Re: New and Perfect Musical Sacle

The Piagui Musical Scale was discovered recently after a research
period of twelve years. The benefit of using this scale is the perfect
harmony shown by any type of chords when listened on the piano and
other instruments tuned to the Piagui Scale. These features contrast
with the imperfect harmony produced by a piano tuned to the Equal
Tempered Scale.
The basic step was to deduce a frequency progression from Do = 1 up to
2Do = 2, containing 612 elements.
The K and P semitone factors replace the only Tempered factor T. Both
factors were deduced by solving the equation K8 P4 = 2; factors K and P
are elements of the 612 relative frequencies with respect to the first
frequency Do = 1 of the progression.
Three Piagui octave options were also determined which work as follows:

(Do=1) x K x K x P x K x K x P x K x K x P x K x K x P = 2

(Do=1) x K x P x K x K x P x K x K x P x K x K x P x K = 2

(Do=1) x P x K x K x P x K x K x P x K x K x P x K x K = 2

The progression components are determined by the cyclical application
of the comma factors M, J and U, where M, named schisma in the XVI
century, with a value of (32805 / 32768) = 1,00112915039, together with
the commas J = 1,0011313711 and U = 1,00121369651, I deduced, produce
all the elements of the progression.

C. Mario Pizarro

On Mon, May 19, 2008 at 8:26 PM, piaguiscale <piagui@...> wrote:
> The Piagui Musical Scale was discovered recently after a research
> period of twelve years. The benefit of using this scale is the perfect
> harmony shown by any type of chords when listened on the piano and
> other instruments tuned to the Piagui Scale. These features contrast
> with the imperfect harmony produced by a piano tuned to the Equal
> Tempered Scale.
> The basic step was to deduce a frequency progression from Do = 1 up to
> 2Do = 2, containing 612 elements.
> The K and P semitone factors replace the only Tempered factor T. Both
> factors were deduced by solving the equation K8 P4 = 2; factors K and P
> are elements of the 612 relative frequencies with respect to the first
> frequency Do = 1 of the progression.
> Three Piagui octave options were also determined which work as follows:
>
> (Do=1) x K x K x P x K x K x P x K x K x P x K x K x P = 2
>
> (Do=1) x K x P x K x K x P x K x K x P x K x K x P x K = 2
>
> (Do=1) x P x K x K x P x K x K x P x K x K x P x K x K = 2
>
> The progression components are determined by the cyclical application
> of the comma factors M, J and U, where M, named schisma in the XVI
> century, with a value of (32805 / 32768) = 1,00112915039, together with
> the commas J = 1,0011313711 and U = 1,00121369651, I deduced, produce
> all the elements of the progression.
>
> C. Mario Pizarro

So how exactly do M, J, and U enter into the calculation of K and P?
You have one equation in two variables and so an infinite number of
possible solutions for K and P. How did you decide on your particular
choice?

David Bowen

--- In tuning@yahoogroups.com, "piaguiscale" <piagui@...> wrote:
>
> The Piagui Musical Scale was discovered recently after a research
> period of twelve years. The benefit of using this scale is the
perfect
> harmony shown by any type of chords when listened on the piano and
> other instruments tuned to the Piagui Scale. These features
contrast
> with the imperfect harmony produced by a piano tuned to the Equal
> Tempered Scale.
> The basic step was to deduce a frequency progression from Do = 1 up
to
> 2Do = 2, containing 612 elements.
> The K and P semitone factors replace the only Tempered factor T.
Both
> factors were deduced by solving the equation K8 P4 = 2; factors K
and P
> are elements of the 612 relative frequencies with respect to the
first
> frequency Do = 1 of the progression.
> Three Piagui octave options were also determined which work as
follows:
>
> (Do=1) x K x K x P x K x K x P x K x K x P x K x K x P = 2
>
> (Do=1) x K x P x K x K x P x K x K x P x K x K x P x K = 2
>
> (Do=1) x P x K x K x P x K x K x P x K x K x P x K x K = 2
>
> The progression components are determined by the cyclical
application
> of the comma factors M, J and U, where M, named schisma in the XVI
> century, with a value of (32805 / 32768) = 1,00112915039, together
with
> the commas J = 1,0011313711 and U = 1,00121369651, I deduced,
produce
> all the elements of the progression.
>
> C. Mario Pizarro
>

Since it's "perfect", it must have a "neutral" third, the kind that
makes beautiful women in the Balkans throw their heads back and lift
their palms to the sky. How about some frequency ratios, cent values,
or highly poetic yet highly accurate descriptions like "the same as
Paul and John sang on the first beat of measure two, only one
Ringotone flat"?

-Cameron Bobro

On Tue, May 20, 2008 at 7:38 AM, Cameron Bobro <misterbobro@...> wrote:
> Since it's "perfect", it must have a "neutral" third, the kind that
> makes beautiful women in the Balkans throw their heads back and lift
> their palms to the sky. How about some frequency ratios, cent values,
> or highly poetic yet highly accurate descriptions like "the same as
> Paul and John sang on the first beat of measure two, only one
> Ringotone flat"?

I agree with Cameron; I can't make head or tail of this. What are K
and P? To me it looks just like one realization of the "diminished"
temperament, and I can't see how it's more "perfect" than any other
temperament.

Keenan

> So how exactly do M, J, and U enter into the calculation of K and
P?
> You have one equation in two variables and so an infinite number of
> possible solutions for K and P. How did you decide on your
particular
> choice?
>
> David Bowen
>--------------------------------------------------------------------
-

(1) I decided to develope a progression of consonant and relative
frequencies with respect to Do = 1 up to 2Do = 2 starting with the
comma or schisma M = (32805/32768) = 1,00112915039. this comma was
deduced in the XVI century by Zarlino; it is the smallest consonance
that can be distinguished by the ear.
Each relative frequency of the progression was named cell. At the
beginning of the progression I had to deduced two additional commas
J and U whose values are:
J = 1.0011313711….
U = 1.001213696511…
With the aforementioned commas the progression from Do = 1 up to 2Do
= 2 contained 612 cells (cell # 612 = 2Do = 2). Each one was
determined by multiplying the preceding cell by the comma that is
aligned to the cell to be known ( M or J or U ).
It was noticed that the first 100 cells contain the relative tone
frequencies of the extended scales of Pythagoras and Aristoxenus
(bc) (21 tones per octaves each) emerged with exactness, so these
tone frequencies were the references for expanding the progression
up to cell # 612. The mentioned three commas that work cyclically
produced all the tones of both extended scales a long the whole set
of 612 cells.

(2) Actually, the two equations with 4 variables are
Km Pn = 2 (m and n are exponents)
m + n = 12 (m and n are integer unknown numbers to be determined;
the sum m + n gives the number of tones of the Piagui octave).
K and P are the semitone factors that replace the only Tempered T = 2
(to the exponent 1/12) = 1.05946309436….

(3) SPECIAL WAY FOR SOLVING TWO EQUATIONS WITH 4 VARIABLES ( K, P,
m, n) the K and P values must be near to the Tempered T because the
sound discrepancy between Tempered and chord sounds are supposed to
be small so K , P and T should have about the same magnitude.
It was also assumed that K and P are cells of the progression.
According to the preceding paragraph I took 6 cells that showed
positive and negative differencies with respect to T. The T Tempered
factor is not a cell of the progression. The six experimental cells
are

CELL # COMMA F(M,J,U) DECIMAL VALUES
48 M M30J16U2 1.0558784008
49 J M30J17U2 1.05707299111
50 J M30J18U2 1.05826893295
51 M M31J18U2 1.05946387773
52 M M32J18U2 1.06066017178
53 M M33J18U2 1.06185781663
T = 1.05946309436….

By mathematical steps and using these 6 figures in 9 auxiliary
equations, 9 results like the following figuries

(#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

ln = neperian logaritm

Eight solutions for n are not integer numbers. But solution number
#5 gives n = 4, therefore m = 8 because m + n = 12.
The semitones K and P of the Piagui Musical Scale 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.

C. MArio Pizarro

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:
>
> On Tue, May 20, 2008 at 7:38 AM, Cameron Bobro <misterbobro@...>
wrote:
> > Since it's "perfect", it must have a "neutral" third, the kind
that
> > makes beautiful women in the Balkans throw their heads back and
lift
> > their palms to the sky. How about some frequency ratios, cent
values,
> > or highly poetic yet highly accurate descriptions like "the same
as
> > Paul and John sang on the first beat of measure two, only one
> > Ringotone flat"?
>
> I agree with Cameron; I can't make head or tail of this. What are K
> and P? To me it looks just like one realization of the "diminished"
> temperament, and I can't see how it's more "perfect" than any other
> temperament.
>
> Keenan
>
-------------------------------------------------------------------
(1) I decided to develope a progression of consonant and relative
frequencies with respect to Do = 1 up to 2Do = 2 starting with the
comma or schisma M = (32805/32768) = 1,00112915039. this comma was
deduced in the XVI century by Zarlino; it is the smallest consonance
that can be distinguished by the ear.
Each relative frequency of the progression was named cell. At the
beginning of the progression I had to deduced two additional commas
J and U whose values are:
J = 1.0011313711….
U = 1.001213696511…
With the aforementioned commas the progression from Do = 1 up to 2Do
= 2 contained 612 cells (cell # 612 = 2Do = 2). Each one was
determined by multiplying the preceding cell by the comma that is
aligned to the cell to be known ( M or J or U ).
It was noticed that the first 100 cells contain the relative tone
frequencies of the extended scales of Pythagoras and Aristoxenus
(bc) (21 tones per octaves each) emerged with exactness, so these
tone frequencies were the references for expanding the progression
up to cell # 612. The mentioned three commas that work cyclically
produced all the tones of both extended scales a long the whole set
of 612 cells.

