I spent the better part of Saturday trying to create a xenharmonic wiki page for 94edo that I thought would pedagogically helpful, but I lost hours of work in a site crash and ended up deciding a blog post here might be better in the long run. This might not be super formal or super thorough, but I’d like to write something about what I’ve been doing looking into this tuning system over the last few years.
I first discovered 94edo after spending some time with both 41edo and 53edo, and being fairly pleased with both of them, but realising sometimes they were a little extreme sometimes, and it would be nice to have a middleground. Even though they seem incredibly similar for someone used to courser grids, from a JI perspective, one can compare how they do on ratios involving different prime numbers. Both absolutely nail approximating pythagorean tuning (primes 2 and 3), 41edo tunes prime 5 a bit too low, 7 ever so slightly flat, and 11 a bit sharp, and prime 13 is much too high to really lock into place, whereas 53’s prime 5 is really good, 7 is a bit too high, 11’s way too low, and it locks prime 13 well into place, ever so slightly flat. 41+53=94 seemed to make a lot of sense, as I could keep most of what I knew about either system, and the approximations almost all balanced out to some very nice interval territory.
Checking out its list of intervals I was struck by how close things were to just intonation, even of much higher prime numbers than I was expecting, and found it was the first edo system to have 23-limit consistency (only beaten by edos about three times its size). Since looking into JI ratios approximated by 50edo (with its meantone fifths), and later, 46 and 63 (with their gentle fifths), I’d discovered I liked having an approximation of prime 23 to play around with, and I began to really like the 23-prime limit as a rough way to analyse the pitch/interval continuum. I’ll save the numbers for the bottom of the post for those interested – a few Scala printouts with interval differences in steps and cents.
But back to the more musical side of things, because I already pretty much knew how 41edo and 53edo worked, it was great being able to do the same kinds of tricks here. I had treated both in the past as a big chain of fifths, and found that pythagorean intervals appeared first, then pental/prime-5, then septimal/prime 7. The 12-fifth (pythagorean) comma became incredibly useful, as it represented the difference between these different qualities within the same broad interval class (e.g. major third).
Let’s give a more concrete example. Here’s a chain of forty fifths/53 tones, symmetrical about D (though we probably won’t need all of them)
Ebbbb-Bbbbb-Fbbb-Cbbb-Gbbb-Dbbb-Abbb-Ebbb-Bbbb-Fbb-Cbb-Gbb-Dbb-Abb-Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#-Fx-Cx-Gx-Dx-Ax-Ex-Bx-F#x-C#x-G#x-D#x-A#x-E#x-B#x-Fxx-Cxx
Phew. Some of these have a lot of sharps and flats, but I want to keep this usable for 41, 53, and 94edo, so I’m not making a circle just yet. However, we can simplify things if we know what the pythagorean comma (+12 fifths) does. It also represents raising by 80:81, the syntonic/pental comma, and also 63:64, the septimal comma, and it is one degree of both 41edo and 53edo, and therefore two degrees of 94edo. I will use the symbols / and \ to mean raising and lowering respectively by this multi-purpose comma. Let’s get rid of some of those double and triple flats and sharps. Notice that going down 12 fifths from B gets us a comma lower, to Cb. Let’s respell this as \B [the current convention is for comma accidentals to come after the regular note name, but I have started writing them beforehand, to make naming chords and scales heaps easier]. In the Sagittal microtonal notation system, this symbol is a shorthand for the left side of a downward pointing arrow, \!, which Sagispeak calls “pao” [p for pental and ao for down], vs /|, “pai” [pental-high]. Okay sorry, got bogged down in semantics. Let’s respell.
\\C-\\G-\\D-\\A-\\E-\\B-\\F#-\\C#-\\G#-\Eb-\Bb-\F-\C-\G-\D-\A-\E-\B-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-/F-/C-/G-/D-/A-/E-/B-/F#-/C#-/G#-//Eb-//Bb-//F-//C-//G-//D-//A-//E. A wee bit easier to look at. Notice also that \F#=Gb, \C#=Db, \G#=Ab, \D#=Eb and \A#=Bb, and so on.
