EDOs to ETs

Approaches to Connecting EDOs to Temperaments

"Equal temperaments" (ETs), also known as "rank-1 temperaments", are temperaments which map JI intervals of a given prime-limit or subgroup to iterations of a single generator. "Equal divisions of the octave" (EDOs) are exactly what they sound like, a division of a Just 2/1 ratio into some number of equal parts. An equal temperament is defined by a p-limit val for some odd prime p, but an EDO is defined as a 2-limit val: that is, by only the number of steps to an octave, and nothing else.

Support for higher-rank temperaments

One approach, which could be referred to as the "top-down" approach, is to begin with a higher-rank temperament, and temper out additional commas until a rank-1 temperament is reached. If we assume that prime 2 is tuned Just, the mapping to 2 in the rank-1 temperament's val gives us the number of an EDO. For example, if we have a rank-1 val of <22 a b c...z|, we know this is 22-EDO, because prime 2 is mapped to 22 steps of the generator. By tempering different commas (or sets of commas), we can find different ETs that support any given higher-rank temperament.

The bottom-up approach

Since there are infinitely-many temperaments that could be used as starting-points to reach a rank-1 temperament, but only a finite number of possible sounds in an EDO, another approach to tempering is to start with the intervals available in an EDO and determine by sound the JI ratios which are most plausibly approximated. There are a variety of ways to approach this. To begin with, we might start by assuming a prime limit p to start with, and then calculate the patent val for the given EDO based on that p-limit. From the patent val, we can derive the p-limit commas tempered out in the EDO in a straightforward way. Of course, we may also use non-patent and higher-error vals if we desire, which will lead to a different set of commas being tempered out. But in any case the advantage of this approach is that it makes it straightforward to compare the performance of various ETs within a pre-defined prime limit. However, this approach does not tell us which JI ratios would are being most strongly suggested by the sound of each interval in the EDO. As a result, we may overlook EDOs with solid JI approximations based on a p-limit JI subgroup, because they perform poorly in the full p-limit. We may also end up saying that some tempered intervals "approximate" JI intervals to which they bear little sonic resemblance. An example of this would be treating 8-EDO as a 5-limit temperament, which leads (for instance) to mapping JI intervals of 5/4 and 4/3 both to the interval of 450 cents.

Thus, another possible approach is to map each interval in the EDO to the simplest rational interval it could be said to approximate, or in other words to find the JI subgroup which best fits the EDO. From this, we can figure out the temperament by seeing which ratios end up equated by that mapping. For example, we might map 10-EDO as follows:

Degree	0	1	2	3	4	5	6	7	8	9	10
Cents	0	120	240	360	480	600	720	840	960	1080	1200
Ratio	1/1	15/14	8/7	5/4	4/3	7/5	3/2	8/5	7/4	13/7	2/1

These ratios generate the 2.3.5.7.13 subgroup.

This is step 1, actually, because we have only one ratio mapped to each degree. In step 2, we add more by finding the intervals between the original intervals, as well as the intervals that appear by multiplying ratios, and end up with something like this:

Degree	0	1	2	3	4	5	6	7	8	9	10
Cents	0	120	240	360	480	600	720	840	960	1080	1200
Ratio	1/1	15/14, 16/15, 21/20, 14/13	8/7, 9/8, 7/6	5/4, 6/5, 16/13	4/3, 21/16, 64/49	7/5, 10/7	3/2, 32/21, 49/32	8/5, 5/3, 13/8	7/4, 16/9, 12/7	13/7, 28/15, 15/8, 40/21	2/1

From this list, we can then figure out the commas that are tempered out simply by finding the differences between all the ratios that are mapped to the same degree. Alternatively, we can use the subgroup val for the generated subgroup, in this case 2.3.5.7.13, and from that find a basis for the commas as 25/24, 28/27, 40/39 and 50/49. One drawback to this entire approach is that if the ratios chosen in step 1 have are not approximate with very high accuracy, when we get to step 2 we may still end up conflating ratios which bear little sonic resemblance. You can see in the above chart that 3\10 approximates 5/4, 6/5, and 16/13, and when tuned Just, all three of these ratios are quite sonically distinct.

Another approach is to ignore individual intervals and instead concentrate on the largest and lowest subgroup of the harmonic series that can be well-approximated in the EDO. This approach has not yet been mathematically formalized, so deciding which subgroup of the harmonic series is the "best fit" for an EDO is currently somewhat a matter of personal discretion. One interpretation of 10-EDO based on this method, however, would be to treat it as a 2.7.13.15 subgroup, as 10-EDO provides very convincing matches to all of those harmonics. This leads to the following mapping:

Degree	0	1	2	3	4	5	6	7	8	9	10
Cents	0	120	240	360	480	600	720	840	960	1080	1200
Ratio	1/1	14/13, 15/14, 16/15	8/7, 15/13	16/13	169/128, 64/49	91/64	49/32, 256/169	13/8	7/4, 26/15	15/8, 13/7, 28/15	2/1

Unlike in the intervalic approach, this mapping ends up equating intervals which are much closer together, and thus has a lower "error" from JI; in other words, the difference in sound between the tempered intervals and the ratios they are said to approximate is comparatively lower than in the intervalic approach. On the other hand, the ability of 10-EDO to suggest or imply harmonic identities based on the 5-limit is obscured here, since we don't end up treating (say) 720 cents as an approximation to 3/2, or 360 cents as an approximation to 5/4.