(2) Actually, the two equations with 4 variables are
Km Pn = 2 (m and n are exponents)
m + n = 12 (m and n are integer unknown numbers to be determined;
the sum m + n gives the number of tones of the Piagui octave).
K and P are the semitone factors that replace the only Tempered T = 2
(to the exponent 1/12) = 1.05946309436….

(3) SPECIAL WAY FOR SOLVING TWO EQUATIONS WITH 4 VARIABLES ( K, P,
m, n) the K and P values must be near to the Tempered T because the
sound discrepancy between Tempered and chord sounds are supposed to
be small so K , P and T should have about the same magnitude.
It was also assumed that K and P are cells of the progression.
According to the preceding paragraph I took 6 cells that showed
positive and negative differencies with respect to T. The T Tempered
factor is not a cell of the progression. The six experimental cells
are

CELL # COMMA F(M,J,U) DECIMAL VALUES
48 M M30J16U2 1.0558784008
49 J M30J17U2 1.05707299111
50 J M30J18U2 1.05826893295
51 M M31J18U2 1.05946387773
52 M M32J18U2 1.06066017178
53 M M33J18U2 1.06185781663
T = 1.05946309436….

By mathematical steps and using these 6 figures in 9 auxiliary
equations, 9 results like the following figuries

(#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

ln = neperian logaritm

Eight solutions for n are not integer numbers. But solution number
#5 gives n = 4, therefore m = 8 because m + n = 12.
The semitones K and P of the Piagui Musical Scale 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.

After reading the book where 34 piagui chord graphs are compared
with the corresponding Tempered graphs, you will see that the Piagui
responses are aesthetical displays while the tempered responses are
disordered.

C. Mario Pizarro

piaguiscale wrote:
>> So how exactly do M, J, and U enter into the calculation of K and > P?
>> You have one equation in two variables and so an infinite number of
>> possible solutions for K and P. How did you decide on your > particular
>> choice?
>>
>> David Bowen
>> --------------------------------------------------------------------
> -
> > (1) I decided to develope a progression of consonant and relative > frequencies with respect to Do = 1 up to 2Do = 2 starting with the > comma or schisma M = (32805/32768) = 1,00112915039. this comma was > deduced in the XVI century by Zarlino; it is the smallest consonance > that can be distinguished by the ear. Right, that's what we'd normally call a schisma, and so not a comma. 1.954 cents.

> Each relative frequency of the progression was named cell. At the > beginning of the progression I had to deduced two additional commas > J and U whose values are: > J = 1.0011313711�. > U = 1.001213696511�

Why?

Anyway, J is 1.958 cents and U is 2.100 cents. So these are approximate schismas.

> With the aforementioned commas the progression from Do = 1 up to 2Do > = 2 contained 612 cells (cell # 612 = 2Do = 2). Each one was > determined by multiplying the preceding cell by the comma that is > aligned to the cell to be known ( M or J or U ). > It was noticed that the first 100 cells contain the relative tone > frequencies of the extended scales of Pythagoras and Aristoxenus > (bc) (21 tones per octaves each) emerged with exactness, so these > tone frequencies were the references for expanding the progression > up to cell # 612. The mentioned three commas that work cyclically > produced all the tones of both extended scales a long the whole set > of 612 cells. Why 612? There are about 614.2 schismas to an octave, so you're rounding off to a multiple of 12?

> (2) Actually, the two equations with 4 variables are > Km Pn = 2 (m and n are exponents)
> m + n = 12 (m and n are integer unknown numbers to be determined; > the sum m + n gives the number of tones of the Piagui octave). > K and P are the semitone factors that replace the only Tempered T = 2
> (to the exponent 1/12) = 1.05946309436�. This means you're looking for a 12 note MOS with semitones called K and P.

> (3) SPECIAL WAY FOR SOLVING TWO EQUATIONS WITH 4 VARIABLES ( K, P, > m, n) the K and P values must be near to the Tempered T because the > sound discrepancy between Tempered and chord sounds are supposed to > be small so K , P and T should have about the same magnitude. > It was also assumed that K and P are cells of the progression. > According to the preceding paragraph I took 6 cells that showed > positive and negative differencies with respect to T. The T Tempered > factor is not a cell of the progression. The six experimental cells > are I can't make any sense of this.

> CELL # COMMA F(M,J,U) DECIMAL VALUES > 48 M M30J16U2 1.0558784008
> 49 J M30J17U2 1.05707299111
> 50 J M30J18U2 1.05826893295
> 51 M M31J18U2 1.05946387773
> 52 M M32J18U2 1.06066017178
> 53 M M33J18U2 1.06185781663
> T = 1.05946309436�.

The "COMMA" column is deriving these semitones in terms of the schismas above. So cell 48 is 30M + 16J + 2U (M, J, and U as pitch differences in e.g. cents rather than frequency ratios). I don't get the decimal values exact but they're near enough. I still have no idea why these values are special or what the letters after the cell numbers mean.

> By mathematical steps and using these 6 figures in 9 auxiliary > equations, 9 results like the following figuries
> > (#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
> > ln = neperian logaritm Huh? What are a and b? And what's a "neperian logaritm"?

> Eight solutions for n are not integer numbers. But solution number > #5 gives n = 4, therefore m = 8 because m + n = 12. That'll be the diminished temperament Keenan identified.

> The semitones K and P of the Piagui Musical Scale have been sought > and worked out: > K = Kb = (Cell No. 52) = (9/8)1/2 = 1.06066017178

That's the square root of 9:8 or 101.955 cents.

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

Aha! 8:9 times the 4th root of 2. 96.090 cents.

> K8 P4 = [ (9/8)1/2 ]8 [ (8/9)( 21/4 ) ]4 = 2 Tells us octaves are pure by the looks of it.

> 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. 21/12 is a way of writing the twelfth root of 2. The rest is some form of technobabble for something like diminished temperament. An open choice of K and P arrangements suggests the 12 note scale needn't be an MOS.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> >
> > ln = neperian logaritm
>
> Huh? What are a and b? And what's a "neperian logaritm"?

"ln". It's called neperian logarithm in some places. "Napier's
constant" is another name, or "Euler's number", for e, also where I
live.

-Cameron Bobro

Cameron Bobro wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > > >>> ln = neperian logaritm >> Huh? What are a and b? And what's a "neperian logaritm"?
> > "ln". It's called neperian logarithm in some places. "Napier's > constant" is another name, or "Euler's number", for e, also where I > live. Apparently so, as a synonym for "Napierian logarithm" because Napier sometimes called himself "Neper". It looks like "logaritm neperian" is valid in a language called "piemont�isa" (Piedmontese?):

