Tricycle notation is a way of annotating conventional music notation to get more pitch precision. It's a fairly simple idea that I found accidentally when working on tripod notation. Here, I'll introduce tricycle notation on its own terms and add lots of complications to make it look forbidding and mathematical.
Tricycle notation takes the conventional pitch names as used in staff notation or jianpu and firstly interprets them according to meantone temperament. Then, it divides the octave into three wheels, and places each pitch on a wheel. Each pitch of a 5-limit major or minor triad sits on a different wheel. Indicating the wheels means you can interpret chords in terms of 5-limit just intonation.
When it's obvious what wheel a pitch sits on, you don't need to indicate the wheel. The wheels move, so the obvious interpretation might change as a piece progresses. In addition, because there are only ever three wheels, syntonic pitch drift eventually takes you back where you started. This means the notation allows for locally precise notation of just intonation, but is globally imprecise with respect to pitch drift.
You can probably think of naming schemes for three things from mythology or history or popular culture. For now, I number them from 1 to 3. The pattern is cyclic, so there's no significance to which wheel you start with. I start with C on the middle of wheel 1.
I also use fingering indications to show the wheels in Lilypond. This is good because it doesn't mean inventing any new typesetting, and real fingerings aren't likely to be relevant in contexts where you want to indicate precise just intonation. In plain text, this means a C♯ on the first wheel is written "C♯-1". You can imagine other ways of writing the wheels, like colored or shaped noteheads.
Finally, tricycle notation assumes 225:224 is tempered out. This makes it a notation for the 7-limit planar temperament called 7-limit marvel.
Each interval has a wheel number: the number of wheels it moves you up. Here are some simple ratios along with their meantone names and wheel numbers.
|Ratio||Name||Wheel Number||Alternative Name|
|16:15||minor second||0||diatonic semitone|
|15:14||minor second||0||septimal semitone|
|10:9||major second||0||minor tone|
|9:8||major second||1||major tone|
Although there's some overlap, there's a strong tendency for the wheel number to increase as an interval gets bigger, until the wheel number gets to three at which point it loops round to zero.
The three wheels are ordered and in a cycle. This leads to a standard mathematical structure called "modulo arithmetic". Here are the basic rules:
If you want to move a pitch up by an interval, and you know the wheel number of the interval, you add the wheel number of the interval to the pitch's wheel, and if the result is more than three then you take away three. This gives you the wheel of the new pitch. For example, consider raising G-3 by a perfect fifth. A perfect fifth has a wheel number of 2. 3 + 2 = 5. 5 is more than 3 and 5 - 3 = 2. The result is D-2.
The structure of intervals in tricycle notation matches that of a mixed group. Meantone, like all regular temperaments, is a finite abelian group, which means you form intervals from a basis of two intervals (say, tone and semitone) with as many of either as you like.
The wheel numbers form a cyclic group known as the torsion subroup. Cyclic groups are often used in group theoretic models of equal temperaments such that they repeat about the octave. The wheel numbers also repeat every octave, and so are related to three equal divisions of the octave.
This leads to tricycle notation involving periodicity blocks that "have torsion". The commatic unison vectors are 225:224 and (81:80)³ or 531441/512000. Interpreting such "torsional temperaments" is tricky, and the simplest theories declare them to be ill-formed. Here, though, is a case where a temperament with torsion is meaningful.
