94edo, how to deal with a big schismatic chain of fifths

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):

Absolute notation Relative notation Classic ratios (5-prime limit) Septimal ratios (7-prime) 11-prime and above
//E //M2     15/13
//A //P5     20/13
//D //P1     33/32
//G //P4     11/8
/B#=//C /A6=//m7     11/6, 117/64
/E#=//F /A2=//m3     11/9, 39/32
/A#=//Bb /A5=//m6 81/50*   44/27, 13/8
/D#=//Eb /A1=//m2 27/25*   13/12
/G#=//Ab /A4=//d5 48/25*   13/9
/C#=//Db /M7=//d8 36/25* 27/14 52/27
/F#=//Gb /M3=//d4 32/25* 9/7  
/B /M6   12/7  
/E /M2   8/7  
/A /P5   32/21  
/D /P1 81/80 64/63  
/G /P4 27/20    
B#=/C A6=/m7 9/5    
E#=/F A2=/m3 6/5    
A#=/Bb A5=/m6 8/5    
D#=/Eb A1=/m2 16/15 15/14  
G#=/Ab A4=/d5 64/45 10/7  
C#=/Db M7=/d8 243/128   19/10, 36/19
F#=/Gb M3=/d4 81/64   19/15, 24/19
B M6 27/16   22/13, 32/19
E M2 9/8    
A P5 3/2    
D P1 1/1    
G P4 4/3    
C m7 16/9    
F m3 32/27   19/16, 13/11
Bb m6 128/81   19/12, 30/19
Eb m2 256/243   20/19, 19/18
\G#=Ab \A4=d5 45/32 7/5  
\C#=Db \M7=d8 15/8    
\F#=Gb \M3=d4 5/4    
\B=Cb \M6=d7 5/3    
\E=Fb \M2=d3 10/9    
\A \P5 40/27    
\D \P8 160/81 63/32  
\G \P4   21/16  
\C \m7   7/4  
\F \m3   7/6  
\\A#=\Bb \\A5=\m6 25/16* 14/9  
\\D#=\Eb \\A1=\m2 25/24* 28/27 27/26
\\G#=\Ab \\A4=\d5 25/18*   18/13
\\C#=\Db \\M7=\d8 50/27*   24/13
\\F#=\Gb \\M3=\d4 100/81*   27/22, 16/13
\\B=\Cb       18/11, 64/39
\\E=\Fb       12/11, 128/117
\\A       16/11
\\D       39/20
\\G       13/10
\\C       26/15

* = maps to these ratios using the patent val, but not actually the closest approximation to these ratios in 94edo, hence consistency breaking above the 23-odd limit. Perfectly valid for 41 and 53edos, and I still use them in 94 even though they’re not the closest approximation to JI, as their component parts are tuned very close and breaking the chain would stand out more than a little beating on an already rather dissonant interval.

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.

41_94edo_axiskeyboard_names_cents_for_blog

 

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)

a brief introduction to tuning, numbers, history

so i’ve decided to write a blog about something i’m passionate about. it’s not so obvious what i should post here, who my intended audience is and all that so i guess i’ll start with some basics.

intonation in music had been a defining feature for millenia, before music could play played one had to ask, with what pitches? before one can think about melody, or harmony, or voice leading, or voicing, one must first decide on pitch.

it seems in our modern world pitch is given too little attention, with the most obvious step being selecting one from twelve, the twelve notes found on almost all instruments and in most musicians’ heads as the only notes, the ‘right’ notes, handed down through the ages from the hands of the musical gods. this is rubbish.

those twelve notes, twelve relationships, that have come to define so much about modern music, are an invention, one simple cold calculated solution to the problem of which notes to choose, and where to stop, in defining a musical universe that is practical to play around in. twelve equal divisions of the octave: 12EDO.

all in all it’s a rather good system for its simplicity, with twelve equally spaced pitches we get only twelve unique interval classes: a minor second, a major second, a minor third, a major third, a very nearly perfect fourth, an augmented fourth, a very nearly perfect fifth, a minor sixth, major sixth, minor seventh, major seventh, and perfect octave. we are taught in modern theory that these twelve intervals are enharmonically equivalent to other intervals, that for example an augmented fourth is ‘the same’ as a diminished fifth, because they share the same key on the instrument, they are the same pitch even though we might call them different names or give them a different symbol on the page. that we should try to spell according to the classical rules of harmony and voice leading, but that in practice it’s all the same, a diminished fourth IS a major third, and an augmented unison is a minor second, and so on and so forth…

but where did these interval classes come from? how long have we had this conflation of perhaps twenty-one different categories (or more) being crammed into a grid of twelve notes?

there was a time only a few hundred years ago when a G# could never be an Ab, where no two notes were ‘equivalent’ in the modern sense.  if you wanted an augmented fourth and not a diminished fifth, that’s exactly what you got, not some middle-of-the-road approximation of both.