http://pms.wikipedia.org/wiki/Leonhard_Euler

You learn something every day! Maybe there is a language where you can say "neperian logaritm" but I don't know what it is.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> piaguiscale wrote:
> >> So how exactly do M, J, and U enter into the calculation of K
and
> > P?
> >> You have one equation in two variables and so an infinite
number of
> >> possible solutions for K and P. How did you decide on your
> > particular
> >> choice?
> >>
> >> David Bowen
> >> ----------------------------------------------------------------
----
> > -
> >
> > (1) I decided to develope a progression of consonant and
relative
> > frequencies with respect to Do = 1 up to 2Do = 2 starting with
the
> > comma or schisma M = (32805/32768) = 1,00112915039. this comma
was
> > deduced in the XVI century by Zarlino; it is the smallest
consonance
> > that can be distinguished by the ear.
>
> Right, that's what we'd normally call a schisma, and so not
> a comma. 1.954 cents.
>
> > Each relative frequency of the progression was named cell. At
the
> > beginning of the progression I had to deduced two additional
commas
> > J and U whose values are:
> > J = 1.0011313711….
> > U = 1.001213696511…
>
> Why?
>
> Anyway, J is 1.958 cents and U is 2.100 cents. So these are
> approximate schismas.
>
> > With the aforementioned commas the progression from Do = 1 up to
2Do
> > = 2 contained 612 cells (cell # 612 = 2Do = 2). Each one was
> > determined by multiplying the preceding cell by the comma that
is
> > aligned to the cell to be known ( M or J or U ).
> > It was noticed that the first 100 cells contain the relative
tone
> > frequencies of the extended scales of Pythagoras and Aristoxenus
> > (bc) (21 tones per octaves each) emerged with exactness, so
these
> > tone frequencies were the references for expanding the
progression
> > up to cell # 612. The mentioned three commas that work
cyclically
> > produced all the tones of both extended scales a long the whole
set
> > of 612 cells.
>
> Why 612? There are about 614.2 schismas to an octave, so
> you're rounding off to a multiple of 12?
>
> > (2) Actually, the two equations with 4 variables are
> > Km Pn = 2 (m and n are exponents)
> > m + n = 12 (m and n are integer unknown numbers to be
determined;
> > the sum m + n gives the number of tones of the Piagui octave).
> > K and P are the semitone factors that replace the only Tempered
T = 2
> > (to the exponent 1/12) = 1.05946309436….
>
> This means you're looking for a 12 note MOS with semitones
> called K and P.
>
> > (3) SPECIAL WAY FOR SOLVING TWO EQUATIONS WITH 4 VARIABLES ( K,
P,
> > m, n) the K and P values must be near to the Tempered T because
the
> > sound discrepancy between Tempered and chord sounds are supposed
to
> > be small so K , P and T should have about the same magnitude.
> > It was also assumed that K and P are cells of the progression.
> > According to the preceding paragraph I took 6 cells that showed
> > positive and negative differencies with respect to T. The T
Tempered
> > factor is not a cell of the progression. The six experimental
cells
> > are
>
> I can't make any sense of this.
>
> > CELL # COMMA F(M,J,U) DECIMAL VALUES
> > 48 M M30J16U2 1.0558784008
> > 49 J M30J17U2 1.05707299111
> > 50 J M30J18U2 1.05826893295
> > 51 M M31J18U2 1.05946387773
> > 52 M M32J18U2 1.06066017178
> > 53 M M33J18U2 1.06185781663
> > T = 1.05946309436….
>
> The "COMMA" column is deriving these semitones in terms of
> the schismas above. So cell 48 is 30M + 16J + 2U (M, J, and
> U as pitch differences in e.g. cents rather than frequency
> ratios). I don't get the decimal values exact but they're
> near enough. I still have no idea why these values are
> special or what the letters after the cell numbers mean.
>
> > By mathematical steps and using these 6 figures in 9 auxiliary
> > equations, 9 results like the following figuries
> >
> > (#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
> >
> > ln = neperian logaritm
>
> Huh? What are a and b? And what's a "neperian logaritm"?
>
> > Eight solutions for n are not integer numbers. But solution
number
> > #5 gives n = 4, therefore m = 8 because m + n = 12.
>
> That'll be the diminished temperament Keenan identified.
>
> > The semitones K and P of the Piagui Musical Scale have been
sought
> > and worked out:
> > K = Kb = (Cell No. 52) = (9/8)1/2 = 1.06066017178
>
> That's the square root of 9:8 or 101.955 cents.
>
> > P = Pb = (Cell No. 49) = (8/9) ( 21/4 ) = 1.0570729911
>
> Aha! 8:9 times the 4th root of 2. 96.090 cents.
>
> > K8 P4 = [ (9/8)1/2 ]8 [ (8/9)( 21/4 ) ]4 = 2
>
> Tells us octaves are pure by the looks of it.
>
> > 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.
>
> 21/12 is a way of writing the twelfth root of 2. The rest
> is some form of technobabble for something like diminished
> temperament. An open choice of K and P arrangements
> suggests the 12 note scale needn't be an MOS.
>
>
> Graham
>
------------------------------------------------------------------
------------------------------------------------------------------
INTRODUCTION
I first got involved with musical scales after reading W. T.
Bartholomew's "Acoustics of Music", Joaquin Zamacois', "Teoría de la
Música", M. Caillaud's, "Notions D'Acoustique", Julian
Carrillo's "Sonido 13" and other fascinating works. Authors
usually start analyzing the theme by studying the works of
Pythagoras and Aristoxenus, Zarlino, William Holder, Ramos de
Pareja, Delezenne and other well known researchers on the
improvement of musical scales.
All of us pay well-deserved homage to Pythagoras for his heptatonic
scale set down about 550 BC. Nearly 250 years later, Aristoxenus
conceived a second heptatonic scale, also known as the scale of
physicists and geometricians. The work of these great researchers
enabled the incipient audience of that era to listen to melodies
that had only seven pitches per octave. Then, at the beginning of
the second millenium, sharps and flats were added to the heptatonic
scales extending them to twenty-one pitches per octave, based on the
Pythagoras and Aristoxenus scales. At about 1560, Zarlino set out
the "commas", such as (81/80) = 1.0125 and Dr. Delezenne deduced the
chromatic semitone 135/128. In 1985, the smallest comma known was
used: 1.00112915 = (32805 / 32768) = M, obtained by dividing the
Delezenne and Pythagoras semitones, that is (135/128) ÷ (256/243),
which played a fundamental role in the establishment of an eminently
good scale. Considering the transcendental contributions of Zarlino
and Delezenne, they must also be included in our homage to the
masters.
In 1957, I had the opportunity to talk to one of the greatest guitar
players of our time, Andrés Segovia, in Lima during his concert tour
period. I asked him if he was satisfied with the quality of guitar
chords.
He wanted to know why I asked this question. I told him that I had
detected discords when playing the guitar, and that I could perceive
a slight incoherence in the tone frequencies. He replied that he
did not approve such chords either, but that it was an old problem
and he hoped to see it solved some day.
Twenty-five years later, I decided to study the harmony problem. I
found interesting information in books from the National Library,
but solving the problem seemed to require about as much work as
finding the Philosophers' stone. In 1482 Ramos de Pareja proposed a
dodecatonic scale, named "Tempered", as a practical solution to
imperfect chords. The sole semitone factor T of the Tempered scale
set the same frequency relation between any two consecutive notes
within the octave. J. S. Bach introduced the Tempered scale in 1722.
From 1994 to 1997, I wrote to manufacturers of musical instruments
abroad, trying to convince them of the advantages of using a piano
tuner ruled by the new scale. I also sent them some computer-
plotted graphs for comparing Tempered and "Piagui" chords, a term
used for the new intonation. Some of them replied, congratulating
me on my endeavor to resolve the harmony problem, but that was all.
At that time, I realized that the design of electronic organ
keyboards and electronic tuners interrupted the use of all the
hardware in their designs and products. Most manufacturers were
already using software.
After long years of research, I finally decided to set down my
findings and started writing this book in June 2002. I apologize
for including so many mathematical reasonings. I had to avail
myself this opportunity to provide the full and complete details on
the subject.
A well known analyst once stated, somewhat
pessimistically: "Musical scale is not one, not natural or even
founded necessarily on laws of constitution of musical sound, but
very diverse, very artificial and very capricious".
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 October 4th, 2003, the first string of the available Piagui
guitar was tuned to obtain a 440 Hz corresponding to the note La.
Then, by audible comparisons, the twelve reference tones of the
middle octave on a grand piano were tuned and the tones extended
throughout the keyboard. When the pianist Luis E. Colmenares-
Perales played the first of the classic works listened to that
evening, he said enthusiastically: "Not only the harmony but
everything is clearly more pleasant than what is heard on a piano
tuned to the Tempered scale".
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
required by the brain to classify the harmony. Adagios, nocturnes,
serenades and many, but not all, classical and selected pieces of
music show the difference in the new musical system. However,
Chopin's Fantasia Impromptu and Rimsky-Korsakov's Flight of the
Bumblebee do not demonstrate this difference.
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.
Mankind has sought and achieved, over millenia, the perfection of
almost all things that are linked to him. Musical harmony, with
which we have lived for many centuries, was achieved and had only a
small degree of imperfection left to solve. From now on, we can say
that this problem no longer exists.

Regards
MARIO PIZARRO
ELECTRONIC ENGINEER
(PIAGUISCALE)

