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.

 

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)

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what music theory do i need to start tuning?

First posted 22/09/2018, edited 7/11/2018

You don’t really need to know anything to start, many have gone into the field with only their ears and instrument and just does what feels the most natural. Sometimes what turns out to be the most natural is similar to what we’ve grown up with, or similar to the simple fractions our ears are so good at identifying, so called “just intonation”. Sometimes a xenharmonic explorer tunes up something totally unexpected and full of the unfamiliar. Some musicians (even some working at a very high level) are averse to theory, but I think it helps to know at least a little bit of what we are doing here, in order to learn from each other, share ideas, and grow all kinds of new music that will enrich future generations.

I thought I’d start with some basics. In order to keep them applicable to microtonal music only a couple of definitions need to be broadened, so the musical alphabet, notes, accidentals, scales and keys is a good place to start.

Most of the world shares musical traditions based around 5-note (pentatonic) and 7-note (heptatonic) scales, and an octave framework. Even in most modern music, these are the two most popular numbers for notes per octave in melodies and harmonies in a huge number of musical styles. You may even ask “why octaves?”, but I will just say for now that humans on average have an amazingly accute understanding of the octave as a reference, and the concept of octave equivalence is pretty well ingrained in us. Some xenharmonic musicians have gone against that, but that is a rabbit hole for another day.

Although pentatonic systems have been huge for melody for thousands of years, most musical cultures have developed a heptatonic system for talking about, counting, singing, and reading, musical notes. As the concept of zero wasn’t huge in many parts until quite a bit later, musicians tended to label the notes in their scale starting with the first one and proceeding through to the last one in each octave, where the scale pattern might be able to be seen repeated up and down the range of the instrument, or where certain scale degrees were flexible to permit subtle changes necessary for particular nuances. Many cultures used some kind of alphabet or syllable system to count notes in each octave. The most popular systems around the world are presently {A, B, C, D, E, F, G, A}, the “letter system” and {Do, Re, Mi, Fa, Sol, La, Ti/Si, Do}, “solfege”, with {Sa, Re, Ga, Ma, Pa, Dha, Ni, Sa}, “sargam” also another notable example particularly across the Indian subcontinent. We could also write {1, 2, 3, 4, 5, 6, 7, 8}, which is the basis for interval naming systems in most languages, where notes a step apart are said to form a 2nd, e.g. A-B or Do-Re or Sa-Re, and distances increase with each scale degree, e.g. C-G, Fa-Do, and Ga-Ni all being 5ths (if you miscount, make sure you’ve counted both endpoints – this is the same system of counting places in a race, but not the same counting system we use to work out, say, a difference in age).

Let’s relate these systems to each other so readers from slightly different backgrounds can still make use of the same logic. The letter system {A, B, C, …} is fixed – that is, the letters are not thought to change pitch depending on the music. However, the other three systems are usually thought of as moveable, so that the first note name lines up with the tonic or home note or pitch centre in the music. Where those systems are tied down to a fixed pitch, e.g. fixed solfege, Do is usually made equivalent to C.

Do    =   Sa     =  1
Re    =   Re/Ri =  2
Mi    =   Ga     =  3
Fa    =   Ma     =  4
Sol   =   Pa      =  5
La    =   Dha   =  6
Ti/Si =   Ni      =  7
[Do   =   Sa      = 8, although considering the octave the start of the next set is also just fine]

The exact pitch of these musical notes can depend on the instrument, technique and style, but even more important than that is the scale or key where they’re located. In most contemporary music which uses the letter system, letters are combined with sharp (♯) and flat (♭) to show these differences, and while some solfege systems use change in vowel quality to reflect these changes, they are generally not reflected in sargam, so I’d like to stick to the letter system as the quickest and arguable most universal way to refer to musical notes (at least in English).

We can make two of the most common musical scales from around the world if we take the interval of a 5th, for example between D and A, and continue to chain 5ths from either side. Taking our chain to five notes we might have C-G-D-A-E, an example of a pentatonic set, which we could rearrange in pitch order starting from any of the notes, i.e. C-D-E-G-A, D-E-G-A-C, E-G-A-C-D, G-A-C-D-E, or A-C-D-E-G. Notice how we get 5 modes out of a 5-note scale. This is a great set of notes in most tunings, as long as there is something like a 5th (otherwise we will end up with rather odd things happening, and the pitch order of our notes won’t be preserved, e.g. A might be higher than C or lower than G). The tuning range for 5ths where this works, i.e. the tuning forms a constant structure is rather large, the 5ths can be as small as half an octave, or as large as two-thirds of an octave. The convention is to write these octave fractions using a backslash, i.e. 1\2 and 2\3, as a regular slash is reserved for the ratios of just intonation, where for example 2/1 would be a perfect octave above the tonic 1/1, and 3/2 would be a perfect fifth above the tonic, with the occasional convention that inverse ratios represent pitches below the tonic.

[I’m probably using way too much jargon. Some of these things are easily searched online, but usually it’s just easier to ask people directly, as the usage isn’t totally standardised, but we are trying to create a new musical language that adequately deals with the intricacies of tuning. You can find me here, on facebook, or you can ask the Xenharmonic Alliance facebook group if you have any questions]

If we continue our chain of fifths out to 7 notes, F-C-G-D-A-E-B, we end up with a heptatonic scale with 7 modes, each starting on a different letter, most commonly called Ionian/Major (C), Dorian (D), Phrygian (E), Lydian (F), Mixolydian/Dominant (G), Aeolian/Natural minor (A) and Locrian (B).

Now where tuning comes into this is, these seven letters need not be equally spaced, and the fifths may not all be the same size. If they are, then we end up with a circle of 7 fifths, and our tuning is 7-tone equal temperament (7TET). Here, taking us up or down 7 fifths returns us to the same place, so there is no need of sharp or flat symbols, and there are only 7 possible intervals within the span of an octave. However, most of the world doesn’t use 7TET.

Let’s expand our chain of fifths from 7 notes to 12. Using our usual method of going down a fifth from F we get B, but we already have a note called B (7 fifths above), so we will call this one “B-flat”, or B♭. This 8 note set was used for a while in medieval music, thought of as 6 fixed scale degrees with the B♭/B flexible depending on the direction of the melody, and what fitted best, and giving rise to a few more options in terms of harmony. Another fifth down from B♭ and we get E♭. Taking our chain further in the other direction, and we encounter a similar problem, a fifth up from B seems to be F, so we call our new F, “F-sharp”, or F♯, and we continue with C♯ and G♯. Now we have 12 notes: E♭-B♭-F-C-G-D-A-E-B-F♯-C♯-G♯. This has been one of the most popular sets of notes for western music involving melody and harmony, in a variety of tunings over the last 650 years or so. It was also popularised as a good layout on keyboards, and the layout on 12-note-per-octave keyboards everywhere is called Halberstadt after an organ in Halberstadt from 1361 using the layout, with the 7 letters as larger keys towards the front, and the 5 accidentals (notes involving a sharp or flat) as smaller raised keys towards the back. Many of us now know the keyboard with each octave laid out as 7 white and 5 black keys.

