Re: More questions on scales

Alain Naigeon wrote:

>> Later in history one wanted a different sound of some chords, 
>> experimenting with the major [third] 5/4 in Just Intonation
>>   http://en.wikipedia.org/wiki/Just_intonation
>>   http://en.wikipedia.org/wiki/31_equal_temperament
> 
> When reading another message, I fisrt thought it was a typo
> (and should be fair enough to still think so) : but 5/4 is the
> major third, and the major tones you're refrring to are 9/8
> and *10/9* - don't you agree?

Yes, in Just, the major third M3 is 5/4 and the major second M2 or 
simply M is 9/8, so the distance between M2 and M3 is (5/4)/(9/8) = 
10/9, which is not M2 but somewhat smaller. So, Just is not a true 
diatonic scale as it is built up by three intervals - the D minor chord 
does not have a perfect fifth 3/2. And the Western key signatures are 
not valid for such a scale if interpreted literary as pitches. 31-ET 
tempers that difference between 9/8 and 10/9 out, so that the key 
signatures interpreted as picthes become valid again.

I want to avoid the common "whole tone" and "semitone", as in for 
example the close Pythagorean approximation 53-ET, M = 9 and the minor 
second m = 4 tone steps. I just encountered a case where one instead 
divided the 12-ET whole tone in 9 steps, leaving the 12-ET semitone, os 
that one instead a different 108-ET (54 = 6*9 for the whole tone 
division, and another factor 2 to get a multiple of 12 so that the 
semitone fits in). It is like microtones for example in Arab music some 
call "quarter tones", but that tend to lead to the misunderstanding that 
they are 1/2 12-ET tone step or semitone - one agreed already in 1932 
they are not.
http://en.wikipedia.org/wiki/Arab_music#More_notes_used_than_in_Western_scale

   Hans Aberg