proportional/spatial + symbolic notation

[cmloegcmluin]

I was doing some research this weekend to help me contextualize Sagittal with respect to other pitch notations. In particular I became interested in proportional notation, sometimes also called spatial notation. This does not have to do with notating the 3D location of musical sounds in space* but with placing music symbols on the page such that their distances are proportional with respect to a musical property. I have most commonly seen this done horizontally, for duration, e.g. a note which lasts twice as long as another should have twice as much space between itself and the next note. But the concept may also be applied vertically to pitch, e.g. a major third should appear on the page slightly closer to the fifth while a minor third should appear on the page slightly closer to the root. I first came across such a thing in the JI journal 1/1, in the work of David Canright, where he calls it "rational notation".



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:



to 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:

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.

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.

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!

[yahya.abdal-aziz]

To start near the end of your post: Another excellent quality of the piano-roll notation (which I think of as a "player piano", or pianola, notation, having when a child delighted in my grandparent's instrument of that type) is this:

• that it shows both the start and the end of each note.

As to vertical height corresponding proportionately to pitch, we are of course talking of a logarithmic scale, since - even in the inaccurate heights accorded to pitches in the traditional notation, every octave is always the same total count of lines and spaces. The inaccuracies have two sources:

• Every two successive nominals have the same height difference on any staff, whether they are tones or semitones.

• Any "great staff" (system of two staves, one treble above one bass) has an often arbitrary amount of space between the bottom line of the top staff and the top line of the bottom staff. Specifically, the height of the gap between A3 (the A below middle C) and E4 (the E above middle C) is usually much greater than that between A2 and E3 on the bass staff, or equivalently, between A4 and E5 on the treble. Years ago, I remember asking why the ledger line for middle C wasn't equidistant from both staves, while still just one space away from either. Or, putting this another way, why isn't the grand staff an 11-line staff, with the middle (C) line perhaps dotted, dashed, or coloured red?

An accurately-logarithmic vertical pitch scale would probably suit dodecaphonists quite well, and could easily be implemented with a set of seven lines (and the spaces between them) for every octave, with the first and seventh line perhaps marked out specially, say in red or with a slightly thicker line, e.g. the [dodecaphonists']

• treble staff would consist of seven lines C4 D4 E4 F#4 G#4 A#4 C5;

• bass staff of C3 D3 E3 F#3 G#3 A#3 C4;

• upper treble staff of C5 D5 E5 F#5 G#5 A#5 C6;

• lower bass staff of C2 D2 E2 F#2 G#2 A#2 C3;

with an extra set added above or below as necessary to cover the performers' entire instrumental ranges. Logically these staves would simply use an octave number as clef. Such a great staff, of 6n+1 lines to span any n octaves, every 6^th line being notably different, would have the advantage that no , , or accidental is necessary to correctly notate any atonal dodecaphonic music. Plus, it would serve as a useful pitch notation for most solo diatonic music - easy cases being pentatonic and non-modulating tonal music.

But bearing in mind the principle you enunciated, that "the space between each pair of staff lines spans two nominals, since one nominal lives in the space between them", a dodecaphonist who eschewed or ignored all harmonic relation between notes with different names, thus preserving only the ancient principle of octave equivalence, would probably want their 12 notes per octave to each have its own, equally important and unadulterated by accident[al], nominal; say, O P Q R S T U V W X Y Z. And each nominal would have its own position on the staff, or place. Then the dodecaphonic [sub-]staff for octave n would contain the lines, in ascending order:

• O_n Q_n S_n U_n W_n Y_n O_n+1

and enclose the spaces:

• P_n R_n T_n V_n X_n Z_n.

Mind you, I'm not sure that many dodecaphonists even exist in the early 21^st century, let alone any total serialists! And a system of 12-place staves (as here described) that might suit them very well would, ipso facto, be unsuited to a 19-EDO (¿novodecaphonist?) composer or performer, or to any other d-EDO (of d degrees per octave) if d wasn't a factor or multiple of 12. Also, for notating pentatonic music, the 12-place staff would be overkill.

But would the 12-place staff system perhaps be more useful for tonal music than Traditional Notation (TN)? I can see musicians rapidly developing an intuition for which places (note positions) are in a given key or scale, and which are altered places; even which alterations are ornamental and which represent modulations. For many years, I've made quick (tonal) melody sketches as simple proportional line graphs, drawn freehand (and thus subject to scribal error!) with pencil or pen, similar in shape and proportion to "pianola" notation, consisting only of horizontal line segments of various lengths but no appreciable thickness. Supposing one had a ready supply of 12-place music paper to hand, one could easily jot down ideas with any writing instrument, even a chisel-point felt-tip marker.