Group theory tells us that a mixed group only has a single torsion subgroup. If you think you have two torsion subgroups (say you formed the union of two mixed groups) you can replace them with a single cyclic group whose cycle length is the product of the two being unified. Tricycle notation involves the union of the group of meantone names and the group of wheel numbers. It is possible to think of the meantone names as a mixed group. If you measure meantone using diatonic and chromatic steps, you could consider either to repeat about the octave. For example, make the diatonic names cyclic so that a C is a C is a C and use the number of chromatic steps to measure the interval in an octave-specific way. This doesn't lose scope because there's only a finite set of names within the octave if you restrict yourself to double sharps and flats, and a cyclic groups of note names is adequate for this. So, a cyclic group of 7 note names unifies with a cyclic group of 3 wheels to give a new torsion subgroup for tricycle notation with 21 elements. I can't find my book on group theory to see how this works, and I don't think it would be at all useful anyway because it's simpler to think of the torsion group of three wheel numbers combining with a rank 2 free subgroup corresponding to meantone temperament. It's an interesting complication to note though and will appeal to people who like bringing concepts from group theory into models of musical pitch.
Here's a I-IV-ii-V-I progression in C major showing pitch drift:
The last chord has drifted to C on wheel 3, as well as being a different inversion. In fact, this highlights a problem with voice leading in three part just intonation harmony that I might explain more. For now, let's add an ephemeral fourth part.
Now there's a chord sequence that returns to where it started in meantone, but drifts by a syntonic comma in just intonation. If it went around 3 times, it would end up on the same wheel it started on, but (if tuned to strict just intonation) it would sound lower.
If you go straight from I to ii, there aren't any pitches held over so you have the option of jumping to a "bad" fourth.
With explicit wheel indications, you can make it clear what D minor chord you want.
A dominant seventh chord includes a minor seventh. As such, its just tuning is ambiguous, and can be clarified with wheel indications. Voice leading naturally leads to a seventh with a wheel number of two, hence a ratio of 16:9.
The F held from the subdominant strengthens the cadence. The root is reinforced by a step on both sides on the spiral of fifths. A ratio of 9:5 would be simpler for the seventh, though, and leave the dominant seventh chord composed of a major and two minor thirds with 5-limit tuning. Making the wheels explicit guides you towards a way for this to work, and leads to a ii-V7-I progression.
This includes minor chord voicings that would be frowned on in common practice harmony, and are only included as examples. Take heart, though, that the ii is voiced as 6:10:12:15 giving simple ratios.
Tritone substitutions are a harmonic technique where you replace a dominant seventh chord with another a tritone away. The new chord can be interpreted as 4:5:6:7, and for proper meantone spelling should be written with an augmented sixth rather than a minor seventh. This exploits marvel temperament (225:224 tempered out), and ties in with the old rule that an augmented sixth chord resolves downward by a minor second. It happens to be a common progression that can be tuned with 7-limit chords such that the roots move by an interval with seven-ness, which some people consider to be an exceptional microtonal trick.
It also turns out that tritone substitutions resolve the pitch drift problem, so you can substitute I-IV-ii-V-I into this
and spice the chords up, confident that that notation will find a way for it to work in marvel temperament.
The usual way of writing just intonation is to start with a white-note scale tuned to a specific just intonation scale and adding accidentals for comma shifts. There are two problems with this approach that tricycle notation solves:
Tricycle notation solves these problems by replacing a fixed white note scale with three arbitrary pitch regions. Where the JI interpretation is obvious, you don't need to do anything special. (We could claim people have been writing in tricycle notation for hundreds of years if we were being silly.) As the pitch drifts, you only need to make a few wheel locations explicit to show what's going on, and because there are only ever three wheels the written complexity doesn't increase with drift.
Tricycle notation is also uniquely suited to adaptive tuning schemes, where the performers (or a computer) make small adjustments to correct for pitch drift. Because there's no fixed reference pitch, there's nothing to hide when the pitch drifts.
This might all be thoroughly unintuitive for real performers. I don't like to speculate on this because history shows that it's difficult to predict what new things people will come to find intuitive. So if you think this is a terrible idea and nobody will understand it, the best thing is to teach some performers the notation and show us all how much difficulty they would have with it.
Tricycle should work well with computer notation systems. There are simple algorithms that can determine the implicit wheels and lead to strictly tuned JI or marvel pitches. But because existing software isn't written around these principles, they're quite difficult to adapt. I'm working on an implementation in Lilypond but haven't finished yet.