With 12 notes we have many more than 7 intervals, but instead, even if each of the eleven fifths in the chain is the same size, we will have 23 sizes of interval, each generatable by stacking fifths (our system’s generating interval or generator) along with reducing by octaves (our system’s period). The concept of a chain of pitches generated by any two unique intervals is sometimes called a linear tuning or a rank-2 or 2D tuning, as it takes two dimensions to lay out all the possible pitches of such a sytem, although the octave component is often ignored and the pitches displayed on a line (like I have already been doing in this post). Here are our intervals.

augmented unison (A1) from C, Eb, F, G, Bb ——————– +7 fifths
minor second (m2) from C#, D, E, F#, G#, A, B —————– -5 fifths
major second (M2) from C, D, Eb, E, F, F#, G, A, Bb, B —— +2 fifths
augmented second (A2) from Eb, F, Bb ————————— +9 fifths
diminished third (d3) from C#, G# ———————————- -10 fifths
minor third (m3) from C, C#, D, E, F#, G, G#, A, B ———— -3 fifths
major third (M3) from C, D, Eb, E, F, G, A, Bb —————— +4 fifths
augmented third (A3) from Eb ————————————— +11 fifths
diminished fourth (d4) from C#, F#, G#, B ———————- -8 fifths
perfect fourth (P4), from C, C#, D, E, F, F#, G, G#, A, Bb, B -1 fifth
augmented fourth (A4) from C, D, Eb, F, G, Bb ————— +6 fifths
diminished fifth (d5) from C#, E, F#, G#, A, B —————– -6 fifths
perfect fifth (P5) from C, C#, D, Eb, E, F, F#, G, A, Bb, B — +1 fifth
augmented fifth (A5) from C, Eb, F, G, Bb ———————- +8 fifths
diminished sixth (d6) from G# ————————————– -11 fifths
minor sixth (m6) from C#, D, E, F#, G, G#, A, B ————— -4 fifths
major sixth (M6) from C, D, Eb, E, F, G, A, Bb, B ————- +3 fifths
augmented sixth (A6) from Eb, Bb ——————————– +10 fifths
diminished seventh (d7) from C#, F#, G# ———————– -9 fifths
minor seventh (m7) from C, C#, D, E, F, F#, G, G#, A, B — -2 fifths
major seventh (M7) from C, D, Eb, F, G, A, Bb —————- +5 fifths
diminished octave (d8) from C#, E, F#, G#, B —————— -7 fifths
perfect octave (P8) everywhere ————————————- 0 fifths

Some of these might look unusual to a musician just getting into more advanced harmony, but all of these, and sometimes a few more, pop up all over the place in the corpus of music written down up to the present time, especially those taking less than 7 fifths to generate.

How are our 12 notes and 23 resultant intervals to be tuned?

There are huge number of possible solutions, but I shall take us through three of the most important tuning ranges when dealing with 12-note chains of fifths like this.

If one tuned the fifths to be as harmonious as possible, one would probably find that each perfect fifth had its lower and upper notes’ frequencies in a ratio of 2:3. A 2:3 fifth is often called a just or pure fifth, and a chain of these 2:3 ratios (with a bit of 1:2 octave reduction when necessary) gets us a famous tuning we now know as Pythagorean, after Pythagoras, who was famous for discovering or rediscovering the rational proportions involving the prime numbers 2 and 3 that made harmonious sounds, for example the 1:1 perfect unison, 1:2 perfect octave, 1:3 perfect twelfth, 2:3 perfect fifth, 1:4 double octave, 3:4 perfect fourth, 3:8 perfect eleventh, etc. When two 2:3 perfect fifths are stacked one ends up with (2*2):(3*3) = 4:9, a rather sonorous major ninth.When four 2:3 fifths are stacked one gets 16:81, to most ears a rather large and active major third plus two octaves. Bringing these ratios within the octave for easy comparison with each other we have an 8:9 major second, a 64:81 major third, and a 3:4 perfect fourth. One can tell just looking at the numbers that the major third is going to be acoustically more complex and therefore less stable, but it lends itself very nicely to resolutions to a simpler 2:3 fifth or 3:4 fourth, as in much early European harmony. Continuing, one finds a minor second (-5 fifths) with a ratio of 243:256, slightly smaller than the standard half-step or semitone on today’s guitars and pianos, but rather lovely as a melodic movement or for a brief moment of tension before release. If we go a bit further the other way, we find a different type of semitone, the chromatic semitone or augmented unison, with a ratio of 2048:2187 (+7 2:3, fifths, reduced by several octaves). We find the size of this semitone is almost exactly in the ratio of 5:4 with the minor second, so we could, like others have done before, measure intervals using these small parts, called commas, or kommas, with 4 parts to the minor second, 5 parts to the augmented unison, 9 parts to the major second, 13 parts to the minor third, etc, and realise that an octave would be 53 parts, meaning even if we continued our chain endlessly to generate new Pythagorean ratios, the 53-tone equal temperament would still be a very good approximation, and for most purposes is the best simple model for Pythagorean intonation.

Another option would be to aim for sweeter, simpler major thirds. The simplest ratio in the major third region is 4:5, one comma narrower* than the Pythagorean 64:81 major third. Say we like these thirds a lot. And we want to make them our default major third. Since we have 4:5 one comma narrower than 64:81, and both span 4 perfect fifths, if we narrow each fifth by 1/4 of that comma, then our originally 64:81 major third becomes 4:5. This tuning is known as (1/4-comma) meantone, and it ruled most of the western world of music between the 15th and 18th centuries, with some usage stretching even further. It was the default tuning for most keyboard instruments, as its thirds and sixths were sweet and close to simple ratios (4:5, 5:6, 3:5, 5:8) it gave sonorous harmonies, although bare fifths were less pleasant than in Pythagorean. if we look at the semitones in meantone, their sizes are the other way around, with the augmented unison smaller, around 2/5-tone, and the minor second around 3/5-tone, their difference being the -12 fifth diminished second, here being equivalent to the ratio 125:128, which we can think of as an octave less three 4:5 major thirds. In ratio arithmetic, this is (1:2)x(5:4)x(5:4)x(5:4)=125:128.

*actually a syntonic, as opposed to Pythagorean, comma, but their difference is less than two cents, almost imperceptible.