The way they handled the 7-prime limit was virtually identical (though while 41 has magic temperament and 53 has hanson, 94 has neither of those), but I had to be a little careful about primes 11 and 13, as those more distant intervals had slightly different possible mappings, that is, I could approximate them in different places in the chain of fifths depending on the exact tuning. Maybe a table could be helpful… Ah, doesn’t look like you can do tables on WordPress… Here’s one I made on Word (hopefully it will transfer over okay):
The above ratios pretty much work in all three tunings, which is pretty awesome. But 94 tones per octave is way too many to think about, so I thought about some clever ways to chop it down to size. While making a subset of the tones you want the most is great, it often leads to inconsistent mappings and weird stuff going on as you transpose through different keys. I wanted something that felt playable in every key. And so I found that by cutting the chain of fifths at 53 tones one could wrap it into a circle with the remaining fifth only one degree (~12 cents) narrow, still plenty usable. This is a great tuning, however I went with something even smaller, and snipped the chain at 41 tones, leaving the remaining fifth one degree wide. This meant I got all of the wonderful pure sounding intervals I wanted in close keys, and a slight variation (one degree) in size as I modulated into distant keys. For example, middle thirds go between ~9:11 (when mapped as AA2) and ~13:16 (when mapped as ddd5), and this also means that some keys have the large minor second usually ~15:16 (when mapped as A1) much closer to 13:14 (when mapped as ddd4), which can be fun. It’s a little bit like a well-temperament in that it circulates, and you know what kind of intervals to expect from the same mappings, but where the circle closes you get a nice little colour shift. Great if that’s what you want, not so great if you want a nice classic 5-limit JI major sound in the key of \Db major, and you end up with ~370c thirds, a 485c fourth, 868c sixths and 1072c sevenths.
Here’s the 41-tone tuning (from C):
1/1 25.5324 63.8291 89.3615 114.8939 140.4263 178.7230 204.2554 229.7878 268.0845 293.6169 319.1493 344.6817 382.9784 408.5108 434.0432 472.3399 497.8723 523.4047 548.9371 587.2338 612.7662 638.2986 676.5953 702.1277 727.6601 765.9568 791.4892 817.0216 842.5540 880.8507 906.3831 931.9155 970.2122 995.7446 1021.2770 1046.8094 1085.1061 1110.6385 1136.1709 1174.4676 2/1
Here’s a picture of my keyboard (only a two octave range, but all of the pitch classes show up at least once, and some of them wrap around nicely, as they would in 41edo). You can fill in the rest of the pitch names and cent values if you want. I had enough Paint’ing. Ah, oops. Just realised I should have also labeled C/ 26 as B#. Plus I broke my own rule and put the comma accidentals after the note names… Oh well.
The 714 cent “wolf” fifth is between //C and \Ab, all other fifths are 702 cents.
Before I get into too many nitty gritty numbers, here’s some music I made using the above keyboard. All improvised and unedited, just an axis running through Pianoteq:
https://archive.org/details/4191edoPlayingWithThe13Limit https://archive.org/details/4194EDOBosanquetAxis18thAug20181FeelingSadButWarmingUp https://archive.org/details/41-94edo09sept2017
