# Talk:159edo notation

## Atoms of the language

As I understand it, there are Syntonic commas integrated into the symbols, currently only combinations with five base symbols (, , , , ), as well as Septimal comma symbols (, , , ). If we add symbols for 1/4 and 3/4 accidentals and the darts, we can combine them in the table. I'm now working on it... --Xenwolf (talk) 08:26, 4 October 2020 (UTC)

- Aura, I saw that you were as confused as I about the wide lines above. It results from the image dimensions, the browser automatically adjusts the line height to the upper and lower extrema. BTW: I have 2 files out of 16 ready (the 0.5 sharp and 1.5 sharp, which are easy to extent by the synto-arrows), but I won't upload them until I have finished them all, the darts will most likely follow the septi-flags regarding their vertical position... --Xenwolf (talk) 17:32, 4 October 2020 (UTC)

- Thanks for telling be about those wide lines... Oh, and it's good to see that you've got some more stuff done. For the record, the darts following the septi-flags in terms of their vertical position actually makes sense, though to be fair, once that's done, we ought to tell Kite about our design for the darts, as I suspect that these darts are the accidentals that are best used in Ups and Downs notation on scores. --Aura (talk) 17:48, 4 October 2020 (UTC)

- A dart is 1\159 or approximately 1/3 of a Syntonic comma or 1/4 of a Septimal comma. All of them are not exact, but in good approximation. I think you'll find all the symbols you need in the Category:159edo notation, I have to pause now. --Xenwolf (talk) 22:01, 4 October 2020 (UTC)

## Dart symbol(s)

I like the idea of universal usability indicating some edo step or double step. The graphical symbols themselves are the least sophisticated glyphs I made, IOW they are still more like a *sketch*. Maybe the distance between the double darts is a bit too small. Also the angle isn't carefully designed as are the "rounded" edge parts. --Xenwolf (talk) 20:36, 5 October 2020 (UTC)

- I'm glad you like the idea of the darts being universally usable- I mean, they are nothing more than better versions of the glyphs of ups and downs notation after all... As for the distance between the double darts being two small, that really depends on the height of the triple dart glyphs, as the triple dart glyphs should have the same height as a natural accidental. However, you are absolutely right about the rounded edges of the glyphs, those really do need fixing... --Aura (talk) 20:50, 5 October 2020 (UTC)

- One more thing... the angle of the darts is something that should be decided by Kite, as he's the one who first came up with Ups and Downs notation, and besides, he would need to know about the stylistic upgrade anyway. --Aura (talk) 21:10, 5 October 2020 (UTC)

Okay, so, after what Kite said, and given my thoughts on the Rastma being a potential JI basis for the single step of 159edo, I'm thinking we ought to replace the dart glyphs entirely- thus separating this system from the Ups and Downs notation system proposed by Kite- and use a different combining glyph to represent the Rastma as a single step of 159edo, with the double step and quadruple step of 159edo being represented by syntonic-modified accidentals that are further modified by the combining element representing the Rastma. On another note, I think we can relegate the Septimal comma symbols and 13-limit accidentals to being optional, that way, people who want to use a simpler version of this system can do so. If need be, we can always ask the people who make new versions of Helmholtz-Ellis notation to separate the syntonic comma modifiers from the other accidental symbols, and we can go from there. --Aura (talk) 14:57, 7 October 2020 (UTC)

## Kite's proposal for 159edo notation

I prefer a simpler notation, with as few extra accidental pairs as possible. You have 3 pairs, plus the half-sharp/half-flat, so really 4 pairs. One could do single-pair, only ups and downs (or darts as you say), but one would need at least septuple ups, and in practice octuple or more. I think that's impractical, so I propose a double-pair notation.

I also like to have all accidentals be pronounceable, so that one can not only write down the music but also talk about the music. That lets me name chords, which in some genres is as important as writing down the music.

I would notate 159edo with a combination of ups/downs and lifts/drops. The latter (shall we call them slants?) are written / and \. The ups and downs are used as in 53edo, so one up is 3 edosteps. One lift is 1 edostep.