So why doesn’t everybody still tune their keyboard to Pythagorean, or meantone? The main reason is the intervals formed by bridging the distant ends of the chain of fifths – since we don’t necessarily have a circle, but rather a 12-note chain, as spelled out earlier: Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, and so the interval from G#-Eb, looking to many modern musicians like a perfect fifth spelled differently, is actually a different interval with a different sound, a diminished sixth. In Pythagorean, this interval is a comma (or roughly 1/53 of an octave) narrower than a perfect fifth, so is mostly unsuitable for stable harmonies, though one might be able to slip one in very briefly in passing without too many musicians noticing. In meantone, the diminished sixth is sharp of a perfect fifth by almost twice that amount, and the interval from G#-Eb in meantone was often called a “wolf” fifth – perhaps you can imagine why.

Creating a circle of twelve fifths and making sure that all intervals and keys were usable was one of the major developments in tuning theory in the 18th century. A 12-tone circulating system with sufficiently acceptable intervals in all twelve tonal centres came to be known as a well-temperament, and most well-temperaments have a range of key colours that lie somewhere between meantone and Pythagorean – that is, some with slightly narrow fifths and rich, warm major thirds, and others with strong pure fifths and more active, energetic major thirds. Even if an interval is spelled as a diminished sixth, it will still function well as a perfect fifth, and so we have an enharmonic equivalence, e.g. between C#-Db, D#-Eb, etc, or more generally, +12 fifths = a perfect unison (octave reduced).

Knowing about these three historically popular 12-note tunings teaches us about a few concepts which are really helpful for exploring tuning theory:
1) ratios (and the harmonic series, which I didn’t go into, but with which all ratios can be derived)
2) regular temperament
3) irregular temperament
4) mapping, e.g. of abstract intervals to ratios, M3=4:5 in meantone, or physical mapping of pitches on a keyboard
5) enharmonic equivalence, e.g. P5=d6 in a well-temperament.
6) different interval sizes within a sort of fuzzy “bucket”, e.g. d6<P5 in Pythagorean.

If any of these approaches are carried beyond twelve notes or applied to different numbers of pitches, or to different ratios, they are amazing resources, and can be far more exciting than the now-familiar 12-tone equal temperament, while still giving the musician the ability to use most of the knowledge they’ve gained about other tuning systems or music in general.

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” diesis, Dydimic 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.

non mean tones

This is my first post in a rather long time. But since that last post i’ve been spending most of my time dealing with tuning systems that don’t support meantone.

What does it mean for a tuning not to support meantone? Simply, that four perfect fifths up doesn’t give you the best approximation to the fifth harmonic 5/1 (or octave reduced to 5/4). In a system of Pythagorean tuning (with pure 3/2 fifths, 701.955c), four fifths up gives you (3/2)^4=81/16, which though fairly close to 5/1, isn’t the best approximation if you have a system of 9 or more tones. In fact, a regular 12-tone chain of pure fifths will offer 4 sweet low thirds much closer to the 5/4 thirds of meantone, but they are spelled as diminished fourths, tuned as 8 fifths down, instead of four fifth up. How close to 5/4? Just a tiny comma of 32805/32768, or 1.95c, called the schisma, away from 5/4. In these cases where eight fifths down gives the best approximation to 5/4, the temperament is called “schismatic” as it tempers out this schisma.

A 12-note chain from Eb up to G# will give these diminished fourths at B-Eb, F#-Bb, C#-F and G#-C, with the remaining 8 major thirds tuned as 81/64s (also called ‘ditones’, as they are double the regular 9/8 tone). But if we extend our pythagorean cycle out to 53 tones, we can close the spiral of fifths without any wolf intervals. Once again, like we did with a 31-tone meantone circle, we can equally temper our 53-tone Pythagorean circle without much damage at all, giving very near pure thirds in all 53 keys, as well as pure fifths, and now very close to pure 6/5 minor thirds too. Because we have a pure octave, virtually pure third harmonic and very-close-to-pure fifth harmonic, 53EDO is a brilliant closed system that can represent almost all ratios in the 5-limit. Though as it also contains very good thirteenth and nineteenth harmonics, and a seventh harmonic that is only 4.76c sharp, it can be thought of as representing the full 2.3.5.7.13.19 subgroup with very good accuracy.

However 53 is a lot of tones, so we should go back to our pure fifth chain and find other places we can close the circle without much damage. As it turns out, 12, 29, and 41 show up as great options before we hit 53.

We already know about 12, but as it tempers 4 fifths up the same as 8 fifths down (i.e. equates major third and diminished fourth), it won’t give us a better approximation of 5/4 than the regular third, and so it works in practice very similar to most meantones.

29 has great fifths just slightly wide of pure, but this leads to an eightfold flattening of the syntonic third (=diminished fourth), and so instead of 5/4 (386.31c) we end up with 372.41c, a large middle third very close to 26/21, but really too small to hear it as a major third, or at least one drastically flat.

41, however, provides with a great system that excels in its approximations to ratios of both 3 and 7, as well as a very acceptable 5, 11 and 13. However, it goes even further to give 19, 29 and 31 with great accuracy. 17 and 23 only have errors of 12.12 and 13.64c respectively, both less than the error on 5/4 in well-loved 12EDO! We could probably go even higher, but a 31-limit seems high enough for intervals that are really appreciable to the ear.

The only thing that bugs me a little about 41 is the ~6c error on the schismatic major and minor thirds. Sure, 6c might not seem like much (“6% of a semitone?…!”) but it’s enough to give a wee bit of tension, a push towards the “middle third” region, and create a wee bit of beating that can be a little unpleasant in certain timbres. Still, in terms of efficiency and sheer cool, 41 is a great tuning. Probably my favourite non-meantone EDO of practical size (as in, all notes being available/playable on instruments). The ratios of seven sound particularly good, with the 1-step comma ~= the septimal comma 64/63, the 2-step small minor second a near-28/27, a wonderful small semitone, and 7/6, 9/7, 21/16, 7/5 and their inverses, all great sounding intervals in their own right.

Where 5/4 is reached by tuning 8 fifths down and 7/4 is reached by 14 fifths down, we get the Garibaldi temperament (named after Eduardo Sábat-Garibaldi, who promoted a schismatic-style 53-tone system where major thirds are lowered, minor raised a comma from their Pythagorean positions, and sevenths reached by similar means, lowering the regular minor seventh by a comma, building several “Dinarra” guitars on which to showcase this tuning system). Both 41 and 53EDO support Garibaldi, and it’s a very simple bridge into the 7-limit using essentially a 3-limit (pure fifths-based) system. However, it’s not the only [simple] solution outside of JI and meantone.