It's possible to resolve ambiguities without giving a name to each wheel, or remembering the default wheel for any given pitch at a given time. You can do this by adding symbols for "a bit sharper than you might expect" and "a bit flatter than you might expect". No symbols mean "use the expected tuning". Sometimes there won't be a clear expected tuning, and most of the time there'll only be two plausible wheel allocations, so the relative wheel symbols aren't notating comma shifts relative to an expected pitch. They can still inherit accidentals designed for comma shifts, though, and will sometimes act like comma shifts relative to a fixed scale but more often be subtly different in a confusing way.
The expected wheel assignment algorithm may or may not pay attention to accidentals. It would certainly depend on the musical context, in particular the other pitches currently being played. Let's assume it ignores accidental and show the relative wheel interpretations for C major (and no other context).
|Pitch||Flatter alternative||Expected wheel||Sharper alternative|
I'll use ↑ for "sharper than you might think" and ↓ for "flatter than you might think". With these symbols, it's possible to give chords unambiguous tunings. For example, a D minor chord sharing a pitch with the expected F major tuning can be written "D↓ F A" to make it clear that the D is lower than you might think. A G major chord could be "G B D↑" but the context would probably make that clear anyway.
Say you want to use a C major chord with a Pythagorean tuning because you like it and don't care about 5-limit hegemony. You can call it "C E↑ G". There's no need to specify the wheels of C and G because they're obvious. If you used note names based on a Pythagorean diatonic along with a kind of musica ficta system for assuming 5-limit tunings, it wouldn't be possible to say when you really meant a Pythagorean third like this.
Another existing form of notation is to use comma shifts relative to a fixed 5-limit scale, usually chosen as C major with D↑ (9/8). This would keep D minor as "D↓ F A" and a Pythagorean C major as "C E↑ G". But it means you have to remember that the fifth between D and A is a wolf, and should be played as such if there are no contradictory comma shifts. If you use a music ficta rule to say that there are no wolf fifths, suddenly it's impossible to say when you really want a wolf. With relative tricycle notation, all you do is specify the unexpected tunings of both pitches, for example "D↑ A↓". A fifth written like this will always be a wolf, and a fifth with relative wheel symbols pointing in the same direction should always be a 3:2. A fifth without relative wheel symbols will probably be a 3:2 as well. You don't need to remember a special scale context to decide this.
Relative wheel symbols look like the clearest way of notating tricycle notation for human performers. We can't be sure until human performers try using them, though.
You can extend tricycle notation to the 11-limit by, for example, tempering out 385:384 to give 11-limit marvel. This doesn't give any interesting complications, though. It's also nice to make use of half-sharps as available in Tartini-Couper and other "quartertone" notations that also work with 31-equal and might even have been designed for it. It's fairly obvious how to use half-sharps for the new 11-limit intervals, but not so obvious what wheels they should go on, so let's think about that.
The obvious part is that 11:8 should be written as a super-fourth, or a half-sharp above a perfect fourth. The question is whether to make the super-fourth cover two wheels, like a major third or perfect fourth, or three wheels like a perfect fifth. To resolve this, let's look at the 11:9 neutral third. Half-sharps make this bisect the perfect fifth, and leave it equidistant between major and minor thirds. Should we define it to cover two wheels like major and minor thirds? I say "yes" because the wheel structure gets too fragmented otherwise. The cost of this decision is that we lose the ability to distinguish 11:9 from 27:22 which implies 243:242 being tempered out. I accept this penalty. The consequence is that 11:8 has a wheel number of two, like a fifth.
The mathematical result is that half-sharps and half-flats that don't change the wheel lead to shifts of either 55:54 (the amount by which 11:9 is wider than 6:5) or 45:44 (the amount by which 5:4 is wider than 11:9). The two ratios being treated the same leads to 243:242 being tempered out.