There are only two sizes for each of the 41 interval classes, and as this scale is a constant structure, the mapping remains very consistent, but depending on the key one has a low or high variant tuning of the interval, as explained earlier. But in order to show the interval sizes available in all 41 keys to give you an idea of how the tuning works as a modulating system, here’s a table of the intervals available from each pitch (thanks Scala!). Unfortunately 41 intervals do not fit on one or even two lines, so this may be tricky to read (spoiler alert: cent value overload)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 1/1 : 25.5 63.8 89.4 114.9 140.4 178.7 204.3 229.8 268.1 293.6 319.1 344.7 383.0 408.5 434.0 472.3 497.9 523.4 548.9 587.2 612.8 638.3 676.6 702.1 727.7 766.0 791.5 817.0 842.6 880.9 906.4 931.9 970.2 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1174.5 2/1 25.5 : 38.3 63.8 89.4 114.9 153.2 178.7 204.3 242.6 268.1 293.6 319.1 357.4 383.0 408.5 446.8 472.3 497.9 523.4 561.7 587.2 612.8 651.1 676.6 702.1 740.4 766.0 791.5 817.0 855.3 880.9 906.4 944.7 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 63.8 : 25.5 51.1 76.6 114.9 140.4 166.0 204.3 229.8 255.3 280.9 319.1 344.7 370.2 408.5 434.0 459.6 485.1 523.4 548.9 574.5 612.8 638.3 663.8 702.1 727.7 753.2 778.7 817.0 842.6 868.1 906.4 931.9 957.4 983.0 1021.3 1046.8 1072.3 1110.6 1136.2 1161.7 1200.0 89.4 : 25.5 51.1 89.4 114.9 140.4 178.7 204.3 229.8 255.3 293.6 319.1 344.7 383.0 408.5 434.0 459.6 497.9 523.4 548.9 587.2 612.8 638.3 676.6 702.1 727.7 753.2 791.5 817.0 842.6 880.9 906.4 931.9 957.4 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1174.5 1200.0 114.9 : 25.5 63.8 89.4 114.9 153.2 178.7 204.3 229.8 268.1 293.6 319.1 357.4 383.0 408.5 434.0 472.3 497.9 523.4 561.7 587.2 612.8 651.1 676.6 702.1 727.7 766.0 791.5 817.0 855.3 880.9 906.4 931.9 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 140.4 : 38.3 63.8 89.4 127.7 153.2 178.7 204.3 242.6 268.1 293.6 331.9 357.4 383.0 408.5 446.8 472.3 497.9 536.2 561.7 587.2 625.5 651.1 676.6 702.1 740.4 766.0 791.5 829.8 855.3 880.9 906.4 944.7 970.2 995.7 1034.0 1059.6 1085.1 1123.4 1148.9 1174.5 1200.0 178.7 : 25.5 51.1 89.4 114.9 140.4 166.0 204.3 229.8 255.3 293.6 319.1 344.7 370.2 408.5 434.0 459.6 497.9 523.4 548.9 587.2 612.8 638.3 663.8 702.1 727.7 753.2 791.5 817.0 842.6 868.1 906.4 931.9 957.4 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1161.7 1200.0 204.3 : 25.5 63.8 89.4 114.9 140.4 178.7 204.3 229.8 268.1 293.6 319.1 344.7 383.0 408.5 434.0 472.3 497.9 523.4 561.7 587.2 612.8 638.3 676.6 702.1 727.7 766.0 791.5 817.0 842.6 880.9 906.4 931.9 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1136.2 1174.5 1200.0 229.8 : 38.3 63.8 89.4 114.9 153.2 178.7 204.3 242.6 268.1 293.6 319.1 357.4 383.0 408.5 446.8 472.3 497.9 536.2 561.7 587.2 612.8 651.1 676.6 702.1 740.4 766.0 791.5 817.0 855.3 880.9 906.4 944.7 970.2 995.7 1034.0 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 268.1 : 25.5 51.1 76.6 114.9 140.4 166.0 204.3 229.8 255.3 280.9 319.1 344.7 370.2 408.5 434.0 459.6 497.9 523.4 548.9 574.5 612.8 638.3 663.8 702.1 727.7 753.2 778.7 817.0 842.6 868.1 906.4 931.9 957.4 995.7 1021.3 1046.8 1072.3 1110.6 1136.2 1161.7 1200.0 293.6 : 25.5 51.1 89.4 114.9 140.4 178.7 204.3 229.8 255.3 293.6 319.1 344.7 383.0 408.5 434.0 472.3 497.9 523.4 548.9 587.2 612.8 638.3 676.6 702.1 727.7 753.2 791.5 817.0 842.6 880.9 906.4 931.9 970.2 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1174.5 1200.0 319.1 : 25.5 63.8 89.4 114.9 153.2 178.7 204.3 229.8 268.1 293.6 319.1 357.4 383.0 408.5 446.8 472.3 497.9 523.4 561.7 587.2 612.8 651.1 676.6 702.1 727.7 766.0 791.5 817.0 855.3 880.9 906.4 944.7 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 344.7 : 38.3 63.8 89.4 127.7 153.2 178.7 204.3 242.6 268.1 293.6 331.9 357.4 383.0 421.3 446.8 472.3 497.9 536.2 561.7 587.2 625.5 651.1 676.6 702.1 740.4 766.0 791.5 829.8 855.3 880.9 919.1 944.7 970.2 995.7 1034.0 1059.6 1085.1 1123.4 1148.9 1174.5 1200.0 383.0 : 25.5 51.1 89.4 114.9 140.4 166.0 204.3 229.8 255.3 293.6 