Tuning various edos by ear
Ever since I started playing music I've wondered why we have 12 notes per octave, but normally only use 7 together at a time. What made these particular combinations sound good together and not others? Was it a natural result of mathematics that music form scales of this number, or an artifact of the way the human ear and mind works? After many years of gradually discovering the history of pythagorean tunings, their tempering, various folk scales, meantone commas, and all that jazz that gradually extended the number of notes instruments could handle before the rise of electronic tuning made it possible to evenly tune 12 notes an equal distance apart, and sharps & flats became genuinely enharmonic, I'd say probably not. 12 was not an inevitable number, merely a convenient one due to it's good approximation of 3/2.
So I decided to look at the alternatives. If you tune by ear to their best simple ratio, or a simple tonal diamond of two of their best ratios, which numbers come closest to being an equal division of the octave without needing any further tempering, and which are dramatically out. If you found yourself magically transported to another world and had to create musical instruments from the local materials without any electronic reference to help you even everything out, which number scales would be easy, and which would be hard? This is not a judgement on how good particular EDO's actually sound compared to others of course, which is highly subjective, but merely their ease of creation. Let's run the numbers.
List of deviation from true equal division of various EDO's by tuning to their best ratios
|Best ratio(s)||Absolute error (cents)||Relative Error (percent)|
|14||11/9 x7 by 9/7 x2||33.805||39.446%|
|15||8/7 x5 by 6/5 x3||44.022||55.0175%|
|16||11/6 x8 by 7/4 x2||10.633||14.177%|
|18||7/6 x9 by 9/8 x2||5.5439||8.3159%|
|20||13/8 x10 by 9/5 x2||7.1526||11.9211%|
|21||11/9 x7 by 7/4 x3||37.2448||65.1816%|
|22||9/7 x11 by 11/10 x2||14.168||25.9746%|
|24||11/6 x8 by 11/8 x3||7.41566||14.8313%|
|27||7/6 x9 by 13/8 x3||9.466||21.2985%|
|30||13/8 x10 by 9/8 x3||12.5689||31.422%|
|32||12/11 x8 by 16/15 x4||6.76555||18.04147%|
|33||9/7 x11 by 9/5 x3||13.966||38.4078%|
|34||13/9 x17 by 5/4 x2||23.106||65.4672%|
|36||7/6 x9 by 3/2 x4||7.4989||22.49674%|
Best ratio(s):N/A Absolute error:0 Relative Error:0
Any note is always in tune with itself, unless it has inharmonic elements that mean it isn't. Nothing much I can say here.
Best ratio(s): 7/5, Absolute error: 17.487c Relative Error: 2.914%
Why you would want to get a √2 as precisely tuned as possible without other reference points is beyond me, but to do so, you just need to familiarise yourself with the sounds of 7/5 and 10/7, then move very gently between them until it sounds as bad as possible. It's not as if anyone will care if you get it slightly wrong.
Best ratio(s): 5/4 Absolute error: 41.058c Relative Error: 10.264%
Ah yes, the lesser diesis, the bane of people trying to stack major thirds. A very noticeable wolf note people have spent a lot of time and effort on various ways of tempering out. This is definitely not the way to go about it.
Best ratio(s): 5/3 Absolute error: 62.565c Relative Error: 20.855%
Ah yes, the greater diesis, the bane of people trying to stack minor thirds. Once again, not a very convincing system on it's own.
Best ratio(s): 7/4 Absolute error: 44.129c Relative Error: 18.387%
Once again, even the closest simple ratios are way out, giving us a very obvious comma. Trying to get 5 equidistant notes without electronic reference is yet again going to be a nuisance involving lots of tedious back and forthing that will inevitably wind up heavily tempered anyway, as many primitive systems around the world demonstrate.
