What are the characteristic intervals or scales of 13-limit harmony and beyond?

Question

I've been studying and really getting into the Extended Just Intonation music of composers like Ben Johnston and Kyle Gann. I've started trying to explore the sort of intervals and scales that are available to a composer as they move to different p-limit ratios. 3-limit is largely characterized by its perfect 5ths and 4ths as well as its fairly dissonant thirds. 5-limit, of course, is characterized by its 5:4 and 6:5 thirds and new scale step sizes like 16:15 and 10:9. These are all pretty obvious to anyone regardless of their interest in EJI.

As I explore 7-limit, it seems like one class of intervals that become available are what I've seen described as "consonant dissonances." Here I'm thinking of the 7:5 tritone and the 7:4 seventh in particular. Some scale step sizes include 15:14 half steps and 8:7 whole steps.

11-limit seems to open up a large class of "neutral" intervals that are more-or-less in between major and minor, e.g. the 11:9 3rd, 11:6 seventh, 18:11 sixth. Neutral steps like 33:32 quarter steps and the 11:10 and 12:11 neutral seconds also become available.

I'm going into this much detail, because this is the sort of info I'd love to have for 13-limit and higher, but I've completely hit a wall. I'm not finding discussions of characteristic 13-limit intervals or scales. Obviously, I could just start trying every ratio involving 13 multiples, but I'd love more guided suggestions about specific intervals or classes of intervals to listen for. Bonus points for 17-limit, 19-limit and other examples.

Answer 1

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.

Answer 2

I'm not going to lie, Just Intonation is not my forte, and I hate math, so I'll spare most of the number-talk.

From what I've read from a few difference sources (some of which are outline below), there are a few reasons why there is little discussion / application of extended-limit ratios:

• In Harry Partch's landmark text The Genesis of Music, he simply suggests that he stopped calculating interval limits beyond 11 for purely aesthetic reasons, suggesting that further limits were quite reasonable to be used. However, I believe he does provide charts with calculated intervals including 11-limit.

• Some musicians and theorists, such as the one cited here suggest that as the limit increases, one of two things tend to happen: 1.) Certain ratios can be reduced to smaller ratios, and therefore fall into a different limit class. 2.) Intervals that cannot be reduced tend to sound closer and closer to approximations of the 12-ET (Equal Temperament) system, and are thus unhelpful harmonically. Further discussion on the prevalence and application of upper-limit intervals may be found here.

I would suggest looking specifically at Pg.52-53 for a possible corroborating explanation for the breakdown of larger limit intervals. Pg.74-81 Offers some cursory discussion of 11, 13, and higher-limit intervals.

From a comment from this link, David B. Doty's book The Just Intonation Primer appears to be a valuable resource with extended discussion on 13 - 19 limit intervals and primes beyond.

In summation, I think it is important to note that while most of the sources I've referenced are amateur, I think it is curious that their reasoning for lack of EJI exploration and analysis is separate but comparable.

I hope that helps you get going in the right direction.