46EDO provides a very interesting alternative, where fifths are only 2.39c sharp of pure, and all tones can be reached by chaining these fifths. Looks very much like our Pythagorean circle tunings above right? This time, however, 8 fifths down gets us a neutral third around 21/17, and 14 fifths down gets us to 939.13c, or the best approximation of 12/7. As much better approximations to both 5/4 and 7/4 exist, this is not Garibaldi, nor schismatic at all. 7/4 is in fact reached by 15 fifths UP, or as a double augmented fifth if tuning by fifths, while 5/4 is 21 fifths up… However, 46EDO also affords simpler routes to the higher partials, with 13/8 tuned as a regular augmented fifth (8 fifths up), 11/8 tuned as an augmented third (11 fifths up) and 23/16 tuned as an augmented fourth (6 fifths up), which makes playing in the 2.3.(7.)11.13.23 subgroup pretty easy in a Bosanquet-like setup.

Of course there are an infinite number of non-meantone systems worthy of explorations, some without even a proper perfect fifth (such as 11, 13, 16, 18, etc EDOs), or at least on the borders of the usual acceptable range (5, 7 EDOs and their multiples), but some of the EDO systems where, though meantone isn’t supported, theory is fairly transferable between them and a system like 12EDO or Pythagorean, include 17, 22, 27, 29, 34, 36, 41, 46, 53.

I’ll try and getting round to posting some links soon instead of just spilling out words and numbers…

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

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

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

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

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

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

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

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

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

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

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

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

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

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

application

50EDO might seem like an impossibility, another musical universe with nothing tying it to the system most people claim to think in. it shares no common pitches with 12EDO, and has more than 4 times the number of intervals. which means we can’t apply what we already know right? we have to start learning from scratch?

nope. although we spend almost all of our time as musicians locked into something at least close to 12EDO, a lot of musical thinking follows the rules of meantone, and traditional music theory is pretty much based on it. although some music is very free with its enharmonic spelling (like spelling a C# major chord with F standing in for E#), or even benefits from this freedom (so, say, we don’t have to use key signatures with 7 sharps or more), and we can always count things in semitones – our atomic unit of pitch in 12EDO, the traditional theory says to spell correctly, so that a fifth up always looks like a fifth, and so while F is a minor seventh up from G, E# is an augmented sixth, an important difference in terms of tonal directionality, but also traditionally could have been a matter of tuning (although augmented sixth chords mostly came into use when meantone was fizzling out and being replaced by circulating well-temperaments)

‘flats’ being ‘higher than sharps’ in enharmonic pairs a diesis apart (e.g. Db vs C#, F vs E#, Dbb vs C) fits very well with our standard notation, so that ascending intervals almost always look ascending on paper (note however that the interval from Cb-B#, a semitone in 12EDO, is now only a coma of 24c, or half the diesis, which in 12EDO, is tempered out!)

the main thing about meantone though is its mean tones, right? if we accept the major third as needing to be around 5/4 (in our case a smidgen lower at 384c), then we aren’t going to divide it into a larger and smaller tone, but two even tones.

this may seem like common sense, and 12EDO also follows this pattern, but non-meantone systems don’t. Just Intonation, schismic tunings and superpythagorean tunings with a larger tone at 9/8 (204c) or higher will need the additional tone to be around 10/9 (182c) to balance things out. which is fine if you know your way around comma adjustments and make sure you don’t end up with wolves around the place. meaning ‘traditional’ (meantone) chord sequences like ii-V would often require a comma movement from the root of ii up to the fifth of V. the comma represented by “-” here might be between 17c [as in 72EDO] and 55c [as in 22EDO] for example.

a four part ii-V-I on C:

D- D  E-
A- B- C
F   G  G
D- G  C

in meantones of course, this comma is tempered out, so it certainly makes diatonic chord sequences a lot easier.

so far so easy right? but, i hear you saying, surely we’ll run into trouble when we play something which Assumes enharmonic equivalence, like a symmetrical pattern which divides the octave into any number of equal parts, notably 3 or 4 or 6 (factors of 12 which are not factors of 50).

let’s first try the division of the octave into thirds. in 12EDO this would be an augmented chord, which, when inverted retains exactly the same intervals.
with 50, it seems like this is impossible, but instead we just have a few different options.

if we want to repeat the same interval three times, then we won’t land on a perfect octave, but we can land close, or a bit further away and ‘bend the ear’ a little. a classic augmented triad would be a stack of major thirds (16\50 steps), so C, E, G#, leading to the next note being B# (48\50), 48 cents lower than we started. which means if we build major chords on each root, we would go around 25 of the 50 keys (all of the even steps) before getting back to where we started, instead of just three steps C-E-G#-C.

so we could use a slightly larger interval of 17\50, or 408 cents. very close to our 12EDO major third of 400c. but 3 times 17 is 51, so we end up modulating up a comma of 24 cents. this could be very interesting on the ears. C, E+, G#^(Ab), C+. micro-modulations, you might call these. obviously if we keep going, we will get through all 50 keys, since 17 and 50 are coprime.

but if we want to land up on the same C after just three steps (which might be a desired option in a more familiar style), we simply need major thirds of different sizes. which we have, 16 steps (major), 17 steps (greater) and 18 steps (super) [also possibly the greater middle and augmented thirds at 15 and 19 steps respectively if you’re getting really experimental]. once option is 16+16+18 and its permutations, two major thirds with the remaining being super, the only solution in regular 12 tone meantone subsets, e.g. C, E, G#(Abv), C. but of course we also have 16+17+17 and its permutations, which would be the smoothest option for those familiar with 12EDO and thirds close to 5/4. but it would mean if you were elaborating on this three tone structure by splitting it into a six tone [uneven] ‘whole-tone’ scale, or harmonising in triads, you would once again have to choose from a few options, which might be a rather good excercise.

my pick for a harmonically appealing ‘whole-tone’ scale might be C, D, E, F#+(Gb-), Ab, Bb, C, with steps of 8, 8, 9, 9, 8, 8, although anything from 7 to 10 steps might be ‘perceived as a kind of whole-tone’ in this context.

likewise diminished [4-step] and octatonic [8-step] patterns are also assymetrical, but beautiful to play around with the options, with different combinations benefiting from anything from strong harmonic foundation to a smooth melodic contour, or just something a little whacky.
C, D, Eb, F, Gb, Ab, A, B, C (‘classic’)
C, D, Eb-, F, Gb-, Ab-, A+, B+, C (‘quasi-Pythagorean’)
C, D, Ebv, F, Gbv, Abv, A, B^, C (‘sub’)
C, D-, Eb, Fv, Gb-, G#, A+, Bv, C (‘quasi-equal’)
are all lovely rather different octatonics. and there are many more, plus their modes, each has its own place in some musical style or other.

in fact there are multiple reinterpretations of every single scale in 12EDO, every chord, every musical entity. some of them may have to be bent to fit 50’s pitch grid if you like, but there are always options.

