Musicians and composers just getting into the area of microtuning might, in an initial survey of the available resources on tuning around the web, encounter a barrage of intimidating jargon and mathematical concepts. Fortunately, a composer or player need not worry about the math side of things! This article suggests a number of ways to jump right in and start putting the products of tuning theory to work.
Quoted from the Xenharmonic Alliance II, January 21 2013:
Math: Use as much or as little as you'd like
Mike Battaglia wrote:
"Tuning math is complex. But, if you're a composer, you don't need to ever worry about it. You don't ever need to know what a val is, or what a wedgie is, or what an epimorphic scale is, if you don't want. You don't need to learn how matrix multiplication works or deal with any of that. Those things are useful mathematical constructions which make it easier to communicate, program, and reason about the topic at hand, but they're not anything that a musician should ever, in general, have to know about to write microtonal music.
If, on the other hand, you DO want to learn tuning math, a basic undergraduate linear algebra course will enable you to understand about 90% of it if we phrase it a certain way.
But if you don't, then you don't need to. Just ask here for a tuning system and we'll recommend you like a hundred to use, along with some associated scales, and some suggestions for using them. Then it's up to you to go nuts and write. That's all it requires."
Keenan Pepper wrote:
"And the way to really understand regular temperaments without bothering with all the math is:
- Learn a reasonable amount about just intonation. Things like "prime limit" and "otonality" shouldn't scare you. Being familiar with the whole 11-limit tonality diamond is a good thing.
- For each temperament you're interested in, learn how the different tempered JI intervals relate to each other in that temperament. For example, in porcupine, two 11/10s equals one 6/5, and that interval (11/10) also represents 12/11 and 10/9 at the same time.
- You can express this in a complicated mathematical way with linear alegbra, but that's completely unnecessary if you understand it in a musical way as relationships between intervals you're already familiar with."
Mike Battaglia wrote:
"For starters, you might want to mess around with what's called "porcupine" temperament in 22-EDO. The base diatonic-sized scale is (as steps out of 22-EDO) 4 3 3 3 3 3 3, and you can chromatically alter anything in that scale you want by 1 step out of 22. For instance, if you flat the 7, you get the scale 4 3 3 3 3 2 4, which is nice because it has a 4:5:6:7:9:11 chord in it. There's another "superdiatonic" scale at 1 3 3 3 3 3 3 3 which you can morph the above into if you want, and a 15-note chromatic scale at 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2; feel free to not stick dogmatically to these exact scales but to change them as you desire.
If you don't mind a bit less accuracy in representing harmonic ratios, the above temperament also exists in 15-EDO, except the diatonic-sized scale is now 3 2 2 2 2 2 2, the superdiatonic is 1 2 2 2 2 2 2 2, and the chromatic scale is all of 15-EDO. 15-EDO also has a really nice "symmetrical diatonic" scale, called the Blackwood-10 scale, which is 2 1 2 1 2 1 2 1 2 1. Play around with major and minor chords in this scale a bit, and transpose them up and down the shape of the scale, and you'll see why it's so magical - the circle of fifths is only five steps long."
World Music Scales
If you don't care about harmonic ratios at all, but perhaps instead care about world music, then you might enjoy 16-EDO, which contains in it a wonderful extension of both gamelan music and Chopi timbila music; 5 2 2 5 2 is a great pelog and 3 3 3 4 3 works as slendro.
Both of these scales can be "extended" into "diatonic"-sized versions, such as the 2 3 2 2 3 2 2 mavila-7 scale, named after the Chopi village which used a very similar scale to tune its xylophones. This scale is really nuts because of its "anti-diatonic" structure; the 675 cent fifths are so flat that four of them gives you a 300 cent minor third instead of the more familiar major third. As a result, there's now two large seconds and five small seconds, four major thirds and three minor thirds, etc, which is the complete opposite of what you're used to. There's even a 2 2 2 1 2 2 2 2 1 9-note superdiatonic scale to play with if you want, and of course you should always feel free to chromatically alter things!