Best ratio(s): 9/8 Absolute error: 23.460c Relative Error: 11.73%
A dramatic improvement, due to the closeness with which 6 9/8's approximate an octave. It's still not that great compared to some higher EDO's, but at least it's a start. What's more, unlike 12EDO, where the wolf note has the same error, but is very jarring in a chord, in 6 it merely makes the final step in the chain a very mildly flat 10/9, noticeably different, but also a consonant interval, and the two major thirds that cross it very slightly flat 5/4's, noticeably sweeter sounding than the other 4. This gives the scale a definite tonal center that makes it noticeably more pleasant than true 6EDO whatever key you play in. So this is a success in it's own way, but still not quite what we're looking for.
Best ratio(s): 11/9 Absolute error: 31.856c Relative Error: 18.583%
Yet again, no nearby ratio is even close to being circulating, and even the closest one is fairly complex. This makes it increasingly apparent that the choice of 7 notes for a lot of scales is more a property of the way the human brain works (and our memory in particular) than any inherent one of acoustics. This is an area aliens with different hearing and mental capacities would definitely wind up developing differently even if listening to the same notes and timbres.
Best ratio(s): 11/6 Absolute error: 5.096c Relative Error: 3.397%
Now this is much more like it. Stacking 8 12/11's, or their simpler and easier to distinguish octave inverted version gets you 8EDO with an error of just over 5 cents, pretty unnoticeable against the general limits of ear tuning precision. If you want an accurate 2 or 4EDO, it's easier to tune this then only play a subset of the notes. Now if only it were more pleasant sounding in general. Uniform and easy to construct does not always mean good.
Best ratio(s): 7/6 Absolute error: 1.8318c Relative Error: 1.378%
Even better than 8, stacking 9 7/6's gets you 9EDO. (and also a much more accurate 3EDO in the process) The error is unnoticeable, the interval much simpler and easier to pick out, and 9 sounds considerably more consonant in general than 8. It might not have quite enough notes for complex harmony, particularly as it lacks a good 5th, but as a melodic system, it has more than enough resources to occupy you for years. If you're starting from scratch, and want to create an equal tuning system without electronic reference, this is definitely the one to go for first.
Best ratio(s): 13/8 Absolute error: 5.2766c Relative Error: 4.3971%
A little trickier to tune and higher error, but 10EDO is still pretty indistinguishable from a stack of octave reduced 13th harmonics, and also gives you a good 5EDO into the bargain. At this point you have a functional, if not great 5th, and can combine it with the 2nd, 3rd or 4th to get distinct flavours of harmony. You can even extend it with 7ths, 9ths, etc, and all the intervals will remain tolerably consonant. Well worth considering.
Best ratio(s): 9/7 Absolute error: 14.0795c Relative Error: 12.902%
On the other hand, while 9/7 looks like a smaller fraction than 13/8, it's much harder to create and hear as a consonance than a divisor of a pure harmonic, and even if you do, you'll still get a noticeable comma if you stack it. And that's 11EDO's best interval! You're not forming this one without an electronic tuner anytime soon.
Best ratio(s): 3/2 Absolute error: 23.46c Relative Error: 23.46%
Ah yes, the dreaded pythagorean comma. The source of centuries of trouble because ancient musicians decided to create a system that favours 3/2's over all other intervals, and then found that there were flaws in that system. There have been many books written on how that developed over the years, the many experiments in just scales and meantone commas that were tried, and why we wound up with the compromise we have today, that does 3/2s, 4/3's, and 9/8's well, 5/4 and 6/5 recognisably but with noticeable error that is mainly tolerable because they're in the opposite direction & cancel each other out, and higher order ratios not at all. There's no way I could recount all of that, but this definitely shows that 12 is not the most effective choice for an equally divided scale. If we had to start all over again with the benefit of hindsight, things would develop very differently.
Best ratio(s): 9/5 Absolute error: 28.7467 Relative Error: 31.1415%
The best interval has a slightly higher error than 12, and you need to stack one more to get back to the starting point, making the comma quadratically more noticeable, and any tempering of the system involve even trickier decisions. Since it sounds generally worse than 12, and lacks the convenient divisability to split the commas in various ways, it's easy to see why this one never caught on.