if theoretically correct spelling is preserved in a piece of 12EDO music, then there should be no problem whatsoever in adapting it to 50. the main consideration which might come up is which kind of augmented, diminished, and ‘dominant’ seventh chord to use.

due to harmonically consistent spelling we can assume the basic major and minor triads to be 0-384-696c (16,29 steps) and 0-312-696c (13,29 steps) respectively. but for dominant sevenths, we can choose 1 of 3 notes for the seventh if the chord, as well as alter the third, depending in how much tension or stability we want.

i would class dominant sevenths as chords of (16/17), 29, (40/41/42) steps, with other alterations making the chord a bit unique to just call a seventh. first we have the classic or diatonic seventh 0-384-696-1008, built from the classic diatonic scale. the seventh is rather tense and high (diminished fifth against the major third) in comparison to the smooth triad, so it’s great as a tense moment in diatonic contexts. particularly in earlier styles where smooth triads are the normal and sevenths are an occasional heightened dissonance. next we have the sub seventh, 0-384-696-960, the closest 50 has to 4:5:6:7, straight from the harmonic series, more concordant and rooted, without much need to resolve to anything. this is my favourite in most contexts, especially modern harmonic styles, but the seventh may be a little low melodically. third is the lesser seventh, 0-384-696-984, with the familiar symmetrical 600c ‘tritone’ between third and seventh. this would my pick for baroque and classical style repertoire (and possibly also more modern styles) where the seventh is a regular occurrence, but still needs to be resolved.

next time i’ll try and post a concrete example of this stuff in practice, by taking a 12EDO, Pythagorean or well-tempered piece and ’50-ifying’ it, where one can choose to preserve harmonic mood and shape or create a kind of parallel universe, where consonance, dissonance and voice leading can be very different indeed from how they were originally intended.

the third(s). vertical harmony

we hear major and minor thirds every day. almost any harmonised melody in existence is bound to contain a bunch of thirds or sixths (their inverse), and as people we seem to like the sound of them, gentle, easy, warm. if you get a few musicians or singers in a room together and give them a simple tune to play around with, one of the most natural things to do would be to stick some parallel thirds or sixths above it,or below, sometimes major, sometimes minor, to fit the tonality.

when these kinds of things happen, when musicians just go with the flow and whip some harmony out, blending melody and harmony into something really tasty, tuning happens almost by accident. good tuning. easy tuning. natural tuning. if there’s a major third, especially if there’s only two parts, it’ll be very close to a 5/4 (386 cents). it’s just what happens. it’s how most of us sing, especially when we’re not thinking too hard. it’s what happens when a string player double-stops when playing solo, because any ‘unnatural’ third will grate on the ears.
most of the time, smooth harmony means smooth, small numbers. and 5/4 definitely fits that bill.

while most string players’ perception may be that the ‘correct’ intonation of a major third might be at or very close to 81/64 (around 408 cents), four perfect fifths up of 3/2 each (minus two perfect octaves), and most instrumentalists who don’t think too much about tuning might say 400c (the one major third available in 12EDO, 4 semitones up) is the correct, or ‘only’ major third, the reality of music practice begs to differ. even within ‘classical’ music circles major thirds are frequently lowered to 390 cents or so, and if a stable major third or triad is needed, especially when sustained, even a third of 390 cents will stick out as ‘wrong’, the 5/4 comes into play. when you have a vertical major third, it’s very easy to play 5/4, but playing 400c or 81/64 or an ‘expressive’ 410 or 415 cent third is remarkably difficult.

it’s especially true of singing. the 5/4 is to most ears the only natural major third that comes to mind, though many might not have thought about it, the note on that piano or guitar you’re practicing with is the wrong note. try this:

play or get someone else to sing a drone on a pitch from 12EDO. find something mid-range. listen to the drone. now sing a major third above it, and hold it there. is it stable? is it warm? is it the right third? if you like the sound of it, check it against the major third on your 12EDO instrument, piano, guitar (as long as it’s been carefully tuned), or electronic instrument. what you should notice, is that the third on the instrument is not the third you want, not the one you’ve been singing, but around 1/7 of a semitone sharp. this is an experiment i should have got you to do right from my first post, before i delved into numbers and history and terminology. something fundamental to a lot of tuning practice. much of the reason many shy away from 12EDO, it just doesn’t have all of the right notes. 50, on the other hand, has really close approximations to quite a few of them.

[bonus experiment: try a minor seventh over a drone. see if you can get it to lock into tune, a powerful, almost otherworldly sound. a little lower than you might expect. this time you’ll find your 12EDO instrument is 31 cents (very nearly 1/3 of a semitone) sharp of the 7/4 you’re singing. owwww. although the temperament on 5 might go unnoticed by some, 12EDO really does not represent intervals of 7.]

ok so back to the thirds. if we agree on 5/4 being appropriate in most situations as the ideal major third (with 50EDO’s approximation @ 384c, from now on, simply the major third, only just over 2 cents flat), then what’s our ideal minor third? this question is perhaps a little harder to give one answer, there are three simple minor thirds that could all be useful in different contexts, as well as a few other far more complex ones. 6/5, 7/6, 19/16, 13/11, 32/27, 20/17…

6/5 (around 316c) is the classic choice, with an odd-limit of 5 (nice and simple). it is the 3/2 complement of 5/4, meaning if you set your fifth to the simplest ratio possible (3/2), and divide that into a 3/2, the remainder will be 6/5, either at the top (major chord, harmonics 4:5:6), or at the bottom (minor triad, harmonics 10:12:15 or subharmonics 6:5:4) of the major third. this minor third is brilliant when accurate, but tends to be  little discordant when not approximated well (e.g. off by more than 5-10 cents or so). in 50EDO, the minor third @312c is within our tolerance, and sounds pretty great.

but even though one might unconsciously go for the 5/4 major third even in an unaccompanied melody, minor thirds are often a bit lower than 6/5, which might have something to do with our familiarity and even preference for a tone close to 12EDO’s minor third. which is actually ridiculously close to 19/16. though people often try to explain 12EDO as a 5 prime-limit system with rather large deviations, it might be more helpful to think of the primes 2, 3, 17, 19 (with an absence of both 5 and 7):
1/1 17/16 [or more accurately, 18/17] 9/8 19/16 24/19 4/3 17/12 3/2 19/12 32/19 16/9 32/17 [17/9] 2/1, giving an error of between about 1 and 4.5 cents.

so why do we like 19/16? the numbers aren’t as small as 6/5 or 7/6 or even 11/9, and the prime limit is much higher (19)…
the answer is likely difference tones and what some call virtual fundamental.

