Re: Preceding message

To the tuning list,

The progression of musical cells was derived in 1983, it is a chain of an unlimited number of octaves each containing 612 cell frequencies. Any cell frequency is the product of M, J or U commas by the preceding cell value. To reduce product errors 11 decimal digit cells were used, common fractions had to be set aside. Actually, (9/8)^6 is the cycle that comprises six segments of 9/8 which are coupled geometrically. The octave 2 arises twelve cells before cell Nº 624.

The progression is able to produce 12 tone scales according to a pair of adders like 50/52, 49/53 provided their sum gives 102. Their values should be about 50.

When the pair is 49/53, the C# frequency with values comprised within 1 and 2 to be detected in the progression is given by the first number (49), it means that C# frequency corresponds to cell # 49 whose consonant frequency is 1.05707299111. The second adder 53 plus the preceding 49 makes 102 so cell # 102 = 1.12246370821 is D.frequency.

The first adder 49 works again, it is summed to 102 to give (102 + 49) = 151 so cell # 151 (1.1865234375) is the Eb note frequency.

Similarly, 151 summed to the second adder 53 gives 204, so cell # 204 = 1.25991918672 is the E tone frequency.

F tone frequency corresponds to cell # (204 + 49) = Cell # 253 = 1.3318294975 and so forth.

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As an example, 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.

Therefore, a consonant 12 TET cells scale has been drawn from the progression. Should M, J or U values or 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'?. I think it was done by the cell exactness, the correct comma sequence and their values.

May be the tuning list is not interested on this subject since only a few members requested the progression. The 12 TET cells scale shown below is a reason to investigate on this matter. By now, there is no explanation on what principles were used by the M, J and U commas to settle such abilities.

I would be pleased to send you the progression (7 pages) as soon as you request it.

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 FREQ...FREQ. ......(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...2.0000000000..1

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