50EDO and why it kicks ass, types of thirds, a meantone series

For a year or so, I got heavily into meantone tunings, tuning my piano in a twelve note subset Eb to G#, realising with a set like that one can not only play beautifully rich major, minor, diminished and augmented chords in a number of keys but also those fun septimal chords, sub(minor) and super(major) triads and sevenths, and even making sense of the wolf

Doing a really DIY job of refretting an acoustic guitar to 31EDO and playing around with a 19 tone subset on keyboard (Gb to B#) made me realise that a lot of what makes meantone great happens over great distances in the spiral (or 31-tone circle), and that 12 notes didn’t really cut it if you wanted to do interesting stuff in several keys, without running into wolves by playing the ‘wrong’ enharmonic.

Falling in love with 31EDO and the closely related 31-tone quarter-comma meantone, I started feeling like this was a system I could settle down in so to speak, to musically exist within 31EDO as most musicians live within the bounds of 12EDO, it has a larger variety of concordant intervals than 12, and represents the 7 prime-limit incredibly well for its size, as well as a few good ratios of 11.

It’s hard to ask much more from a meantone. We have perfect or near-perfect 5 and 7 identities (the major third and sub seventh or augmented sixth), a fifth which is not too flat (19EDO’s fifth is about the limit before I start disiliking fifths altogether), and multiple different interval categories.

As well as minor and major we get sub (lower than minor or perfect, higher than diminished) and super (higher than major or perfect, lower than augmented), as well as middle, what a lot of musicians call the ‘neutral’ intervals, represented by quarter tones in standard current music practice

I really should have gone with an ascending sequence of meantones, starting with 12EDO (or 7 then 12 if you’d like), and ascending through 19 before we get to 31, 50, 81… This is part of what’s known as Kornerup’s sequence, which, adding the previous numbers gets us closer and closer to a Golden meantone division, where several intervals are in the Golden Ratio with one another. But it’s also a good way to build up an understanding of larger and larger meantone systems, through diatonic, chromatic, enharmonic and beyond.

12 we know (I hope), with 2 kinds of third-based tonalities in the basic sense, minor and major, although they are respectively 16 and 14 cents away from what we might call ideal thirds (6/5 and 5/4). 19 separates the enharmonic pairs of 12, provides distinct 1/3 and 2/3 tones for the augmented unison and minor second respectively, and allows us new sub and super sounds, which being closer to ratios of 13 than 7 sound pretty whacky, but are fun nevertheless. Major thirds are just a little too flat for my liking, minor thirds are pretty much bang on 6/5. 31 gives us very accurate ratios of 5 and 7, some of 11, and represents 5 different tonalities: sub(minor) [rather accurate 7/6], minor [1/4 comma flat of a 6/5], middle [very close 11/9], major [pretty much bang on], super(major) [not quite so accurate but still a similar flavour to 9/7]. It does an ok job at being an 11-limit system, but most of the time it’s better treated as staying within the 7-limit, where it does an excellent job.

So what more could we want? Why go past 31?

Because 50 has things that 12, 19 and 31 can’t do. It’s a meantone that can represent any interval at all with a maximum error of 12c, but represents a good majority of important intervals even up into much higher primes 11, 13, 17, 19, 23, etc, with very good accuracy. It’s a meantone that can almost sound like just intonation if you want it to, or can sound rather unfamiliar, yet with mostly very logical tonal connections.

Chords have a lovely warmth to it due to rather good the accuracy on all thirds, it is ideal for a lot of early music, modal or otherwise, and might provide a rather good way to analyse a large number of intervals within a meantone context, a master system if you will, with which to learn about and use new interval classes, as well as give flexibility to transform pre-existing structures and create entirely new ones within a 50-tone intonational universe.

I’m not saying it’s the be all and end all of tunings, but if meantone logic appeals to you, you couldn’t do too much better.

Here is a table of intervals, including cent values, interval names and note names. I don’t think the accidental symbols display properly on smartphones but hopefully it’s sweet on computers.

One important thing is out selection of thirds, and how close they are to certain ratios. As well as minor and major within about 4 and 2 cents of 6/5 and 5/4 respectively, and sub and super thirds within a couple cents of the septimal 7/6 and 9/7, we have a lesser and greater third in between, which might well be more familiar to those practiced in Pythagorean or expressive (slightly raised from Pythagorean) intonation, the kind practiced by most string soloists and generally thought of as ideal melodic intonation. But now instead of one middle third we get two, one closer to minor and one closer to major, both of which are familiar to the musics of Persia and the Middle East, but less so to much of western music, and might be viewed at first as a supraminor and submajor. Finally, we have two outliers which don’t at first look or sound like thirds, the diminished and augmented, or to use some newly coined terms, arto and tendo (from the Latin contract and expand), lower than sub and higher than super. I was skeptical of these two at first (they’re actually also enharmonically equivalent to the super second and sub fourth respectively!), but now really playing around with them and using them in the harmonic context of 15/13 and 13/10, I’ve realised what great little intervals they are.

Which all means we have a grand total of 10 types of thirds, 10 basic tonalities with which to build chords, scales, and music of many moods. [I made this little demo track, overdriven to demonstrate how different types of thirds in 50EDO can give us a strong difference tone bass line. Here I’m using 8 out of 10 different thirds once each in a chord sequence] This of course also holds for their inverse sixths, and since all thirds and sixths are rather close to simple just ratios, we get lovely in-tune 2 part harmony in thirds and sixths [with 3-part harmony often happening as a result of strong difference tones]. After getting really familiar with the thirds and sixths of 50, 31’s minor third seems pretty sour and 19’s major third just a bit wrong…

As 50EDO is pretty much the musical universe I’m operating in at the moment, there’s a lot that I want to do with it. But I wanted to introduce its features and get you all accustomed to it little by little first, before trying to explain the complex stuff like enharmonic modulations, irregular mappings to imitate other regular tunings, hyperchromatic and commatic voice leading, as well as something I’d like to focus on in future: bringing jazz and gospel harmony rooted in 12EDO into the universe of 50, and seeing what magic might happen

2 thoughts on “50EDO and why it kicks ass, types of thirds, a meantone series

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