difference tones are present every time two or more frequencies/pitches interfere, and are simply pitched at the difference between the pitches. if we have two tones at 440Hz (concert A4) and 528Hz (a little above concert C5), then the difference tone will be 528-440=112Hz, or around F2. in frequency ratios, we can represent 440:528 as 5:6, so the difference tone will be 6-5=1, or the fundamental of the series where 440Hz is the 5th partial and 528Hz the 6th. so the dyad 5:6 gives the triad [1:]5:6 in the first order, then [1:]{1:}{4:}{5:}5:6 when including second order difference tones. difference tones generally get fainter as order increases so there’s not usually much point looking beyond the first order.

so a 5:6 minor third actually has a virtual fundamental a major third below the lower tone, as it forms a full 4:5:6 major chord through its difference tone. which means when we have a full triad of 10:12:15, we get [2:3:5:]10:12:15, or octave reduced to [8]:10:12:15, a major 7th chord, because of the added difference tone a major third below. which means we have somewhat of a conflict of roots, e.g. if we had C minor we’d also be hearing a low Ab.

the 19:16 third, however, is a little more harmonically coherent if you will, with the lower pitch. [3:]16:19 means the difference tone is on a low dominant bass note, much like a classic V64 chord before a resolution to V53 — and we can do this if we move from 1/1 and 19/16 to 15/8 and 9/8, with the difference tone sustaining the bass, even though the dominant note is not present in either dyad!!! so when we add the fifth to our 19-limit minor third dyad, we reinforce that fifth already present way down at the bottom of the chord, and thus a minor triad like 16:19:24 is fairly strong [although we also get a bit of a clash of thirds with [3: fifth][5: major third][8: octave]16 octave:19 minor third:24 fifth

***NOTE: one of the few things about 50EDO i don’t like is its lack of an accurate 19:16. at 297.5c it can be represented almost equally well (or badly depending on your viewpoint) by the lesser third of 288c or the minor third of 312c, although other ratios of 19 such as 19/18 (as 96c), 24/19 (as 408c), 19/14 (as 528c), and their inverses,

the other obvious third is the sub or subminor third, at or nearby 7:6 (264c in 50EDO), which due to a first order difference tone a fifth below the ‘root’ of the chord leads to an implied [1:]6:7, or a 1-5-sub7 triad (perhaps we could call this a power seventh?).

finally the last simple third is the 9/7 (approximated well by 50EDO’s 432c), the super or supermajor. which can represent a rooted tonality but is much more intuitive to use as a harmonic extension on a different root, most obviously the sub seventh below the bottom of the dyad, as implied by difference tones, e.g. D and F#^ implying a possible root of E^ for a 1-sub7-9 triad, or the fifth above that for a sub triad, e.g. B^ D F#^

once we have these thirds we’ve at least got what we need for simple and strong vertical harmony, with each type of third pointing towards a certain ideal 3 part harmony based on difference tones: root position triads, 64 minors, major 7s, sub minors and sub7s, sub9s…

next time i’ll try and cover the not-so-obvious, or not-so-simple thirds in a bit more detail. these others aren’t particularly recognised in western theory, and we might have a tough time understanding or fighting with difference tones, but they’re a bunch of fun to play around with and allow us more tonal identities than we might be used to

leading tones. semitones and other divisions.

all regular meantones have rather low major sevenths, in comparison to our modern 12EDO or Pythagorean tuning, as a result of flattened fifths combined with low (closer to optimal 5/4) major thirds, or alternatively, because the flatness of the the fifth is multiplied 5 times, e.g. F-C-G-D-A-E. many people see this as a problem, a reason to choose a well-temperament or closer-to-pure-fifth tuning over meantone, so that those major sevenths can be closer to the octave, a smaller leading tone providing not only a bit of tension to keep things moving but a rather nice close to phrases, small melodic minor seconds (100c or less, say) contrasting possibly with a reasonably large whole step of perhaps 200 or 210 or even 220 cents.

small step resolutions are very inviting, heightening dissonance before a cadence, and bending the listener’s ear just a little before resolving. historically many sizes of minor second have been used, and even today with 12EDO as a standard there is some debate on how high leading tones should be to provide just the right feel.

my view is that it depends on context. most people have overlooked – or simply not had enough experience playing and listening to – proper low leading tones, being the default option in meantones. those of 1/5-comma meantone at a pure 15/8 (1088.3c), 1/4-comma at 1082.9c, or 2/7-comma at 1079.1c, all rather different from say 12EDO’s semitone. to my ears, these are just right as major sevenths in most music without too much movement, a lazy large minor second up to the octave is rather restful, warm and means a pretty well-tuned dominant chord if resolving a perfect cadence. a lot of musicians argue that dominant chords and especially dominant sevenths should be anything but restful, with a sharpened third and possibly even seventh heightening tension, an active dominant being more effective than a static one. but i myself love the sound of leading tones in the above meantone systems, with the 120c seconds of 50EDO being particularly effective, especially if the tonic is sounded against it or in close proximity (e.g. in a pre-emptive resolution ala Purcell, or in a trill)

but of course this isn’t just about harmony, the high leading-tone aesthetic has a lot to do with melodic intonation, where a large minor second (say, 110c or more) will supposedly ‘stick out’ and weigh the phrase down. larger minor seconds are perhaps heavier in that respect. but how small is a good small second? The 100c semitones of 12EDO? 90c Pythagorean minor seconds? 84c (21/20)? 71c (25/24, or roughly the minor second of 17EDO)? 63c (28/27, really beautiful septimal minor second, made much use of in that La Monte Young piece i posted the other day)? 56c (22EDO’s minor second)? surely we’ve already surpassed what we can call a minor second, right? not so. i’ve found even 48c makes a great small second, just less than a quarter tone with regards to 12EDO. where does this interval show up? in 50EDO of course, along with a great selection of other minor seconds.

50EDO’s intervals in the minor second range are the following:

48c (arto or diminished second), 72c (sub second), 96c (lesser second), 120c (minor second), and if you’re adventurous, 144c (lesser middle second)

one really cool thing about meantones is that while their regular minor second will be on the large side, they will always have a smaller interval available in the form of the augmented unison, which often also represents the septimal minor or sub second. so although 120c is considerably large, we can go for a more regular semitone (this time it is actually half of the mean tone) at 96c, or use the sub second to give that 28/27 feeling <3, or we can actually use the diesis at 48c as a leading tone, rather effectively.

which means if we are resolving some kind of dominant cadence, a 120c minor second will give us a very close to pure major chord on the V, the 96c lesser second will give us a Pythagorean or greater third (408c, very close to 12EDO at 400, and very active in comparison), the sub second at 72c will give us 432c (a super third representing 9/7, an altogether different mood, which some might call ‘high tension’, though it’s actually much more stable than the greater third), and finally the diminished second, when used as a second, will mean we get an augmented third on the V chord, looking rather like 10:13:15, a tendo triad rather than a usual major.

