A JI scale from an adder series

To tuning yahoogroups

I thought that frequency series in the C/2C range where one adder establishes its components cannot determine musical micro-intervals and scales, instead I believed that only geometric progressions might define the tone frequencies.
A few days ago I concluded that I was wrong, a series can do that provided the adder has the needed value.

When I was checking a device, the number 0.0151515... was the solution of a technical problem and decided to know more about this decimal figure. It carried me to [(2^6)(3)(5)]^(-1) = 1/[(2^6)(3)(5)] = 0.00104166666.....= (1/960).
I used this decimal number as a series adder without the premise of expecting musical tones along its expansion.

It was a surprise to notice that all or most JI tones appeared in the developed series, so I started to analyze it and prepared the attached excel pages.

I think it is an important finding no matter if the JI scales are not welcome scales as declared some musicians.

There are some interesting features:

-- Semitone factor values (Ratios between two consecutive tones) are rather uncommon for me excepting 16/15 that links E/F and B/2C, (15/8), (5/3), (3/2), (4/3), (5/4) and (9/8).

-- Semitone factors comply with the following sequence:
A B C D (16/15) A B C D A B (16/15).

-- The number of adders comprised in the range C to E equals 60 while from E to A this number is 80; from A to B it is 100 and from B to 2C there are 120 adders, in this case there is a total increase of 120 x 0.0010416666.. = 0.125 regarding B = 1,875 = (15/8). Therefore 0.125 + 1.875 = 2 = 2C.

Similarly, since in the semitone interval C to C# there are 60 adders and C = 1, the frequency increase is given by 60 x 0.0010416666... = 0.0625. Hence, C# = 1 + 0,0625 = 1.0625 = (17/16) and so forth.

--- Other features, please examine the attached series in excel where I give more details.

May be this subject is important as a source of new musical parameters.

Thanks

Mario Pizarro

piagui@ec-red.com
Lima, July 13, 2009