Tim and using higher segments of the harmonic series to form "ideal" scales

Tim>"(1/3  2/3)   3/3  4/3  5/3  6/3  =   6/6  7/6  8/6  9/6  10/6  11/6 
12/6  = 12/12  13/12  14/12  15/12 16/12  17/12  18/12  19/12  20/12
 21/12  22/12 23/12  24/12 = 24/24 etc"

   I have long believed that the most efficient scales are really just based on higher parts of the harmonic series with many common factors.

  7-tone JI diatonic is based on the 24th partial harmonic series: 24/24 27/24 30/24 32/24 36/24 40/24 45/24 (48/24)...and 24 has many factors including 2,3,4,6,8, and 12.  Some other good harmonic series partials include 18 (factors 2,3,6,9), 30 (factors 2,3,5,6,10,15) and 36 (factor 2,3,4,6,9,12,18).  What these all have in common is factors of both 2 and 3...though you can also try for numbers with 5 and 2 as factors IE 20, which has 2,4,5, and 10 as factors.

  It seems to me that, over about the 24th harmonic series partial as a starting point (meaning harmonics 36 to 72)...the number of factors available per range experiences diminishing marginal returns.  For 18 and 20, you get 4 factors...for 24,
you get 6 (significantly more)...for 36, you get 7 (barely any more)...and for 48 (factors 2,3,4,6,8,12, and 24) you get 8 factors (barely any more and the range needed to get 8 factors is twice as large as the range needed to get 6!).

   Do any of these rest of you have any ideas which ranges of numbers (particularly of around 48 or under IE 48 to 96th harmonics) have the most common factors?

Hi Michael, good post, good question...I lean towards the music more than the math, thus I really do try to keep it simple. My thoughts on this: every great musician needs a rocket scientist for additional guidance systems (hehe.)
 
BTW,  I wanted to throw in a little more info on the Natural 3 whole tones scale, so look for my edit
thanks
Tim
 
--- On Tue, 7/5/11, Michael <djtrancendance@...> wrote:

From: Michael <djtrancendance@yahoo.com>
Subject: [tuning] Tim and using higher segments of the harmonic series to form "ideal" scales
To: tuning@yahoogroups.com
Date: Tuesday, July 5, 2011, 9:47 AM

Tim>>>>    "(1/3  2/3)     3/3  4/3  5/3  6/3   
 where 3 is the Base or number of natural whole tones
 
     6/6   7/6   8/6   9/6   10/6   11/6   12/6  
 where natural half tones are nested into the 3 Base scale         
 
These values converts to  1    7/6   4/3   3/2     5/3   11/6     2 
 
note how 3/2 and 4/3 allow for modulation into the common nhs scale
 
12/12   13/12   14/12   15/12  16/12   17/12   18/12   19/12   20/12   21/12   22/12 23/12   24/12      where quarter tones are nested into the 3 base  scale and convert to
 
1   13/12   7/6   5/4    4/3    17/12   3/2   19/12    5/3   21/12   11/6   23/12    2
 
Did you see how 5/4 , (otherwise known as a whole tone), becomes a "quarter tone" ????
 
We have a long way to go to fully exploit the vastness of our natural music systems
 
Have fun
simpletim  >>>> 7/5/11 ( the day not a ratio stream , hehe )

   I have long believed that the most efficient scales are really just based on higher parts of the harmonic series with many common factors.

  7-tone JI diatonic is based on the 24th partial harmonic series: 24/24 27/24 30/24 32/24 36/24 40/24 45/24 (48/24)...and 24 has many factors including 2,3,4,6,8, and 12.  Some other good harmonic series partials include 18 (factors 2,3,6,9), 30 (factors 2,3,5,6,10,15) and 36 (factor 2,3,4,6,9,12,18).  What these all have in common is factors of both 2 and 3...though you can also try for numbers with 5 and 2 as factors IE 20, which has 2,4,5, and 10 as factors.

  It seems to me that, over about the 24th harmonic series partial as a starting point (meaning harmonics 36 to 72)...the number of factors available per range experiences diminishing marginal returns.  For 18 and 20, you get 4 factors...for 24,
you get 6 (significantly more)...for 36, you get 7 (barely any more)...and for 48 (factors 2,3,4,6,8,12, and 24) you get 8 factors (barely any more and the range needed to get 8 factors is twice as large as the range needed to get 6!).

   Do any of these rest of you have any ideas which ranges of numbers (particularly of around 48 or under IE 48 to 96th harmonics) have the most common factors?

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