I've experimented, before this, with a 4-line staff per octave, with two note positions in the space between each line pair, so that a note is always in touch with a line; either (1) hanging from, (2) sitting astride or (3) sitting on top of the wire, er … line. Another system distinguishes three places per space and three per line (or equivalently, if one draws a supplementary line in the middle of each space, just three places per line, with the notehead either astride the line symmetrically, or just touching the line next above or below but mostly astride the line asymmetrically). But with these kinds of system, I find my positioning too inaccurate when I write in haste, and on the whole prefer only one note position per space or line. Or, as one may do in haste, simply go commando and do without any visible support!

To conclude, I welcome discussion of any proposal to reduce the cognitive burden on performers by amending the staves we use, as well as the accidentals.

[cmloegcmluin]

To start near the end of your post: Another excellent quality of the piano-roll notation (which I think of as a "player piano", or pianola, notation, having when a child delighted in my grandparent's instrument of that type) is this:

• that it shows both the start and the end of each note.

Ah, yes, great point!

As to vertical height corresponding proportionately to pitch, we are of course talking of a logarithmic scale, since - even in the inaccurate heights accorded to pitches in the traditional notation, every octave is always the same total count of lines and spaces.

Yes, exactly.

The inaccuracies have two sources:

• Every two successive nominals have the same height difference on any staff, whether they are tones or semitones.

Yup, this inaccuracy was my main concern.

• Any "great staff" (system of two staves, one treble above one bass) has an often arbitrary amount of space between the bottom line of the top staff and the top line of the bottom staff. Specifically, the height of the gap between A3 (the A below middle C) and E4 (the E above middle C) is usually much greater than that between A2 and E3 on the bass staff, or equivalently, between A4 and E5 on the treble. Years ago, I remember asking why the ledger line for middle C wasn't equidistant from both staves, while still just one space away from either. Or, putting this another way, why isn't the grand staff an 11-line staff, with the middle (C) line perhaps dotted, dashed, or coloured red?

Really?! I didn't even realize that was a problem. I would have assumed the great staff would have been essentially an 11-line staff with the middle (C) line altered or omitted, as you describe.

Regarding your other suggestions: I find those all very interesting. And I'd like to see some examples of your experiments! I never formally studied music myself, and personally I tend to write really harmonically obscure music which has no relation to a circle of fifths or octave... so I have no particular love for the traditional 5-line staff; that is to say, I'm all for propositions to start-from-scratch and find the perfect solution freed of historical burden, or to find a separate perfect substrate for each given musical style. But in this topic I'm trying to challenge myself to work within tradition, to just propose a little reform that the average reader of traditional notation wouldn't balk at. It sounds like you're on board with my proposal in general though; don't see any major issues with it?

[yahya.abdal-aziz]

As far as I understand it, I think your proposal to use vertical pitch height more rationally (logarithmically) does make sense.

… But why not just correct the irregularity? …

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.

I should confess that when I jot down a melodic contour, without benefit of lines, although I certainly do use longer lines for longer note durations, they're not proportionately so - in fact, they're probably closer to logarithmic! So e.g. if the pulse of the motif is a sixteenth note, that will have the shortest line; an eighth note will be longer, but less than double that for the sixteenth; and so progressively through longer durations. Of course, the lengths aren't exactly constant for each duration, since this is just "taking notes"! I'll try to post a clear illustration of this. I do think that logarithmic note durations make more sense than strictly proportional ones; just as for pitch height.

As for me posting examples - back at ya!

I don't want to move notes around within the spaces between lines, because that hinders the flow of notating music. Look at the sketchbooks of any professional musician - it's all shorthand. Hmm, perhaps that's another topic: what simplifications or accommodations should we make when using Sagittal at speed?

Anyway, I think that - in terms of perceptual load - it's only practical to write, and read back, notation on lined staves when we place noteheads at just two locations relative to the lines: either on the line (straddling it symmetrically), or between two lines. That is, the centre of every notehead should be exactly on a line or at the centre of the space between two (staff or leger) lines. Another way of thinking about this is the following:

• Take any staff of equally spaced lines;

• in the space between each pair of lines, imagine that we rule another feint line; then

• every notehead should have its centre on a staff line or an imaginary line.