`
`

0 natural 1 / lift 2 ^\ updrop // (double-lift) 3 ^ up 4 ^/ uplift 5 ^^\ double-updrop ^// (up double-lift) 6 ^^ double-up vvv# triple-downsharp 7 ^^/ double-uplift vvv/# triple-down liftsharp 8 ^^^\ triple-updrop ^^// (double-up double-lift) vv\# double-down dropsharp 9 ^^^ triple-up vv# double-downsharp 10 ^^^/ triple-up lift vv/# double-down liftsharp v\\# (down double-drop sharp) 11 v\# downdrop sharp 12 v# downsharp 13 v/# downlift sharp \\# (double-drop sharp) 14 \# dropsharp 15 # sharp

`
`

Notes flatter than natural can be deduced by symmetry, i.e. C \C v/C vC v\C etc. Notes beyond sharp just run through the same list, but adding "sharp": sharp, liftsharp, updrop sharp (or double-lift sharp), upsharp, uplift sharp... going to double-sharp eventually.

Instead of ^^^ one could put an actual numeral 3 right on the score, like ^3. If someone actually used just darts and no slants, they would really need to write ^7 and not ^^^^^^^.

The spectrum of qualities looks like this:
`
`

0 m minor 1 /m liftminor 2 ^\m updrop minor 3 ^m upminor 4 ^/m uplift minor 5 v\~ downdropmid 6 v~ downmid 7 v/~ downliftmid 8 ^\~ updropmid 9 ^~ upmid 10 ^/~ upliftmid 11 v\M downdrop major 12 vM downmajor 13 v/M downlift major 14 \M dropmajor 15 M major 16 /M liftmajor 17 ^\M updrop major 18 ^M upmajor 19 ^/M uplift major etc.

`
Thus 4:5:6:7:9:11 = P1 vM3 P5 v\m7 M9 v/~11 = C vE G v\Bb D ^^/F = Cv9(v\7)v/~11 = C down-9 downdrop-7 downlift-mid-11
`

I like using both ups/downs and lifts/drops for larger edos only when the edo is multi-ring (the circle of 5ths doesn't include every note) and each ring requires ups and downs. For example 205edo is 5 rings of 41edo, but 124edo is not 4 rings of 31edo. The lifts and drops label the rings. In 159edo, there's a lift ring, a drop ring, and a plain ring. The lift ring is also a double-drop ring.

Now this notation may not be what you are looking for, because it isn't based on HEJI. If you want to stick with HEJI, I understand. But I must say I dislike using a half-sharp symbol for 7\159 when the sharp symbol is 15\159. Counter-intuitive. I also dislike that ^ means 1\159 but an up-arrow attached to a sharp or flat or natural means 3\159.

As far as the exact shape of the down, I like the letter v in a narrow sans serif font like arial narrow. Sleek and compact. I really dislike any serifs on it. Now here you can get into a debate about how Western music notation has a certain look, like the treble clef being a stylized letter G, and microtonal accidentals should have the same look. Eh, I just don't agree.

It's hard to get the caret as skinny as the letter v. So if you make actual image files, I'd say for the up, just rotate the arial narrow v 180 degrees.

I sort of like the way you stack two ups or downs on top of each other. But when there is a chord in close position on the staff, and one note is double-down, it can appear as if two adjacent notes each have a down. Another problem is that the ups or downs can't get really skinny. I would prefer two skinny ups side-by-side over two wide ups one above the other. Takes up the same amount of space, and is far less likely to be misinterpreted. And you can't type the stacked glyph, and I think all notation should be typeable. So all in all, I'm against the idea, but not strongly against it. I think stacking three ups is definitely going too far.

In general I don't like combining accidentals into complex glyphs, like the arrows on the sharp that HEJI has. I prefer multiple accidentals in a row, analogous to letters forming words. I also prefer accidentals not get too tall, since it makes it hard to notate close-voiced chords.

Anyway, just my thoughts. :) *unsigned contribution by: TallKite (Talk) 04:02, 7 October 2020 (UTC)*

- Thanks for taking the time to respond. Perhaps we can file this under other ideas for notating 159edo... Yes, I do fancy the idea of sticking to HEJI and Mainstream Quartertones- at least for the main accidentals- as they are more likely to be familiar to people, and for the record, I
*can*actually pronounce note names such as "D-Sesquiflat". Nevertheless, it's good to see that we both agree that there's a need for more than one set of accidental symbols on larger EDOs. I realize you dislike the idea of the demisharp and demiflat symbols being mapped to 7 steps when sharp and flat are mapped to 15, and you have every right not to like that, but dare I point out that we have a similar problem in 5-limit Just Intonation where two 25/24 chromatic semitones add up to less than a proper whole tone, let alone a proper double sharp, so that objection doesn't exactly sit well with me. On another note, now that you have pointed out the problems with stacking the darts, as well as the mismatch between the darts and the combined glyphs, I think that means we'll just have have to get more creative. --Aura (talk) 04:59, 7 October 2020 (UTC)

