Two equal tempered scales

The two equal tempered scales.

The scale shown below corresponds to pair 51/51. Note that Eb, F# and A tone frequencies of 12 TET and the corresponding cell tones that work in the progression coincide. The remaining tone comparisons show very small differences which can be neglected.

A consonant 12 TET cells scale has been drawn from the progression. Should M, J, U values and their sequence are not the ones that work in this set, a useless progression would be the result.

How this pair (51/51) succeeded in constructing the 12 TET scale and the 12 tone scales obtained by using the pairs 49/53, 50/52 ?. The job was done by the progression cell exactness, the M, J, U correct sequence, their numbered position and the pair numbers. However, since the purpose of the progression expansion was mainly the finding of the probable route followed by frequencies starting with Do = 1 to get well known consonances along the route, several attempts were done with one, two and three commas before the schisma M and the deduced J,U commas gave the solution

The 12 TET cells scale and the 12 T equal tempered scale shown below give a reason to investigate on this matter. By now, there is no explanation on what principles were used by M, J and U commas to settle such abilities.

Thanks

Mario Pizarro

piagui@...

Lima, May 05

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Pair 51/51........(A)........(B)

ADDER ....N0TE...PROGRESSION ...12 TET TONE

...Cell Nº .....CELL FREQUENCY..FREQUENCY..(A)/(B)

...# 0....C..1...........1..........1

51...# 51..C#...1.0594638777.1.0594630943..1.000000739

51...# 102..D...1.1224637082..1.1224620483..1.000001479

51...# 153..Eb.1.1892071150.1.1892071150 ..1

51...# 204..E....1.2599191867.1.2599210498...0.999998521

51...# 255..F..1.3348388671.1.3348398541...0.999999261

51...# 306..F#.....1.4142135623.1.4142135623..1

51...# 357..G..1.4983081847.1.4983070768..1.000000739

51...# 408..Ab...1.5874033994.1.5874010519..1.000001479

51...# 459..A...1.6817928305.1.6817928305.1

51...# 510..Bb...1.7817948013.1.7817974362..0.999998521

51...# 561..B...1.8877472295.1.8877486253..0.999999261

51...# 612..2C...2.0000000000..2.000000000...1

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