319.1 344.7 383.0 408.5 434.0 459.6 497.9 523.4 548.9 587.2 612.8 638.3 663.8 702.1 727.7 753.2 791.5 817.0 842.6 880.9 906.4 931.9 957.4 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1161.7 1200.0 408.5 : 25.5 63.8 89.4 114.9 140.4 178.7 204.3 229.8 268.1 293.6 319.1 357.4 383.0 408.5 434.0 472.3 497.9 523.4 561.7 587.2 612.8 638.3 676.6 702.1 727.7 766.0 791.5 817.0 855.3 880.9 906.4 931.9 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1136.2 1174.5 1200.0 434.0 : 38.3 63.8 89.4 114.9 153.2 178.7 204.3 242.6 268.1 293.6 331.9 357.4 383.0 408.5 446.8 472.3 497.9 536.2 561.7 587.2 612.8 651.1 676.6 702.1 740.4 766.0 791.5 829.8 855.3 880.9 906.4 944.7 970.2 995.7 1034.0 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 472.3 : 25.5 51.1 76.6 114.9 140.4 166.0 204.3 229.8 255.3 293.6 319.1 344.7 370.2 408.5 434.0 459.6 497.9 523.4 548.9 574.5 612.8 638.3 663.8 702.1 727.7 753.2 791.5 817.0 842.6 868.1 906.4 931.9 957.4 995.7 1021.3 1046.8 1072.3 1110.6 1136.2 1161.7 1200.0 497.9 : 25.5 51.1 89.4 114.9 140.4 178.7 204.3 229.8 268.1 293.6 319.1 344.7 383.0 408.5 434.0 472.3 497.9 523.4 548.9 587.2 612.8 638.3 676.6 702.1 727.7 766.0 791.5 817.0 842.6 880.9 906.4 931.9 970.2 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1174.5 1200.0 523.4 : 25.5 63.8 89.4 114.9 153.2 178.7 204.3 242.6 268.1 293.6 319.1 357.4 383.0 408.5 446.8 472.3 497.9 523.4 561.7 587.2 612.8 651.1 676.6 702.1 740.4 766.0 791.5 817.0 855.3 880.9 906.4 944.7 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 548.9 : 38.3 63.8 89.4 127.7 153.2 178.7 217.0 242.6 268.1 293.6 331.9 357.4 383.0 421.3 446.8 472.3 497.9 536.2 561.7 587.2 625.5 651.1 676.6 714.9 740.4 766.0 791.5 829.8 855.3 880.9 919.1 944.7 970.2 995.7 1034.0 1059.6 1085.1 1123.4 1148.9 1174.5 1200.0 587.2 : 25.5 51.1 89.4 114.9 140.4 178.7 204.3 229.8 255.3 293.6 319.1 344.7 383.0 408.5 434.0 459.6 497.9 523.4 548.9 587.2 612.8 638.3 676.6 702.1 727.7 753.2 791.5 817.0 842.6 880.9 906.4 931.9 957.4 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1161.7 1200.0 612.8 : 25.5 63.8 89.4 114.9 153.2 178.7 204.3 229.8 268.1 293.6 319.1 357.4 383.0 408.5 434.0 472.3 497.9 523.4 561.7 587.2 612.8 651.1 676.6 702.1 727.7 766.0 791.5 817.0 855.3 880.9 906.4 931.9 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1136.2 1174.5 1200.0 638.3 : 38.3 63.8 89.4 127.7 153.2 178.7 204.3 242.6 268.1 293.6 331.9 357.4 383.0 408.5 446.8 472.3 497.9 536.2 561.7 587.2 625.5 651.1 676.6 702.1 740.4 766.0 791.5 829.8 855.3 880.9 906.4 944.7 970.2 995.7 1034.0 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 676.6 : 25.5 51.1 89.4 114.9 140.4 166.0 204.3 229.8 255.3 293.6 319.1 344.7 370.2 408.5 434.0 459.6 497.9 523.4 548.9 587.2 612.8 638.3 663.8 702.1 727.7 753.2 791.5 817.0 842.6 868.1 906.4 931.9 957.4 995.7 1021.3 1046.8 1072.3 1110.6 1136.2 1161.7 1200.0 702.1 : 25.5 63.8 89.4 114.9 140.4 178.7 204.3 229.8 268.1 293.6 319.1 344.7 383.0 408.5 434.0 472.3 497.9 523.4 561.7 587.2 612.8 638.3 676.6 702.1 727.7 766.0 791.5 817.0 842.6 880.9 906.4 931.9 970.2 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1174.5 1200.0 727.7 : 38.3 63.8 89.4 114.9 153.2 178.7 204.3 242.6 268.1 293.6 319.1 357.4 383.0 408.5 446.8 472.3 497.9 536.2 561.7 587.2 612.8 651.1 676.6 702.1 740.4 766.0 791.5 817.0 855.3 880.9 906.4 944.7 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 766.0 : 25.5 51.1 76.6 114.9 140.4 166.0 204.3 229.8 255.3 280.9 319.1 344.7 370.2 408.5 434.0 459.6 497.9 523.4 548.9 574.5 612.8 638.3 663.8 702.1 727.7 753.2 778.7 817.0 842.6 868.1 906.4 931.9 957.4 983.0 1021.3 1046.8 1072.3 1110.6 1136.2 1161.7 1200.0 791.5 : 25.5 51.1 89.4 114.9 140.4 178.7 204.3 229.8 255.3 293.6 319.1 344.7 383.0 408.5 434.0 472.3 