And given that we can easily rule thicker and thinner lines, why not also consider using a staff with alternating thick and thin lines, placing a note on a thin line rather than using an accidental? We would thus trade a little more accuracy in vertical positioning for a little less complexity in symbolic modification: More lines, but fewer accidentals. But there has to be a balance point, because it would be unworkable to go to either extreme: of having a distinct height for every pitch possible in every tuning, and consequently no accidentals whatsoever; or having just one height for all pitches and an accidental encoding even the nominal and octave. How many more lines could we handle comfortably? I don't know; we should experiment a little. It's also great to have most solo instruments use just one staff, supplemented by a few leger lines on occasion. Even the three staves used routinely by an organist take quite a bit of training and practice to sight-read proficiently. Here the performer has to coordinate two hands, two feet and sight inputs with the aural outputs; quite a cognitive load, greater than for performers on most other instruments.

[cmloegcmluin]

As far as I understand it, I think your proposal to use vertical pitch height more rationally (logarithmically) does make sense.

Groovy!

I think that - in terms of perceptual load - it's only practical to write, and read back, notation on lined staves when we place noteheads at just two locations relative to the lines: either on the line (straddling it symmetrically), or between two lines. That is, the centre of every notehead should be exactly on a line or at the centre of the space between two (staff or leger) lines.

Oh, so maybe you disagree with one part of my proposal? This part:

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.

I certainly agree that we shouldn't try to position pitches proportionally with respect to the microtonal accidentals — i.e. that pitch information should be indicated only symbolically — but it seems like it would be inconsistent of us if we didn't also proportionally space the faint/imaginary staff line (as you describe it, and I like that description) that is somewhere between the two solid/real staff lines, since that information is also 3-limit.

[Dave Keenan]

I agree with Yahya that your proposed unevenly-spaced lines (or more-to-the-point, unevenly-spaced staff positions) won't work very well with symbols designed for evenly-spaced lines.

But there are certainly staff proposals out there, whose intent is to linearise pitch vertically (where pitch is log of frequency), as the piano-roll does for 12edo.

I too enjoyed my grandparents' pianola as a child.

Chromatic staff proposals have been around for a long time and are directly related to the piano roll, as they have a line for every second note of 12edo.
http://musicnotation.org/

Aaron Hunt's Megastaff has a line for every second note of 41-edo, and redefines the sharp and flat symbols as steps of 205edo = 5 × 41edo (approx 6 cents).
https://musictheory.zentral.zone/huntsystem1.html

Sagittal can work fine with either of these systems. But when I suggested to Aaron Hunt, many years ago, that he could use sagittals for 1 and 2 steps of 205edo, e.g. and  , instead of redefining sharps and flats, he couldn't see the point.

One way to use the Chromatic staff with Sagittal would be to treat the staff positions as a 12-note chain of fifths, say Eb Bb F C G D A E B F# C# G#. These 12 names would be treated as compound-nominals. Sharp and flat symbols would not be used on the staff. Instead, Pythagorean-comma symbols would be used to extend the chain of fifths. It could not use multi-shaft sagittals unless we redefined the additional shafts as representing Pythagorean commas rather than apotomes when on a Chromatic staff.

Of course, this would only be strictly linear in pitch when the notational fifth is the 12edo fifth, but it will be more nearly linear for JI and other edos than the standard staff, which is linear only for the extreme fifths of 7edo.

[cmloegcmluin]

I agree with Yahya that your proposed unevenly-spaced lines (or more-to-the-point, unevenly-spaced staff positions) won't work very well with symbols designed for evenly-spaced lines.

Fair enough.

The fact that I don't see something like my proposal anywhere in this Music Notation Project's galleries is discouraging. I didn't realize how many approaches were out there already for grappling with these shortcomings! I did see one proposal which had the black lines for black keys, and spaces scaled as to whether they fit one or two white keys, which is kind of interesting. But there's a lot of other shenanigans going on with it; I don't like any of these that bring notehead shape or filling into play because I'd rather leave those up to representing other information (e.g. in one of my pieces I change the notehead shape to a right-triangle when it's √2 the duration it would be normally). I was really looking for a minimal-impact improvement to standard notation.

But I really like this chromatic staff proposal that they have on their front page. I think I'd like to use it myself, despite it being a more dramatic impact, more on par with the stuff Yahya was suggesting for me. Unfortunately MuseScore, the software I use for producing sheet music, does not yet support it (and it doesn't look like there are plans to do so anytime soon).

I don't recognize any microtonal peeps in their MNMA members list, but I am heartened to see that their criterium #14 recognizes microtonal needs.

...when I suggested to Aaron Hunt, many years ago, that he could use sagittals for 1 and 2 steps of 205edo, e.g. and  , instead of redefining sharps and flats, he couldn't see the point.

