls+kirnberger+atom&hl=de&ct=clnk&cd=2&gl=de
>
> Even the later Farey's units work the same way as in Kirnberger's
case:
> Lit: Mark Lindley's article "Stimmung und Temperatur"
> pp. 317-319
> SIGMA := schisma = 5*3^8/2^15
> middle_Unit: "f" = 3^37/25/2^54
> tiny_unit: "m"= 3^84*5^12/2^161
>
> A 20th century rediscovery of the "m"-unit is documented in:
> /tuning-math/message/16962
>
> Mark judges on p.317 about Farey's 1806 5-limit-base rediscovery:
>
> "125 Jahre früher wäre das eine brilliante Entdeckung gewesen"
> tr:
> '125 years earlier that would had been a brilliant discovery'
>
> There's no need for any other irrational
> base-system for 5-limit intervals.
>
> As far as i can see at the moment:
> Nothing new from Peru.
>
> A.S.
>
-------------------------------------------------
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.
Ki------ ------ln Ki ------12ln Ki
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.

THE PIAGUI MUSICAL SCALE
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.

T = 21/12 = 1.05946309436

Let Pa = (48) Ka = (51)
Pb = (49) Kb = (52)
Pc = (50) Kc = (53)

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.
---- Ki --- --- ln Ki---------- 12 ln Ki
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

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.

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.

FIRST SEQUENCE OR PIAGUI I
NOTE SF --- CELL No. RELATIVE FREQUENCY --- FREQUENCY (Hz)
C ------1 ------1 --------------------- 261.6255 -- *
C#------K------ 52 1.06066017178 = K ------277.4958
D ------K ----- 104 1.125 ------ = K2 ----294.3287 **
Eb------P ----- 153 1.189207115 -- = K2P ----311.127 -- *
E------ K ----- 205 1.26134462288 = K3P ----330
F------ K------ 257 1.33785800438 = K4P 350.0178
F#------P------ 306 1.41421356237 = K4P2 369.9944 *
G ------K------ 358 1.5 ------ = K5P2 392.4383 **
Ab------K------ 410 1.59099025767 = K6P2 416.2437
A------ P------ 459 1.68179283051 = K6P3 440 *
Bb------K------ 511 1.7838106725 = K7P3 466.6905
B K------ 563 1.89201693432 = K8P3 495
2C P------ 612 2 ------------ = K8P4 523.2511 *

* Tempered and Piagui I frequencies
** Pythagorean frequencies
SF = Semitone Factor

Eight tones of any of the three new octaves differ from the tempered
ones; the other four have identical values (C, Eb, F#, A). Eight of
the Piagui I frequencies are higher than their corresponding values
in the tempered octave. Four of the Piagui II are higher while the
other four are lower and eight Piagui III pitches are even lower.
Each sequence begins with keynote C making successive groups of KKP,
KPK and PKK. It follows that if keynote is changed to C# whose
frequency is 277.4958 Hz, the chromatic scale is made up by sequence
KPK. Likewise, when it is changed to D whose frequency is 294.3287
Hz, the PKK sequence works in this new chromatic scale. Thus,
chromatic scales of the three sequences work simultaneously.