497.9 523.4 548.9 587.2 612.8 638.3 676.6 702.1 727.7 753.2 791.5 817.0 842.6 880.9 906.4 931.9 957.4 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1174.5 1200.0 817.0 : 25.5 63.8 89.4 114.9 153.2 178.7 204.3 229.8 268.1 293.6 319.1 357.4 383.0 408.5 446.8 472.3 497.9 523.4 561.7 587.2 612.8 651.1 676.6 702.1 727.7 766.0 791.5 817.0 855.3 880.9 906.4 931.9 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 842.6 : 38.3 63.8 89.4 127.7 153.2 178.7 204.3 242.6 268.1 293.6 331.9 357.4 383.0 421.3 446.8 472.3 497.9 536.2 561.7 587.2 625.5 651.1 676.6 702.1 740.4 766.0 791.5 829.8 855.3 880.9 906.4 944.7 970.2 995.7 1034.0 1059.6 1085.1 1123.4 1148.9 1174.5 1200.0 880.9 : 25.5 51.1 89.4 114.9 140.4 166.0 204.3 229.8 255.3 293.6 319.1 344.7 383.0 408.5 434.0 459.6 497.9 523.4 548.9 587.2 612.8 638.3 663.8 702.1 727.7 753.2 791.5 817.0 842.6 868.1 906.4 931.9 957.4 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1161.7 1200.0 906.4 : 25.5 63.8 89.4 114.9 140.4 178.7 204.3 229.8 268.1 293.6 319.1 357.4 383.0 408.5 434.0 472.3 497.9 523.4 561.7 587.2 612.8 638.3 676.6 702.1 727.7 766.0 791.5 817.0 842.6 880.9 906.4 931.9 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1136.2 1174.5 1200.0 931.9 : 38.3 63.8 89.4 114.9 153.2 178.7 204.3 242.6 268.1 293.6 331.9 357.4 383.0 408.5 446.8 472.3 497.9 536.2 561.7 587.2 612.8 651.1 676.6 702.1 740.4 766.0 791.5 817.0 855.3 880.9 906.4 944.7 970.2 995.7 1034.0 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 970.2 : 25.5 51.1 76.6 114.9 140.4 166.0 204.3 229.8 255.3 293.6 319.1 344.7 370.2 408.5 434.0 459.6 497.9 523.4 548.9 574.5 612.8 638.3 663.8 702.1 727.7 753.2 778.7 817.0 842.6 868.1 906.4 931.9 957.4 995.7 1021.3 1046.8 1072.3 1110.6 1136.2 1161.7 1200.0 995.7 : 25.5 51.1 89.4 114.9 140.4 178.7 204.3 229.8 268.1 293.6 319.1 344.7 383.0 408.5 434.0 472.3 497.9 523.4 548.9 587.2 612.8 638.3 676.6 702.1 727.7 753.2 791.5 817.0 842.6 880.9 906.4 931.9 970.2 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1174.5 1200.0 1021.3: 25.5 63.8 89.4 114.9 153.2 178.7 204.3 242.6 268.1 293.6 319.1 357.4 383.0 408.5 446.8 472.3 497.9 523.4 561.7 587.2 612.8 651.1 676.6 702.1 727.7 766.0 791.5 817.0 855.3 880.9 906.4 944.7 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 1046.8: 38.3 63.8 89.4 127.7 153.2 178.7 217.0 242.6 268.1 293.6 331.9 357.4 383.0 421.3 446.8 472.3 497.9 536.2 561.7 587.2 625.5 651.1 676.6 702.1 740.4 766.0 791.5 829.8 855.3 880.9 919.1 944.7 970.2 995.7 1034.0 1059.6 1085.1 1123.4 1148.9 1174.5 1200.0 1085.1: 25.5 51.1 89.4 114.9 140.4 178.7 204.3 229.8 255.3 293.6 319.1 344.7 383.0 408.5 434.0 459.6 497.9 523.4 548.9 587.2 612.8 638.3 663.8 702.1 727.7 753.2 791.5 817.0 842.6 880.9 906.4 931.9 957.4 995.7 1021.3 1046.8 1085.1 1110.6 1136.2 1161.7 1200.0 1110.6: 25.5 63.8 89.4 114.9 153.2 178.7 204.3 229.8 268.1 293.6 319.1 357.4 383.0 408.5 434.0 472.3 497.9 523.4 561.7 587.2 612.8 638.3 676.6 702.1 727.7 766.0 791.5 817.0 855.3 880.9 906.4 931.9 970.2 995.7 1021.3 1059.6 1085.1 1110.6 1136.2 1174.5 1200.0 1136.2: 38.3 63.8 89.4 127.7 153.2 178.7 204.3 242.6 268.1 293.6 331.9 357.4 383.0 408.5 446.8 472.3 497.9 536.2 561.7 587.2 612.8 651.1 676.6 702.1 740.4 766.0 791.5 829.8 855.3 880.9 906.4 944.7 970.2 995.7 1034.0 1059.6 1085.1 1110.6 1148.9 1174.5 1200.0 1174.5: 25.5 51.1 89.4 114.9 140.4 166.0 204.3 229.8 255.3 293.6 319.1 344.7 370.2 408.5 434.0 459.6 497.9 523.4 548.9 574.5 612.8 638.3 663.8 702.1 727.7 753.2 791.5 817.0 842.6 868.1 906.4 931.9 957.4 995.7 1021.3 1046.8 1072.3 1110.6 1136.2 1161.7 1200.0 2/1
This is kind of crazy. The 23-odd limit diamond has 118 distinct ratios in it (more even than in 94edo), and yet 94edo consistently approximates the whole thing very well.
