**(note: I caught a mistake in the file, fixed it, and uploaded a new version)**

Please find my scale I call "Yer" above, notated in eleven different Sagittal notations. I've come to find that working with the Sagittal notation system is as much an art as it is a science, especially when working with a strange-ish scale such as Yer, so I welcome any and all feedback on my aesthetic and technical choices, especially if anyone has an opinion about which method I should settle on. I think this study may also be of general use for understanding and comparing the various notations of the Sagittal system. At the very least I hope this post is of some interest to folks and not just a bunch of nonsense.

To be clear, this amount of effort is not required or normal to notate one's music in Sagittal. Another reason I undertook this study was to prepare myself for implementing a Sagittal notation calculator which is capable of automating many of the tasks I did manually here.

Initial struggle with Yer

I started down this path of obsession with Sagittal when I decided that both of the two microtonal notation systems I understood already — Ben Johnston's and EHEJIPN — were unsatisfactory for notating Yer. Because Yer is an Euler-Fokker genus of four primes, one of its pitches is the combination of every one of these four primes, and so in both of these notations that pitch requires an overwhelming

*four*accidentals. I was seeking a notation system which could capture the pitch information of Yer in a more compact way, and fortunately one of Sagittal's selling points, according to the Xenharmonikôn article introducing it to the world, was just that:

So I thought I would give it a try.One very important characteristic of the Sagittal system is the avoidance of the clutter resulting from multiple side-by-side microtonal symbols, such as occur in the notational systems of Bosanquet, Ben Johnston, and Paul Rapoport, and most recently in the "Extended Helmholtz-Ellis JI Pitch Notation" of Marc Sabat and Wolfgang von Schweinitz.

Sagittal was designed to best suit the most popular pitches, and rightfully so. Unfortunately for me, however, Yer does not use these more popular pitches, and as a consequence, Sagittal's JI notation does not precisely represent most of them. That said, at the highest level of precision it affords, it approximates them within fractions of a cent, so its accuracy was not my concern. My issue rather was with how the symbols did not visually represent the harmonic composition of each pitch. I was losing out on one of the expressive powers of the Sagittal system: the flags its symbols are comprised from encode pitch information — not only by their physical size corresponding to the size of the alteration, but by their shapes encoding certain prime numbers — so if one commits to learning what these flags represent, even if one hasn't memorized the totality of a symbol, one can infer the nature of its pitch alteration from the elements it consists of. So in fact, the situation was worse than me simply missing out on the power of this aspect of Sagittal; I felt this aspect was amounting to noise, interfering with what my notation

*was*trying to say.

Yer is an EFG of high primes — 11, 13, 17, and 19. Sagittal's defining commas are 23-limit, so limit wasn't the problem. Sagittal does actually have commas for many combinations of primes 11 through 19, however, mostly it has them in terms of translating from one to the other, e.g. 11:19, 13:17; Yer on the other hand always has them on the same side, e.g. 11.19, 13.17.

Yer's raison d'être is alienation from the lower primes 3, 5, and 7. Almost all of Sagittal's commatic alterations make liberal use of 3, because the Sagittal system is built upon chains of fifths. So I would even say that Sagittal's symbology was acting even worse than noise: it was antithetical to the ethos of my scale.

That being the case, I resigned myself to accepting this cost as a necessary trade-off for a practical notation. Readers of my notation could at the very least leverage the remarkably readable and distinct shapes Sagittal provided for each of the pitch alterations, to be able to rapidly memorize their association with cents deviations. The notation would certainly not help them learn to think of the pitches in terms of their true underlying JI content, but they could probably due my music justice. In any case, it'd be better than mere cents deviations to notate the pitch alterations.