12EDO is a tuning that optimises the tuning of fifths, their sums, and their octave inversion, fourths. its ancestor is a tuning known as Pythagorean, though Pythagoras was evidently not the first person to come up with it. The tuning uses one ratio as its generator, the ratio 3 to 2, meaning when you go up a 3/2 the higher note’s frequency is 3/2 times that of the lower note. in musical terms, 3/2 is a perfect fifth. Really perfect. beatless. beautiful. every third harmonic of the lower tone lining up with every second of the higher note, it’s a focussed powerful sound, very familiar to most traditional musics, and our modern 12EDO version of 7 ‘semitones’ is pretty darn close, with an error of less than 0.28%. if we take a chain of fifths downwards and upwards from any arbitrary pitch, we end up with a chain like this, each step to the right going up a perfect fifth:

32/243-16/81-8/27-4/9-2/3-1/1-3/2-9/4-27/8-81/16-243/32-729/64

or in musical notation, taking the starting frequency 1/1 as C, we have:

Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#

although Db is now like six octaves below F#, so let’s bring the whole scale within one octave [doubling a frequency brings it up an octave, halving it brings it down] so that each ratio is between 1 (the initial frequency) and 2 (the octave above)

256/243-128/81-32/27-16/9-4/3-1/1-3/2-9/8-27/16-81/64-243/128-729/512, rearranging from smallest to largest (i.e. lowest to highest in pitch):

1/1-256/243-9/8-32/27-81/64-4/3-729/512-3/2-128/81-27/16-16/9-243/128

C-Db-D-Eb-E-F#-G-Ab-A-Bb-B.

Look familiar? twelve tone chromatic scale. the original, formalised actually well before Pythagoras in Ancient China (read the intro to Harry Partch’s Genesis of a Music for a more detailed and possibly humorous run-down on the history of tuning).

here though, the steps are Not equal, the augmented unison steps are larger than the minor seconds. and we can’t change the spelling willy-nilly without changing the ratios. F# means 6 fifths up from C, or 729/512, whereas Gb would be 6 fifths down, or 1024/729. different notes, Gb being less than a quarter of a modern semitone lower than its enharmonic partner. if we want to play F# with a fifth above it, and we’ve only got the twelve keys above, we have to use Db a the top, which at almost a quarter of a semitone flatter than a perfect fifth, doesn’t sound too pretty.

if we continue this chain we run into bundles of these close pairs, which means we either need an instrument with 53 keys (where the notes almost exatcly line up – see http://en.wikipedia.org/wiki/53_equal_temperament), or we need to stick to a set of twelve notes so we don’t wind up with any pairs. the clever thing is, if we narrow each fifth just a tad, the sharps will drop and the flats raise, so that if we set our fifths to the twelfth root of 2, as they are in 12EDO, all of the pairings merge to a single note

the perfect fifth is what gives most modern harmony its power, the reason power chords in rock and pop and fourth harmonisations in jazz are popular today is probably because the 4/3 fourth and 3/2 fifth are very well represented in 12EDO, just as the fifth was given extreme importance prior to the renaissance period. 3/2 is a beautiful stable consonance.

but it is not the only beautiful stable interval. ratios of the magic numbers 2 and 3 reigned supreme throughout a lot of early musical history, but 5 soon came to be seen as warm, emotive, and important in expressing human drama in a way ratios of 3 and 2 couldn’t do so easily. what’s special about 2, 3, and 5? they’re all primes, and though there are differing opinions on their fundamentality, primes are important building blocks for rational harmony.

if you’ve ever heard overtone singing in a tonal context, or tried tuning a fretless instrument playing solo, or heard any good a capella quartets or choirs or small string groups, you probably already know the sound of 5. 5/4 is the major third, the Perfect major third, the one most people will sing or play instinctively because in most contexts it sounds miles better than the 81/64 major third you get by going up four perfect fifths, or even the major third in 12EDO, which people keep telling us is ‘right’ and ‘in tune’…

so how about we substitute all those big ugly numbers out of our old scale with some simpler warmer harmonies? let’s draw up a little lattice with 3/2 fifths going horizontally and 5/4 major thirds vertically:

5/3   5/4  15/8 45/32
A-       E-    B-      F#-
4/3    1/1  3/2    9/8
F         C      G       D
16/15 8/5 6/5  9/5
Db+   Ab+ Eb+ Bb+        or, arranged by pitch:

1/1-16/15-9/8-6/5-5/4-4/3-45/32-3/2-8/5-5/3-9/5-15/8
C       Db+    D    Eb+ E-    F      F#-     G    Ab+  A-   Bb+  B-

still a 12-tone chromatic scale, but less even, with 34 different intervals…! a whole bunch of flavours to try!

i feel like i should stop here, didn’t quite expect to write that long of a first post intro, but hopefully it gives you a little taste of what numbers might mean in music, and perhaps the inkling that 12EDO might not be the ideal musical universe after all