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)

Getting back on the band wagon. Starting a new blog.

Wow, it’s been a long time since I wrote a post on here.

Over the intervening months and years I have thought several times about writing a book, or several books, and have made some small attempts towards that goal, but thinking more and more about the age we’re living in, the way we communicate and consume media leans heavily in favour of those which are internet friendly, bite-sized, and interactive. So I’m deciding to start a blog, which will hopefully document what I would like to do as a music educator. Some of it will be focused on my piano teaching, and what I do day to day (I teach in a few schools at the primary, intermediate and high school level, but more and more of my work is as a private teacher in students’ homes), and some of it will be focused on my xenharmonic/microtonal pursuits – what I have learned about tuning, how it applies to music, how musicians of any level can use it to make interesting music and to connect the dots between music, art, science, creation and analysis.

I’d like to explore both the fundamental elements of music (for beginner and intermediate musicians) and also the more specialised area of tuning, how it affects melody, harmony, timbre, technique, interpretation, style, notation, instrument design, and general musical thinking. It is interesting going down the road of teaching music from a tuning-agnostic, or tuning-variable point of view. If we are fully agnostic, it’s pretty hard to allow the use of any exact interval, but to appeal to the largest number of people and to make most of the huge corpus of existing music available for us to relate to I will be using some structures that most practicing musicians already know, and simply expanding their meanings to include what might happen outside of 12-tone equal temperament (12TET).

I’m not into telling people who have studied for years to “wipe the slate clean”, and so most of what I do microtonally can be understood through the lens of the everyday musician, or at least, I’d like to think so. I also want to make this intuitive to grasp, and interesting enough to spark creative exploration on the part of the reader (you). Feel free to ask any questions. I’m not much of a writer, so if there’s anything I can clarify, let me know. I have read some fantastic books which deal at least a little with tuning and how it can affect other musical elements. I’d like to try and inspire in a reader a fraction of what Helmholtz’ On the Sensations of Tone or Partch’s Genesis of a Music or Mathieu’s Harmonic Experience inspired in me, and try to stay true to my own direction at the same time. This is probably not a how-to guide on tuning of every kind, but hopefully it will be enough of an introduction to ideas which can be interwoven with common practice music theory.

Before I start my first post proper, maybe I should try to answer the question, why bother with microtonality? Many practicing musicians hear words like microtonal or alternative tunings and think something along the lines of I’m doing pretty well with the tuning I’ve got, why would I want anything different? It already sounds fine, and I wouldn’t know what to do with any extra notes

Thinking about tuning is just like thinking about any other element of music, it opens up a whole lot of doors into expression, the sorts of sounds and structures one can use, and I think it deserves as much respect as other variables which are taken for granted – style, form, timbre, texture, rhythm, etc. Not many musicians would submit themselves to an entire career playing music of one rhythm, or of one texture, or using one timbral colour, but so many musicians never have the doors opened to tunings outside 12TET, to the universe of other possibilities. It’s like the 12TET system is a planet, supporting all sorts of landscapes and forms of life, but it is still just one equal temperament (among the infinite, let alone the families of linear and higher dimensional regular temperaments, and all their irregular cousins, and of course, just intonation and its countless possibilities for pitch). Who says other planets out there don’t support life?

So many interesting things can be done only in particular tunings, and some of that has been very well explored in 12TET. Most of these features are not taught to musicians as features of the tuning system, but as musical facts. Here are a couple of them:

  • a chain of 12 perfect fifths is equivalent to 7 perfect octaves, so a “circle” of twelve equally sized perfect fifths can be made, and modulations can be made around the circle. F# and Gb major for example, become the same pitch, and therefore the same key, and the difference between them is lost, except in the notation.
  • a chain of 3 major thirds lands us at a perfect octave. This means we can have symmetrical augmented harmonies and melodies.
  • a chain of 4 minor thirds lands us at a perfect octave. This means we can have symmetrical diminished harmonies and melodies.
  • stacking two identical major seconds lands us on the sweetest major third, close to the simple ratio 4:5.
  • in a dominant seventh chord, the interval between the third and seventh is equivalent to its inversion, the interval between the seventh and third. This allows progressions where tritones hold and the rest of the chord moves around, and allows tritone substitution without modulation.

Following these four harmonic paths in just intonation leads to four distinct pitches, and none of those are the same as where we started (for those interested, the small resultant intervals have names too – respectively, the Pythagorean comma, the diesis, “major” diesisDydimic or syntonic comma, and the jubilisma, though I had to look up that last one!). More importantly, there are tuning systems that share these features, and we can group them in families, e.g. those which make the above intervals into unisons, or “temper out” those intervals. The temperament families referred to above would be pythagorean (12, 24, 36, 48, 60, 72, 84, 96), augmented (3, 6, 9, 12, 15, 18, 21, 24), diminished (4, 8, 12, 16, 20, 24, 28, 32), meantone (12, 19, 26, 31, 33, 43, 45, 50), and pajara (10, 12, 22, 24, 32, 34, 44, 46). I decided to give a few equal temperaments belonging to each family so you get the idea that these features are adopted on some planets, but certainly not all. Laws governing musical life are not the same everywhere in the universe. And in fact any musical universe can be dreamed up, based on any rules, or even a lack of such rules, to give any kind of imaginable structure or chaos.

There is also the (not so small) matter of sound. This is the real point for most microtonalists. Numbers are there for those interested, and I can certainly get carried away with them (as seen above) but music really should be about sounds, and the expressive capabilities gained when tuning is a flexible parameter are not insignificant. One can hear the subtle differences between string quartets playing with near-pure fifths and wide major thirds (often sought after for expressive melodic playing) vs those playing with near-pure thirds (great for music full of rich major and minor harmonies), or how a harpsichord reacts differently in 12TET vs in a good meantone or well-temperament for period repertoire. More often than not one also achieves different kinds of timbres, different kinds of harmonies, different kinds of melodies when playing for extended periods in different tunings.

Sorry, got a bit carried away on this first post. Will try to make the next one a bit more focused on exploring one topic at a time. Might write a bit about the three keyboard layouts I use, the reasons we have them and why they’re important, at least in my musical explorations.

Thanks all for following along. Hopefully back soon.