C. MARIO PIZARRO

🔗piaguiscale <piagui@...>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> Something new from Peru -
>
> http://www.geocities.com/piaguiscale/page.html
>
> This seems to be singing the merits of an unequal circular tuning
> with 8 pure fifths, where the octave is divided into 4 equal minor
> thirds. Logically, this should mean that there are 4 quarter-
ditonic-
> comma tempered fifths distributed equally round the circle.
However
> the author is frustratingly vague in his explanations.
>
> Why this should be musically beneficial is not really explained.
> There are a lot of graphs but what they represent is never quite
> defined.
> The 'motivation' is a division of the octave into 612 (unequal)
> parts. If anyone can understand what he's doing you have more
> patience than me...
>
> ~~~T~~~
>
> -------------------------------------------------
> Una progresión de frecuencias consonantes relativas a Do=1 fue
> deducida empleando las consonancias naturales de las escalas de
> Pitágoras, de Entonación Justa y las derivadas de ambas. El
análisis
> reconoció la articulación cíclica de tres "commas" y definió los
> valores de 612 frecuencias relativas o células de la progresión,
> distribuidas cíclicamente entre Do=1 y 2Do=2. Las commas que
> intervienen a modo de factores en la progresión son las
siguientes:
>
> M = [(38 x 5) / 215] = [(32805 / 32768)] = 1.00112915039062
>
> J = [(225x21/4)/(313x52)] = [(33554432x21/4)/39858075]
=1.001131371103
>
> U = [(212x 52 x 31/2) / 311] = [(102400 x 31/2)/177147] =
> 1.0012136965066
>
> La progresión tiene 624 células y se expresa como sigue:
>
> (QRQQROQ)(QRQQROQ)(QRQQROQ)(QRQQROQ)(QRQQROQ)(QRQQROQ)= (9/8)6,
> donde (QRQQRoQ) =
>
> (MMJJMMMMJJMMMMJJMM) (JJMMUUMMJJMMMMJJ)
>
> (MMJJMMMMJJMMMMJJMM) (MMJJMMMMJJMMMMJJMM)
>
> (JJMMMMJJMMUUMMJJ)(MMJJMMMMJJMMMMJJMM)= (9/8) = 1.125
>
> Se consideró la posibilidad de que dos factores de
> escalamiento K y P operan apropiadamente en la octava del nuevo
> sistema musical en lugar del único factor T de la escala temperada
y
> este supuesto resultó acertado.
>
> Operaciones matemáticas permitieron deducir los
valores
> de K y P que forman parte de las 612 células de la progresión como
se
> detalla en el libro "The Piagui Musical Scale: Perfecting Harmony"
>
> K = 1. 06.. (Mayor que
> T) P = 1. 05.. (Menor
que
> T)
>
> El análisis y desarrollos que se detallan en el libro
> condujeron a satisfacer la cuadratura de la octava, como lo
> demuestra la relación siguiente:
>
> K8 P4 = 2
>
-------------------------------------------------------
-------------------------------------------------------

Hi Tom,
All the tone frequencies of the three Piagui scale octave variants
are linked by perfect fifths and perfect fourths. Four octaves were
drawn on the square perimeter where you can see four links of
perfect fifths and five links of perfect fourths per octave. A total
of nine perfect links per octave. Nevertheless when all the major
and minor Piagui and Tempered triads are drawn by using an special
software the Piagui responses are aesthetical 100% contrasted with
the caotic responses of the Tempered scale.
Some of grand pianos in Lima were tuned to the Piagui I variant.
I will try to copy below the explained links.
My name is Mario Pizarro; on the internet I used C. Mario Pizarro
Age: 74
Profession: Electronic Engineer.
I hope to insert the mentioned graph.
Mario Pizarro
Lima, May 27
-----------------------------------------------------
-------------------------------------------------------
PERFECT FIFTHS AND FOURTHS IN PIAGUI SCALES

THE COMPUTER DIDN´T ACCEPT THE DIAGRAM.

SEE BELOW.

Perfect fifth
Perfect fourth

Probably, man's common sense expected perfect links in the octaves
over the centuries. The precise and suitable values of K and P
produce these remarkable results.
The combined sounds produced by adding C + G or C + F, where G and F
are the perfect fifth and perfect fourth respectively with respect
to note C, are detected not only by our brain but also through
computer-printing responses. Thus, the mentioned perfect intervals
are the basic contributions, classifying the Piagui scales as the
best way to perfecting harmony. This technical procedure is applied
on triad evaluations where three tone frequencies are added as
discussed in Chapt

---------------------------------------------------------
GRAPH No. 3
TEMPERED CHORD WAVE PEAKS
C Major = C + E + G

GRAPH No. 4
PIAGUI I CHORD WAVE PEAKS C Major = C + E + G

THE ONLY WAY TO GET THE DIAGRAMS AND MORE: To buy it.