This looks like it works well enough, and allows you to notate 11-limit music with reasonable accuracy. If I start with 11-limit ratios and use linear algebra to convert them into half-sharp notation plus wheel indications, I find some results end up implying other intervals being tempered out, reducing the overall accuracy. I think this comes from three syntonic commas being tempered out, and there are 11-limit intervals equivalent to three syntonic commas that also get tempered out. I haven't worked out the details yet. It looks like you can get a long way with the accuracy of only 225:224 and 243:242 being tempered out.
A C harmonic-eleventh chord is now written C-1 E-2 G-3 A♯-1 C-1 F+-3 with "+" as the symbol for a half-sharp.
Now that we've got half-sharps and half-flats, how about using them to write 7:4 as a subminor seventh instead of an augmented sixth? I think this is a good idea, and will generally allow multiple ways of writing intervals, which is a good thing for the same reason that enharmonic equivalences get used in conventional notation. The question is what wheel to put the 7:4 subminor seventh on. I'll go through arguments for the two obvious spellings before announcing the decision I made a long time ago.
The first way, then, is to make both pitches of an 8:7 sit on the same wheel whether it's spelled as a supermajor second or a diminished third. This means the half-sharp widens a 10:9 major second into an 8:7. The mathematical consequence of this is that "half-sharp" on the same wheel refers to 36:35. Making 36:35 equivalent to 45:44 means tempering out 176:175 leading to würschmidt temperament
The other way is to make 8:7 written as a supermajor second cover two wheels like major thirds and perfect fourths. This looks like an unnecessary complication because it means the number of wheels covered by a ratio depend on its spelling, and also that 8:7 written this way has a wheel number of one but 7:6 written consistently has a wheel number of zero, giving the impression that 8:7 is the larger interval. But let's follow the logic anyway, to comply with my pre-conceived notions. 8:7 is now on the same wheel as 9:8, so the half-sharp on the same wheel means an interval of 64:63. It also happens that a half-sharp changing the wheel refers to 36:35, which is nice because this interval has a larger span and expanding by a wheel generally makes an interval a little bigger (while holding the other spelling constant). It also means that 64:63 is the same kind of thing as 54:55 and 45:44 and 36:35 is the same kind of thing 33:32 (the half-sharp that makes an 11:8 super-fourth larger than a 4:3 perfect fourth). It also happens that making 55:54 and 64:63 equivalent implies tempering out 385:384 and this leads to miracle temperament.
My carefully concealed opinion is that miracle temperament is really good so if there's a plausible argument for tricycle notation becoming a way of notating miracle, let's go for that. This is interesting because previous notations for miracle didn't look anything like this.
A G harmonic-eleventh chord can now be written G-3 B-1 D-2 Ft-2 G-3 C+-2 with "t" as the symbol for a half-flat.
Here's another list of intervals including alternative spellings for the sevens.
|Ratio||Name||Wheel Number||Alternative Name|
|16:15||minor second||0||diatonic semitone|
|15:14||minor second||0||septimal semitone|
|10:9||major second||0||minor tone|
|9:8||major second||1||major tone|
Note that when there are two intervals with the same spelling the larger one (as a pitch difference) has the next wheel number. This preserves the idea that moving to the next wheel works like raising by a comma.
To convert from a ratio to a meantone spelling and a wheel number, you can factorize and use a temperament matrix. There are two different mappings for the different ways of spelling intervals involving factors of seven. The first is called spectacle and corresponds to 7:4 as an augmented sixth.
The other mapping is called mirwomo and maps 7:4 to a subminor seventh.
To deduce the spelling of a ratio-space vector, convert it by one of these matrices. The diatonic steps tell you how many times to increment the pitch's letter, in alphabetic order modulo 7. Alternatively, add one to get the "nth" name, for example there are two diatonic steps to a third. Divide the number of subchromatic steps by two to get the number of piano keys to move up by. If there are an odd number of subchromatic steps, you have a "sub" or "super" interval.