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> piaguiscale wrote:
> >> So how exactly do M, J, and U enter into the calculation of K
and
> > P?
> >> You have one equation in two variables and so an infinite
number of
> >> possible solutions for K and P. How did you decide on your
> > particular
> >> choice?
> >>
> >> David Bowen
> >> ----------------------------------------------------------------
----
> > -
> >
> > (1) I decided to develope a progression of consonant and
relative
> > frequencies with respect to Do = 1 up to 2Do = 2 starting with
the
> > comma or schisma M = (32805/32768) = 1,00112915039. this comma
was
> > deduced in the XVI century by Zarlino; it is the smallest
consonance
> > that can be distinguished by the ear.
>
> Right, that's what we'd normally call a schisma, and so not
> a comma. 1.954 cents.
>
> > Each relative frequency of the progression was named cell. At
the
> > beginning of the progression I had to deduced two additional
commas
> > J and U whose values are:
> > J = 1.0011313711….
> > U = 1.001213696511…
>
> Why?
>
> Anyway, J is 1.958 cents and U is 2.100 cents. So these are
> approximate schismas.
>
> > With the aforementioned commas the progression from Do = 1 up to
2Do
> > = 2 contained 612 cells (cell # 612 = 2Do = 2). Each one was
> > determined by multiplying the preceding cell by the comma that
is
> > aligned to the cell to be known ( M or J or U ).
> > It was noticed that the first 100 cells contain the relative
tone
> > frequencies of the extended scales of Pythagoras and Aristoxenus
> > (bc) (21 tones per octaves each) emerged with exactness, so
these
> > tone frequencies were the references for expanding the
progression
> > up to cell # 612. The mentioned three commas that work
cyclically
> > produced all the tones of both extended scales a long the whole
set
> > of 612 cells.
>
> Why 612? There are about 614.2 schismas to an octave, so
> you're rounding off to a multiple of 12?
>
> > (2) Actually, the two equations with 4 variables are
> > Km Pn = 2 (m and n are exponents)
> > m + n = 12 (m and n are integer unknown numbers to be
determined;
> > the sum m + n gives the number of tones of the Piagui octave).
> > K and P are the semitone factors that replace the only Tempered
T = 2
> > (to the exponent 1/12) = 1.05946309436….
>
> This means you're looking for a 12 note MOS with semitones
> called K and P.
>
> > (3) SPECIAL WAY FOR SOLVING TWO EQUATIONS WITH 4 VARIABLES ( K,
P,
> > m, n) the K and P values must be near to the Tempered T because
the
> > sound discrepancy between Tempered and chord sounds are supposed
to
> > be small so K , P and T should have about the same magnitude.
> > It was also assumed that K and P are cells of the progression.
> > According to the preceding paragraph I took 6 cells that showed
> > positive and negative differencies with respect to T. The T
Tempered
> > factor is not a cell of the progression. The six experimental
cells
> > are
>
> I can't make any sense of this.
>
> > CELL # COMMA F(M,J,U) DECIMAL VALUES
> > 48 M M30J16U2 1.0558784008
> > 49 J M30J17U2 1.05707299111
> > 50 J M30J18U2 1.05826893295
> > 51 M M31J18U2 1.05946387773
> > 52 M M32J18U2 1.06066017178
> > 53 M M33J18U2 1.06185781663
> > T = 1.05946309436….
>
> The "COMMA" column is deriving these semitones in terms of
> the schismas above. So cell 48 is 30M + 16J + 2U (M, J, and
> U as pitch differences in e.g. cents rather than frequency
> ratios). I don't get the decimal values exact but they're
> near enough. I still have no idea why these values are
> special or what the letters after the cell numbers mean.
>
> > By mathematical steps and using these 6 figures in 9 auxiliary
> > equations, 9 results like the following figuries
> >
> > (#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
> >
> > ln = neperian logaritm
>
> Huh? What are a and b? And what's a "neperian logaritm"?
>
> > Eight solutions for n are not integer numbers. But solution
number
> > #5 gives n = 4, therefore m = 8 because m + n = 12.
>
> That'll be the diminished temperament Keenan identified.
>
> > The semitones K and P of the Piagui Musical Scale have been
sought
> > and worked out:
> > K = Kb = (Cell No. 52) = (9/8)1/2 = 1.06066017178
>
> That's the square root of 9:8 or 101.955 cents.
>
> > P = Pb = (Cell No. 49) = (8/9) ( 21/4 ) = 1.0570729911
>
> Aha! 8:9 times the 4th root of 2. 96.090 cents.
>
> > K8 P4 = [ (9/8)1/2 ]8 [ (8/9)( 21/4 ) ]4 = 2
>
> Tells us octaves are pure by the looks of it.
>
> > 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.
>
> 21/12 is a way of writing the twelfth root of 2. The rest
> is some form of technobabble for something like diminished
> temperament. An open choice of K and P arrangements
> suggests the 12 note scale needn't be an MOS.
>
>
> Graham
>
-----------------------------------------------------------------
-----------------------------------------------------------------

Chapter III
THE WIDE SPECTRUM OF
NATURAL CONSONANCE
III. 1 THE NATURAL PROGRESSION OF MUSICAL CELLS
Since the imperfection of the tempered scale is responsible for
discords, as acknowledged by analysts and musicians, then musical
instrument manufacturers should have the option of using a better
intonation in order to perfect their instruments' musical
expression.
Probably, a set of natural consonant frequencies could be deduced
from the scientific scales of Pythagoras, Just Intonation and their
extended scales. Such a set might include a progression of commas
ruled by mathematical and cyclical features in order to obtain the
information needed to establish a new musical scale of twelve
pitches per octave, to generate the expected harmony.
A study of consonance derived from the Pythagoras and Just
Intonation scales and the extremely useful comma sequence, permits
detection of 612 relative frequencies comprised between C = 1 and
octave 2C = 2.
The successive application of the three types of commas M, J and U
showed a set of 612 relative frequencies with respect to C = 1.
These frequencies are termed cells whose values depend solely on
numbers 2, 3 and 5. Among the cells, the ten frequencies comprised
in both heptatonic scales appear with high accuracy as well as the
remaining frequencies included in the extended scales of Pythagoras,
Just Intonation and those involved in musical elements for the first
time.

The M, J and U values were deduced by using the heptatonic semitones
256/243 and 16/15 as well as notes of the Pythagoras and Aristoxenus-
Zarlino Extended Scales. This origin gives commas, cells and
segments, the scientific features that will set out the musical
roots to perfecting harmony.
Successive cycles of 9/8 were detected in the progression of
relative frequencies. If C is the base, the first segment ends on
9/8. The second one starts at 9/8 and goes up to (9/8)2. The third
segment covers the interval (9/8)2 to (9/8)3 and so on. Each
segment contains 104 commas, with M as the majority of them. Fewer
type J follow, and only four type U.
All cells are distributed symmetrically with respect to the square
root of the product of the segment's initial and the final cell.
For example, the symmetrical center of the first segment (SC1) is
given by [1 x (9/8)]1/2. The second symmetrical center
(SC2) is given by [(9/8) (9/8)2]1/2, the third one
(SC3) by [(9/8)2 (9/8)3]1/2 and so on.
The comma sequence in the first segment is enough to derive the
sequence and values of the remaining cells up to the octave 2. The
sixth segment gives the final cell (9/8)6 = 2.02728652954 which is
equal to 2B# of the Pythagoras extended scale. When reduced to the
first octave by dividing by 2, it gives another cell as well. The
M, J and U commas are the exclusive functions of 2, 3 and 5:

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

Long ago, the quotient (32805 / 32768) = [(38 x 5) / 215] =
1.00112915039 = M was deduced and termed "comma" or "schisma", the
smallest consonance that can be distinguished by ear, according to
the analysts. In 1986, the analyses assumed that at least one of
the commas deduced centuries earlier, had to be one of the links in
a set of micro-consonance. It was a crucial and correct assumption.
The chromatic interval (9/8) ÷ (16/15) = (135 / 128) = 1.0546875 was
proposed by physicist Delezenne who used the Zarlino semitones
mentioned by J. Zamacois in his work "Teoría de la Música". This
value was derived from the natural intervals 9/8 and 16/15 of the
Pythagoras and Just Intonation scales respectively. The relation
between the Delezenne chromatic semitone 135 / 128 and the
Pythagoras semitone 256 / 243 gives the basic consonance M:
(135 / 128) / (256 / 243) = (32805 / 32768) = 1.00112915039 = M.
This was the origin of the fundamental comma M.
The numerical procedure to define the 624 cells is easily understood
when one cell of the progression is taken as an example, such as
Cell No. 21, having a value of M13 J8 = 1.024. When the M comma
that produces Cell No. 22 multiplies 1.024, the result is:
Cell No. 22 = (Cell No. 21 x M)
= [1.024 (38 x 5) / 215] = M14 J8 = 1.02515625.
In order to simplify the tabulation of 624 cells, the relative
frequency values, expressed in terms of 2, 3, 5 have been omitted.
In the following chapters, several tables concerning the Natural
Progression of Musical Cells, the new musical scale and other data
regarding harmony, make a study of the contents of this work a hard
job. It was not considered advisable to suppress any table and/or
mathematical discussions involved with musical elements and the new
scale harmonies that should be set out here. Other subjects related
to the proposal had to be set aside in order to avoid tiring the
reader with the innumerable data.
The following data are derived from the Natural Set of Relative
Frequencies which, in its turn, arises from the scientific scales of
Pythagoras and Aristoxenus-Zarlino. The data given in several pages
detail the authentic roots of the art of music.

Regards

MARIO PIZARRO
ELECTRONIC ENGINEER
(PIAGUISCALE)