of course anther option is to change the position of the dominant, so that placing it on the greater fifth (720c) will provide a pure third when we have a melodic semitone, and a super third where we have the melodic diminished/arto second in the melody. one more degree up at 744c almost doesn’t look like a fifth (being about 1/2 way in between 12EDO’s fifth and minor sixth), but it represents the regular wolf in meantone, e.g. G# to Eb, or B to Gb, and in 50EDO, this interval is actually a lot more consonant (at least to my ears) than it is in usual varieties like 1/4 comma or 31EDO. a cadence of the kind Vicentino was very fond of (in, or very close to, quarter-comma) might mean the bass descending from the ‘wolf’ super fifth to the tonic, while the leading tone on the super major seventh provides a pure third to the bass, while at the same time keeping the voice leading smooth with only a small 72c sub second. a very nice melodic minor second indeed, 3/8 of a 50EDO mean tone. in my 50EDO notation, we might have (in Bb major):

Cantus: Bb—-A^-Bb
Bass:     Bb-G-F^-Bb

or in four parts, a little more elaborate possibly not quite Vicentino’s style any more):

Cantus: Bb-Bb—–A^———-Bb
Alto:        D–D–Eb-D#^-E^-F^-F
Tenor:     F–G——A^—C^—–D
Bass:      Bb-G——F^———–Bb

although this is just a I-vi-V(4-3)-I progression, it’s got a bit of attitude. the dominant chord is raised by the diesis which means when we tie the Eb (minor sixth above G) over to the dominant, we get a sub seventh instead of a minor seventh, which being enharmonically equivalent to an augmented sixth (respelled F^-D#^) we can resolve it upwards just for fun (historically incorrect voice-leading but this is just a [bad] experiment really), the very smooth passing major seventh to octave on F^ leading us not to Bb^ but to Bb, possibly a very unexpected jump, though back right where we started. good luck singing that alto part…

a better example might be a modernised medieval style cadence to an open fifth and octave, with ‘double leading tones’: sharpened fourth and seventh ascending to the fifth and octave, and a lower voice moving from second down to the tonic. on D we might have:

Cantus: D—–C#^(tr.)-D
Alto:       G-A–G#^——A
Bass:      G-Fv-E———D

the only problem with this kind of voicing is that the fifth at 696c is a little flat when left open (without a third in the chord), the ear notices it’s not quite a perfect fifth in certain timbres, so either choose your timbres carefully, or maybe just don’t use it as a closing sonority, unless you want that hint of instability. filling in the chord would give us a more stable sound, but it would kind of kill the style we’ve got going.

but back to minor seconds, if you want to do something like 12EDO, or want the ability to be a little ambiguous, use the (here 96c) semitone, which can stand in for a whole bunch of different ratios, just like in 12EDO, it can mean the ability to go in just about any direction, it’s a bit tense (because it’s not really close to simple consonant ratios). we still haven’t had an example with the semitone so why don’t we try a more harmonically adventurous one like this:
F>Fsub7(b9)>Eb-m7>D+M13>D+sub7(b9)>Eb/G>F/G>G

F Gb-               >Gb-            >F#+                       G    A  B
A     C Db- Ebv Eb-     Db->C#+  D+      Eb >Eb F   G
C     Ebv    C      Db- C Bb-   B+ A+ Bb^ C-    Bb C  D
F                          Eb-               D+                         G      >G

ok that got way outta hand. but hopefully it at least helped demonstrate the ability of semitones to be ambiguous, and allow not only semitonal but greater tone modulations, as the semitone is a comma-inflected interval (either a comma-raised augmented unison or a comma-lowered minor second), comma-modulations are very easy, and I could just have easily moved to G+ instead of G (as the bass would suggest perhaps I should have, instead i held over the Eb, the b9 of D+ to give a minor sixth against G). there are just so many possibilities.

to be honest i’m usually much more concerned with harmonic events than melodic voice-leading, so i might use all of the different intervals in the ‘minor second’ range at different times, for different purposes, when i’m playing particularly involved passages. for me, the most obvious uses would be:

minor second: as  regular diatonic step, whenever playing a modal line

lesser second (semitone): as a trill, usually above the target pitch but can work from below too, and the most obvious ‘flat nine’, closest to the 17th harmonic

sub second: whenever resolving sub and super intervals: e.g. a sub seventh to a major sixth, a sub sixth to a fifth, a super seventh to an octave, a super third to a fourth.

arto second: when resolving arto and tendo intervals, or when doing something whacky, like moving between sub and super intervals- super sixth to sub seventh, sub third to super second, or when moving between regular and sub/super- fourth to super fourth (as in 4/3-11/8), or minor third to sub third (as in 6/5 to 7/6).

 

hardly any of this is going to make much sense without hearing it. for those of you bearing with me, thank you. but do some experimenting, find a way to play these things for yourself (my favourite is the programs Scala + pianoteq). i’ll try and make some demos in the next wee while to show off different kinds of resolutions using these steps and other intervals, but my main point in this post was that there are always options, and having many intonational possibilities even with just one chord sequence makes the world of harmony that much more exciting.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

fifths. the spiral. circles. temperament.

just for anyone skeptical that a system of thirty one tones to the octave is ridiculous or impractical, keyboard (and other) instruments were built in the 16th, 20th and 21st century, and the Fokker organ in Amsterdam continues to see the performance of new compositions and old, and stands as a symbol of the thirty one tone movement. http://www.huygens-fokker.org/instruments/fokkerorgan.html

some other keyboard instruments to tickle your fancy
http://www.h-pi.com/eop-keyboards.html

and of course guitars
https://www.facebook.com/Swordguitars

thirty one tone music is really out there, you just have to know where to look…

anyway, on to today’s main topic: the spiral of fifths.

so, i’m guessing, if you’re a modern musician, you’ve been taught that fifths are arranged in a circle of twelve notes, and if you go up or down twelve fifths from any note you end up back where you started right? twelve fifths up from Ab is Ab? right?

this is a modern take on ancient theory, to suit the 12-notes-per-octave agenda, equal or otherwise. in truth fifths should be arranged in a spiral, regardless of how they are tempered.

as we saw in the thirty one tone post yesterday, the more fifths we go up, the more sharps in the key, and the more fifths we go down, the more flat. we should name the keys accordingly, and not assume that any pitch in the spiral is enharmonically equivalent to any other pitch.

so if you wanted to go up and down in fifths from C, you’d have:
…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…

this spiral continues infinitely in both directions, at least in theory, regardless of the size of the fifths. it might help to associate the height and length of the spiral with pitch at that point, so that if we travel up one node, we are always going a fifth higher in pitch, and down three nodes means going down three fifths, or a major thirteenth.