Here's the spectacle mapping involving chromatic steps
And here's a similar mapping for mirwomo.
E+--G♯--B+--D♯------A♯------E♯------B♯ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ E---G+--B---D+--F♯--A+--C♯--E+--G♯--B+--D♯------A♯------E♯------B♯ \ / \ / \ / \ / \ / \ / \ / \ / \ \ / \ / \ / \ / \ / \ / \ / \ / \ \ / \ / \ / \ / \ / \ / \ / \ / \ Et--G---Bt--D---F+--A---C+--E---G+--B---D+--F♯--A+--C♯--E+--G♯--B+--D♯------A#---- / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ E♭--Gt--B♭--Dt--F---At--C---Et--G---Bt--D---F+--A---C+--E---G+--B---D+--F♯--A+--C♯ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / ----G♭------D♭--Ft--A♭--Ct--E♭--Gt--B♭--Dt--F---At--C---Et--G---Bt--D---F+--A---C+ \ / \ / \ / \ / \ / \ / \ / \ / \ \ / \ / \ / \ / \ / \ / \ / \ / \ \ / \ / \ / \ / \ / \ / \ / \ / \ F♭------C♭------G♭------D♭--Ft--A♭--Ct--E♭--Gt--B♭--Dt--F---At--C \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / F♭------C♭------G♭------D♭--Ft--A♭--Ct 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
In fact, this doesn't show 7-limit pitches in the middle of the triangles because they would end up on the wrong wheels however you spell them. So it isn't at all specific to miracle, but I saved it until now anyway. Some pitches are equivalent by miracle rules, for example Ft-2 and E♯-3 or B+-2 = C♭-1.
If you start somewhere in the middle, you can probably move to any other pitch on this lattice without ambiguity if you give the wheel indications. If you keep expanding the lattice, it will eventually become ambiguous whether you're supposed to jump left or right.
Here's the chain of secors. Each pitch name sits on an appropriate wheel.
D♯ E F G♭ G+ A- A♯ B C D♭ 1 1 1 1 2 2 3 3 3 3 D+ Et Ft F♯ G A♭ A+ Bt Ct 1 1 1 2 2 2 3 3 3 C♯ D E♭ E+ F+ Gt G♯ A B♭ 1 1 1 2 2 2 3 3 3 B+ C+ Dt D♯ E F G♭ G+ At 1 1 1 2 2 2 2 3 3 A♯ B C D♭ D+ Et Ft F♯ G A♭ 1 1 1 1 2 2 2 3 3 3 A+ Bt Ct C♯ D E♭ E+ F+ Gt 1 1 1 2 2 2 3 3 3 G♯ A B♭ B+ C+ Dt D♯ E F G♭ 1 1 1 2 2 2 3 3 3 3 G+ At A♯ B C D♭ D+ Et Ft 1 1 2 2 2 2 3 3 3 F♯ G A♭ A+ Bt Ct C♯ D E♭ 1 1 1 2 2 2 3 3 3 E+ F+ Gt G♯ A B♭ B+ C+ Dt 1 1 1 2 2 2 3 3 3
This loops round, so you can't notate all these pitch clases at the same time without ambiguity. But it does show that you get quite a lot of distinct miracle pitches.
I think this is a 31 pitch-class "canasta" scale.
B♭ Bt B B+ Ct C C+ C♯ D♭ Dt D D+ D♯ E♭ Et E E+ 1 1 1 2 1 1 2 2 1 2 2 2 3 2 2 3 3 Ft F F+ F♯ G♭ Gt G G+ G♯ A♭ At A A+ A♯ 2 3 3 3 3 3 3 1 1 3 1 1 1 2
Note that each pitch name is distinct even without the wheels. This is because the wheel-less notation is already good for 31-equal.