--- In tuning@yahoogroups.com, "piaguiscale" <piagui@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > piaguiscale wrote:
> > >> So how exactly do M, J, and U enter into the calculation of K
> and
> > > P?
> > >> You have one equation in two variables and so an infinite
> number of
> > >> possible solutions for K and P. How did you decide on your
> > > particular
> > >> choice?
> > >>
> > >> David Bowen
> > >> --------------------------------------------------------------
--
> ----
> > > -
> > >
> > > (1) I decided to develope a progression of consonant and
> relative
> > > frequencies with respect to Do = 1 up to 2Do = 2 starting with
> the
> > > comma or schisma M = (32805/32768) = 1,00112915039. this comma
> was
> > > deduced in the XVI century by Zarlino; it is the smallest
> consonance
> > > that can be distinguished by the ear.
> >
> > Right, that's what we'd normally call a schisma, and so not
> > a comma. 1.954 cents.
> >
> > > Each relative frequency of the progression was named cell. At
> the
> > > beginning of the progression I had to deduced two additional
> commas
> > > J and U whose values are:
> > > J = 1.0011313711….
> > > U = 1.001213696511…
> >
> > Why?
> >
> > Anyway, J is 1.958 cents and U is 2.100 cents. So these are
> > approximate schismas.
> >
> > > With the aforementioned commas the progression from Do = 1 up
to
> 2Do
> > > = 2 contained 612 cells (cell # 612 = 2Do = 2). Each one was
> > > determined by multiplying the preceding cell by the comma that
> is
> > > aligned to the cell to be known ( M or J or U ).
> > > It was noticed that the first 100 cells contain the relative
> tone
> > > frequencies of the extended scales of Pythagoras and
Aristoxenus
> > > (bc) (21 tones per octaves each) emerged with exactness, so
> these
> > > tone frequencies were the references for expanding the
> progression
> > > up to cell # 612. The mentioned three commas that work
> cyclically
> > > produced all the tones of both extended scales a long the
whole
> set
> > > of 612 cells.
> >
> > Why 612? There are about 614.2 schismas to an octave, so
> > you're rounding off to a multiple of 12?
> >
> > > (2) Actually, the two equations with 4 variables are
> > > Km Pn = 2 (m and n are exponents)
> > > m + n = 12 (m and n are integer unknown numbers to be
> determined;
> > > the sum m + n gives the number of tones of the Piagui octave).
> > > K and P are the semitone factors that replace the only
Tempered
> T = 2
> > > (to the exponent 1/12) = 1.05946309436….
> >
> > This means you're looking for a 12 note MOS with semitones
> > called K and P.
> >
> > > (3) SPECIAL WAY FOR SOLVING TWO EQUATIONS WITH 4 VARIABLES (
K,
> P,
> > > m, n) the K and P values must be near to the Tempered T
because
> the
> > > sound discrepancy between Tempered and chord sounds are
supposed
> to
> > > be small so K , P and T should have about the same magnitude.
> > > It was also assumed that K and P are cells of the progression.
> > > According to the preceding paragraph I took 6 cells that
showed
> > > positive and negative differencies with respect to T. The T
> Tempered
> > > factor is not a cell of the progression. The six experimental
> cells
> > > are
> >
> > I can't make any sense of this.
> >
> > > CELL # COMMA F(M,J,U) DECIMAL VALUES
> > > 48 M M30J16U2 1.0558784008
> > > 49 J M30J17U2 1.05707299111
> > > 50 J M30J18U2 1.05826893295
> > > 51 M M31J18U2 1.05946387773
> > > 52 M M32J18U2 1.06066017178
> > > 53 M M33J18U2 1.06185781663
> > > T = 1.05946309436….
> >
> > The "COMMA" column is deriving these semitones in terms of
> > the schismas above. So cell 48 is 30M + 16J + 2U (M, J, and
> > U as pitch differences in e.g. cents rather than frequency
> > ratios). I don't get the decimal values exact but they're
> > near enough. I still have no idea why these values are
> > special or what the letters after the cell numbers mean.
> >
> > > By mathematical steps and using these 6 figures in 9 auxiliary
> > > equations, 9 results like the following figuries
> > >
> > > (#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
> > >
> > > ln = neperian logaritm
> >
> > Huh? What are a and b? And what's a "neperian logaritm"?
> >
> > > Eight solutions for n are not integer numbers. But solution
> number
> > > #5 gives n = 4, therefore m = 8 because m + n = 12.
> >
> > That'll be the diminished temperament Keenan identified.
> >
> > > The semitones K and P of the Piagui Musical Scale have been
> sought
> > > and worked out:
> > > K = Kb = (Cell No. 52) = (9/8)1/2 = 1.06066017178
> >
> > That's the square root of 9:8 or 101.955 cents.
> >
> > > P = Pb = (Cell No. 49) = (8/9) ( 21/4 ) = 1.0570729911
> >
> > Aha! 8:9 times the 4th root of 2. 96.090 cents.
> >
> > > K8 P4 = [ (9/8)1/2 ]8 [ (8/9)( 21/4 ) ]4 = 2
> >
> > Tells us octaves are pure by the looks of it.
> >
> > > 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.
> >
> > 21/12 is a way of writing the twelfth root of 2. The rest
> > is some form of technobabble for something like diminished
> > temperament. An open choice of K and P arrangements
> > suggests the 12 note scale needn't be an MOS.
> >
> >
> > Graham
> >
> -----------------------------------------------------------------
> -----------------------------------------------------------------
>
> Chapter III
> THE WIDE SPECTRUM OF
> NATURAL CONSONANCE
> III. 1 THE NATURAL PROGRESSION OF MUSICAL CELLS
> Since the imperfection of the tempered scale is responsible for
> discords, as acknowledged by analysts and musicians, then musical
> instrument manufacturers should have the option of using a better
> intonation in order to perfect their instruments' musical
> expression.
> Probably, a set of natural consonant frequencies could be deduced
> from the scientific scales of Pythagoras, Just Intonation and
their
> extended scales. Such a set might include a progression of commas
> ruled by mathematical and cyclical features in order to obtain the
> information needed to establish a new musical scale of twelve
> pitches per octave, to generate the expected harmony.
> A study of consonance derived from the Pythagoras and Just
> Intonation scales and the extremely useful comma sequence, permits
> detection of 612 relative frequencies comprised between C = 1 and
> octave 2C = 2.
> The successive application of the three types of commas M, J and U
> showed a set of 612 relative frequencies with respect to C = 1.
> These frequencies are termed cells whose values depend solely on
> numbers 2, 3 and 5. Among the cells, the ten frequencies
comprised
> in both heptatonic scales appear with high accuracy as well as the
> remaining frequencies included in the extended scales of
Pythagoras,
> Just Intonation and those involved in musical elements for the
first
> time.
>
> The M, J and U values were deduced by using the heptatonic
semitones
> 256/243 and 16/15 as well as notes of the Pythagoras and
Aristoxenus-
> Zarlino Extended Scales. This origin gives commas, cells and
> segments, the scientific features that will set out the musical
> roots to perfecting harmony.
> Successive cycles of 9/8 were detected in the progression of
> relative frequencies. If C is the base, the first segment ends on
> 9/8. The second one starts at 9/8 and goes up to (9/8)2. The
third
> segment covers the interval (9/8)2 to (9/8)3 and so on. Each
> segment contains 104 commas, with M as the majority of them.
Fewer
> type J follow, and only four type U.
> All cells are distributed symmetrically with respect to the square
> root of the product of the segment's initial and the final cell.
> For example, the symmetrical center of the first segment (SC1) is
> given by [1 x (9/8)]1/2. The second symmetrical center
> (SC2) is given by [(9/8) (9/8)2]1/2, the third one
> (SC3) by [(9/8)2 (9/8)3]1/2 and so on.
> The comma sequence in the first segment is enough to derive the
> sequence and values of the remaining cells up to the octave 2.
The
> sixth segment gives the final cell (9/8)6 = 2.02728652954 which is
> equal to 2B# of the Pythagoras extended scale. When reduced to
the
> first octave by dividing by 2, it gives another cell as well. The
> M, J and U commas are the exclusive functions of 2, 3 and 5:
>
> 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
>
> Long ago, the quotient (32805 / 32768) = [(38 x 5) / 215] =
> 1.00112915039 = M was deduced and termed "comma" or "schisma", the
> smallest consonance that can be distinguished by ear, according to
> the analysts. In 1986, the analyses assumed that at least one of
> the commas deduced centuries earlier, had to be one of the links
in
> a set of micro-consonance. It was a crucial and correct
assumption.
> The chromatic interval (9/8) ÷ (16/15) = (135 / 128) = 1.0546875
was
> proposed by physicist Delezenne who used the Zarlino semitones
> mentioned by J. Zamacois in his work "Teoría de la Música". This
> value was derived from the natural intervals 9/8 and 16/15 of the
> Pythagoras and Just Intonation scales respectively. The relation
> between the Delezenne chromatic semitone 135 / 128 and the
> Pythagoras semitone 256 / 243 gives the basic consonance M:
> (135 / 128) / (256 / 243) = (32805 / 32768) = 1.00112915039 = M.
> This was the origin of the fundamental comma M.
> The numerical procedure to define the 624 cells is easily
understood
> when one cell of the progression is taken as an example, such as
> Cell No. 21, having a value of M13 J8 = 1.024. When the M comma
> that produces Cell No. 22 multiplies 1.024, the result is:
> Cell No. 22 = (Cell No. 21 x M)
> = [1.024 (38 x 5) / 215] = M14 J8 = 1.02515625.
> In order to simplify the tabulation of 624 cells, the relative
> frequency values, expressed in terms of 2, 3, 5 have been omitted.
> In the following chapters, several tables concerning the Natural
> Progression of Musical Cells, the new musical scale and other data
> regarding harmony, make a study of the contents of this work a
hard
> job. It was not considered advisable to suppress any table and/or
> mathematical discussions involved with musical elements and the
new
> scale harmonies that should be set out here. Other subjects
related
> to the proposal had to be set aside in order to avoid tiring the
> reader with the innumerable data.
> The following data are derived from the Natural Set of Relative
> Frequencies which, in its turn, arises from the scientific scales
of
> Pythagoras and Aristoxenus-Zarlino. The data given in several
pages
> detail the authentic roots of the art of music.
>
> Regards
>
> MARIO PIZARRO
> ELECTRONIC ENGINEER
> (PIAGUISCALE)
>
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

To Graham Breed
David Bowen

I think that if you read carefully the message # 76868 you will
understand the details of the Piagui Musical Scale. Since I had to
send manymessages to the tuning group I am a little confused and
don´t remember to whom I have responded his mesage. Could you ask
the same questions?.
It is a pity that I can not send diagrams from my book.