From Scala:
Step size is 12.76596 cents 1: 1/1: 0: 0.0000 cents diff. 0.000000 steps, 0.00000 cents 2: 24/23: 6: 76.5957 cents diff. 0.228348 steps, 2.91509 cents 3: 23/22: 6: 76.5957 cents diff. -0.028251 steps, -0.36066 cents 4: 22/21: 6: 76.5957 cents diff. -0.308734 steps, -3.94129 cents 5: 21/20: 7: 89.3617 cents diff. 0.383403 steps, 4.89451 cents 6: 20/19: 7: 89.3617 cents diff. 0.043945 steps, 0.56100 cents 7: 19/18: 7: 89.3617 cents diff. -0.332236 steps, -4.24131 cents 8: 18/17: 8: 102.1277 cents diff. 0.248556 steps, 3.17307 cents 9: 17/16: 8: 102.1277 cents diff. -0.221507 steps, -2.82775 cents 10: 16/15: 9: 114.8936 cents diff. 0.247716 steps, 3.16233 cents 11: 15/14: 9: 114.8936 cents diff. -0.356353 steps, -4.54919 cents 12: 14/13: 10: 127.6596 cents diff. -0.050029 steps, -0.63867 cents 13: 13/12: 11: 140.4255 cents diff. 0.145141 steps, 1.85287 cents 14: 12/11: 12: 153.1915 cents diff. 0.200097 steps, 2.55443 cents 15: 23/21: 12: 153.1915 cents diff. -0.336986 steps, -4.30195 cents 16: 11/10: 13: 165.9574 cents diff. 0.074668 steps, 0.95322 cents 17: 21/19: 14: 178.7234 cents diff. 0.427348 steps, 5.45551 cents 18: 10/9: 14: 178.7234 cents diff. -0.288290 steps, -3.68031 cents 19: 19/17: 15: 191.4894 cents diff. -0.083679 steps, -1.06824 cents 20: 9/8: 16: 204.2553 cents diff. 0.027049 steps, 0.34532 cents 21: 26/23: 17: 217.0213 cents diff. 0.373490 steps, 4.76796 cents 22: 17/15: 17: 217.0213 cents diff. 0.026208 steps, 0.33458 cents 23: 8/7: 18: 229.7872 cents diff. -0.108637 steps, -1.38686 cents 24: 23/20: 19: 242.5532 cents diff. 0.046417 steps, 0.59256 cents 25: 15/13: 19: 242.5532 cents diff. -0.406382 steps, -5.18786 cents 26: 22/19: 20: 255.3191 cents diff. 0.118614 steps, 1.51422 cents 27: 7/6: 21: 268.0851 cents diff. 0.095112 steps, 1.21420 cents 28: 20/17: 22: 280.8511 cents diff. -0.039733 steps, -0.50724 cents 29: 13/11: 23: 293.6170 cents diff. 0.345238 steps, 4.40730 cents 30: 19/16: 23: 293.6170 cents diff. -0.305186 steps, -3.89599 cents 31: 6/5: 25: 319.1489 cents diff. 0.274765 steps, 3.50765 cents 32: 23/19: 26: 331.9149 cents diff. 0.090362 steps, 1.15356 cents 33: 17/14: 26: 331.9149 cents diff. -0.330144 steps, -4.21461 cents 34: 28/23: 27: 344.6809 cents diff. 0.323461 steps, 4.12929 cents 35: 11/9: 27: 344.6809 cents diff. -0.213622 steps, -2.72709 cents 36: 16/13: 28: 357.4468 cents diff. -0.158666 steps, -2.02553 cents 37: 21/17: 29: 370.2128 cents diff. 0.343669 steps, 4.38727 cents 38: 26/21: 29: 370.2128 cents diff. 0.036504 steps, 0.46601 cents 39: 5/4: 30: 382.9787 cents diff. -0.261240 steps, -3.33499 cents 40: 24/19: 32: 408.5106 cents diff. 0.318711 steps, 4.06865 cents 41: 19/15: 32: 408.5106 cents diff. -0.057470 steps, -0.73366 cents 42: 14/11: 33: 421.2766 cents diff. 0.295209 steps, 3.76863 cents 43: 23/18: 33: 421.2766 cents diff. -0.241873 steps, -3.08775 cents 44: 9/7: 34: 434.0426 cents diff. -0.081587 steps, -1.04154 cents 45: 22/17: 35: 446.8085 cents diff. 0.034934 steps, 0.44598 cents 46: 13/10: 36: 459.5745 cents diff. 0.419907 steps, 5.36052 cents 47: 30/23: 36: 459.5745 cents diff. -0.032892 steps, -0.41990 cents 48: 17/13: 36: 459.5745 cents diff. -0.380173 steps, -4.85328 cents 49: 21/16: 37: 472.3404 cents diff. 0.122162 steps, 