if fifths are equal to 3/2, we have a Pythagorean system with each 12 fifths taking us up around 23c in pitch, so that in ‘enharmonic pairs’, the ‘sharper’ of the two will be higher in pitch when both notes are taken within the same octave. e.g. B# higher than C, G# higher than Ab.

if fifths are equal to 700c, as in 12EDO, then each twelve fifths we go up will take us up 0 cents in pitch, i.e. we ain’t goin’ anywhere honey . but i said the spiral was infinite didn’t i? well, the note names should take in to account where you are in the spiral, so a fifth up from D# has to be A#, not suddenly Bb. even though 12EDO tempers the two notes to the same pitch, we still spell according to the spiral. we still spell the scale of F# major as F# G# A# B C# D# E# F#, though many musicians would much prefer seeing the more familiar Bb, Eb and F substituted for those i ain’t got no damn E# key on my piano notes.

so spelling is according to the spiral, but most of the time we’re actually hearing a circle of twelve fifths right? just like the textbook theory? well yeah, in what are called circulating systems the spiral has been cut off at some point (enough double sharps already!), and turned into a circle.

in 12 note circulating tunings, including 12EDO and any of the many well-temperaments, we cut the spiral off at 12 pitches, effectively making 12 fifths up exactly the same as 0 fifths up. which means probably flattening at least a few of them a little bit from 3/2 in order to avoid a wolf between the top and bottom of the spiral, e.g. between what would have been G# and Eb, a diminished sixth, if we take the 12 note chain Eb Bb F C G D A E B F# C# G#.

this alteration of a perfect interval, in this case 3/2, in order to merge two notes into one, is called temperament, and we see it all the time. almost all instruments in the modern world are tempered, mostly equally. we can flatten fifths slightly so that 12 fifths make seven perfect octaves (as in 12EDO), or a little bit more so that four fifths makes a double compound 5/4 major third (as in quarter-comma meantone, with 31EDO very close), or even more so that three make a compound 5/3 major sixth (as in 1/3-comma meantone, 19EDO very close). or we can widen intervals, widen our fifths rather dramatically so that four make a double compound 9/7 super third (22EDO gets close to this).

of course, temperament doesn’t just apply to fifths. we’re used to 3 major thirds adding up to an octave, but this is a particular tempered worldview. in regular theory, three major thirds makes an augmented seventh. which is only ever equal to an octave when major thirds are tempered to 400c each. in most systems three major thirds will either come out flatter (if major thirds are less than 400c) or sharper (when they’re bigger) than an octave. if they’re just 5/4s, then we end up 41 cents below an octave, just like in 1/4 comma meantone. which makes progressions like Ab-C-E-G# rather exciting!!!

four minor thirds? a diminished ninth! four 6/5 thirds gets us to (6/5)^4=1296/625, or 62.5 cents sharper than an octave. ouch. in diatonic tunings (built in chains of fifths), this means when fifths are equal to 700c, augmented sevenths and diminished ninths are equal to octaves, when fifths are less than 700c (think meantones), augmented sevenths are lower and diminished ninths are higher, and when fifths are greater than 700c (for example Pythagorean or super-Pythagorean, superpyth for short), augmented sevenths are higher and diminished ninths are lower. phew.

so why is this important?

the concept of a chain of a single generator plus the period of an octave (known as linear [rank-two] tunings) is pretty fundamental to scale building . a lot of the time this means generating scales by fifths as we’ve done here, known as syntonic tuning systems, where the fifths could be 700c or 703c or 696c or 720c or 686c (if you just love that howl).

of course we could have three generators (rank-three), for example 2/1, 3/2 and 5/4, which would give us an infinite lattice of what is known as 5-limit just intonation, all ratios containing the products and divisors of primes 2, 3 and 5. but most of the time we’d choose a set of them as our scale, like this twelve note scale i posted earlier:
1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8, or the 22 shrutis of the Indian Classical tradition:
1/1 256/243 16/15 10/9 9/8 32/27 6/5 5/4 81/64 4/3 27/20 45/32 729/512 3/2 128/81 8/5 5/3 27/16 16/9 9/5 15/8 243/128 2/1

or rank-four, with say, 2/1, 3/2, 7/4 and 11/8, to give us a rather interesting set of pitches, or rank n, but rank one should do for now. and in any case, syntonic tunings are probably the best way to explain important families of tuning suitable for almost all musics of the western tradition, and to build on what many of you already know about music theory.

if you’re wondering what some good circulating fifth systems are, you can start by looking at equal temperaments which have a useable fifth and approximate other colourful intervals along the chain. or simply take a chain of fifths of any size you like and see what you find.

5 and 7 EDO, with fifths of 720c and 686c respectively, will probably be at the limits of tolerance to what you might call a useable fifth. so they might provide an upper and lower bound on some tunings to explore.

check out this page too
http://en.wikipedia.org/wiki/Syntonic_temperament

from smallest to biggest, a good selection of equal temperaments [along with a good selection of musical tastings] might be:

685.7c: 7EDO [this album is one of the coolest things i’ve heard, definitely not an ideal tuning but very interesting]
https://archive.org/details/Knowsur-NanaWodori

690.9c: 33EDO [not really recommended, but what the hell]
http://soonlabel.com/xenharmonic/archives/1767

692.3c: 26EDO [i’m not really familiar with this one, a little strange]
http://midiguru.wordpress.com/2013/01/06/public-rituals/

694.7c: 19EDO [this one’s become rather popular, great for guitar, close to 1/3 comma]
http://www.broadlandsmedia.com/microstick/music/

696.0c: 50EDO [just amazing, my favourite, close to the 2/7 comma of Zarlino]

696.8c: 31EDO [funkadelic, popular, about as many frets as you can fit on a guitar, close to 1/4 comma]

697.7c: 43EDO [decent meantone, very close to 1/5 comma]

698.2c: 55EDO [Mozart’s tuning, great Baroque meantone close to 1/6 comma]

http://tonalsoft.com/monzo/55edo/55edo.aspx

700.0c: 12EDO [your old friend] [no links necessary]

701.9c: 53EDO [absolutely wonderful for primes 2.3.5.13, also as a master system for Indian musics]

702.4c: 41EDO [like Pythagorean, but with more attitude]

[also the basic layout for this beast of a keyboard http://www.h-pi.com/TPX28buy.html]

704.3c: 46EDO [great fun, just a little twisted]

705.8c: 17EDO [awesome as a guitar tuning]
http://micro.soonlabel.com/0-hosted-albums/heptadecaphilia.html

709.1c: 22EDO [another rather popular one, very odd at first, but a gem]

711.1c: 27EDO [interesting…]

720.0c: 5EDO [great pentatonic, just a bit bizarre]
https://soundcloud.com/search?q=5EDO

fwoah! that should do it for now. enjoy the links. next time i might spend a bit of time talking about my favourite of these, 50EDO, and why it’s so damn cool.