Regards

MARIO PIZARRO
ELECTRONIC ENGINEER
(PIAGUISCALE)

piaguiscale wrote:

> I think that if you read carefully the message # 76868 you will > understand the details of the Piagui Musical Scale. Since I had to > send manymessages to the tuning group I am a little confused and > don�t remember to whom I have responded his mesage. Could you ask > the same questions?.
> It is a pity that I can not send diagrams from my book.

I have the messages from you but I don't have the time to read them carefully. I'll get back to you when I have some questions.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> piaguiscale wrote:
>
> > I think that if you read carefully the message # 76868 you will
> > understand the details of the Piagui Musical Scale. Since I had
to
> > send manymessages to the tuning group I am a little confused and
> > don´t remember to whom I have responded his mesage. Could you
ask
> > the same questions?.
> > It is a pity that I can not send diagrams from my book.
>
> I have the messages from you but I don't have the time to
> read them carefully. I'll get back to you when I have some
> questions.
>
>
> Graham
>
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Graham,
Piaguiscale is my ID and not my name. I would appreciate if you use
my real name.
Regarding the term "carefully" I wrote in my message, I admit that
it was not a happy term. I apologize.
Regards

Mario Pizarro
Lima, June 8
< piagui@... >
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

ANALYSES OF TEMPERED
AND PIAGUI CHORDS
When successive notes of a piece of music are played rapidly, the
human brain has insufficient time to discern perfect or imperfect
consonance and harmony. In these cases, there is no difference
between the Tempered and the Piagui Scales. Perhaps the least time
needed to evaluate a harmony is about 0.5 seconds.
Several triads of the three new intonations have the same absolute
frequency components (Hz). Following are some of the identical
triads.
Piagui I - C Major = C + E + G = Piagui II - C Major
Piagui I - Eb Major = Eb + G + Bb = Piagui II - Eb
Major
Piagui I - F# Major= F# + Bb + 2C# = Piagui II - F# Major
Piagui I - A Major = A + 2C# + 2E = Piagui II - A Major
Piagui I - C minor = C + Eb + G = Piagui II - C minor
Piagui I - Eb minor= Eb + F# + Bb = Piagui II - Eb minor
Piagui I - F# minor= F# + A + 2C# = Piagui II - F# minor
Piagui I - A minor= A + 2C + 2E = Piagui II - A minor
As Piagui I and Tempered components of E Major are 330, 392.4383,
495 Hz and 329.6276, 391.9954, 493.8833 Hz respectively, typical
discrepancies of 0, + 0.113% and + 0.226% are obtained by dividing
each frequency given in Table XIV by its corresponding Tempered
value as follows: [(330 x 100) / (329.6276)] = + 0.113%
[(392.4383 x 100) / (391.9954)] = + 0.226%
[(495 x 100) / (493.8833)] = + 0.226%

NOTE SYMBOL PIAGUI I TEMPERED DISCREPANCY %*
DO -----C ----- 261.6255(Hz)--- 261.6255(Hz)--- 0
DO# --- C# ---- 277.4958 ------ 277.1826 ------ + 0.113
RE ---- D ----- 294.3287 ------ 293.6648 ------ + 0.226
Eb --- Eb ---- 311.127 ------- 311.127 ------- 0
MI ---- E ----- 330 ----------- 329.6276 ------ + 0.113
FA ---- F ----- 350.0178 ------ 349.2282 ------ + 0.226
FA# --- F# ---- 369.9944 ------ 369.9944 ------ 0
SOL --- G ----- 392.4383 ------ 391.9954 ------ + 0.113
LAb -- Ab ---- 416.2437 ------ 415.3047 ------ + 0.226
LA ---- A ----- 440 ----------- 440 ----------- 0
Tib --- Bb ---- 466.6905 ------ 466.1637 ------ + 0.113
TI ---- B ----- 495 ----------- 493.8833 ------ + 0.226
2DO --- 2C ---- 523.2511 ------ 523.2511 ------ 0

NOTE SYMBOL PIAGUI II TEMPERED DISCREPANCY %*
DO -----C------ 261.6255(Hz)--- 261.6255(Hz)----0
DO# ----C# ---- 277.4958 ------ 277.1826 ------ + 0.113
RE -----D ----- 293.3333 ------ 293.6648 ------ – 0.113
Eb -----Eb ---- 311.127 ------- 311.127 ------- 0
MI----- E ----- 330 ----------- 329.6276 ------ + 0.113
FA ---- F ----- 348.834 ------- 349.2282 ------ – 0.113
FA# --- F# ---- 369.9944 ------ 369.9944 ------ 0
SOL --- G ----- 392.4383 ------ 391.9954 ------ + 0.113
Ab ---- Ab ---- 414.836 ------- 415.3047 ------ – 0.113
LA ---- A ----- 440 ----------- 440 ----------- 0
Bb ---- Bb ---- 466.6905 ------ 466.1637 ------ + 0.113
TI ---- B ----- 493.3259 ------ 493.8833 ------ – 0.113
2DO---- 2C ---- 523.2511 ------ 523.2511 ------ 0
* With respect to Tempered frequencies

NOTE SYMBOL PIAGUI III TEMPERED DISCREPANCY %*
DO -----C ------261.6255(Hz)--- 261.6255(Hz)--- 0
DO#-----C#----- 276.5573------ 277.1826------ – 0.226
RE------D------ 293.3333------ 293.6648------ – 0.113
Eb------Eb----- 311.127 ------- 311.127 --------0
MI------E------ 328.8839 ------ 329.6276 ------ – 0.226
FA------F------ 348.8341------ 349.2282------ – 0.113
FA#-----F#----- 369.9944------ 369.9944------ 0
SOL-----G------ 391.1111------ 391.9954------ – 0.226
Ab------Ab 414.836 ------ 415.3047------ – 0.113
LA------A------ 440 ---------- 440 ---------- 0
Bb------Bb 465.1121------ 466.1637 ------ – 0.226
TI------B------ 493.3259------ 493.8833 ------ – 0.113
2DO-----2C 523.2511------ 523.2511------ 0
* With respect to Tempered frequencies

In Chapter VIII, chord graph evaluations will be made by examining
the chord waves and chord wave peaks of all twenty-four major and
minor triads of both musical systems. The Piagui graph responses
will show aesthetic displays during a one second period for the
triads whose third component is a perfect fifth (3/2) = 1.5. The
triads with keynotes D, F, Ab, B, whose third components are
concordant fifths (1.4949269), gave also perfect responses, the
aesthetic displays are the features of this group of major triads,
while Dm, Fm, Abm, Bm responses are better than the tempered ones.
As a result, twenty perfect triads and four concordant ones make a
total of twenty-four basic triads.
The Piagui I responses of Major triads C, Eb, F# and A are similar
and present 0, + 0.113% and + 0.113% discrepancies with respect to
corresponding Tempered tone frequencies. Consequently, only C Major
triad will be plotted and discussed.

If t1 and t2 are constant, a triad is composed as follows:
sin (2O f1t) + sin [2O f2 (t – t1)] + sin [2O f3 (t – t2)]
The above sum of sinusoids can be plotted to get a chord wave. In
this way plotting reveals whether the frequency components –with
values of four or six place numbers– are the proper ones to give the
right harmony, depending on ratios f3 / f2 and f2 / f1.
Comparisons between Tempered and Piagui triads must be made by
computer and printer for any possible discussion of both kinds of
harmony.
Since the chord wave graphs were not precise enough to reach any
clear and definite conclusions regarding harmony, another type of
graph had to be used: chord wave peaks. In this way, comparison
resulted from an inspection of both Tempered and Piagui triads.
Plotting chord waves and their peaks is probably the only way to
encompass the object of harmony comparisons. Through graph
comparisons, peaks of the Piagui chord waves appear to satisfy the
visual aesthetics, while tempered responses do not. This method was
chosen to provide a visualization of harmonious and discordant
triads.
In one second, an examination of Tempered and Piagui chord wave
peaks showed irregular and aesthetic displays respectively, which
correspond to flawed and true harmonies.
In some cases, when looking at a chord wave response, the real
harmony makes aesthetic wave areas that can be traced periodically,
while the chord wave of an imperfect harmony describes disorderly
and non-periodic waves. In order to analyze clearly both types of
responses, Tempered and Piagui chord waves and chord wave peaks,
were drawn to attain definite conclusions regarding just and flawed
harmony.
The influence of chord wave peaks on the human ear has probably been
studied, as many cases reveal the harmony or discordance of several
sorts of chords if they are evaluated by their chord wave peak
responses.
When an instrument emits three frequencies, each component disturbs
the atmospheric pressure of the environment independently of the
other two components. The foregoing sum of three sinusoids is the
mathematical expression of a triad, where it is assumed that wave
amplitudes are equal to one. There are an infinite number of
combinations of the three components, which depend on t1 and t2.
Anyway, in all infinite cases the triad sound is the same. One of
them, the particular case of t1 = t 2 = 0 is given by:
sin (2O f1 t) + sin (2O f2 t) + sin (2O f3 t)
For instance, Piagui I - G Major may be expressed as follows:
G Major = (G + B + 2D)
= sin [2O (392.4383) t]+ sin [2O (495) t] + sin [2O(2 × 294.32875) t]
The following page shows the sum of three sinusoidal components that
gets the chord wave of G Major in only 5.096 milliseconds. The
graph shows how computer and printer draw a chord wave.
The Tempered and Piagui I frequencies of a particular octave are
given in Table XIV on page 101. These data are used to get chord
waves of major and minor triads and their chord wave peaks.
Actually, the sinusoidal components of any chord of a musical
instrument are combined at random. The preceding sum of three
sinusoids is an unlikely case, since normally each sinusoidal wave
works independently. One particular case where two components are
delayed with respect to the keynote is given here and plotted on
Graphs Nos. 14 and 16. The example is Piagui I - G Major.
G Major = sin [2O (392.4383) t] + sin [2O (495) (t – 0.002)]
+ sin [2O (588.6575) (t – 0.004)]
The first sinusoidal component leads the second and third
components. By examining the graphs, we see that delays do not
affect the periodicity of the complex wave nor the aesthetic
response of chord wave peaks. By applying the same delays to the
Tempered chord, another shape of non-aesthetic display is revealed.
THE COMPONENTS OF PIAGUI I – G MAJOR