Bummer about that. At least you tried.

One way to use the Chromatic staff with Sagittal would be to treat the staff positions as a 12-note chain of fifths, say Eb Bb F C G D A E B F# C# G#. These 12 names would be treated as compound-nominals. Sharp and flat symbols would not be used on the staff. Instead, Pythagorean-comma symbols would be used to extend the chain of fifths. It could not use multi-shaft sagittals unless we redefined the additional shafts as representing Pythagorean commas rather than apotomes when on a Chromatic staff.

Of course, this would only be strictly linear in pitch when the notational fifth is the 12edo fifth, but it will be more nearly linear for JI and other edos than the standard staff, which is linear only for the extreme fifths of 7edo.

Both Evo (Mixed) and Revo (Pure) flavors of 12-Relative (Trojan) notation could be used with a chromatic staff without any changes.

But for EDO notations, Prime Factor notation, and JI notation, you're right. If I'm understanding correctly, the Pythagorean comma (3C) symbol would be used to convert G to A, right (well, in that case, you'd use ). Any double sharp or double flat is only one 3C away from one of these 12 compound-nominals.

This would not constitute a categorical stylistic offense if using Evo flavor, since in that flavor you're accepting of polysymbolic notation (potentially two instead of maximum one accidental per note). However, I note that the 3C may not always locate itself in the same position as the sharp/flat would have (i.e. between the >3-limit sagittal and the nominal) due to the smallest-to-the-left principle; many common sagittals like  ,  ,  , and are larger than the 3C, and something feels awkward to me about making a sandwich of >3-limit information between two pieces of 3-limit information.

In the Prime Factor notation, it somehow seems less bothersome, because the ordering of the alterations by cents size scrambles the ordering of the primes already, so it's just one more prime (3) which is out of order. However, something that might bother people is that it reverses a design principle of the determination of Prime Factor symbols: usually you drop the accents on a symbol if no lower prime has taken the unaccented symbol yet, so theoretically 3 should get  , not  , but then you've left the 5C out of luck because it doesn't have any accents. Although I guess the 3C is just exceptional all around because you wouldn't find it by using the same process as the others (it's not the Olympian symbol for the JI ratio 3/1), so all bets are off.

For the Revo flavor, your proposal to interpret extra shafts as 3C instead of apotomes is really interesting. I note, however, that in the standard staff, a single extra shaft doesn't represent an apotome; it actually represents a  , or 11L . When you have two extra shafts, then you could say together they represent an apotome (or that the 2nd extra shaft represents a , or 11M).

I guess if we want to take Sagittal with chromatic staff more seriously, we should break discussion out to its own topic.

[Dave Keenan]

For the Revo flavor, your proposal to interpret extra shafts as 3C instead of apotomes is really interesting. I note, however, that in the standard staff, a single extra shaft doesn't represent an apotome; it actually represents a  , or 11L . When you have two extra shafts, then you could say together they represent an apotome (or that the 2nd extra shaft represents a , or 11M).

Thanks for the correction. Yes, it would be two extra shafts that corresponded to a 3-comma.

I haven't really thought through any of this. It was pretty much off-the-cuff.

[cmloegcmluin]

Oh I see. Using 2 shafts to represent the 3C seemed bad to me at first, because what if you ever needed to shift by the 3C twice? But, as I said in my previous post, you'd never need to shift by it more than once, so using 2 shafts for it is fine. I think that's better because for any given 1-shaft symbol its 3-shaft symbol exists, and same for between 2- and 4- shaft symbols, but not so between 1- and 2- or 2- and 3- and 3- and 4-. The only thing I don't love about this is that || means 114¢ on a normal staff and only 23¢ on a chromatic staff. But that's probably fine. I know this is all quite off the cuff. But very interesting to think about.

[yahya.abdal-aziz]

In your research on alternative notations, is any member familiar with the books written by Gardner Read on the subject? See the Wikipedia article on Gardner Read, Bibliography section:https://en.wikipedia.org/wiki/Gardner_Read#Bibliography, which includes these titles:

-- (1978). Modern Rhythmic Notation. Indiana University Press. ASIN 0253338670
…
-- (1987). Source Book of Proposed Music Notation Reforms. Greenwood Press. ISBN 031325446X
-- (1990). Twentieth Century Microtonal Notation. Greenwood Press. ISBN 0313273987
-- (1998). Pictographic Score Notation. Greenwood Press. ISBN 0313304696

among others.

And are there any more recent surveys? - since these are all over 20 years old. And Read won't be updating his titles, as he passed away in 2005; tho' others might.