- I must point out one more thing, and that is the fact that 11-limit quartertones are actually useful in modulation to keys that are not on the same circle of fifths, while being simultaneously within an unnoticeable comma's distance from the quartertone of 24edo, which many musicians outside this community are familiar with, hence the rationale behind me elevating the status of the 11-limit, and using mainstream quartertone accidentals to represent it. --Aura (talk) 05:45, 7 October 2020 (UTC)

## Formal article structure

For those who are interested and have the technical skills to compile a book of the wiki pages they are interested in, it is certainly helpful if all wiki pages are structured uniformly with regard to the heading levels. For this very reason, Wikipedia makes sure that the headings of article contents always start at h2. The only h1 heading of an article is the name/title (or lemma). In the finished book an article then corresponds quasi to a chapter.

Short version: wiki articles should resist `= Such Titles =`

. --Xenwolf (talk) 15:56, 7 October 2020 (UTC)

- Is it me or do the
`= h1 Titles =`

on the Quartismic family page actually have a different size from the article title? --Aura (talk) 16:13, 7 October 2020 (UTC)

- It's exactly the same problem there. Well, the family pages started like this on the wikispaces-based wiki. I'd try to reduce these flaws step by step. There are also articles that I would better not touch in this respect, because they are a already a book by their own, for instance Harmonic Entropy, but the vast majority of articles would clearly benefit from a consistent structure. --Xenwolf (talk) 16:21, 7 October 2020 (UTC)

## My Second Idea for a Notation System

Okay, after talking with Sam, I now think we need to make another notation system. Truth be told, I've been trying to make a notation system that is accessible for people with a background in 24edo like myself this whole time, and this is the main reason for my notation systems being as complicated as they are. Nevertheless, I'm hoping to make things as intuitive as possible as well. I'm thinking that this new system should keep the traditional symbols for naturals, sharps, flats, double sharps and double flats, and that we need to make two different versions of each of the quartertone accidentals to reflect how the rastma is not tempered out, and that some of these quartertone accidentals can only be lowered, while the others can only be raised.

One set of quartertone accidentals has the two broad cross strokes of the traditional demisharp accidental set closer together than those of the sharp accidental while keeping the same traditional "demiflat" accidental, while the "sesquisharp" accidental and the "sesquiflat" accidental in this set are similarly redesigned, and modify the base tone by an interval consisting of an apotome and a 33/32 quartertone. The second set of quartertone accidentals are also based on the traditional quartertone accidentals, and the second "demisharp" and "sesquisharp" accidentals has the two broad cross strokes set further apart than those of the sharp accidental, while the corresponding "demiflat" and "sesquiflat" accidentals each have a narrow cross stroke on their stems- the "demisharp" and "demiflat" accidentals in this set modify the base tone by a 709/704 interval, while the "sesquisharp" and "sesquiflat" accidentals in this set modify the base tone by an interval consisting of an apotome and a 709/704 interval. Furthermore, since the rastma itself is a potential JI basis for the single step of 159edo, and since combined symbols need to be deciphered, I think we need to make two new sets of symbols- one set for modification by one or two syntonic commas based on the combining arrows from Helmholtz-Ellis notation, and another pair of symbols to represent modification by a rastma. --Aura (talk) 21:46, 10 October 2020 (UTC)

- How many glyph boxes will be needed? In the first approach, there was a maximum of two boxes (both of them optional): the first box holds a quartertone (or classic) accidental optionally combined with syntonic commas, the optional second box holds either septimal comma glyphs or darts. --Xenwolf (talk) 19:43, 13 October 2020 (UTC)

- At present, I'm thinking there should probably be two- one to hold both the classic and quartertone accidentals, and one to hold accidentals which modify by combinations of the syntonic comma and the rastma. With the syntonic comma modifiers being simultaneously separated from the classic and quartertone accidentals and combined with the rastmic modifiers to form single glyphs, that renders a total of 25 different glyphs as opposed to the 45 in the first system. --Aura (talk) 20:13, 13 October 2020 (UTC)

- A number of them are indeed already there- the natural, the sharp, the flat, the double sharp and the double flat. We'd need to remake the quartertone accidentals as per my descriptions of them during the course of this conversation, with the end result being that instead of there being nine main accidentals in the first box, there would be thirteen accidentals total. Furthermore, we can derive at least four of the accidentals for the second box directly from the syntonic arrows of HEJI- albeit they need to be larger- so that's pretty simple. --Aura (talk) 20:34, 13 October 2020 (UTC)

- The problem, according to what Kite has said, is that these quartertone accidentals by themselves don't add up intuitively- for instance, two modifications by don't add up to a modification by but rather fall short of by a rastma. Don't get me wrong, we can keep all four of the quartertone accidentals, but we still need to create four more on account of the rastma not being tempered out- that is, we need a distinction between "wide" and "narrow" quartertones.