VI. 1 THE MAJOR SCALE
The scale of C Major is made up of the following notes:
C D E F G A B 2C
1 2 3 4 5 6 7 8
(2) (2) (1) (2) (2) (2) (1)
It is useful to note the numbering placed below each note of the
scale as a guide to constructing all sorts of chords. The number
of semitones between two consecutive notes is indicated in
parentheses and applies to all major scales.
C is the keynote, E is third, F is fourth and G is fifth and so on.
The notes of all remaining major scales are named in the same way.
The components of major triads are 1, 3, 5 (the keynote, third and
fifth) while the minor triads are constructed by 1, b3, 5 where the
third is lowered one semitone.
The tones that form major and minor triads of the Piagui I scale and
their frequency discrepancies with respect to the Tempered ones are
given below.
The following pages give the discrepant percentages of chord tone
frequencies of the Piagui scale when compared to the corresponding
tempered chord tones. As noted, three degrees of discrepancy given
by 0, ± 0.113% and ± 0.226% mark the difference between Tempered and
Piagui chords. Despite the fact that these discrepancies are very
small, they make important changes in harmony as shown by chord wave
peaks given in Chapter VIII.
MARIO PIZARRO
ELECTRONIC ENGINEER

I have repeatedly asked how the circle of fifths looks in this tuning,
i.e. what is the temperament in each fifth (never mind the 612
separate cells!) - I have been unable to get any straight answer to
this very simple question. Good luck.

What I *think* the answer is:

D-A, B-F#, G#-Eb, F-C are tempered 1/4 Pythagorean comma; all the
other fifths are pure.

(Of course one can transpose this temperament up or down a fifth to
get three possibilities in total.)

I would not be surprised if someone had already written this
temperament down in the 18th century... anyway it is generally a good
thing to be able to describe your result in a simple and obvious way
rather than taking up hundreds of lines of mathematical obfuscation.
~~~T~~~

--- In tuning@yahoogroups.com, "piaguiscale" <piagui@...> wrote:
>
>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> I have repeatedly asked how the circle of fifths looks in this
tuning,
> i.e. what is the temperament in each fifth (never mind the 612
> separate cells!) - I have been unable to get any straight answer to
> this very simple question. Good luck.
>
> What I *think* the answer is:
>
> D-A, B-F#, G#-Eb, F-C are tempered 1/4 Pythagorean comma; all the
> other fifths are pure.
>
> (Of course one can transpose this temperament up or down a fifth to
> get three possibilities in total.)
>
> I would not be surprised if someone had already written this
> temperament down in the 18th century... anyway it is generally a
good
> thing to be able to describe your result in a simple and obvious
way
> rather than taking up hundreds of lines of mathematical
obfuscation.
> ~~~T~~~
>
>
> --- In tuning@yahoogroups.com, "piaguiscale" <piagui@> wrote:
> >
> >
>
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

To: < tuning@yahoogroups.com >
From: Mario Pizarro, ID: piaguiscale
Date: June 9, 2008

Tom Dent wrote some opinions about the Piagui Musical Scale and I
should respond to < tuning@yahoogroups.com >.
--Sincerely, I do not understand and cannot explain how the circle
of fifths looks in the Piagui scale (If there is that circle in my
scale). To ignore a subject is not a sin.
-- Some days ago, Tom Dent explained that to obtain the circle of
fifths I can start from C = 1 and determine the fifth G = 1,5. Again
the fifth of G that gives D = 2,25 , again the fifth of D to get
2,25 x 1,5 = 3,375 whose reduction gives 1,6875 = Note A of the
Pythagoras scale. The first three tones C, G, and D= 2,25 are Piagui
tones but the last fifth (A = 1,6875)is Pythagorean. ¿ WHAT AM I
DOING IN THE PYTHAGORAS SCALE ?.
-- He also wrote that he put a very simple question. The point is
that I do not see how to continue from a pithagorean tone and why
the Piagui frequencies have to obey predetermined factors like 1,5.
These relations correspond to the Pythagoras criteria. All the
Piagui tone frequencies are related by perfect fifths and perfect
fourths as I showed in one of my messages.
--Tom Dent advises that "NEVER MIND THE 612 SEPARATE CELLS". These
words are equivalent to the phrase: "NEVER MIND THE BEATS OF YOUR
OWN HEART". He is against the 612 consonant cells of the progression
despite this set of frequencies established the K and P semitone
factors and the three variants of Piagui octaves; a ciclical
development where the chisma M together with the commas J and U fill
the octave with exactness. As a matter of fact cell # 612 = 2 is
equal to M376 X J212 X U24.
-- Sooner or later this progression of musical cells will be
acknowledged in the acoustics of physics.
-- Tom Dent uses tabulations like logarithmic tables, I know that he
is a scientific then ¿ Why he suggests to ignore the progression
cells?.
-- Since he has the three sets of Piagui octaves in Hz and in
relative values respect to D= 1, he can derive the circle of fifths,
I don´t mind.
-- Nobody has deduced in the past the perfect musical scale. The
Piagui octaves do it.

MARIO PIZARRO
< piagui@ec-red.com >
Lima, June 9

--- In tuning@yahoogroups.com, "piaguiscale" <piagui@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> >
> > I have repeatedly asked how the circle of fifths looks in this
> tuning,
> > i.e. what is the temperament in each fifth (never mind the 612
> > separate cells!) - I have been unable to get any straight answer
to
> > this very simple question. Good luck.
> >
> > What I *think* the answer is:
> >
> > D-A, B-F#, G#-Eb, F-C are tempered 1/4 Pythagorean comma; all the
> > other fifths are pure.
> >
> > (Of course one can transpose this temperament up or down a fifth
to
> > get three possibilities in total.)
> >
> > I would not be surprised if someone had already written this
> > temperament down in the 18th century... anyway it is generally a
> good
> > thing to be able to describe your result in a simple and obvious
> way
> > rather than taking up hundreds of lines of mathematical
> obfuscation.
> > ~~~T~~~
> >
> >
> > --- In tuning@yahoogroups.com, "piaguiscale" <piagui@> wrote:
> > >
> > >
> >
> xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> To: < tuning@yahoogroups.com >
> From: Mario Pizarro, ID: piaguiscale
> Date: June 9, 2008
>
> Tom Dent wrote some opinions about the Piagui Musical Scale and I
> should respond to < tuning@yahoogroups.com >.
> --Sincerely, I do not understand and cannot explain how the circle
> of fifths looks in the Piagui scale (If there is that circle in my
> scale). To ignore a subject is not a sin.
> -- Some days ago, Tom Dent explained that to obtain the circle of
> fifths I can start from C = 1 and determine the fifth G = 1,5.
Again
> the fifth of G that gives D = 2,25 , again the fifth of D to get
> 2,25 x 1,5 = 3,375 whose reduction gives 1,6875 = Note A of the
> Pythagoras scale. The first three tones C, G, and D= 2,25 are
Piagui
> tones but the last fifth (A = 1,6875)is Pythagorean. ¿ WHAT AM I
> DOING IN THE PYTHAGORAS SCALE ?.
> -- He also wrote that he put a very simple question. The point is
> that I do not see how to continue from a pithagorean tone and why
> the Piagui frequencies have to obey predetermined factors like
1,5.
> These relations correspond to the Pythagoras criteria. All the
> Piagui tone frequencies are related by perfect fifths and perfect
> fourths as I showed in one of my messages.
> --Tom Dent advises that "NEVER MIND THE 612 SEPARATE CELLS". These
> words are equivalent to the phrase: "NEVER MIND THE BEATS OF YOUR
> OWN HEART". He is against the 612 consonant cells of the
progression
> despite this set of frequencies established the K and P semitone
> factors and the three variants of Piagui octaves; a ciclical
> development where the chisma M together with the commas J and U
fill
> the octave with exactness. As a matter of fact cell # 612 = 2 is
> equal to M376 X J212 X U24.
> -- Sooner or later this progression of musical cells will be
> acknowledged in the acoustics of physics.
> -- Tom Dent uses tabulations like logarithmic tables, I know that
he
> is a scientific then ¿ Why he suggests to ignore the progression
> cells?.
> -- Since he has the three sets of Piagui octaves in Hz and in
> relative values respect to D= 1, he can derive the circle of
fifths,
> I don´t mind.
> -- Nobody has deduced in the past the perfect musical scale. The
> Piagui octaves do it.
>
> MARIO PIZARRO
> < piagui@... >
> Lima, June 9
>
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
CORRECTION:
C = 1 INSTEAD OF D = 1.
The following paragraph contains an error:
-- Since he has the three sets of Piagui octaves in Hz and in
relative values respect to D= 1, he can derive the circle of fifths,
I don´t mind.
---------------

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> I have repeatedly asked how the circle of fifths looks in this
tuning,
> i.e. what is the temperament in each fifth (never mind the 612
> separate cells!) - I have been unable to get any straight answer to
> this very simple question. Good luck.
>
> What I *think* the answer is:
>
> D-A, B-F#, G#-Eb, F-C are tempered 1/4 Pythagorean comma; all the
> other fifths are pure.
>
> (Of course one can transpose this temperament up or down a fifth to
> get three possibilities in total.)
>
> I would not be surprised if someone had already written this
> temperament down in the 18th century... anyway it is generally a
good
> thing to be able to describe your result in a simple and obvious
way
> rather than taking up hundreds of lines of mathematical
obfuscation.
> ~~~T~~~
>
>
> --- In tuning@yahoogroups.com, "piaguiscale" <piagui@> wrote:
> >
> >
>
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

To: < tuning@yahoogroups.com >
From: Mario Pizarro, ID: piaguiscale
Date: June 9, 2008

Tom Dent wrote some opinions about the Piagui Musical Scale and I
should respond to < tuning@yahoogroups.com >.
--Sincerely, I do not understand and cannot explain how the circle
of fifths looks in the Piagui scale (If there is that circle in my
scale). To ignore a subject is not a sin.
-- Some days ago, Tom Dent explained that to obtain the circle of
fifths I can start from C = 1 and determine the fifth G = 1,5. Again
the fifth of G that gives D = 2,25 , again the fifth of D to get
2,25 x 1,5 = 3,375 whose reduction gives 1,6875 = Note A of the
Pythagoras scale. The first three tones C, G, and D= 2,25 are Piagui
tones but the last fifth (A = 1,6875)is Pythagorean. ¿ WHAT AM I
DOING IN THE PYTHAGORAS SCALE ?.
-- He also wrote that he put a very simple question. The point is
that I do not see how to continue from a pithagorean tone and why
the Piagui frequencies have to obey predetermined factors like 1,5.
These relations correspond to the Pythagoras criteria. All the
Piagui tone frequencies are related by perfect fifths and perfect
fourths as I showed in one of my messages.
--Tom Dent advises that "NEVER MIND THE 612 SEPARATE CELLS". These
words are equivalent to the phrase: "NEVER MIND THE BEATS OF YOUR
OWN HEART". He is against the 612 consonant cells of the progression
despite this set of frequencies established the K and P semitone
factors and the three variants of Piagui octaves; a cyclical
development where the schisma M together with the commas J and U fill
the octave with exactness. As a matter of fact cell # 612 = 2 is
equal to M376 X J212 X U24.
-- Sooner or later this progression of musical cells will be
acknowledged in the acoustics of physics.
-- Tom Dent uses tabulations like logarithmic tables, I know that he
is a scientific then ¿ Why he suggests to ignore the progression
cells?.
-- Since he has the three sets of Piagui octaves in Hz and in
relative values respect to C= 1, he can derive the circle of fifths,
I don´t mind.
-- Nobody has deduced in the past the perfect musical scale. The
Piagui octaves do it.

MARIO PIZARRO
< piagui@... >
Lima, June 9
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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Here are the Scala files:

http://www.box.net/shared/svawfbkocs

Piagui I is the first layout, Piagui II the second, and Piagui III the third.

It is some interesting kind of diminished temperament. Some of the
keys have slightly better fifths than 12-equal at the cost of the
major thirds, some of the keys have slightly better thirds than
12-equal at the cost of the fifths. Either way, none of the notes are
any more than 4 cents away from 12-equal, so I'd be surprised if the
two could really be distinguished in practice.

-Mike

Tom Dent wrote:
> > What I *think* the answer is:
> >
> > D-A, B-F#, G#-Eb, F-C are tempered 1/4 Pythagorean comma; all the
> > other fifths are pure.
> >
> > (Of course one can transpose this temperament up or down a fifth
> > to get three possibilities in total.)
> >
> > I would not be surprised if someone had already written this
> > temperament down in the 18th century... anyway it is generally a
> > good thing to be able to describe your result in a simple and
> > obvious way rather than taking up hundreds of lines of
> > mathematical obfuscation.

One of those two rotations is F W Marpurg's temperament "H" from _Versuch �ber die musicalische Temperatur_ (Breslau, 1776):

G-D, E-B, C#-G#, Bb-F each narrow by 1/4 PC; all other 5ths pure.

Available in J Murray Barbour's article "Irregular systems of temperament", _Journal of the American Musicological Society_ 1:3 (Fall 1948), page 23.

Brad Lehman

Exactly, every one of these scales is just two pure fifths and one
1/4-PC tempered fifth, repeated all the way round the circle of
fifths. I.e. chop the Pythagorean comma into 4 and distribute the
pieces equally round the circle. That's why C-Eb-F#-A are always
equally tempered, in fact every minor third is the same as 12ET. The
peculiar properties of the tuning, if any, must come from the fact
that eight out of 12 fifths are pure.

(Of course, I did always acknowledge the presence of tempered fifths
in the scale. For example C-G-D pure, D-A tempered by PC/4, A-E-B pure
etc. etc.

So I can perfectly well also deal with fractions of a comma in
temperament. All I wanted to know is the quality of the fifths in the
tunings. Which is now quite clear.)

I would urge Mr. Pizarro to learn what the circle of fifths is, it is
a very basic and useful concept that is required by anyone tuning a
piano by ear, and will allow him to invent a large quantity of new
12-pitch tunings very quickly. Building a scale by fifths is a lot
easier than building by semitones.

One may have 612 cells per octave, but what is the point if you only
ever use about 20 of them?
~~~T~~~