Best ratio(s): 11/9 x7 by 9/7 x2 Absolute error: 33.805c Relative Error: 39.446%
If imitating 7EDO without a tuner is hard, 14 is near impossible. None of the intervals are even close to simple ratios. Just to get something within the bounds of propriety I had to use a tonal diamond of two different intervals, and even then it's the worst relative error yet. It would be an enormous waste of time to even attempt it.
Best ratio(s): 8/7 x5 by 6/5 x3 Absolute error: 44.022c Relative Error: 55.0175%
Even more than 14EDO, 15 doesn't approximate anything at all justly, and that, combined with the shrinking of step size makes it the lowest EDO that strays out of the bounds of propriety entirely however I try and divide it. That small circle of blackwood 5ths really needs an electronic tuner to make it properly even and harmonically circulating rather than developing a default root due to the way it's tempered.
Best ratio(s): 11/6 x8 by 7/4 x2 Absolute error: 10.633c Relative Error: 14.177%
A single step of the best odd interval adds more error than all those 12/11's put together, more than doubling the absolute error and quadrupling the relative error due to the smaller step sizes. While still better than the last 5, and the comma is barely on the edge of the hearable range, 16EDO isn't a very pleasant sounding system even when it is in tune anyway. Probably not worth the effort.
Best ratio(s): 13/9 Absolute error: 22.508c Relative Error: 31.885%
A whole bunch of things working against this one. It's prime, so I can't make a tonal diamond with it, and it's closest to just interval is it's tritone, a fairly complex and difficult to tune ratio. Any way you build it, it'll need some serious tempering. It's no wonder that despite it's strong melodic properties, it was overlooked until we had accurate electronic tuning.
Best ratio(s): 7/6 x9 by 9/8 x2 Absolute error: 5.5439c Relative Error: 8.3159%
Basically the same error as constructing 36EDO then only using half of it. Not as perfect or easy as the purity of 9EDO, but still quite acceptable, and much more flexible in it's uses despite still lacking a good 5th.
Best ratio(s): 5/3 Absolute error: 2.81604c Relative Error: 4.4587%
This one shouldn't be any surprise if you've been here a while. 19EDO's essentially pure 6/5, and even easier to hear and tune to octave inversion is one of it's most notable good points, more than making up for it's slightly weaker 5ths than 12EDO. Stacking 19 just minor thirds gives you 19EDO with only the minutest of pitch drift, probably easily outweighed in error by the precision of your hearing and tools. The result of this is that building instruments in 19 completely lacks any of the tempering problems people struggled with for centuries working with scales of 7 or 12 notes. As long as the thirds line up, everything else will sound equally good in any key too. If you're starting from scratch, and want to be able to play recognisable tunes without making all those old instrument-building mistakes along the way, it's both the easiest and one of the best routes to take.
Best ratio(s): 13/8 x10 by 9/5 x2 Absolute error: 7.1526c Relative Error: 11.9211%
As long as you can distinguish between your 9/8's and your 10/9's, 20EDO is also pretty easy to approximate, and doesn't add too much error compared to 10. This makes for the end of another trio of good EDO's which each have excellent, if very different consonances, and manageable levels of complexity.
Best ratio(s): 11/9 x7 by 7/4 x3 Absolute error: 37.2448c Relative Error: 65.1816%
Not a bad 7/4, but still not nearly good enough to form this one properly. Even using it in a grid doesn't get the error within acceptable levels. Accurate whitewood tunings, like blackwood, are the kind of thing that could only develop with modern technology.
Best ratio(s): 9/7 x11 by 11/10 x2 Absolute error: 14.168c Relative Error: 25.9746%
Once again the rather tricky 9/7 from 11EDO is the closest harmony to just, resulting in a quite noticeable comma even if you can get used to that ratio and tune it accurately. Combined with the smaller step sizes, and this one probably isn't worth the effort to temper the errors out without an electronic tuner.