The 3 3 3 4 3 slendro can turn into the 3 3 3 1 3 3 "slendric-6" scale as well, though I prefer the 3 3 1 3 3 3 mode. This scale is nice because it sounds kind of like a "warped diatonic" scale. Chromatically altering things by 2 steps here, instead of 1 step, sounds really nice.
World music fans might also like 17edo, which contains within it a system of scales very similar to the maqamat used in the middle east; 24edo and 31edo contain the same features but with better intonation, albeit at the cost of more notes, but you can always use smaller 17-note subsets if you want.
17-EDO is also notable for containing a very interesting diatonic scale, for which the fifths are slightly sharp from what we're used to (and rather pleasantly, I think). As a result, the "major thirds" in the diatonic scale end up being closer to 9/7 than 5/4, and the "minor thirds" end up being closer to 7/6 than 6/5. Major chords are thus a bit less "sweet" than usual, though I don't mind them much, whereas minor chords are now 6:7:9 and have a sort of "subminor" flavor to them. You can also think about ditching "thirds," and instead harmonizing this scale in terms of 4:6:7 chords, of which there are four instead of three. I really like this because it has a very calm, quiet, meditative sound to it. This is called "superpyth" temperament, short for "superpythagorean," because the fifths are a bit sharp of Pythagorean (instead of a bit flat as usual).
For people looking to mess with higher-limit stuff, 17-EDO is also really notable for giving decent 13-limit harmony, with the caveat that there's no good representation for 5/1 and other ratios containing 5.
A really good paper on how 17-EDO offers a sort of bizarro-universe for the diatonic scale, which we might have gone with historically if things were different, can be found at http://www.anaphoria.com/Secor17puzzle.pdf .
Historical Tunings and Extended Meantone
Some people are really interested in historic tunings for meantone, and perhaps extending them in ways that go beyond the 12edo "closure" that we ended up going with. So if you really want to get away from the notion that C# has to equal Db, and want to explore extended meantone, there's a number of great options for you.
A particularly well-known historical meantone tuning is 1/4-comma meantone, where the 3/2's are a bit flatter than we're used to, but the major thirds are exactly 5/4. What's particularly interesting is that this same tuning turns the augmented sixth, e.g. C-A#, into an almost exact 7/4 - so you can treat it as a 7-limit tuning.
Furthermore, it's also well-known that 31-EDO is almost exactly the same as quarter-comma meantone - it's 1/4.15-comma meantone instead of 1/4-comma meantone, which is almost certainly indistinguishable by the ear in practice. 31-EDO is spectacular in that it supports all of existing common practice music (at least that which doesn't take advantage of C# having to be equal to Db), but offers you pretty well-tuned 11-limit chords as well, if JI crunchiness is your goal.
As mentioned before, all of the "maqamic" stuff from 17-EDO and 24-EDO is also in 31-EDO, but now the major thirds are 5/4. So there's that as well.
Alternatively, you have 19-EDO, which is very close to 1/3-comma meantone, for which the minor thirds are pure 6/5's. This tuning can also play virtually all common practice music. While 31-EDO does a lot better in the 11-limit, 19-EDO still has an OK 7-limit. However, what you lose in intonation, you gain by having a really badass set of novel enharmonic equivalences: instead of C# equals Db, it's now C# equals Dbb. This leads to a really tight-knit "structure" with tons of enharmonic equivalences everywhere, much like in 12-EDO, and unlike 31-EDO, the enharmonic equivalences are really simple and can be made use of.
For instance, there's now an entire circle of major thirds which doesn't close at the octave after 3 iterations, but which does sync up with the perfect twelfth after 5 iterations. Also, the "diesis" between A# and Bb is now the same as the "chromatic semitone" between A# and Ax, which you can exploit to make some pretty neat musical effects. You've also got some very novel scales, such as the "semaphore-9" scale, which is 3 3 1 3 1 3 1 3 1; subminor and supermajor chords are everywhere here.
You can do all sorts of amazing stuff with 19-EDO if you want, and make it sound almost exactly like ordinary 12-EDO music too, but then depart from it into some rather mind-blowing stuff if you want.
