Tricycle notation is a way of annotating traditional notations (staff notation, jianpu) to specify more precise microtonal pitches consistent with just intonation or regular temperaments. It divides the octave into three "wheels" and labels each pitch on one of these wheels. It can be used for analysis, to show what problems there would be rendering a piece in just intonation or a regular temperament, as a practical notation if you can find musicians to read it, or to work with a computer if you can find software that understands it. It also works as a way of coloring keys on a generalized keyboard, which I call a tricolour keyboard. The common theme is taking meantone temperament and relaxing it so that syntonic commas are expressed.
The core paradigm of tricycle notation is that each pitch of a 5-limit (major or minor) triad sits on a different wheel. For example
To show that the wheel numbers are arbitrary, here's C₂ major.
This page mostly uses C as the reference pitch because it's the tonic of the white-note major scale. There's nothing special about C from the perspective of the wheels, so here are some other chords.
Most of the named pitches here are sitting on the same wheel. The main exception is D, which can be either on the first or second wheel. This shows that D is likely to have two different tunings in C major. If you choose chords that work with all their pitches on consistent wheels, then your music will work in just intonation with fixed pitches. If pitches in adjacent chords are on consistent wheels, but more distant chords require inconsistent wheels, then your music will work in just intonation with variable pitches.
Here's C₁ major with more harmonics.
The point here is that a given interval corresponding to the same simplified ratio is always written with consistent spelling and crossing the same number of wheels. So, 5:4 is always a major third that crosses +1 wheel. 6:5 is always a minor third that crosses +1 wheel. 10:9 is a whole tone (major second) that crosses no wheels. 9:8 is a whole tone that crosses +1 wheel. Because of this, tricycle notation allows you to distinguish the different whole tones that get written the same way in standard notation and so resolve ambiguities in extended 5-limit JI.
Here's a new diagram showing how many wheels different intervals cross.
If you ignore the wheel indications, the result is consistent with 5-limit harmony in meantone temperament. Meantone is a historically important temperament class and linked to the evolution of the notation systems used here. However, it isn't something the average musician gets trained in, so most fiddle players don't expect a chromatic semitone to be smaller than a diatonic semitone, or G♯ to be lower than A♭. The first step toward using tricycle notation is learning to make these distinctions in standard notation.
You can assign the C₁ major scale to wheels as follows.
D, the major second of the scale, shows up twice because it's ambiguous and is likely to be tuned two different ways. The scale can also be written in terms of the melodic intervals of 10:9 whole tones (t), 16:15 semitones (s) and 81:80 syntonic commas (q).
I added a subscript to each "D" to clarify which wheel it's on. The other pitches don't need subscripts because their wheel is obvious, and you can take this as the paradigm for the default wheels given the convention of C being on wheel 1.
The tone (t) and semitone (s) don't change wheels. The comma (q) does advance the wheel. The three commas in this scale correspond to the three wheels that cover an octave. You could define tricycle notation by this scale rather than the idea of triads having three pitch classes sitting on different wheels: both definitions are equivalent.
The two scales with different tunings of D could be written
For more clarity, you can specify the initial wheel and every time a wheel changes. This works because melodic steps (diatonic seconds) most commonly don't cross wheels.
Diatonic leaps usually change the wheel in an obvious way, so you can decide if that needs to be made explicit. The only rule for when you can omit a wheel specification is that your intent must be clear to the reader.
As the wheel indicators for a diatonic scale only need to be there to set the initial wheel and specify when to move by a wheel in the same direction of the melody, you could take the presence of any wheel indicator as meaning "stretch this step by a comma". Instead of specifying the wheel, you could have accidentals for "move up a wheel" or "move down a wheel". The accidentals would work very similarly to comma shifts. I think this is a good idea and might be easier to read in performance. I haven't worked out the details so I won't say any more about it here.
Here are some more major scales.