12R

I had planned from the start to also provide absolute cents deviations, but it was not until I actually began placing them on the page that I noticed a problem: I had just written down a cents deviation that was downward, even though my symbol pointed upward. This happened because my pitch was about 297¢, which is below the standard tuning nominal at 300¢, but above the Pythagorean one at 294¢. Notating the cents deviation relative to the Pythagorean (3-limit just) thirds would be the correct way to go given how Sagittal's JI notation works (if you didn't do this, you might even end up with the same Sagittal symbol representing more than one different cent deviation value). I had made this mistake because I just don't compose music rooted in chains of fifths; I don't think that way. Even if my performers did, as I've explained already, I thought of the fifths as antithetical to Yer. I had been willing to accept that they formed the underpinning of the commatic intervals the symbols were based on, but I was unwilling to accept that my performers (many of whom might not even understand JI) would have to learn a Pythagorean diatonic scale and then fight it in order to play my music.

Dave made a compelling point:

Nonetheless, at this point, I felt compelled to try out Trojan, the 12edo relative notation. The symbols deployed in the Trojan notation for Yer would bear no closer of a relation to the underlying truth of the JI structure of Yer, but would at least leverage modern performer's familiarity with standard tuning.Even though Yer tuning doesn't have any pure fifths, it has some near-misses, of 692c, 711c and 718c. Pure fifths are such strong attractors, it could be useful, in effect, to have the notation tell you where they are so you can avoid them.

As I tried out Trojan, I reflected on the first Sagittal notation I ever used, for another custom scale of mine. I had thought of that notation as a Trojan notation at the time, but now that I knew more, I realized it was actually an altogether different kind that Dave Keenan had devised just for me, just for that type of situation. This system, like EHEJIPN and Johnston notation, can give rise to a great proliferation of accidentals for just a single pitch; however, instead of being based on rational intervals, it is based on increasingly small bisections of standard tuning semitones. I went back over this system with Dave, revising it slightly, and we gave it the name Binary 12R notation (originally "multi-Trojan"). I do not think it is the best notation for Yer, but I have provided it for comparison.

I have also notated Yer with Sagittal's Prime Factor notation (originally "multi-Sagittal"), which is its analog to EHEJIPN, insofar as it uses multiple microtonal symbols. Again, I do not think this is the best choice for Yer; avoiding so many symbols is the reason I switched to Sagittal in the first place. I also note that Prime Factor notation, as it tends to involve less reliable counts of fifths in the sum total of its commatic alterations, results in way more sharps and flats in the final notation than monosagittal JI notation.

EDO

Regarding equal divisions of the octave like 12, though, I decided it might be interesting to see how Yer fared by notating with a Sagittal EDO notation for an EDO which closely approximates it. As long as the symbols were approximations anyway, I reasoned, if an EDO happened to give a more intuitive notation, I’d consider it. The smallest EDO which provided a sufficiently close approximation of Yer's pitches was 57, which I thought was an okay choice because it is a multiple of 19, a popular EDO. 57 happens to be one of those EDOs which Sagittal provides two notations for: one based on its native fifth, and another based on the fifth of an EDO which is a multiple of it — in this case, 171edo. So I gave both of them a go.

For the native fifth, using the mixed version of Sagittal (i.e. not rejecting the conventional sharp and flat symbols) results in only needing to learn two different symbols (and their mirrors). However, it also results in conflating several of Yer's pitches which are separated by comma-sized steps. Plus, performers must learn the cents values of the 57EDO fifths chain, which is an even more obscure set of values than the Pythagorean fifths chain (57EDO's chromatic semitone is half the Pythagorean size!).

With the 171EDO subset notation, I could go for the pure version of Sagittal while only requiring performers to learn four different symbols (and their mirrors). It still has the problem where several comma-separated pitches are not distinguished, however, the cents values of the 171EDO fifths chain are much closer to Pythagorean.

Not quite satisfied with either of these 57EDO notations, I looked to a larger EDO that might differentiate these pitches and do a better job approximating them while still doing a good job with the underlying chain of fifths. I found that in 137EDO. However, I soon discovered that 137EDO was not one of the EDOs that had yet been notated by the Sagittal community (each EDO is its own special problem to be solved; they're all related to each other, but if one hasn't been notated yet, it's not necessarily trivial to decide on its symbol set). And so, with a little help from friends on the forum, I came up with something. That notation is included here too.