Best ratio(s): 11/7 Absolute error: 2.69017c Relative Error: 5.15615%
Now here's an interesting turnup. Admittedly 11/7 is a pretty tricky interval to find and familiarise yourself with enough to tune by ear, but if you can, you can recreate 23EDO as an effectively closed system with even more accuracy than 19EDO. A little easier is 17/16, which will still get you a comma of 8.86c, well within tolerances of most people's hearing. A good reminder that while it might not have a great 3/2, this has quite a few excellent higher-order consonances that can be used to form interesting chords, and is very usable as long as you work with it's strengths rather than against them.
Best ratio(s): 11/6 x8 by 11/8 x3 Absolute error: 7.41566c Relative Error: 14.8313%
Not only a far lower absolute error than 12EDO, but a significantly lower relative error as well. It's much easier to create a good approximation of 12EDO by building this pattern of 24 and then only playing half the notes than all the centuries of faffing around with just 3/2's and various kinds of meantone commas. Replacing the 11/8's with 3/2's also works and introduces only a cent more error. In addition, this diamond emphasises that while 24's extra notes may sound dissonant to ears used to 12, it does excellent approximations of 11 based intervals that can be very consonant if used an octave up in extended chords. That's another topic well worth expanding on elsewhere.
Best ratio(s): 5/4 Absolute error: 57.8428c Relative Error: 120.5059%
Like most of the 5's & 7's before it, 25EDO is just bad. The major third may not look too out on paper, but by the time you've used it 25 times, you don't get anything remotely resembling 25EDO. The 13/11 is a little better, but even if you can pick out a complex factorial of two primes like that it still gives you a relative error of 62.5%, also way outside the bounds of propriety. Even more than 15, it's just not an option at all to tune without electronic assistance.
Best ratio(s): 13/12 Absolute error: 2.929183c Relative Error: 6.34709%
Now here's another very interesting system. It may be a fairly narrow interval, but it's still wide enough to be distinguishable, and 26edo gives us an even closer approximation to 13/12 than 9 or 19 do to their best ratio, creating a very satisfying circulating system with an unnoticeable comma. If your hearing isn't precise enough for that, stacking 7/4's is still good enough to give you a relative error about equal to 12edo - usable, but with definite differences in character depending on how it's tempered and what key you play in. So like 23EDO, it may not be the best at simple chords involving 3's and 5's, but it's relatively easy to create without electronic tuning, and has multiple other excellent intervals for you to explore forming chords with. Meanwhile, only using every other note gives you a far better 13EDO than trying to create it with native intervals, just like 24 & 12. It definitely deserves more attention than it's got so far.
Best ratio(s): 7/6 x9 by 13/8 x3 Absolute error: 9.466c Relative Error: 21.2985%
As the next step in the 9EDO family, this once again gives a satisfactory but not outstanding tonal diamond with commas that, while larger than the last two, are still low enough make it usable. You can substitute 6/5 for 13/8, which is a much easier interval to tune too, while only adding just over a cent to the overall error.
Best ratio(s): 14/13 Absolute error: 7.6091c Relative Error: 17.7534%
After all the previous steps in the 7 family were so disappointing, it comes as a pleasant surprise that 28 has a whole range of near just intervals in it. 5/4 and 21/16 are both close enough to be within the bounds of propriety, but the 3 step 14/13 is particularly good. (41/40 is even better, but that's well within the dissonant diesis range, and if your ears can hear a fraction that small as a consonance and accurately tune it, they're far better than mine) As with 12 and 6, if you want to play 7 or 14EDO with actually equal steps, it's easier to tune a stack of these for minimum beating and then only play a fraction of the notes.