- If I understand you correctly, for an EDO that subdivides a the apotome into 15 equal-sized parts, we need "quartertones" that are 7/15 of an aptome but with symbols can be distinguished from 50-cent symbols not being used here? Isn't
*this*the problem: that you wish to include quartertone symbols to ease adapting the system to people familiar with 24edo? But if this is the case, wouldn't it be better to tell them "the truth" about quartertones: that they aren't actually quarter tones? Remember: the semitones in this system aren't 100cents either but we decided to use the exact symbols of 12edo. --Xenwolf (talk) 21:27, 13 October 2020 (UTC)

- If I understand you correctly, for an EDO that subdivides a the apotome into 15 equal-sized parts, we need "quartertones" that are 7/15 of an aptome but with symbols can be distinguished from 50-cent symbols not being used here? Isn't

- I'm perfectly aware of the fact that two "semitones" add up to more than a "whole tone" in this system- the same problem on a different level, as they are
*still*called "semitones" in spite of this- but we seem to have a way of getting around that in this and other systems by means of the double sharp and double flat accidentals... I guess you could say that I now want to come up with a similar solution for these "quartertones". Alternatively, we should not only mention that the "quartertones" aren't exactly "quartertones", but also that the "semitones" aren't exactly "semitones"... --Aura (talk) 21:40, 13 October 2020 (UTC)

- I'm perfectly aware of the fact that two "semitones" add up to more than a "whole tone" in this system- the same problem on a different level, as they are

- Either way you look at it, and regardless of the option we choose, 33/32 is to the 11-limit what the apotome is to the 3-limit in terms of its importance for navigation. It is also true that I have now decided simply to include quartertone symbols to ease adapting the system to people familiar with 24edo. With this in mind, and in light of the fact that the precursor to the prefix "semi-" apparently had the meaning of "half" since before the days of Latin, what do you propose we do? (comment posted and heavily edited by --Aura (talk) 22:28, 13 October 2020 (UTC))

- Those quartertone symbols should only be used in systems that tempers out 243/242, which guarantees the split of apotome into two. Whereas 159edo (or "15eda", to be precise) is essentially different. However, the solution is clearly present: simply use HEJI symbols for 33/32 will suffice for all undecimal work. If another quartertone is ever needed, it may represent 1053/1024 or 27/26 but never to mess with 11-limit anymore. FloraC (talk) 08:02, 14 October 2020 (UTC)

- Flora, I'm afraid it's not that simple, as and are identical, resulting in confusion, while does not suggest the close connection that exists between 33/32 and 1\24 to microtonalists who come from a 24edo background. Furthermore, what do you do when you take something already modified by the apotome and further modify it by 33/32? This is why I was suggesting new quartertone accidentals in the first place. I admit I'm not sure just what the solution is right now, but I do want to get this issue dealt with. --Aura (talk) 14:50, 14 October 2020 (UTC)

Here's an idea of what I was talking about for this idea originally:

I hope this idea make sense... In rastmic systems only the set of quartertone accidentals in the middle are used, while when the rastma is not tempered out, a contrast exists between the top set and the bottom set. --Aura (talk) 15:45, 14 October 2020 (UTC)

In response to your comment here, Xenwolf, I'm glad you think this picture is clear. --Aura (talk) 19:26, 14 October 2020 (UTC)

- Saw all this and thought I'd mention that the idea that there is only one type of semitone is a modern idea arising directly from 12-edo. In Pythagorean tunings, there are two types of semitone. The diatonic semitone is a minor 2nd 256/243, and the chromatic semitone is the augmented unison 2187/2048. The two add up to exactly a whole tone 9/8. The two semitones are obviously notated completely differently, since one is a 2nd and one isn't. Meantone tunings draw the same distinction, but of course the semitones are tuned differently.

- In 5-limit JI, both semitones can be altered by 81/80, making even more semitones, e.g. 135/128, 25/24, 16/15 and 27/25. Each of these is either diatonic or chromatic, and corresponding pairs add up to a whole tone.

- Once you make the distinction between chromatic and diatonic semitones, it becomes clear that the sharp sign that represents the chromatic semitone was never meant to denote exactly half a whole tone. It represents a whole tone minus a minor 2nd.