Custom

I had figured out how to notate an uncharted EDO in Sagittal. But my ultimate trial was yet to come. I set out to design my own JI Sagittal notation. I wondered: what if the underlying commas the symbols represented

*were*the exact ones I needed? Part of the power of Sagittal is its flexibility. Each symbol (plus diacritic, if any, combination) in Sagittal has a default value which it "really represents" in the absence of other specification, but it also has a "capture zone" of secondary values which it can represent on the basis of it notating that pitch better than any other available symbol in the given notation. Sagittal's JI notations come in a series of precision levels: low, medium, high, very high, and extreme (perhaps even one day "insane" level precision will be supported). When a symbol is introduced at a lower precision level than another, this is because in some sense it has more notational "gravity"; it's usually lower-limit, more important, more popular, etc. So what I did was find a set of commas representing Yer's pitches that fit inside the capture zones of symbols at the precision levels they are introduced at. I figured that each of Yer's pitches would turn up at least one comma-sized interval (whether schismina, kleisma, comma, or diesis), and that hopefully once I had these all laid out before me, I might be able to pick one for each such that the resultant set was evenly spaced enough across the range from 0¢ up to a large diesis (~68¢) such that I could assign one Sagittal symbol to each. If I was lucky, I thought, I might even find enough commas such that I'd have enough options to assign the symbols such that I could pick them such that the flag arithmetic worked out, such that the visual elements of the symbols could communicate the underlying truth of Yer's pitches as I had so hoped for at the outset. After some finagling I did get something working. I had to make only one exception. Yer's pitches prominently involve two superparticular commas: the Blumeyer comma, 2431:2432, and the yama comma, 208:209. The ability to assign these two commas to the two different diacritics available in Sagittal would be instrumental in making this dream of a custom Sagittal JI notation for Yer possible. The Blumeyer comma, at 0.71200¢, does fit inside the capture zone of the symbol (whose default value is the 5.7.13-schismina), which ranges from 0.211¢ to 0.773¢; however, the yama comma, at 8.3033¢, does not fit inside the capture zone of the symbol, which is from 1.757¢ to 2.740¢. I fudged this to make my notation work. I'm pretty happy with the results. I didn't so much achieve flag arithmetic, however, I did make it the case that adding a 17 to a pitch almost always mirrors the symbol horizontally (at least the mixed version of the symbol).

As a final stunt, I imagined what a Sagittal notation might look like if I had total control over the validity of flag combinations. Essentially I wanted a notation like Prime Factor, except instead of adding an entire new microtonal symbol per prime, I would merely add a new flag to the one symbol. This notation is not valid Sagittal like the others. I'm satisfied with the economy and expressiveness of this notation with respect to the JI composition. However, I feel there is a critical flaw with this version: even when the constituent flags of a symbol sum to a negative alteration, the symbol still points up. One of the key benefits of using single-microtonal-symbol Sagittal (as opposed to e.g. EHEJIPN) is that it's immediately obvious whether the total alteration is upwards or downwards; without this effect, I am pretty sure this notation is garbage. Nonetheless, I thought it would be interesting to share, if only as a foil.

One thing you may notice is that most of my examples are based on a 1/1 of B. I know this is not the most common choice, but again, Yer is not your average scale. B was chosen for Yer for a similar reason that C, G, D, or A are often chosen for other scales: because it results in the fewest sharps and flats. In the case of the most accurate JI notation, technically I based the scale on a Cb, because according to my calculations this was the Pythagorean nominal which as the root pitch minimized the total error of the default values of the Sagittal symbols relative to Yer’s true pitches. And when I devised my own custom Sagittal JI notation, it worked out that D was the best choice for locating as many pitches as possible within a few thirds of 1/1.

JI

Looking back on it all, I think that probably the best notation for Yer should be one of the JI notations I tried out at the beginning.