Tricycle notation is kind of over-specified in that you make the choice of three divisions of the octave and you get triads and 5-limit diatonics and the smallest number of wheels that let's you express "up" and "down" with minimal ambiguity, all from that one decision. In fact, it's this rule about tuning 5-limit diatonic scales that is where I really started, or more specifically in magic temperament. The link to triads is a good way of thinking about it, but then you think, can there be four wheels for tetrads? And, in fact, that doesn't work because the wheels are really about the different sizes of whole tones in a 5-limit diatonic scale. There are other scales that would lead to different numbers of wheels and I might work them out one day. The diatonic scale is special because it's the basis of very common systems of notations. This is culturally biased but you can't get away from culture.
Because there are only three named wheels, some intervals that should span more than 3 wheels will be equivalent because of the cyclic nature of the wheels. The most obvious example is that three syntonic commas (each 81:80) are equivalent to a unison. This means the indigestible ratio of 531441:512000 appears to be tempered out, but individual commas of 81:80 aren't tempered out. That isn't a theoretical problem: it's reasonable that a chord sequence will drift by a comma and if you repeat it 3 times the notation leaves you back where you started. It's even a good thing because you might want to correct for pitch drift, and not want the notation to get ever more complex as the piece progresses.
A note here on arithmetic. When you were at school, you probably learned how to multiply fractions, and you probably can't remember the details now. It happens that to cube 81:80 you don't have to worry about lowest common denominators because there are no common factors. What you do is multiply 81 by itself twice to get 531441 and multiply 80 by itself twice to get 512000. You might be able to work that out on paper, or even in your head, but you won't: you'll use a calculator. When I show arithmetic here, I don't expect you to work it out on paper. If you want to check it, use a calculator. I use and RPN calculator I wrote in Python but you can use whatever you'll like. You'll find quite a lot of calculator apps that can handle fractions. There also times when I'll show some ratios and say that tempering them out leads to a certain temperament class. I spent a long time writing code that you can use online to do turn ratios into temperament classes. I don't try to work it out on paper and I don't expect you to. Nor do I expect you to know the underlying mathematics.
A common sense rule is that, if you're writing just intonation, it should be possible to tie down each pitch by following each part independently and following the wheel changes. This is something a computer could do for you, so you could write a piece with tricycle notation and convert it to audio or a more conventional notation for just intonation. If you find a D₁ and a D₂ close together in a just intonation score, you can assume the D₂ is a comma higher than the D₁. But if you see a D₁ and a D₂ on different pages, you don't know which is higher in pitch. D₂ might be a comma higher than D₁ or D₁ might be two commas higher than D₂ or there might have been even more pitch drift on the intervening pages. This is why the notation is globally unspecified. The wheels are specifying harmonic relationships, not nailing down pitches.
You don't have to use tricycle notation with just intonation. You can specify a temperament or leave intonation up to the performers. One option is to correct for pitch drift, which means pitches tend towards common values independent of the wheel. This is adaptive tuning, so you can interpret tricycle notation as a system of hints for an adaptive tuning algorithm.
With double sharps and flats, there are 35 written pitches in standard notation (7 × 5). If you multiply that by 3, there are potentially 105 pitches that can be written at any time. In practice it won't be quite that many, because some are ambiguous. For example, with C₁ as your reference pitch, D₁ and D₂ are both possible, so what is D₃? It could be a comma below D₁ or a comma above D₂. Still, there are quite a lot of pitches. If you don't like double sharps and flats, you have 21 basic pitches and 60 or so with wheel annotations, which is still quite a lot. It's enough for fixed pitch scales with the common numbers of 41 or 53 steps to the octave.
Tricycle notation is more of a discovery than an invention and it's specifically tied to 5-limit just intonation. Still, I discovered it in the context of magic temperament (19 & 22) which, along with a lot of good regular temperaments, belongs to the "marvel" family that tempers out 225:224. You could apply your favorite accidentals to get 7-limit just intonation but I like to use tricycle notation assuming 225:224 is tempered out. Marvel is isomorphic to 5-limit just intonation, which means they're the same shape, so you can write music in 5-limit just intonation and interpret it in marvel. This means writing 7:4 as an augmented sixth and putting the 4 and 7 on the same wheel.
