Yahoo Tuning Groups Ultimate Backup tuning Graphs and harmony

To: Mike Battaglia

Mike,

You have forgotten to confirm if you have received some graphs where major and minor triads of Piagui and their corresponding equal tempered triads are shown like photographs of their harmony. Those "harmony photos" are important results of the research.
Since all the Piagui triad responses are periodic and aesthetic while the equal tempered ones are disordered and approaching to chaotic ¿Don`t you think that these features are enough to conclude that Piagui triad harmonies are better than their corresponding equal tempered triads harmony?.
In my opinion, the fact that some fifths deviate from the classic (3/2) = 1,5 and also some of the thirds do not coincide with the expected (5/4) = 1,25, these Piagui deviations intervene in all kind of Piagui triads to produce aesthetic photos so I think that the aesthetic responses should have a higher primacy than the expected frequency ratios when evaluating the harmony produced by both types of scales.
I am attaching the graphs of D Major and G Major and hope that you receive them because normally graphs and diagrams are not accepted by this link.

Please advise.

Regards

MARIO PIZARRO
Lima, June 21, 2008
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GRAPH No. 7
TEMPERED CHORD WAVE PEAKS,,,,,D Major = F# + A + 2D

GRAPH No. 8
PIAGUI I CHORD WAVE PEAKS,,,,D Major = F# + A + 2D

VIII. 6 CHORD WAVE PEAKS OF G MAJOR (G + B + 2D)

TEMPERED,,,,,,,,,,,,,,,,,,,,,,,,,,...PIAGUI I

COMPONENTS,,,,,,,,,,,,,,COMPONENTS,,,,,,,,,,,,,DISCREPANCY (%)

G = 391.9954 (Hz) G = 392.4383 (Hz) + 0.113

B = 493.8833 B = 495 + 0.226

2D = 587.3295 2D = 588.6575 + 0.226

TEMPERED CHORD WAVE PEAKS (GRAPH No. 11)

The non-aesthetic mark distribution shows the imperfection of the tempered G Major. Three mark waves appear to be superimposed, each one varying in size and shape producing considerable distortion.

This response is similar to that shown in Graph No. 3. The frequency chord components give an unequal number of empty areas around the time axis.

The disorderly display follows the same pattern in all tempered major triads, excepting Tempered D Major where the third tone is 2D to get an aesthetic response with different voicing.

GRAPH No. 11
TEMPERED CHORD WAVE PEAKS,,,,,,,,,,G Major = G + B + 2D

PIAGUI I CHORD WAVE PEAKS (GRAPH No. 12)

Three positive and three negative mark waves give an aesthetic display apparently made by a skilled computer programmer.

Piagui I - G Major is a perfect triad. With keynote G as 1, the triad components are 1, K3P, K5P2 and consequently, its responses are similar to those of C Major.

Similar chord wave peaks are given by C# Major, E Major and Bb Major, and so they were not plotted.

GRAPH No. 12
PIAGUI I CHORD WAVE PEAKS,,,,G Major = G + B + 2D

G MAJOR = G + (B)d + (2D)d

TEMPERED,,,,,,,,,,,,,,,,,,,,,,PIAGUI I

COMPONENTS,,,,,,,,,,,,,,,,COMPONENTS,,,,,,,,,,DISCREPANCY (%)

G = 391.9954 (Hz) G = 392.4383 (Hz) + 0.113

B = 493.8833 B = 495 + 0.226

2D = 587.3295 2D = 588.6575 + 0.226

(G Major) Temoered = sin [2 ? (391.9954) t] + sin [2 ? (493.8833) (t - 0.002)]

+ sin [2 ? (587.3295) (t - 0.004)]

(G Major) Piagui I = sin [2 ? (392.4383) t] + sin [2? (495) (t - 0.002)]

+ sin [2? (588.6575) (t - 0.004)]

As explained previously, all chord wave responses are obtained by adding three sinusoidal waves. Each sinusoid has no range at t = 0 seconds. Furthermore, combined sound does not depend on this particular condition and in order to demonstrate that the aesthetic response remains constant regardless of the time delays of the tone components, G Major is plotted with two delayed tones, (B)d and (2D)d. In no case have overtones been considered.

TEMPERED CHORD WAVE PEAKS (GRAPH No. 15)

The non-aesthetic view corresponds to disharmony.

GRAPH No. 15
TEMPERED CHORD WAVE PEAKS,,,,G Major = G + (B)d + (2D)d

PIAGUI I CHORD WAVE PEAKS (GRAPH No. 16)

Delays do not affect the magnificent appearance of the Piagui I response. Looking at the cyclical mark distribution, we conclude that it corresponds to perfect harmony.

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Graph No. 1 up to No. 34 show chord waves and chord wave peaks of triads whose frequency tones are based on keynotes that lie in the middle octave (261.6255 - 523.2511 Hz). In all cases, the Tempered responses exhibit non-aesthetic displays, which are signs of disharmony. In Chapter IX, Graphs Nos. 35 and 36 show Tempered and Piagui I chord wave peak responses respectively. In these cases, the keynote frequencies of G Major (2G + 2B + 4D) are twice the values contained in the middle octave. It can be seen that the degree of discordance of the Tempered triad is greater during the same one second of the display period. As the keynote frequencies go up, the Tempered triad is more discordant.

The question is if the mentioned deterioration that we endure at the present time while the new intonation keeps its chord qualities, might increase the interest of composers to use higher keynote frequencies of the Piagui scale. This way, true harmonious chords would be generated from the upper octaves of the piano keyboard.

GRAPH No. 16
PIAGUI I CHORD WAVE PEAKS...G Major = G + (B)d + (2D)d

Regards
MARIO PIZARRO
Lima, June 21, 2008

Graphs and harmony

🔗Mario Pizarro <piagui@...>

🔗Mike Battaglia <battaglia01@...>