There are also the people who care about none of this, but really just want really well-intoned higher-limit chords. Gene Smith is certainly the person who's explored this style the most, though I also have a few things to say about it.
These people already tend to know a bit of math related to JI, so I'll assume you know what a prime limit is and what ratios are below, as well as having some compositional idea of the sounds you're after.
If EDOs are what you're after, and you want a medium-sized one with good 11-limit stuff, you should try starting with 22-EDO. 22-EDO isn't perfect for the person who really insist in hyper-JI accuracy, but if you want is the usual crunchy 11-limit sound without requiring a lot of notes, it's pretty hard to beat 22-EDO. 4:5:6:7:9:11 chords can sound pretty spectacular, though you can hear the difference from JI if you look for it (or use a really harsh GM timbre which brings that stuff out). 22-EDO has a few interesting ways of tempering intervals together, which can be either a bug or a feature depending on what you want; 6/5 and 11/9 are equated, as are 7/5 and 10/7.
If that's not accurate enough for you, and especially if you want a higher-limit tuning which still works with common practice harmony, it's difficult to beat 31-EDO, as previously mentioned. 31-EDO gives you great 11-limit resources, as well as some decent 13-limit ones as well, it's also very close to the historic 1/4-comma meantone.
Somewhere between 22-EDO and 31-EDO in terms of accuracy is 34-EDO; it's like two copies of 17-EDO, and is as accurate in the 13-limit as 17-EDO is, except now you have 5/4 as well. 7/4 isn't perfectly represented here, but if you're happy with it in 17-EDO, you'll be happy here as well. Peter 'Rush' Kosmorsky and Neil Haverstick have both worked extensively with this tuning. 34-EDO also has very nice sounding 5-limit harmony, in some way made nicer by the slight sharpness of the 3/2; I would suggest people to look at it before demanding 41-EDO or 53-EDO, as I describe below.
If that's still not accurate enough for you, and you insist on using an EDO, you have 41-EDO and 46-EDO, both of which handle the 13-limit very well, especially the latter. 41-EDO has very accurate 3/2's, even more so than 12-EDO, and is a "schismatic" temperament, which says that instead of 5/4 being four 3/2's minus a few octaves, it's now eight 4/3's minus a few octaves. In other words, spelled the usual way, C-E s no longer 5/4, but C-Fb is.
In contrast, 46-EDO has slightly sharp fifths, a bit less sharp than 17-EDO and well within tolerable range. It's also an excellent tuning for "sensi" temperament, which envisions a stack of 9/7's such that two of them makes 5/3, a continuation of this stack hits a number of 7-limit ratios very accurately, and in 46-EDO the 11-limit is supported as well. The sensi-9 scale is 5 5 7 5 5 7 5 7 (and its modes), and you should feel free to use this as a diatonic scale while modulating around and chromatically altering things by 2 steps out of 46 (this is the difference between the large and small step, or 7-5=2).
Once you get to 53-EDO, the 5-limit is so accurate that it's barely distinguishable from JI. If you want almost perfect 5-limit harmony in an EDO, 53-EDO is hard to beat. It also performs extremely well in the 13-limit. 53-EDO, along with 22-EDO and 31-EDO, also supports a very interesting temperament called "orwell," which can be envisioned as a chain of 7/6's, such that two of them makes a pretty decent 11/8, three of them makes a pretty decent 8/5, and seven of them makes a pretty decent 3/1. A good scale to mess around with is 3 2 3 2 3 2 3 2 2 in 22-EDO, or 4 3 4 3 4 3 4 3 3 in 31-EDO. I sort of hear this stack-of-7/6's temperament as being an evil twin to the more lighthearted stack-of-9/7's which is sensi temperament above.
If that's still not enough for you, try taking a look at 72-EDO. 72-EDO basically splits each 12-EDO step into 6 different tiny "shades", and in so doing gives you stunningly pristine 11-limit harmony, as well as decent 13-limit harmony too. 72-EDO is very intuitive for us as 12-EDO listeners: as it combines 24-EDO and 36-EDO, you can sharpen things by 1/6 of a semitone, 1/3 of a semitone, and 1/2 of a semitone, for instance.