Best ratio(s): 11/10 Absolute error: 14.86037c Relative Error: 35.9370%
While 29EDO has a slightly better 5th than 12, it's still not nearly close enough to stay in the bounds of propriety over 29 iterations. Stacking 11/10's at least works, reminding us how good this EDO is for porcupine scales, but the comma is very noticeable and will need tempering. Probably not worth the effort.
Best ratio(s): 13/8 x10 by 9/8 x3 Absolute error: 12.5689c Relative Error: 31.422%
Like 24 to 12, gets considerably better results than trying to create 15EDO natively. Still, both ratios are pretty complex, the comma is very noticeable, by this point the numbers are definitely getting unwieldy, and it still doesn't have a good 5th, so once again, this is not worth the hassle.
Best ratio(s): 5/4 Absolute error: 24.273c Relative Error: 62.704%
While 31EDO's 5/4 is good, it's still not precise enough when you're operating at this kind of level, unlike the best intervals of 19 or 26. Getting an accurate approximation of it without electronic tuning or a complex tonal diamond of tempering is unfortunately not an option.
Best ratio(s): 12/11 x8 by 16/15 x4 Absolute error: 6.76555c Relative Error: 18.04147%
Like all the recent multiples of 2, half of this is once again much better than building 16EDO natively, but this one is also surprisingly usable in itself as well. If you like the very superpythagorean sound, this is much more accurate than 17, 22 or 27.
Best ratio(s): 9/7 x11 by 9/5 x3 Absolute error: 13.966c Relative Error: 38.4078%
While it has a whole tone and tritone less than a cent out, the cumulative error is once again enough that stacking them will break propriety, so a tonal diamond using the best interval from 11EDO is the best approximation of 33EDO, and even that's not great. Definitely one to skip.
Best ratio(s): 13/9 x17 by 5/4 x2 Absolute error: 23.106c Relative Error: 65.4672%
A real disappointment. While 34 is an excellent 5-limit system, none of it's intervals have the level of precision needed to reconstruct it convincingly without electronic tuning. Another write-off.
Best ratio(s): 7/5 Absolute error: 12.068% Relative Error: 35.1980
Not as bad as I expected, due to the closeness it approximates the septimal tritones, but still not good enough to create without serious tempering. As neither 7 or 5 individually have any close approximations to simple ratios, a tonal diamond doesn't work either.
Best ratio(s): 7/6 x9 by 3/2 x4 Absolute error: 7.4989c Relative Error: 22.49674%
Almost identical absolute error to 24EDO, and relative error is still slightly smaller than 12EDO. As it gives you good 7's rather than 11's, much less xenharmonic sounding in general than 24. Both are much easier to recreate by ear than native 12, but as it uses simpler intervals, 36 is particularly easy for it's size. Definitely better than 12 if starting from scratch.
Best ratio(s): 11/8 Absolute error: 1.23622c Relative Error: 3.8119%
Another surprising one that's easy to tune from an octave reduced harmonic, and closes with an error well into the imperceptible range. Even more than 19 or 26, any comma you hear after tuning 37 successive 11/8's will be far more a matter of your own limits to tuning precision than the error inherent in the system. I intended to stop at 36, but this turned out to be so good I had to include it.
Unsurprisingly, 9 and 19 are the clear winners overall, being buildable from the simplest ratios, and having some of the lowest errors, which I already knew before, but this also revealed a few unexpectedly good alternatives that are improvements if you can distinguish and tune to more complex ratios. Sometimes doubling makes things much more than twice as bad, and sometimes it's actually easier to get an evenly spaced small scale by building one considerably larger and then only using a subset. It is particularly interesting that neither 7 or 12 are good numbers for an equally spaced scale on their own, while 24, 28 and 36 are substantial improvements. There's also definite islands of good and bad numbers, and it would be interesting to see how that extends up into higher EDO's if anyone has the time to crunch the numbers. Hope this has been interesting and helped reveal a bit more about the mathematics that music is built upon.