🔗Andreas Sparschuh <a_sparschuh@...>

🔗piaguiscale <piagui@...>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> Something new from Peru -
>
> http://www.geocities.com/piaguiscale/page.html
>
> This seems to be singing the merits of an unequal circular tuning
> with 8 pure fifths, where the octave is divided into 4 equal minor
> thirds. Logically, this should mean that there are 4 quarter-
ditonic-
> comma tempered fifths distributed equally round the circle.
However
> the author is frustratingly vague in his explanations.
>
> Why this should be musically beneficial is not really explained.
> There are a lot of graphs but what they represent is never quite
> defined.
> The 'motivation' is a division of the octave into 612 (unequal)
> parts. If anyone can understand what he's doing you have more
> patience than me...
>
> ~~~T~~~
>
> -------------------------------------------------
> Una progresión de frecuencias consonantes relativas a Do=1 fue
> deducida empleando las consonancias naturales de las escalas de
> Pitágoras, de Entonación Justa y las derivadas de ambas. El
análisis
> reconoció la articulación cíclica de tres "commas" y definió los
> valores de 612 frecuencias relativas o células de la progresión,
> distribuidas cíclicamente entre Do=1 y 2Do=2. Las commas que
> intervienen a modo de factores en la progresión son las
siguientes:
>
> M = [(38 x 5) / 215] = [(32805 / 32768)] = 1.00112915039062
>
> J = [(225x21/4)/(313x52)] = [(33554432x21/4)/39858075]
=1.001131371103
>
> U = [(212x 52 x 31/2) / 311] = [(102400 x 31/2)/177147] =
> 1.0012136965066
>
> La progresión tiene 624 células y se expresa como sigue:
>
> (QRQQROQ)(QRQQROQ)(QRQQROQ)(QRQQROQ)(QRQQROQ)(QRQQROQ)= (9/8)6,
> donde (QRQQRoQ) =
>
> (MMJJMMMMJJMMMMJJMM) (JJMMUUMMJJMMMMJJ)
>
> (MMJJMMMMJJMMMMJJMM) (MMJJMMMMJJMMMMJJMM)
>
> (JJMMMMJJMMUUMMJJ)(MMJJMMMMJJMMMMJJMM)= (9/8) = 1.125
>
> Se consideró la posibilidad de que dos factores de
> escalamiento K y P operan apropiadamente en la octava del nuevo
> sistema musical en lugar del único factor T de la escala temperada
y
> este supuesto resultó acertado.
>
> Operaciones matemáticas permitieron deducir los
valores
> de K y P que forman parte de las 612 células de la progresión como
se
> detalla en el libro "The Piagui Musical Scale: Perfecting Harmony"
>
> K = 1. 06.. (Mayor que
> T) P = 1. 05.. (Menor
que
> T)
>
> El análisis y desarrollos que se detallan en el libro
> condujeron a satisfacer la cuadratura de la octava, como lo
> demuestra la relación siguiente:
>
> K8 P4 = 2
>
-------------------------------------------------------------------
-------------------------------------------------------------------

Hi Tom,

I didn´t get your reply. Last message I sent you: May 27, 08:31 PM.
Below you have the PIAGUI I tone frequencies as functions of PERFECT
FIFTHS AND PERFECT FOURTHS.

The progression Cells Nos. 0, 153, 306, 459 and 612 give the
frequencies of the following tones in the octave: C, Eb, F# and A.
Each 153 cells make one fourth of the octave (612 = Note 2Do = 2

C = 1, (

C# = F#/1,3333...= 1,06066017178..

D = G/1,3333... = 1,125

Eb= (2)1/4 = 1,189207115...

E = (A/1,3333,,,,)= 1,26134462288....

F = (Bb/1,3333,,,)= 1,33785800438....

F# = (2)1/2 = 1,41421356237....

G = (C)(1,5)= 1,5

Ab= (C#)(1,5)= 1,59099025767

A = (2)3/4 = 1,68179283051

Bb = (Eb)(1,5)= 1,7838106725

B = (E)(1,5)= 1,89201693433

2C = 2

-----------------------------
I hope to get your news soon

I remain

MARIO PIZARRO
MAY 29, 2008
04:10 pm
LIMA- PERU

🔗piaguiscale <piagui@...>

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> I was sent a link to this page:
>
> http://www.relfe.com/solfeggio.html
>
> Does anyone in the tuning list have any particular insights into
what
> this tuning system is all about?
>
> Any comments appreciated.
>
> Thanks
>
> Charles Lucy
> lucy@...
>
> - Promoting global harmony through LucyTuning -
>
> for information on LucyTuning go to:
> http://www.lucytune.com
>
> For LucyTuned Lullabies go to:
> http://www.lullabies.co.uk
>

--------------------------------------------------------------
--------------------------------------------------------------
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.

ln Pa = ln 1.0558784008 = 0.05437302789
ln Pb = ln 1.0570729911 = 0.05550375948
ln Pc = ln 1.05826893295 = 0.05663449107

T = 21/12 = 1.05946309436

Let Pa = (48) Ka = (51)
Pb = (49) Kb = (52)
Pc = (50) Kc = (53)

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.
---- Ki --- --- ln Ki---------- 12 ln Ki
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

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

* Tempered and Piagui I frequencies
** Pythagorean frequencies
SF = Semitone Factor

MARIO PIZARRO
(PIAGUISCALE)
ELECTRONIC ENGINEER

What on earth .. ?

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