Identifying the 7th harmonic with an augmented sixths means that augmented sixth chords and tritone substitutions from traditional harmony can be brought into the 7-limit. This ties in with my idea of evolutionary harmony. If the wheels are correct, it means the chords are closer than meantone, but it doesn't guarantee just intonation.
As 14, 15, 16 are all on the same wheel and separated by diatonic semitones, 16:15 and 15:14 must be the same. That shows 225:224 is tempered out because 15² = 225 and 16 × 14 = 224.
Missing from that diagram is 64:63 which is a doubly diminished second or enharmonic diesis (C to D♭♭) and goes down a wheel as the pitch goes up. 64:63 is a comma smaller than 36:35, which is also a diesis and has a wheel count of 0. The only way for 64:63 to be smaller is if it has a wheel count of -1. This means that the wheel goes down when the pitch goes up from 9:8 (D) to 8:7 (E𝄫). That's inelegant but necessary: it stands out on a score and tells you something funny is going on.
There's a run of +1 intervals from 9:8 to 4:3 that I like. They're equally spaced in magic temperament, which means consistent with 19-equal and all written with single sharps and flats. If you add a perfect fifth (C₁ and G₃ in the example) we could call it a set of generalized triads. It covers the usual 5-limit major and minor, sus chords, and the intermediate 9-limit pair 6:7:9 and its reverse.
Note that 21:20 comes up as a chromatic semitone changing wheels. There's a general rule that the "♯" and "♭" symbols keep you on the same wheel unless something exceptional happens. A 21:20 step is one of those exceptional things. 21:20 is 13.8 cents (126:125) sharp of 25:24. They can be considered equivalent but in this case they're treated as being a comma apart. The wheel annotations are there to make the distinction clear.
25:21 is an alias of the Pythagorean minor third of 32:27. This pair again shows that 225:224 is tempered out.
Because I found tricycle notation when working with magic temperament, I also use it with a consistent 11-limit mapping. This works where 11:8 is an augmented fourth (or tritone) and crosses 1 wheel. You might think that 11:8 is a bit small for a tritone, but it does work out with some temperament.
Here, 9, 10, 11 are all on the same wheel and 10:9 and 11:10 are both whole tones (major seconds). That means 10:9 and 11:10 are equivalent and so 100:99 is tempered out. The arithmetic here is quite simple: 10 × 10 = 100 and 9 × 11 = 99. This is a way of bringing 11-limit harmony into an extended 5-limit framework, which is important, because 11-limit intervals tend to be described as "neutral" or "quartertones" filling in the gaps of the conventional framework. I have more to say about this on a page on dominant sevenths and my YouTube channel. If 100:99 and 225:224 are tempered out together, the result is called apollo temperament .
Note that the interval 21:22 rises in pitch and on the staff when the wheel number goes down. This is a valid use of a "backwards" interval because it's written as a diatonic semitone, but it's smaller than the usual 16:15 or 15:14, which keep to the same wheel, so the change has to be less than 0 wheels.
This makes good use of the notation where most single sharps and flats are used. There are some equivalences within the 15-limit because 100:99 and 225:224 are tempered out.
11-limit intervals tend to fall in the gaps between standard interval. The 11th harmonic, or 11:8, is a poor fit for an augmented fourth, although it works that way with apollo, and an equally poor fit for a perfect fourth. What we really need is a half-sharp symbol so that 11:8 can be a half-augmented fourth. Half sharps are generally associated with quartertones, but they don't need to divide the whole tone into four equal parts and they're perfectly consistent with meantone tuning, in which case the perfect fifth gets divided into two equal parts which we could call "mohajira" and associate with 31 divisions of the octave. (Sometimes half sharps are called "quarter sharps". That's just wrong. I'm not having anything to do with that.)
