Marpurg and Piagui I I scales

To Brad Lehman,

Last two days I studied the coincidence between the Marpurg and the Piagui I I scale. I used semitone factors and the progression of musical cells to derive the three variants of Piagui scales and these tools didn´t help when trying to use them on a matter that involves cents. Then I followed the advise given by TOM DENT who recommended to work with cents and this facilitated the verification of equality between both scales.

I have consulted to the editor of "The Piagui Musical Scale: Perfecting Harmony" about the use of the Piagui I I scale, now that we know that F. W. Marpurg proposed the same scale in 1776 and a writer published in 1948 eleven Marpurg scales and one of them coincides with the Piagui I I variant. The representative of the editor in the United States responded that the three Piagui scale variants are protected by the Library of Congress Control Number:2003195201 and no part of this book may be transmitted or the information contained may be availed by any means without written permission from the author. In my opinion, since there is not an important difference between the Piagui I I and the equal tempered intonation, it is expected that musicians will prefer the use of the well known equal tempered scale.

Below I detail the procedure employed to demonstrate that both scales are the same; at first sight they look quite different and I had thought that nobody could deduce the same scale.

As you wrote, the arrangement of all the notes by fifths is as follows:

p n p p n p p p n p n ,,,,,narrow(n) and pure (p) fifths

C, G, D, A, E, B, F#, C#, G#, Eb, Bb, F

0, 6, 0, 0, 6, 0, 0, 6, 0, 0, 6, 0

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, .

p = 702 cents ---------- n = 702 - 6 = 696 cents

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(C=0 cts), (G=702 cts), (D=198 cts), (A=900 cts), (E=402 cts), (B=1098 cts), (F#=600 cts)

(C#=102 cts), (G#=798 cts), (Eb=300 cts); (Bb=1002 cts), (F=498 cts), (2C= 1200 cts)

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ORDAINED TONES IN THE OCTAVE:

(C=0 cts), (C#=102 cts), (D=198 cts), (Eb=300 cts); (E=402 cts), (F=498 cts) (F#=600 cts)

(G=702 cts), (G#=798 cts), (A=900 cts), (Bb=1002 cts), (B=1098 cts), (2C= 1200 cts)

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The Piagui II semitone factors are ordained this way: C=1xKxPxKxKxPxKxKxPxKxKxP=2C

Assuming that C = 600 Hz, Marpurg and Piagui scales are the same if the twelfth frequency is equal to 2C = 1200 Hz:

600 x K x P x K x K x P x K x K x P x K x K x P x K = 1200 Hz

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Sorry for the delay.

MARIO PIZARRO

piagui@...

Lima, June 24, 2008