I took three different approaches to a JI notation for Yer. The simplest approach uses only up to the medium level precision's symbols. This entails a total of 8 symbols (using the mixed version helps here) to learn. Many intervals are consistently notated (in terms of the visual flag arithmetic), the pitches that are separated by comma-sized steps are clear to find, and the nominals are fairly well-distributed (i.e. we don't end up with 6 pitches based off B and none off E or anything like that). However, while this simple JI notation does not conflate any pitches, it does conflate some intervals, i.e. many symbols represent more than one alteration from the chain of fifths. If one's scale involves n different values of deviation from the chain of fifths, is it acceptable to have anything other than n different symbols (setting aside questions of pure vs. mixed versions of Sagittal for now)? As far as I know this is an open question.

The accurate approach of JI I provided as another extreme example to foil the others against. This degree if accuracy is clearly not worth the requisite sacrifices. As I mentioned, according to my calculations, I ran the numbers and found that it was not possible for the extreme level of precision to do any better than this at approximating Yer's pitches. Cb is tied with two other choices of a nominal set to 1/1 for providing the most accurate possible representation of the pitches, with 8 of the 16 of them exact, and only a total of 0.633 cent error across all 18 (I chose Cb from these three because it was most similar to B, which was the choice for most of the other notations I tried). Pretty impressive. The drawbacks of this accuracy should be immediately apparent: many more shafts to the symbols, many more diacritics, a different symbol for pretty much every pitch, many places where a pitch will be higher than another but use a nominal lower than other (the accidentals cause them to cross), and of course there is the matter of even the pitch representing 1/1 requiring an accidental, etc.

Finally, for my balanced approach to JI notation: this uses the extreme JI precision level, with some thinking attributed to the choices among the 2-3 options available for each pitch (based off different Pythagorean nominals). So it uses extreme accuracy, but doesn't go so nuts over accuracy that it garbles all the symbols and such like the most accurate version possible does. Another benefit is that it uses the same accidentals for 11, 17, and 19 as are used in the Prime Factor Sagittal notation. For this notation I also decided to experiment with dropping diacritics in place while providing a key. According to Dave:

More on that topic can be found here: viewtopic.php?f=6&t=15)....although the precise sagittal notation of some commas may require diacritics (accent marks), it is quite permissible (and indeed suggested) to omit them from the staff when (a) your precise tuning is described elsewhere, and (b) this would not lead to a failure to distinguish different pitches.

I note that a standard format for a key has not yet been defined. I wasn't totally sure how to approach it. In particular, would readers assume if I gave an alias for a symbol that it would apply to its vertical mirror as well? If so, I couldn't simplify to because Yer also includes a . The same issue applies to Yer for and , as well as to and (which have two different diacritics). I ended up including only the aliases for symbols without this problem, and oriented the symbol downwards if that was the one situation where the alias applied.

I believe this key is important to include, because sometimes after dropping accents, the bare symbol is no longer the best approximation of the pitch; for example, Yer's pitch 13⋅17 has the accidental at the very high precision level. But if you were to drop the diacritic and leave only , that symbol could be interpreted as a merely high precision level symbol, but at the high precision level, 13⋅17 is better approximated by than , so if one did not specify that was standing in for , one could end up with a subpar approximation.

Perhaps the key should also be expected to state which type of Sagittal notation is being used. Although I know that part of the beauty of the Sagittal system is its interoperability, that it is not critical to be explicit about this, since the symbols mean about the same thing in any context.

Conclusion

So I guess this was all also a sort of stress test for Sagittal. I know Sagittal is meant to be open for extension, to adapt to the needs of its community, as long as core principles are understood and respected. I hope I have done a good enough job respecting that ethos here. It’s clear that Sagittal is solving a fascinatingly complex problem and this has given me a ton of respect for how much vision, ingenuity, and elbow grease has gone into it. I still feel I’ve only begun to scratch the surface — I don’t completely have my head around the periodic table of EDOs on the main page, and have only the slightest inkling of the decision making process that went into winnowing down the commas that now make up the approximately 233EDA (where the "A" stands for apotome = chromatic semitone) of the extreme JI precision. But I look forward to participating in this community and continuing to learn. I believe Sagittal has achieved its goal to design the most powerful and flexible microtonal notation system, and I plan to use it for my music moving forward.

Well, that’s all I have to say about this. Again, I welcome any and all critique of this effort.