The interval of 11:9 gives an 11-limit neutral third. This is roughly in between a major and minor third. As tricycle notation puts a whole range of thirds all on the same wheel, it's simplest to call 11:9 a neutral third crossing +1 wheels. That's all we need to define the wheel mapping.
Standard character sets haven't embraced half sharps and half flats so I'm using "‡" for a half sharp and "t" for a half flat. I'm keeping the "marvel" approximation for the 7-limit.
This wheel assignment means 11:10 crosses a wheel and 12:11 stays on the same wheel. This makes sense because 11:10 and 12:11 are both neutral seconds and 11:10 is a larger interval than 12:11.
Now, because the notation gives us more precision, the 22nd harmonic is in the right place.
This gives a unique mapping to all 11-limit intervals which makes it suitable for notating just-intonation with minimal ambiguity. It still requires temperament though. Firstly, 225:224 is still tempered out and 16:15 and 15:14 are still indistinguishable. Secondly, the 11:9 neutral third is defined to equally divide the perfect fifth, which implies 243:242 is tempered out. The arithmetic is 3:2 ÷ 11:9 ÷ 11:9 = 243:242.
Tempering out 225:224 and 243:242 together gives spectacle temperament. It can get to with 3.4 cents of just (for 9:8) and most intervals are closer. This is close enough to be a guide for flexible pitched instruments trying to find just intonation.
Because we're now using both double sharps and half sharps, there are now 45 (9 × 5) written pitches. The tricycle annotations bring that to up to 135 (45 × 3) pitches at any time. That's adequate for 72 note scales (and 72-equal tempers out both 225:224 and 243:242).
You could redefine the wheels so that 11:9 sits on the same wheel and 27:22 crosses +2 wheels. This would mean melodic steps are less likely to sit on the same wheel, but the wheel assignments would be the same as for apollo temperament above and that wouldn't be a disaster. I think I worked out the details once and didn't like it but I don't remember why. It surely would work and would allow more precise distinctions to be made.
If we have half-sharps and half-flats available, there's a temptation to use them for the intervals of 7, and to have 7:4 as a half-diminished seventh instead of an augmented sixth. This means it can be written as C to B♭t instead of C to A♯. If the 8:7 sits on the same wheel, and is combined with augmented sixths with the same wheels, then the result is würschmidt temperament which is fine but I won't cover it more here.
The alternative is to make 8:7 cover +1 wheels and 7:4 cover +2 wheels. This shows that the wheel allocations depend on how you spell an interval: they aren't inherent to the harmonic. We've already seen the 11th harmonic falling on different wheels with different notations.
With this spelling, you can spot the seventh degree of a harmonic seventh chord because either it has a half sharp or half flat applied, or it's the only pitch that doesn't have a half sharp or half flat applied.
With this change, all the intervals of 7 end up on different wheels to where they were before.
All these ratios are notated distinctly. To work out what temperament is implied, we need to get two half sharps to add up to a whole sharp or chromatic semitone. A half sharp is 64:63. A whole sharp is 25:24. 64:63 × 64:63 ÷ 25:24 = 32768:33075 so a comma of 33075:32768 is tempered out. That leads to a temperament class called labizoyo . It can get to within 3.4 cents of just intonation so it's definitely workable.
The heading here is miracle, so you won't be surprised if I say that this ratio is tempered out in miracle temperament. More than that, the TOP tuning comes out identical for both of them, so there isn't much to lose by tempering out the simpler comma 225:224 and using both 7-limit mappings simultaneously. (The TE tunings are different and there is a small gain in optimising for the 7-limit minimax rather than TOP but I don't have that wired up yet.) That means the wheel an interval ends up on depends on how you write it which shows that the wheels are about notated intervals, not the inherent nature of intervals. As miracle works in the 11-limit, you can also use this notation for it, and it tempers out 243:242 so the "spectacle" mapping above will work.