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Mario Pizarro
Aug-17-2005, 16:28
I have set in my book: The Piagui Musical Scale: Perfecting Harmony, new concepts of the scientific basis of a new musical scale.
The smallest comma M= (32805/32768)=1.00112915039 and the new J and U ones I detected, define a progression of 612 relative frequencies from note Do = 1 up to 2Do=2.
The treatment is based on limited information from the ancient scales, as well as on data contained in the progression for detecting K and P semitone factors that rule the harmony of the Piagui scale and replace the Tempered T.
The combined work of K and P can set the needed twelve-tone frequencies for any octave. Their precise and suitable values determine that perfect fifths and perfect fourths link all tone frequencies of the piano keyboard in cycles. These remarkable results made possible the attainment of the best expressions of harmony.
The graphs given in the book show the quality of harmony and let the evaluation of all major and minor triads of both, Tempered and Piagui scales. It also deals with the application of the new scale to the piano, electronic organ, guitar and electronic tuner.
The equal-tempered octave is ruled by 1xTxTxTxTxTxTxTxTxTxTxTxT = 2 while
1 x K K P K K P K K P K K P = 2 corresponds to the Piagui octave. The values of semitone factors K and P are also deduced in the book.
Sufficient data are given for tuning the piano to the top quality musical scale.
The Piagui scale is not an invention; it is a discovery that resolves the slight discordance problem of the equal-tempered scale.
C. Mario Pizarro
piagui@ec-red.com

Frederik Magle
Aug-17-2005, 16:55
Welcome to this forum Mario Pizzaro!
That's a very interesting subject, but please note that we do not allow advertising (especially not in a first post) according to the posting guidelines (http://www.magle.dk/music-forums/information-announcements/announcement-posting-guidelines-rules.html). So I have removed the link in your post. Feel free to add more information to this thread (by clicking reply) that does not require buying the book https://www.magle.dk/ubbthreads/images/graemlins/smile.gif

Mario Pizarro
Aug-19-2005, 19:40
The opinions of some noted researchers regarding the equal-tempered scale are given here.
--"Equal temperament enables us to play equally well, or perhaps we should say equally badly, in all keys".
--"Thus the tuning of a tempered instrument is in reality a process of controlled mistuning".
The Tempered scale attains octave quadrature by means of simple progression, with no link to the scientific consonance that distinguishes the Pythagoras and Just Intonation scales. The majority of audiences do not detect its slight imperfection. If they have not heard better chords there is no reason to complain or to expect harmonious ones.
The equal-tempered octave is ruled by 1xTxTxTxTxTxTxTxTxTxTxTxT = 2 while
1 x K K P K K P K K P K K P = 2 corresponds to the Piagui octave. The values of semitone factors K and P are also deduced in my book "The Piagui Musical Scale: Perfecting Harmony". The new scale is derived from the Pythagoras and Just Intonation extended scales.
The book also contains many graphs that "show" and evaluate the quality of harmony of all major and minor triads of both,Tempered and Piagui scales. It also deals with the application of the new intonation to the electronic organ, guitar and electronic tuner.
Sufficient data are given for tuning the piano to the top quality musical scale.
The Piagui scale is not an invention; it is a discovery that resolves the slight discordance problem of the equal-tempered scale.
C. Mario Pizarro
piagui@ec-red.com

