100edo, a modern “well-temperament” or two

sorry for the massive break. not really sure why i didn’t write anything earlier. having become really familiar with 50edo, i looked into 100edo not too long ago for several reasons: obviously it keeps of 50edo’s intervals, and adds steps in between each, meaning twice the number of gradations, interval colours. but splitting an interval in half doesn’t mean much unless you get some nice new dyads and chords, preferably those that approximate harmonic and subharmonic ratios. 100edo is pretty brilliant at that.

splitting our 24c comma in half we get what we might call a semicomma of 12c, which could well be smaller than what many would call a ‘step’. however, this little 12c step can take us from an average-ish approximation to pretty much bang on just intonation.

it just so happens that 50’s meantone lies almost exactly 12c flat of 9/8, and 10c sharp of 10/9, so we can now get great ratios of 9, essentially fudging 5-limit just intonation using adaptive meantone. along with the very near pure 5, 11 and 13 we already had, this gives us virtually just 10/9, 9/8, 11/9, 13/9 and their inverses 18/13, 18/11, 16/9, 9/5. we can also use the 12c interval to mend over-tempering due to the reasonable flatness of 50’s fifths, approximating 15/8 better as 1092c, 45/32 as 588c. but more than that it gives us a better aproximation of 7, along with very good 17 and 19 to give us a good shot at the full 29-limit. 2.(3).9.11.13.17.19.23.29. and of course any interval can be approximated within 6 cents. what this twelve cent interval is is the difference between a slightly narrow fifth and a slightly wide fifth, both rather acceptable and both incredibly helpful in getting different intervals out of a circle of fifths. which is just what we’re about to show.

but 100 tones? overkill right? pretty much impossible to play on any instrument i know of. unless, of course, we use subsets. and it just so happens there is a wonderfully simple 12-tone subset which will be familiar to those with knowledge of irregular meantone schemes. because when you’ve got two sizes of fifth…

take a chain of 8 regular (696c) fifths, this could be Eb-B or F-C# or whatever suits your fancy. now if you’ve only got twelve tones and continue your regular fifths, you’ll run into a big wolf. so how about making all the remaining fifths wide so we end back where we started? it so happens our new large fifth @ 708c is just the right size, add four of those and we close the circle. very simple, and it looks a bit like meantone. but it circulates in 12 tones, we have good fifths (as good as we can get in 100edo, ~6cents of error) everywhere, we have some of the usual “good keys” and “bad keys” but we get a lot more bang for our buck.

72.0
192.0
288.0
384.0
504.0
576.0
696.0
780.0
888.0
996.0
1080.0
1200.0

with only 12 notes you wouldn’t expect too many intervals. 12edo only has 12. this little set has 44! ranging from rich low meantone to otherworldly septimal harmony, meaning a whole lot more key colour than most of the well-temperaments that were considered ‘good’. a twelve tone set works very well on most instruments as long as you remember that keys have characteristics, if our meantone fifths ran F-C# and our large fifths ran C# back up to F we’d find that C, G and D majors were our usual sweet meantone with thirds of 384 and 312c (~5/4 and 6/5), but travel halfway round to C# and you find the tonic third at 432c (~9/7) and mediant and submediant thirds at 276c (~7/6). then in between quintal and septimal thirds we get keys around the middle with thirds somewhere in the middle, 396 and 408c majors and 300c minors. these are the kinds of thirds we get in usual well-temperaments, but here they represent only the middle of the spectrum. we have four ‘minor’ thirds 276, 288, 300, 312, and 5 majors 384, 396, 408, 420, 432.

if this isn’t enough, simply taking two of these twelve note chains some distance apart will give us some wonderful 24-note systems. the most obvious choice for me would be 48c, the regular meantone diesis. i should mention that this idea (of 8 meantone fifths and 4 larger fifths to close the circle, as well as to displace a second identical circle by the meantone diesis) is not my own, and i learned of it through the wonderful work of Margo Schulter, who had pioneered the idea with Zarlino’s 2/7 comma, of which 50EDO is a bit of a variation on. here is the 24 tone set:

0.0
48.0
72.0
120.0
192.0
240.0
288.0
336.0
384.0
432.0
504.0
552.0
576.0
624.0
696.0
744.0
780.0
828.0
888.0
936.0
996.0
1044.0
1080.0
1128.0
1200.0

it might not look too special, but now we have a whopping 94 intervals. and a LOT of them are far from your regular major and minor. just looking at the thirds we have 240, 252, 264, 276, 288, 300, 312, 324, 336, 348, 360, 372, 384, 396,408, 420, 432, 444, 456, and continuing in 12c increments [as some may consider 228 and 468 to be within the realm of “thirds” too], 19 [or perhaps 21] different thirds, covering the area from 8/7 and 15/13 up through the regular minors, small and large middle thirds, majors, supermajors up through 13/10 and 21/16.

just from 24 tones we got two circulating systems encompassing the entire 100edo gamut except for 6 intervals: 12c, 180c, 516c, 684c, 1020c, 1188c, of which only 180c(~10/9) and its inverse 1020c(~9/5) are any real use. this struck me as possibly the ideal 24-note keyboard tuning for anyone with an interest in meantones pental or septimal, or well temperaments, or middle intervals, or approximating just intonation, or learning about key characteristics in a major way. ideal for everyone who can appreciate fifths with 6 cents of error (myself definitely included)

however, while it’s a great tuning for dual 12-note keyboards or single manual 24-note keyboards where they exist, it doesn’t work well for generalised keyboards, simply because it’s not really a generalisable tuning. it is not generated by, say a series of fifths, but by two partial series of fifths of different sizes. hexagonal layouts (like those on the Axis49 and 64, Chameleon, Microzone, Terpstra) work on the premise of scale generators, with each of the three axes representing one interval, where for example the 12:00 and 4:00 axes added together equal the 2:00 axis (giving a rank-3 matrix when including octaves). the bosanquet layout works particularly well for 50edo with 12:00=3\50 (72c), 2:00=8\50(192c) and 4:00=5\50(120c), but working out a layout for 100EDO, or even for this “well-temperament” to be consistent leads to some not-quite-so-practical solutions. at the moment, the best solution i can see is just to use two 12-tone or one 24-tone keyboard (if you can find one…)

i was going to make this post about things unrelated to 50edo, but i guess that’ll just have to wait for next time…