Other Interesting Tuning Systems

There are so many tuning systems out there that you should look into, so these are a few that I find really interesting/have done work in at some point.

17edo

Aside from 31edo, 17 has to be my favorite tuning system. It has solid harmonies, approximating harmonics up to 13 quite well besides 5, which it completely whiffs on.

Lacking 5 is a big deal, as it's such a fundamental part of standard harmony, but 17's 7-limit intervals overcome this issue. Where 17 really excels though, is melody.

The semitone is a perfect size for driving direction, the diatonic scale is harder while the pentatonic scale is softer, and the three types of seconds, small minor, neutral, and larger major, are very interesting melodically.

On top of that, it has good neutral intervals for its size, and neutral scales are traditionally great melodically.

We can even perfect its harmonies by employing a well temperament. If you don't know what that is, I'll discuss it below, but essentially we just make certain intervals better in certain keys, and other intervals better in others.

In 17, thirds are coincidentally right between a more "gentle" third and a more standard septimal third, so a well temperament has the thirds closer to one extreme in some keys, and the other extreme in others. The "gentle" minor and major thirds are close to 13/11 and 14/11, and the septimal are more like 7/6 and 9/7.

George Secor wrote about one of these well temperaments in his 17-tone Puzzle, and I definitely recommend reading about it if you have the time.

Finally, 17, like 31, is a prime, so you get a wealth of interesting temperaments, unlike 12edo, including Squares, which it shares with 31edo.

Werckmeister III

12-tone well temperaments are also quite useful, and while I've worked with a lot of the less common ones, the traditional Werckmeister III is definitely one of the best. I still prefer septimal meantone, as it allows far more variation in intervals, not just optimizing 5-limit ratios, but for that purpose, Werckmeister III does a pretty damn good job.

The idea is that you make a circle of pure 3/2 perfect fifths to get twelve notes. This is Pythagorean tuning, but the issue is that the fifths don't close at the octave, the result is 23 cents too high.

By cutting a quarter of the difference off of four select perfect fifths (C-G, G-D, D-A, B-F#), we make major thirds in more common keys more pure, without the need for any wolf fifths. Each interval has keys that tune it better, and keys that tune it worse. The common keys like C, G, D, A, and E have great tuning all around, but every key is different, keeping variety that we don't see in equal temperament.

22edo

22edo is the first equal temperament to adequately represent the 11-limit. Additionally, it represents simple harmonic subset chords with low complexity in many of its temperaments.

22 has a sharp fifth that generates 7/6 and 9/7 thirds, as opposed to the 6/5 and 5/4 that we're used to. This is called superpyth temperament, which 17 shares, and 22 does it far better the most

Other temperaments that 22 includes are Machine, which generates a whole tone scale with 4:7:9:11 chords, and Magic, a mainly 5-limit temperament with a major third as a generator.

This is one of the most widely loved EDO systems, and for good reason. There's a hell of a lot to like about it. I still strongly prefer the sound of 31, and find that 31 does a much better job with melodies, along with more accurate harmonies, but 22 has some features that definitely make it worth checking out, and with an 11edo subset, it's a very different world.

41edo

41 is definitely more unwieldy than other systems I like, but it has one purpose that it excels at. 41 represents musical systems from around the world quite well in comparison to other systems.

While 41 definitely doesn't capture the small details of world music systems, and it's not accurate to simplify these cultural scales in this way, 41 has the necessary tools to allow a composer to experiment with them at a more surface level with a higher degree of accuracy than with other systems.

It has neutral intervals and a strong Pythagorean skeleton for Arabic music, a 380 cent major third along with other similar "submajor" intervals and otherwise solid 5-limit JI for Turkish music, Slendric and Mavila temperaments for a rough Slendro and Pelog, the near-7edo scale Tetracot[7] that approximates the slightly assymmetric 7-tone tuning of classical Thai music, 5-limit JI that can fairly accurately represent Indian Carnatic scales, strong Pythagorean scales that are useful for Japanese classical music, and the tools necessary to be used for Western music.

Specifically for Western music, 41 approximates 5-limit JI very accurately, as well as having the option to play a roughly-pythagorean scale with the wolf fifth between common notes like D and A, placing it in such a place that the tuning of nearby 5-limit ratios are more accurate, seen here. The wolf fifth can cause some issues, but it's definitely an option.

Centaur

Centaur is a 12-tone JI scale with a massive amount of variation between keys. It starts with the Ptolemy diatonic scale of 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1, and adds in 21/20 7/6 7/5 14/9 and 7/4. It's a very interesting septimal scale, and you can learn more about it here.

Personally, I'm a fan of taking the scale and tempering out the 225/224 comma. This is because there are a lot of intervals that are slightly off from a simplier JI ratio, and tempering this comma out helps smooth over a lot of those inconsistencies in an already very complex scale. For instance, the fifth between 15/8 and 7/5 being perfect is useful in a major context.

The great thing about this scale is just how much variation there is. There are three wolf fifths, but this is not surprising for a 12-note JI scale. We see major thirds ranging from slightly flat of 5/4 to 9/7, with many in between. Each key sounds very distinct, with rich septimal flavor. It's definitely a scale to look into.

Carlos Alpha

Not all scales need to include octaves, and Alpha is a great example of this. It's approximately 15.39edo, with intervals not repeating at the octave. It's instead close to 9edf, or 9 equal divisions of the perfect fifth.

This has a few effects. For one, 6/5, 5/4, and 3/2 are all extremely accurate for the relative size of the system. Additionally, 8/7 and 11/8 are approximated very accurately, allowing most useful 7- and 11- limit intervals to be reachable within the span of a few octaves, which would be wild for an edo system around this size.

The melodies present in Alpha are quite interesting and different from standard tuning, with a small semitone, a neutral second, and a large whole tone, just like Valentine. Thus, Alpha provides a great balance between harmonic and melodic strength, all for sacrificing the octave.

The disadvantages come in a couple ways. Scale building is difficult, as we're used to having our octave period for any scales we make. Additionally, it's difficult to make music without octaves. If you want a bass instrument playing in a lower register, you'll have to accept that it'll be off from everything else, and the same holds for any instruments playing in different registers, unless you fudge the tuning a bit.

We can actually get the Carlos Alpha scale in 31edo by increasing the diesis size to ~39 cents, which improves the tuning of the 3/2, 6/5, and 11/8, while still giving us a decent octave of ~1209 cents.

This scale comes from the musician Wendy Carlos, who created three of these type of scales, that you can read about here. Alpha is the smallest, and in my opinion most interesting, but Beta and Gamma have their uses too. Beta is close to 19edo in tuning, but with a sharper octave that purifies the fifth and improves on the thirds. Gamma is even denser than 31edo, being close to 34, so if you want a fine and very accurate nonoctave tuning, it's a good one to consider.