# ET to JI

### Equal Temperaments approximated in Just Intonation

## Background

In 1979, in the Spring/Summer issue of "Interval" magazine, I published "Ratios For Equal-Temperment", excerpted below:
Here are several tables I generated to translate tempered scales into close ratio (just intonation) approximations.
These tables were generated using John Chalmers' algorithm for converting decimal fractions ... to ratios. The ratios I give are succesive approximations from that algorithm... Of course there are many other ratios that are close to any given tempered interval, but I just list close approximations, starting with low-number ratios..

## Program

Here is an implementation of the algorithm I used for the above-mentioned article, compiled for MS-DOS: et2ji.exe (30K).

Also, here is the source code in "C": et2ji.c (3K).

Below is the comment header from the source code:
et2ji : to approximate equal tempered tones by just intonation intervals
usage:
et2ji #tones/oct [tolerance(cents) [show all?(Y/N)] ]
the first argument is the number of tones per octave (may be noninteger)
the second argument is the tolerance in cents [optional; default is 1.0]
the third argument [optional; default is No], if present means
show all intermediate approximations as well as best ones
(warning: this may make output lines very long!)
the output, for each tempered tone, gives tone number, cents, and
successive JI approximations with cents error.
NOTE: these JI intervals do NOT include all simple nearby ones.
I got this algorithm from John Chalmers; thanks John!

Finally, here is a sample output, given the command-line input

et2ji 7
7-tone equal temperament: just approximations to 1 cent
1 ( 171.4): 10/9 (+11.0) 11/10 (-6.4) 21/19 (+1.8) 32/29 (-1.0) 53/48 (+0.1)
2 ( 342.9): 5/4 (+43.5) 6/5 (-27.2) 11/9 (+4.6) 39/32 (-0.4)
3 ( 514.3): 4/3 (-16.2) 35/26 (+0.3)
4 ( 685.7): 3/2 (+16.2) 52/35 (-0.3)
5 ( 857.1): 5/3 (+27.2) 18/11 (-4.6) 23/14 (+2.3) 41/25 (-0.7)
6 (1028.6): 9/5 (-11.0) 29/16 (+1.0) 96/53 (-0.1)

*NOTE: All software made available through this page and subsidiary pages is supplied on an "as is" basis, with no warrantees of any kind. The author bears no responsibility for any consequences of using this software.*
last updated 1999 Aug 22

David Canright
-- DCanright@NPS.edu