Mario Pizarro
Aug-19-2005, 19:43
The smallest comma M= (32805/32768)=1.00112915039 together with J=1.0011313711 and U=1.0012136965 determine the tone frequencies of the pythagorean and Just Intonation scales as well as the twelve tone frequencies of the Piagui musical scale.
The new tone frequencies of the Piagui scale were deduced thanks to the data given by the successive comma products of the following progression:
1 x (QRQQRo Q) (QRQQRo Q) (QRQQRo Q) (QRQQRo Q) ...., where 1= C= Do.
Q= (M M J J M M M M J J M M M M J J M M)
R= (J J M M U U M M J J M M M M J J)
Ro = (J J M M M M J J M M U U M M J J) (Inverse sequence of R).
The product of the first 612 commas gives 2 Do = 2.
The first twelve products set the pythagorean comma :
1 x M M J J M M M M J J M M= 1.01364326477
Note D= 9/8= 1.125 of both ancient scales is given by the product of 104 commas comprised in (Q R Q Q Ro Q).
The pythagorean note E = 1.265625= (Q R Q Q Ro Q) (Q R Q Q Ro Q)
Similarly, the remaining tone frequencies of both ancient scales are found in the progression.
The graphs shown in my book "The Piagui Musical Scale: Perfecting Harmony" show and evaluate the quality of harmony of all major and minor triads of both, Tempered and Piagui scales. The book also deals with the application of the new intonation to the electronic organ, guitar and electronic tuner.
The equal-tempered octave is ruled by 1xTxTxTxTxTxTxTxTxTxTxTxT = 2 while
1 x K K P K K P K K P K K P = 2 corresponds to the Piagui octave. The values of K and P are also deduced in the book.
Sufficient data is given for tuning the piano to the top quality musical scale and perfect fifths and perfect fourths link all the Piagui keyboard.
The Piagui scale is not an invention; it is a discovery that resolves the slight discordance problem of the equal-tempered scale.
C. Mario Pizarro
piagui@ec-red.com

Mario Pizarro
Aug-19-2005, 19:45
The progression of 612 relative frequencies, from note Do= 1 up to 2Do= 2 is formed by the successive products of M, J and U commas. It is represented by:
1 x (Q R Q Q Ro Q) (Q R Q Q Ro Q) (Q R Q Q Ro Q) (Q R Q Q Ro Q)...= 2, where
Q= (M M J J M M M M J J M M M M J J M M)
R= (J J M M U U M M J J M M M M J J)
Ro = (J J M M M M J J M M U U M M J J) (Inverse sequence of R).
The successive products of the first 612 commas give 2 Do = 2.
M= (32805/32768)=1.00112915039 (The smallest comma)
J=1.0011313711..
U=1.0012136965..
It was guessed that some progression frequencies may not only be tone frequencies of a new musical scale, but also semitone factors of two consecutive notes of this scale. The names K and P were given to the Piagui semitone factors that rule the new octave. Their values are placed near the Tempered T= 1.059463094.., due to the slight imperfection of the equal-tempered scale.
The equation (K x K x K...m times) (P x P x P....n times) = 2, (First equation), is the one that complies with the octave quadrature, when and only when (m + n) = 12, (Second equation), provided m and n are integer and positive numbers. P must be lower than T and K higher than the tempered factor.
The two equations given above, contain four unknown quantities: K, P, m, n. The additional data required to work out the equations were deduced by using the information contained in the progression, as detailed in my book: "The Piagui Musical Scale: Perfecting Harmony".
The graphs shown in the book show and evaluate the quality of harmony of all major and minor triads of both, Tempered and Piagui scales. The book also deals with the application of the new intonation to the electronic organ, guitar and electronic tuner.
The Piagui octave was finally worked out:
K x K x P x K x K x P x K x K x P x K x K x P = 2 , where m = 8 and n = 4
Sufficient data is given for tuning the piano to the top quality musical scale.
The Piagui scale is not an invention; it is a discovery that resolves the slight discordance problem of the equal-tempered scale.
C. Mario Pizarro
piagui@ec-red.com

Mario Pizarro
Aug-19-2005, 19:55
Mathematical analyses were made to resolve the poblem of the slight discordance produced by the equal-tempered scale. Some audiences who are able to distinguish discords detect small harmony imperfections.
Tempered and Piagui 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 tone components yields non-aesthetic chord wave peaks, except for one diminished chord ( Do diminished, or La diminished, or Fa# diminished, or Mib diminished). These are the origin of the discords that humans have endured since 1722.
My book "The Piagui Musical Scale: Perfecting Harmony" 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.
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.
C. Mario Pizarro
piagui@ec-red.com

Mario Pizarro
Aug-23-2005, 04:26
See the attached information

Mario Pizarro
Aug-23-2005, 04:28
See the attached information.