Thus, it combines the nice 11-limit properties of 24-EDO (sharpen C-F by a quarter tone to get 11/8) and the nice 7-limit properties of 36-EDO (flatten C-Bb by a 1/3 of a semitone to get 7/4), and also gives you great 5-limit properties (flatten C-E by 1/6 of a semitone to get 5/4).
If you like the 72-EDO approach but like 17-EDO, you might try 68-EDO as a similar thing.
For people who don't care at all about EDOs, don't mind unequal chromatic scales and just want good harmony, there's a zillion options for you; Gene Smith will no doubt be able to add a lot to this, although I'm quite certain I've scared him off with this Margo-sized post here.
You have Miracle temperament, which exists in 31-EDO, 41-EDO and 72-EDO, which is generated by a slightly sharp 16/15, so that two makes 8/7, three makes 11/9, five makes 7/5, and six makes 3/2. There's a really nice 10-note "decimal" scale here, which is sssssssssL, or 7 7 7 7 7 7 7 7 7 9 in 72-EDO. There's also a 21-note chromatic-sized scale to use, often called "blackjack". So you might enjoy just working within blackjack as the entire tuning, or perhaps the 31-note "canasta" or 41-note "stud loco" scales if that's not enough for you.
There's also Orwell temperament, which I previously wrote about, which exists in 22-EDO, 31-EDO, and also 84-EDO. Orwell, again, is generated by a stack of 7/6's, detuned slightly to hit a ton of very accurate 11-limit intervals; in 84-EDO, this 7/6 is 19 steps, and as the generator is 19 steps out of 84, or "19\84", the name Orwell was given to it. I'm not an expert on it, but Gene seems to like it, and Andrew Heathwaite and Jason Conklin have both written some interesting compositions in it. There's a great 13 note chromatic scale which you can use.
There's also Valentine temperament, which is basically just Carlos alpha with octaves. The generator is a tempered 25/24, tempered such that nine of them is 3/2; there are 15 and 16-note chromatic scales to use, and you can try using every other note of the tuning if you like.
If you liked the maqam stuff in 31-EDO, but think that 31 is too many notes, you can try the mohajira-17 or mohajira-24 scales, which are generated by a neutral third, two of which is a 3/2. There's also something nice in 31-EDO called "mothra," which is generated by an 8/7, three of which make 3/2, and for which 11-note and 16-note chromatic scales exist.
Finally, if you don't really want THAT much accuracy, and perhaps liked something along the lines of porcupine - but found 22 to be just too many notes - you can always load up the porcupine-15 chromatic scale, and just treat it the same way we treated the meantone-12 chromatic scale historically - good for "most keys", but with a few "wolves!"
There are so many good unequal 11-limit temperaments that it's almost impossible for me to write about them all here; if this is really what you're after, post more about the things that you want specifically, and I'll answer specific questions.
More info pending: Mind-bending tunings, non-octave tunings, subgroup tunings
There are two more groups of people I have yet to address:
1) The people who don't care about any of the above, but just want to demolish their brains with the most ridiculous mind#$@*&ing sounds possible, which are nothing like 12-EDO. You're probably going to like scales that sound totally different from 12-EDO, regardless of what ratios are in them; you're probably going to find deliberately "inaccurate" tunings, like 13-EDO and 16-EDO, to be really interesting. You're probably also going to be down to experiment with nonoctave scales like Bohlen-Pierce. Ron Sword is all over this stuff.
2) The people who do care about some of the above, but don't really care about strict full 11-limit harmony or whatever. They might be happy to play around with the 7-limit, but without any prime 5, or the 11-limit without any prime 7, or maybe just a tuning which has good 4:7:9:11 chords but no good 3/1, etc. EDOs which look totally crappy from other perspectives, like 11-EDO, now look amazing to you once you're shown what resources are in them. You'll also like unequal temperaments such as "slendric," existing in 36-EDO, where three 8/7's makes a 3/2, giving you 4:6:7 chords everywhere with no 5.
I'd like to write something for those people, but I'm beat for now. I'm personally in both of these groups, so I've written something for everyone but me. Till next time..."