You see a similar feature in the "global notation" proposed by Dr. Andrew Killick.
And I'm sure other examples could be turned up.
Initially I dismissed these proportional pitch notation strategies as fundamentally at odds with symbolic strategies, such as Sagittal. However, that was before I had a key insight (while sipping on some Sagittal beer, haha): I had seen proportional and symbolic strategies combined when used horizontally, for duration, so why couldn't they be combined just as readily when used vertically, for pitch?
In fact, the majority of my experience with proportional duration notation had been integrated with the traditional symbolic notation. Turning it on is an easy-to-use feature of many digital music notation applications: simply check a box, and now if a quarter note enforces 24 pixels of space between it and the next note, an eighth note will enforce 12, or a triplet 8, and so forth, but they retain the symbolic indicators of duration of their noteheads, stems, flags, beam numbers, and so forth. The proportionality is not necessarily a replacement for the symbolic notation, but simply complements it, and reinforces it. Taking this:
So what would it look like to take symbolic pitch notation, such as conventional notation, or perhaps Evo'd or Revo'd with Sagittal, and imbue it with proper proportionality? The main problem I'd be seeking to solve is this: that the nominals ABCDEFG are equally spaced on the staff, while in reality there is a greater distance between some of them than others.
I was surprised to find little evidence of any proposition to address this issue. I even found this statement in a paper called "The uses of space in music notation" by John Sloboda, which seems to fly in the face of fact:
Of course I agree with the first sentence. But unless equal spacing of staff-lines makes pitch distance accurate by some other reckoning I don't understand, I have to disagree with the second sentence.Although not logically necessary, it has always been standard practise to make the distances between adjacent staff-lines equal to one another. This has meant that the vertical distance between notes is an accurate measure of the pitch distance between them.
Then I came across this interesting article, which describes that early in the history of the lined staff notation, instead of a clef, certain staff lines were labeled with their letter, and that there was a relevance to which of the two were chosen:
The choice to mark F and C was not an arbitrary one. As we learned in Unit 3, our gamut of pitches is comprised of a series of whole steps punctuated by half steps in certain spots. In our major scale, those half steps occur between 3 and 4, and between 7 and 1. Look back to the letter-name chart at the beginning of this Unit, and you’ll see that 3–4 corresponds to E–F, and 7–1 corresponds to B–C. Medieval musicians marked F and C because they were the upper notes of each half step.
So there was at least some effort to somehow call attention to the points of irregularity. But why not just correct the irregularity? Aside from maybe: irregularly placed staff lines don't look as nice?
Let me describe the end result, in case it's not already clear in your mind (sorry, I've nabbed images for all the things I'm making comparisons to, but haven't taken the time to prepare images of my own proposition yet). To be clear, it's a bit different from hybrid spatial+symbolic notation for duration, because for pitch we'd only be spatializing with respect to the fifths, and then the symbols would take it from there.
The space between each pair of staff lines spans two nominals, since one nominal lives in the space between them. So, the two smaller distances (B to C, and E to F) are always paired with one of the five longer distances (A to B, C to D, D to E, F to G, G to A). In other words, there will be two sizes of space between staff (or ledger) lines, but they won't be directly proportional as are the long and short distances between nominals, rather compounds of those short and long distances: some are short+long, and some are long+long. In 12-EDO, the short distances are 100¢ and the long distances are 200¢. So the pitch distance covered from one line to the next is either 100¢ + 200¢ = 300¢ or 200¢ + 200¢ = 400¢. That is to say that the proportion between the short and long distances between staff lines should be 3:4, or 1:1.333. And the pattern of short and long distances would proceed like this: LssLsLs (you can work it out yourself).
In the case of a justly intonated staff, the short distances between nominals are diatonic semitones of 90.225¢ and the long distances between nominals are Pythagorean whole tones of 203.910¢. In other words the proportion is slightly more lopsided between the short spaces between staff lines and the long spaces between staff lines, being more like 1:1.386 (which is close to 5:7, or even closer to 31:43), because 90.225¢ + 203.910¢ = 294.135¢ and 203.910¢ + 203.910¢ = 407.820¢, and 294.135¢:407.820¢ ≈ 1.386.
The most extreme cases would be the 5n-EDOs for which BC and EF are 0 steps; in these cases, the long staff spaces will be 2x the width of the small.
I think if you did this, you'd also want to place the noteheads at the correct position within the spaces. That is (in JI or an EDO close to just fifths such as 12-EDO) a space between B and D, notes with the pitch C should appear about a third of the way up in the space.
In order to avoid wreaking havoc with symbols (such as Sagittal's!) which are designed carefully with respect to the spacing of staff lines, I think if you do any respacing of staff lines, it should involve strictly increasing the space. That way no element of any musical symbol would fall across a staff line if it hadn't before. Instead, it would be the case that more elements of musical symbols would fall on empty space than they did before. I don't think it would be a perfect solution, but certainly would mitigate its impact. As for clefs, which span the entire height of the staff: I guess just stretch them evenly, even if that means they don't end up looking like they were expressly designed to lay atop an irregularly spaced staff.
Perhaps making this happen would be more work than it was worth. Perhaps it's hopeless perfectionism. Or perhaps there's even some benefit to the irregular spacing I haven't determined. I'm just surprised I can't find any discussion of it. I do think it would be really helpful if standard music notation exhibited one of the excellent qualities of the "piano roll" notation that is often used in DAWs, where nominals are spaced proportionally:
I did also find this patent from 2005 for a "musical notation system ... wherein equal sized pitch intervals are represented by equal sized vertical displacements on a musical staff" which may be of interest to some folks.
*Though I also came across a proposed notation for location of sound within 3D space, and that being another interest of mine, I am already working with its designer on getting it included in SMuFL!