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> Here are the second and third layouts he mentioned:
>
> (Do=1) x K x P x K x K x P x K x K x P x K x K x P x K = 2
> K = 1.06066017178 = 101.955 cents
> P = 1.0570729911 = 96.090 cents
>
> C - 0 cents
> C#/Db - 101.955 cents
> D - 198.045 cents
> D#/Eb - 300 cents
> E - 401.955 cents
> F - 498.045 cents
> F#/Gb - 600 cents
> G - 701.955 cents
> G#/Ab - 798.045 cents
> A - 900 cents
> A#/Bb - 1001.955 cents
> B - 1098.045 cents
> C - 1200 cents
>
>
> (Do=1) x P x K x K x P x K x K x P x K x K x P x K x K = 2
> K = 1.06066017178 = 101.955 cents
> P = 1.0570729911 = 96.090 cents
>
>
> C - 0 cents
> C#/Db - 96.090 cents
> D - 198.045 cents
> D#/Eb - 300 cents
> E - 396.090 cents
> F - 498.045 cents
> F#/Gb - 600 cents
> G - 696.090 cents
> G#/Ab - 798.045 cents
> A - 900 cents
> A#/Bb - 996.090 cents
> B - 1098.045 cents
> C - 1200 cents
>

On Tue, Jun 10, 2008 at 1:43 AM, Mike Battaglia <battaglia01@...> wrote:
> Here are the Scala files:
>
> http://www.box.net/shared/svawfbkocs
>
> Piagui I is the first layout, Piagui II the second, and Piagui III the third.
>
> It is some interesting kind of diminished temperament. Some of the
> keys have slightly better fifths than 12-equal at the cost of the
> major thirds, some of the keys have slightly better thirds than
> 12-equal at the cost of the fifths. Either way, none of the notes are
> any more than 4 cents away from 12-equal, so I'd be surprised if the
> two could really be distinguished in practice.

Thanks for clearing this up, Mike. Until now I had no idea what Piagui
was talking about.

Piagui, what's your response to this statement that your scale is
practically indistinguishable from equal temperament?

Keenan Pepper

> > Piagui I is the first layout, Piagui II the second, and Piagui III
> > the third.
> >
> > It is some interesting kind of diminished temperament. Some of the
> > keys have slightly better fifths than 12-equal at the cost of the
> > major thirds, some of the keys have slightly better thirds than
> > 12-equal at the cost of the fifths. Either way, none of the notes
> > are any more than 4 cents away from 12-equal, so I'd be surprised if > > the two could really be distinguished in practice.
>
> Thanks for clearing this up, Mike. Until now I had no idea what Piagui
> was talking about.
>
> Piagui, what's your response to this statement that your scale is
> practically indistinguishable from equal temperament?

Or a response to the observation that one of those three versions was already published by at least two other people (Marpurg in 1776, Barbour in 1948)? It's a possibly useful temperament; it's just not new. But, the lack of novelty doesn't necessarily diminish its value.

On harpsichord, the two different types of 5ths in it (pure vs 1/4 PC narrow) can *certainly* be distinguished in practice. When 5ths or 12ths that severely tempered get played directly, in music with two-voiced texture, the impurity is obvious. The vigorous vibrato of the beats stands out, in contrast with the steadiness of the pure 5ths.

As Mike mentioned, there are two different sizes of major 3rds in it. Eight of them are 8/11 SC sharp, and the other four are only 5/11 SC sharp.

There are two different sizes of semitone in it: eight of them are 102 cents, and the other four are 96 cents.

Brad Lehman