1.55952 cents 50: 4/3: 39: 497.8723 cents diff. -0.013524 steps, -0.17266 cents 51: 23/17: 41: 523.4043 cents diff. 0.006683 steps, 0.08532 cents 52: 19/14: 41: 523.4043 cents diff. -0.413823 steps, -5.28285 cents 53: 15/11: 42: 536.1702 cents diff. -0.061143 steps, -0.78056 cents 54: 26/19: 43: 548.9362 cents diff. 0.463852 steps, 5.92152 cents 55: 11/8: 43: 548.9362 cents diff. -0.186572 steps, -2.38177 cents 56: 18/13: 44: 561.7021 cents diff. -0.131616 steps, -1.68021 cents 57: 32/23: 45: 574.4681 cents diff. 0.214823 steps, 2.74243 cents 58: 7/5: 46: 587.2340 cents diff. 0.369878 steps, 4.72185 cents 59: 24/17: 47: 600.0000 cents diff. 0.235032 steps, 3.00041 cents 60: 17/12: 47: 600.0000 cents diff. -0.235032 steps, -3.00041 cents 61: 10/7: 48: 612.7660 cents diff. -0.369878 steps, -4.72185 cents 62: 23/16: 49: 625.5319 cents diff. -0.214823 steps, -2.74243 cents 63: 13/9: 50: 638.2979 cents diff. 0.131616 steps, 1.68021 cents 64: 16/11: 51: 651.0638 cents diff. 0.186572 steps, 2.38177 cents 65: 19/13: 51: 651.0638 cents diff. -0.463852 steps, -5.92152 cents 66: 22/15: 52: 663.8298 cents diff. 0.061143 steps, 0.78056 cents 67: 28/19: 53: 676.5957 cents diff. 0.413823 steps, 5.28285 cents 68: 34/23: 53: 676.5957 cents diff. -0.006683 steps, -0.08532 cents 69: 3/2: 55: 702.1277 cents diff. 0.013524 steps, 0.17266 cents 70: 32/21: 57: 727.6596 cents diff. -0.122162 steps, -1.55952 cents 71: 26/17: 58: 740.4255 cents diff. 0.380173 steps, 4.85328 cents 72: 23/15: 58: 740.4255 cents diff. 0.032892 steps, 0.41990 cents 73: 20/13: 58: 740.4255 cents diff. -0.419907 steps, -5.36052 cents 74: 17/11: 59: 753.1915 cents diff. -0.034934 steps, -0.44598 cents 75: 14/9: 60: 765.9574 cents diff. 0.081587 steps, 1.04154 cents 76: 36/23: 61: 778.7234 cents diff. 0.241873 steps, 3.08775 cents 77: 11/7: 61: 778.7234 cents diff. -0.295209 steps, -3.76863 cents 78: 30/19: 62: 791.4894 cents diff. 0.057470 steps, 0.73366 cents 79: 19/12: 62: 791.4894 cents diff. -0.318711 steps, -4.06865 cents 80: 8/5: 64: 817.0213 cents diff. 0.261240 steps, 3.33499 cents 81: 21/13: 65: 829.7872 cents diff. -0.036504 steps, -0.46601 cents 82: 34/21: 65: 829.7872 cents diff. -0.343669 steps, -4.38727 cents 83: 13/8: 66: 842.5532 cents diff. 0.158666 steps, 2.02553 cents 84: 18/11: 67: 855.3191 cents diff. 0.213622 steps, 2.72709 cents 85: 23/14: 67: 855.3191 cents diff. -0.323461 steps, -4.12929 cents 86: 28/17: 68: 868.0851 cents diff. 0.330144 steps, 4.21461 cents 87: 38/23: 68: 868.0851 cents diff. -0.090362 steps, -1.15356 cents 88: 5/3: 69: 880.8511 cents diff. -0.274765 steps, -3.50765 cents 89: 32/19: 71: 906.3830 cents diff. 0.305186 steps, 3.89599 cents 90: 22/13: 71: 906.3830 cents diff. -0.345238 steps, -4.40730 cents 91: 17/10: 72: 919.1489 cents diff. 0.039733 steps, 0.50724 cents 92: 12/7: 73: 931.9149 cents diff. -0.095112 steps, -1.21420 cents 93: 19/11: 74: 944.6809 cents diff. -0.118614 steps, -1.51422 cents 94: 26/15: 75: 957.4468 cents diff. 0.406382 steps, 5.18786 cents 95: 40/23: 75: 957.4468 cents diff. -0.046417 steps, -0.59256 cents 96: 7/4: 76: 970.2128 cents diff. 0.108637 steps, 1.38686 cents 97: 30/17: 77: 982.9787 cents diff. -0.026208 steps, -0.33458 cents 98: 23/13: 77: 982.9787 cents diff. -0.373490 