This is a command-line tool for calculating tuning tables and exporting Scala files of Near-Equal Just Intonation (NEJI) tuning systems
Python 3 (tested on version 3.7.1)
NumPy Python module (tested on version 1.19.4)
To calculate a NEJI tuning, you must provide the following three arguments:
- the EDO you wish to approximate or "un-temper" via just intervals,
- a comma-separated list of 'generators': the integers that can be used for the denominator of the just intervals. No spaces allowed here!
- a name for the generated .scl file.
For example, to calculate the NEJI that approximates 15-EDO using just intervals and has 2, 31, and product combinations (i.e., 2 x 31 = 62) as the denominator, this is the command to run:
python neji_calculator.py 15 2,31 31_neji_15The third argument, "31_neji_15", is the file name. This will export a scala file called "31_neji_15.scl".
It will also print the following information:
N-EDO: 15, Generators: [2, 31], Filename: 31_neji_15.scl
degree, ratio, cents, error (cents from EDO)
0 1 0.000000 0.000000
1 65/62 81.805803 1.805803
2 34/31 159.919837 -0.080163
3 71/62 234.660971 -5.339029
4 75/62 329.546856 9.546856
5 39/31 397.447090 -2.552910
6 41/31 484.026833 4.026833
7 43/31 566.482133 6.482133
8 45/31 645.188143 5.188143
9 47/31 720.471050 0.471050
10 49/31 792.616240 -7.383760
11 103/62 878.765060 -1.234940
12 54/31 960.829430 0.829430
13 113/62 1039.179182 -0.820818
14 59/31 1114.136087 -5.863913
15 2 1200.000000 0.000000
Nudging or tweaking a degree in the tuning can be useful for balancing or centering the tuning. The following example is taken from Zhea Erose's 'Eurybia', which uses 12-note undecimal tuning and has a raised fourth. To nudge, specify the degree you wish to nudge (5 in this example) and by how much.
python neji_calculator.py 12 2,11 eurybia --nudge 5 --by 1writes the Scala file eurybia.scl and prints the following output:
N-EDO: 12, Generators: [2, 11], Filename: eurybia.scl
degree, ratio, cents, error (cents from EDO)
0 1 0.000000 0.000000
1 23/22 76.956405 -23.043595
2 25/22 221.309485 21.309485
3 13/11 289.209719 -10.790281
4 14/11 417.507964 17.507964
5 15/11 536.950772 36.950772 (raised by 1/22)
6 31/22 593.717630 -6.282370
7 3/2 701.955001 1.955001
8 35/22 803.821678 3.821678
9 37/22 900.026096 0.026096
10 39/22 991.164720 -8.835280
11 21/11 1119.462965 19.462965
12 2 1200.000000 0.000000
The '--by' argument may be positive or negative. It adjusts by increments of 1 divided by the product of all the generators (increments of 1/22 in this example).
In the above example, the output values in the denominators of the just ratios were 2, 11, and 22. 2 and 11 were specified as generators. 22, the product of 2 and 11, was added by default. This product-taking default setting leads to closer approximations of the EDO scale.
When you have many generators (5 here),
python neji_calculator.py 12 2,3,5,7,14 basically_12_edo
the resulting scale has ratios with large numerators and denominators:
N-EDO: 12, Generators: [2, 3, 5, 7, 14], Filename: basically_12_edo.scl
degree, ratio, cents, error (cents from EDO)
0 1 0.000000 0.000000
1 89/84 100.099210 0.099210
2 55/49 199.979843 -0.020157
3 874/735 299.866836 -0.133164
4 926/735 399.921532 -0.078468
5 327/245 499.810664 -0.189336
6 99/70 600.088324 0.088324
7 881/588 699.991037 -0.008963
8 4667/2940 800.015175 0.015175
9 412/245 899.835106 -0.164894
10 873/490 999.839886 -0.160114
11 185/98 1100.005940 0.005940
12 2 1200.000000 0.000000
This is because it's taking the product of the generators. What if we don't want simpler ratios than this?
We can turn this product behavior off by using the -l or --limit argument. (This has nothing to do with N-limit just intonation.)
python neji_calculator.py -l 12 2,3,5,7,14 N-EDO: 12, Generators: [2, 3, 5, 7, 14], Filename: 12_neji_2-3-5-7-14_lim.scl
degree, ratio, cents, error (cents from EDO)
0 1 0.000000 0.000000
1 15/14 119.442808 19.442808
2 8/7 231.174094 31.174094
3 6/5 315.641287 15.641287
4 9/7 435.084095 35.084095
5 4/3 498.044999 -1.955001
6 10/7 617.487807 17.487807
7 3/2 701.955001 1.955001
8 8/5 813.686286 13.686286
9 5/3 884.358713 -15.641287
10 25/14 1003.801521 3.801521
11 13/7 1071.701755 -28.298245
12 2 1200.000000 0.000000
This is a tidy scale with many familiar just intervals and no denominators are larger than 14. You can see that 2, 3, 5, 7, and 14 all occur on the denominator. But the products are missing, because they were not explicitly passed to the calculator. The exception in this example is 14, the product of 2 and 7, which occurs in the .scl file, but occurs because it was explicitly passed.
In general, limiting the denominators with the -l setting offers an alternative way to explore tunings in the space between JI and EDO. It is useful when you have many generators but don't want to super closely approximate the EDO.