Here are the 41 pitch classes of "stud loco"
B♭ Bt B B+ B+ Ct C C+ C+ C♯ D♭ Dt Dt D D+ D♯ D♯ E♭ Et E E E+ 1 1 1 1 2 1 1 1 2 2 1 1 2 2 2 2 3 2 2 2 3 3 Ft F F F+ F♯ G♭ G♭ Gt G G+ G+ G♯ A♭ At A A+ A♯ A♯ 2 2 3 3 3 2 3 3 3 3 1 1 3 1 1 1 1 2
The additional pitches make pairs with the same half-sharp names and lie on consecutive wheels.
Here are two versions of the 11-limit tonality diamond.
1/1 G-3 D-2 12/11 At-3 Et-2 11/10 At-1 Et-3 10/9 A-3 E-2 9/8 A-1 E-3 8/7 A+-1 E+-3 or F♭-2 7/6 A♯-1 Ft-2 or E♯-3 6/5 B♭-1 F-3 11/9 Bt-1 Ft-1 5/4 B-1 F♯-3 14/11 B+-1 G♭-2 or F♯+-3 9/7 B+-2 G♭-3 4/3 C-1 G-3 11/8 C+-2 G+-1 7/5 C♯-2 G♯-1 10/7 D♭-1 A♭-3 16/11 Dt-1 At-3 3/2 D-2 A-1 14/9 D♯-2 A♯-1 11/7 D♯-3 A♯-2 or B♭t-1 8/5 E♭-2 B♭-1 18/11 Et-2 Bt-1 5/3 E-2 B-1 12/7 E+-3 or F♭-2 B+-2 or C♭-1 7/4 E♯-3 or Ft-2 B♯-2 or Ct-1 16/9 F-2 C-1 9/5 F-3 C-2 20/11 F+-2 C+-2 11/6 F+-3 C+-3 2/1 G-3 D-2
Harry Partch's "there's nothing special about 43" 43-note scale
G G G+ G♯ A♭ At At A A A+ A♯ B♭ B♭ Bt B B+ B+ B♯ 3 1 1 1 3 3 1 3 1 1 1 3 1 1 1 1 2 2 C C C+ C♯ D♭ Dt D D D+ D♯ D♯ 1 2 2 2 1 1 1 2 2 2 3 E♭ Et E E E+ Ft F F F+ F+ F♯ G♭ Gt G G 2 2 2 3 3 2 2 3 2 3 3 2 2 2 3
and the alternative version from "Exposition on Monophony" 1933.
G G+ G+ G♯ A♭ At At A A A+ A♯ B♭ B♭ Bt B B+ B+ B♯ 3 3 1 1 3 3 1 3 1 1 1 3 1 1 1 1 2 2 C C C+ C♯ D♭ Dt D D D+ D♯ D♯ 1 2 2 2 1 1 1 2 2 2 3 E♭ Et E E E+ Ft F F F+ F+ F♯ G♭ Gt Gt G 2 2 2 3 3 2 2 3 2 3 3 2 2 3 3
This all works quite well in the 11-limit. 13-limit extensions are problematic, though. The simplest way is to call 13:8 an exact neutral sixth, which implies tempering out 144:143 to give a miracle extension called "miraculous". It's reasonably good but doesn't represent 13 very well because 13:8 isn't very close to a true neutral sixth.
You can also set 13:8 to be on a different wheel from 11-limit major, minor, and neutral sixths. This allows more distinction of sixths, but there's a cost in the wheels getting more fragmented, and anyway other commas get implied and you end up with a mess of wheel indications that I don't like at all.
I think the best thing is to assume a simple pitch set, and allow for irregular tunings, for example having neutral sixths unequally dividing major and minor sixths. Either that or add new indications for 13-limit pitches, maybe arrows. But in a lot of cases it'll be simpler to give up on the tricycle concept and go for a traditional 13-limit notation. (This, of course, assumes that we can speak of traditions in 13-limit notation.)
I hope that anybody who came here questioning the usefulness of tricycle notation has had their doubts throughly dispelled.