More temperaments (was: white key scales (Dan, Miracle folk))
Posted By: graham@microtonal.co.uk
Fri May 25, 2001 10:25 am
In-Reply-To: <9ekihc+rtsn@eGroups.com>
Dave Keenan wrote:
> --- In tuning@y..., graham@m... wrote:
> > This must be similar to what Dave Keenan's generator-finding program
> does.
>
> No. It works by brute force. It's an Excel spreadsheet. It simply does
> the same calculation for every generator between 0 to 600c in 0.1c
> increments. It calculates how many generators in a chain (up to some
> maximum) give the best (octave equivalent) approximation for each of
> 1:3, 1:5, 1:7, 1:11 and what the errors are. From those it calculates
> the hexad-width and the MA and RMS errors over all 11-limit intervals
> (consistency is guaranteed). Then it flags those that meet the set
> criteria for hexad-width and errors.
Oh, is that the thing I downloaded? I might have another look if I ever
get Excel installed on this machine. I thought you mentioned it could be
getting only local minima once, which would suggest a larger granularity
and optimising from there.
My program's working now. I rank temperaments by the product of the
number of steps for a complete chord and the minimax error, then show the
10 best ones. Miracle actually comes out as third best for the 11-limit.
I make it a 3.3 cent minimax error. The two better ones require more
notes to get that all-important hexad. You can get to 1.3 cents accuracy
with 49 notes or 2.4 cents with 31 notes.
The simplest 15-limit temperament in the top 10 is defined like this, by
octaves and generators:
( 1 0)
( 2 -1)
(-1 8)
(-3 14)
(13 -23)
(12 -20)
That means a 3:1 is two octaves minus a generator, so the generator is a
fourth. A 5:1 is 8 generators minus an octave. That looks like schismic
to me. It's consistent with 41 and 94-equal. You need 37 notes from a
chain of fourths to get a 15-limit chord, and it's accurate to 4.9 cents.
The number one 15-limit temperament is consistent with 87 and 72, defined
like this:
( 3 0)
( 6 -6)
( 8 -5)
( 8 2)
(11 -3)
(14 -14)
It needs 16 generators for a complete chord, but because the octave
divides into three I think that means 49 notes to the octave. So it's
fairly complex, but accurate to 2.8 cents. I don't plan on tuning it up
any time soon, but if anybody has a huge keyboard and wants to play with
15-limit harmony, well, now you know an accurate temperament exists.
Graham