steps, -4.76796 cents 99: 16/9: 78: 995.7447 cents diff. -0.027049 steps, -0.34532 cents 100: 34/19: 79: 1008.5106 cents diff. 0.083679 steps, 1.06824 cents 101: 9/5: 80: 1021.2766 cents diff. 0.288290 steps, 3.68031 cents 102: 38/21: 80: 1021.2766 cents diff. -0.427348 steps, -5.45551 cents 103: 20/11: 81: 1034.0426 cents diff. -0.074668 steps, -0.95322 cents 104: 42/23: 82: 1046.8085 cents diff. 0.336986 steps, 4.30195 cents 105: 11/6: 82: 1046.8085 cents diff. -0.200097 steps, -2.55443 cents 106: 24/13: 83: 1059.5745 cents diff. -0.145141 steps, -1.85287 cents 107: 13/7: 84: 1072.3404 cents diff. 0.050029 steps, 0.63867 cents 108: 28/15: 85: 1085.1064 cents diff. 0.356353 steps, 4.54919 cents 109: 15/8: 85: 1085.1064 cents diff. -0.247716 steps, -3.16233 cents 110: 32/17: 86: 1097.8723 cents diff. 0.221507 steps, 2.82775 cents 111: 17/9: 86: 1097.8723 cents diff. -0.248556 steps, -3.17307 cents 112: 36/19: 87: 1110.6383 cents diff. 0.332236 steps, 4.24131 cents 113: 19/10: 87: 1110.6383 cents diff. -0.043945 steps, -0.56100 cents 114: 40/21: 87: 1110.6383 cents diff. -0.383403 steps, -4.89451 cents 115: 21/11: 88: 1123.4043 cents diff. 0.308734 steps, 3.94129 cents 116: 44/23: 88: 1123.4043 cents diff. 0.028251 steps, 0.36066 cents 117: 23/12: 88: 1123.4043 cents diff. -0.228348 steps, -2.91509 cents 118: 2/1: 94: 1200.0000 cents diff. 0.000000 steps, 0.00000 cents Total absolute difference : 24.02335 steps, 306.68109 cents Average absolute difference: 0.203587 steps, 2.59899 cents Root mean square difference: 0.245330 steps, 3.13188 cents Highest absolute difference: 0.463852 steps, 5.92152 cents
Even taking it up a couple of notches to the full 27-odd limit diamond (155 intervals) and we still get
Total absolute difference : 37.84087 steps, 483.07495 cents Average absolute difference: 0.244134 steps, 3.11661 cents Root mean square difference: 0.295555 steps, 3.77305 cents Highest absolute difference: 0.681148 steps, 8.69551 cents
So consistency breaks for some intervals, but the approximations are still pretty fine.
Looking into more common sets of intervals, here’s the data on the 15-odd limit (49 intervals, probably one of the best tests for modern JI usage)
Total absolute difference : 9.624905 steps, 122.87114 cents Average absolute difference: 0.196426 steps, 2.50757 cents Root mean square difference: 0.232476 steps, 2.96778 cents Highest absolute difference: 0.419907 steps, 5.36052 cents
And the 13-odd limit (41 intervals)
Total absolute difference : 7.481714 steps, 95.51125 cents Average absolute difference: 0.182480 steps, 2.32954 cents Root mean square difference: 0.217173 steps, 2.77243 cents Highest absolute difference: 0.419907 steps, 5.36052 cents
11-odd (29 intervals)
Total absolute difference : 4.980514 steps, 63.58104 cents Average absolute difference: 0.171741 steps, 2.19245 cents Root mean square difference: 0.204313 steps, 2.60826 cents Highest absolute difference: 0.369878 steps, 4.72185 cents
9-odd (19 intervals)
Total absolute difference : 3.040175 steps, 38.81075 cents Average absolute difference: 0.160009 steps, 2.04267 cents Root mean square difference: 0.203155 steps, 2.59347 cents Highest absolute difference: 0.369878 steps, 4.72185 cents
Thanks for reading (and/or scrolling)