The missing Piagui IV intonation

Dear friends,

Some of you know that three Piagui scales were derived by mathematics as explained in the book "The Piagui Musical Scale - Perfecting Harmony", published in 2004. The research found three scale variants named Piagui I, Piagui II and Piagui III with which the harmony plots given in the book showed clear differences between any variant triad chord and those of the 12 TET scale, however they were tiny detected by aural comparisons. The insufficient scale tone differences which in these two scales reach to only 4 cents make about the same triad sounds when comparing the mentioned scales. Here, a few lines contained in pages 74 and 75 of the book is reproduced to see how the number of Piagui scales was limited to three, an evident error as we shall see it. Now, seven years later, Piagui IV, the fourth variant works with + 4 and minus 4 cent tone discrepances; therefore, it is expected clear chord sound differences as well as aesthetical and periodic chord wave peak responses:

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Whereas Cells Nos. 52 and 49 could be the semitone factors of an optimal musical scale, the tone relative frequencies with respect to note C will depend on their sequence within the octave. The sequences need to be analyzed, since tone frequencies and chords will depend on the K and P arrangements within the octave.

IV. 2 THE K AND P SEQUENCE IN THE NEW OCTAVE

The eight K and four P factors comply with proper distribution within the octave if the same factor sequence is maintained from note C to B. When an arrangement of twelve semitone factors is attempted, the idea of four identical groups arises, since there are four P factors working within the octave.

Elements of each group: K, K, P.

Number of groups = 4

Semitone arrangements: KKP, KPK and PKK.

Following are the three types of Piagui octaves:

I) KKP KKP KKP KKP = 2

II) KPK KPK KPK KPK = 2

III) PKK PKK PKK PKK = 2

Four groups of semitone factors comprised by KKP, KPK and PKK rule the relations of tone frequencies in the new octaves of Piagui I, Piagui II and Piagui III scales respectively. However, the new octaves work simultaneously. In fact, the set KKPKKPKKPKKPKKP shows that Piagui II starts from the second K and Piagui III from the first P.

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The fourth solution which is also ruled by 8 K and 4 P semitone factors is given by the following arrangement:

P K K K K P P K K K K P

C = 1 ------ 0 cents--- 0 cents regarding 12 TET

C# = 1.05707299111 = P = 96.09 cents --- minus 4 cents regarding 12 TET

D = 1.1211..... = P K = 198.045 --- minus 2 cents .....

Eb = 1.1892..... = P (K)^2 = 300 --- 0 cents regarding 12 TET

E = 1.2613..... = P (K)^3 = 401.955 --- + 2 cents ....

F = 1.3378..... = P (K)^4 = 503.91--- + 4 cents ....

F# = 1.4142.....= (P)^2 (K)^4 = 600--- 0 cents regarding 12 TET

G = 1.4949.....= (P)^2 (K)^5 = 696.09 --- minus 4 cents .....

Ab = 1.5856.....= (P)^2 (K)^6 = 798.045 --- minus 2 cents .....

A = 1.6817 .....= (P)^2 (K)^7 = 900--- 0 cents regarding 12 TET

Bb = 1.7838.....= (P)^2 (K)^8 = 1001.955 --- + 2 cents ....

B = 1.8920 .....= (P)^3 (K)^8 = 1103.91 --- + 4 cents ....

2C = 2 = (P)^4 (K)^8 = 1200 cents --- 0 cents regarding 12 TET

The plus and minus 4 cents establish the expected aural differences between the Piagui IV and the 12 TET corresponding chords.

More information will be send to you in a few days.

Thanks

Mario

Lima, March 28, 2011