The first thing to consider for 13-limit is the octave-reduce thirteenth harmonic, 13/8. It is the first sixth that occurs in the harmonic series and comes in at about 840.53 cents. It's pretty close to being smack dab in the middle of the 12tet minor sixth and major sixth. So, like 11-limit, this limit is going to contain some neutral intervals. In fact, most limits beyond 7 are going to land you with plenty of neutral intervals. Some are just more neutral than others.
The second thing I like to consider when exploring a limit is its superparticular intervals. These are also called epimore intervals. They are any interval of the form n+1/n. This is where Stormer's Theorem comes in. It states for that for any given prime number, p, the number of sequential integers that are p-smooth are finite and gives a procedure for finding them. A number is p-smooth if it contains no prime factors larger than p. Smoothness is directly applicable to the limit system, as are consecutive integers to the superparticulars. So with a small tweak of this implementation in Python, we can find all of the superparticulars for a given limit. So as we move forward, keep these in mind:
2 limit: 2/1
3 limit: 3/2, 4/3, 9/8
5 limit: 5/4, 6/5, 10/9, 16/15, 25/24, 81/80
7 limit: 7/6, 8/7, 15/14, 21/20, 28/27, 36/35, 49/48, 50/49, 64/63,
126/125, 225/224, 2401/2400, 4375/4374
11 limit: 11/10, 12/11, 22/21, 33/32, 45/44, 55/54, 56/55, 99/98,
100/99, 121/120, 176/175, 243/242, 385/384, 441/440, 540/539,
3025/3024, 9801/9800
13 limit: 13/12, 14/13, 26/25, 27/26, 40/39, 65/64, 66/65, 78/77,
91/90, 105/104, 144/143, 169/168, 196/195, 325/324, 351/350, 352/351,
364/363, 625/624, 676/675, 729/728, 1001/1000, 1716/1715, 2080/2079,
4096/4095, 4225/4224, 6656/6655, 10648/10647, 123201/123200
So to start off, I've build a scale to explore the limit a bit. This scale was inspired by a middle-eastern scale, the name of which I'm not certain. It is essentially a Dorian mode with the 2nd and 6th lowered by a quarter tone. So I'm implementing this with just intervals. First, we'll start with the basic root, fourth, fifth, and octave.
1/1, 4/3, 3/2, 2/1
Next, we'll add the octave-reduced 13th harmonic and its counterpart second, 13/12. 13/12 can be derived a few ways, but I just got it by dividing by a fifth. 13/8 / 3/2 = 13/12.
1/1, 13/12, 4/3, 3/2, 13/8, 2/1
Now it would be simple enough to add in the 6/5 minor 3rd and the 16/9 Pythagorean 7th, but that leaves us with a problem. It's my personal belief that step-wise melodic intervals are more consonant when they are superparticular, or of the form n+1/n. I'm not sure if there is other research or theory to support this, but it's something I've noticed and gotten good results from. The interval between 13/12 and 6/5 is 72/65. Not really the best sounding interval in the world. Similarly, the interval between 13/6 and 16/9 is 128/117. The best solution to this I found was to use the septimal 3rd and 7th. The interval between 13/12 and 7/6 is the superparticular 14/13. The interval between 13/8 and 7/4 is the same. Looks like we're in business.
1/1, 13/12, 7/6, 4/3, 3/2, 13/8, 7/4, 2/1
I want to run up and down this scale in increasingly larger intervals to see what the relative intervals are. Starting with seconds:
Root: 1/1, 13/12, 7/6, 4/3, 3/2, 13/8, 7/4
Second: 13/12, 7/6, 4/3, 3/2, 13/8, 7/4, 2/1
Interval: 13/12, 14/13, 8/7, 9/8, 13/12, 14/13, 8/7
So we recognize 13/12, but 14/13 is new. And this brings us to a new idea. We can pair together intervals based on the interval that they comprise when combined. In this case, it looks like a septimal third can be divided into two tridecimal whole tones. 13/12 is the major tridecimal whole tone, as it is larger, and 14/13 is the minor tridecimal whole tone, as it is smaller. Next is thirds.
Root: 1/1, 13/12, 7/6, 4/3, 3/2, 13/8, 7/4
Third: 7/6, 4/3, 3/2, 13/8, 7/4, 2/1, 13/6
Interval: 7/6, 16/13, 9/7, 39/32, 7/6, 16/13, 26/21
This reveals a little more. The fifth from 13/12 to 13/8 is broken in thirds by 4/3. These thirds are 16/13 and 39/32. 16/13 is the wider of the intervals, but I hear it as very dark. 39/32 sounds brighter, yet still dissonant. The fourth between 7/8 (The seventh below the root) and 7/6 is broken into a third and a second by 13/12. The second is the familiar 14/13, but the third is 26/21, which is very close to 5/4 and could be considered a consonance. Now to fourths.
Root: 1/1, 13/12, 7/6, 4/3, 3/2, 13/8, 7/4
Third: 4/3, 3/2, 13/8, 7/4, 2/1, 13/6, 7/3
Interval: 4/3, 18/13, 39/28, 21/16, 4/3, 4/3, 4/3
The fourths here combine to create sevenths and in this case, the combined intervals are not a lower limit than the components. 18/13 * 4/3 is 24/13 while 39/28 * 4/3 = 13/7. Both are 13-limit sevenths. You can combined both of the 13-limit fourths to get a septimal one. 18/13 * 39/28 = 27/14. At this point, I'll leave you to do the rest if you feel like. It's a rather tedious process and I don't have much time. You'll also find that you can invert the seconds, thirds, and fourths, to get sevenths, sixths, and fifths.
Now, if you remember all of the superparticular intervals, you'll see that a lot of them are ridiculously small. A lot of the time, they represent differences between higher-limit intervals and their lower-limit counterparts. For example, the difference between a minor sixth (8/5) and a tridecimal sixth (13/8) is 65/64, which is one of the 13-limit superparticulars. The difference between the tridecimal sixth and the major sixth (5/3) is 40/39. This can be done with any interval. The results are not always superparticular,
but they very often are, and I believe this holds significance. I need to stop myself here or I'll keep going on for a long while. But there's certainly a lot of information and I hope you have a happy exploration. Keep in mind that this scale didn't contain any 5-limit intervals or 11-limit intervals, so there's multiple dimensions which I haven't even touched on here. I'd highly recommend building some scales and finding a way to get some audio feedback to hear them. I use Pure Data and a basic synth that I programmed in Python using PyAudio and Pygame. I hope this has given you somewhere to start from if nothing else.