- BTW something similar happened to the term tritone. In modern times, it means exactly half an octave, but in the past it literally meant the aug 4th, which at the time was distinct from the dim 5th.

- Well Kite, applying this logic to quartertones, it's clear that 33/32 is a kind of parachroma, and, if you consider that 256/243 is a 3-limit diatonic semitone, while 33/32 is an 11-limit quartertone, and both the 3-limit and the 11-limit are significant (which, after posting the initial version of this comment, I have confirmed through further investigation and documented my findings on the matter), it makes sense to think of 8192/8019 as being the corresponding paradiatonic interval, as 33/32 and 8192/8019 add up to 256/243... However, it seems the relationships between chromas and parachromas is more complicated, as evidenced by the relationship between 45/44 and 55/54- two other 11-limit parachromas which together add up to 25/24 while differing from each other by a ragisma. As another example, 33/32 has a similar relationship to 729/704, with the two adding up to the apotome. This on its face means that there are already multiple types of parachromatic quartertones. --Aura (talk) 14:17, 15 October 2020 (UTC)

- If I take the logic of what I said in the previous comment, that means that a stack of two 33/32 quartertones, plus 8192/8019 plus 729/704 is a full 9/8 whole tone- that means that four quartertones equals a whole tone... If, on the other hand, the rastma is taken from 729/704 and added to 8192/8019, the result is a third 33/32 quartertone plus 4096/3993- another paradiatonic quartertone. This would actually make the addition much more straightforward, as a stack of three 33/32 quartertones plus 4096/3993 is also 9/8. So yes, four quartertones *do* in fact add up to a whole tone. --Aura (talk) 15:45, 15 October 2020 (UTC)

- It does, but I should point out that the vertical lines on all three demisharp accidentals are actually supposed to be the same. The same thing is true of the vertical lines on all three sesquisharp accidentals, as well as the the stems of the various demiflat and sesquiflat accidentals. However, that's not to say that the vertical lines in the demisharp and sesquisharp accidentals all share a length for their vertical lines, however. --Aura (talk) 21:06, 15 October 2020 (UTC)

- Glad you like it! It's actually just cut/copy & paste work, even if done in Inkscape. I was interested in the appearance within staff context and indeed: it seems to work! Also the horizontal and vertical positioning is done by hand and naked eye. A clean design would take a bit longer. BTW also the other glyphs I made are not really clean. --Xenwolf (talk) 21:20, 15 October 2020 (UTC)

- This proposal is basically equivalent to having one accidental for half-apotome and another for 33/32, so, yes, I see it works for most equal temperaments. Better if they appear more distinct glyph-wise. FloraC (talk) 06:15, 16 October 2020 (UTC)

- Thanks for your thoughts, Flora. Besides the minor flaws (lenghts of verticals) the sharps seem okay to me, but the flats are harder. I'm not sure if sticking with the b-shape isn't the fundamental problem here. But you are experienced with fonts maybe you are the right person for the details. Maybe having three lines is for 4 accidentals each is not ideal. I did a tiny improvement right now to set the single accidental glyphs apart from each other (maybe a browser refresh will be needed). --Xenwolf (talk) 06:50, 16 October 2020 (UTC)

- For the record, if there are issues with the flat accidentals, I'd have to say that the thickness and angle of the extra strokes on the stems of the narrow and wide flat accidentals is most likely to be the problem. Perhaps those cross strokes ought to be thicker and should set at 60-degree angles from the vertical rather than 45-degree angles. The reason I say this is that those extra strokes are what make the extra flat accidentals stand out from one another, while the shape of the rest of the accidental is what makes them recognizable as flats in the first place. --Aura (talk) 14:17, 16 October 2020 (UTC)

- Please have a look on the new version of File:Quarter-accidentals-narrow-rastmic-wide.png (maybe a browser refresh is needed). I made the corresponding vertical lines the same length, the cross strokes are thicker with ends that reflect the vertical direction again. --Xenwolf (talk) 21:58, 16 October 2020 (UTC)

- Yeah, I think that will work as long as the positioning is right. While you're at it, the strokes of all the demisharp and sesquisharp accidentals should also be fixed to have the same angle and thickness as those of the sharp symbol. Again, thank you very much for taking your time to help me do this. --Aura (talk) 22:33, 16 October 2020 (UTC)

- Right.

- Oh, and I should also point out that for EDOs where the rastma is not tempered out, but the half-apotome also exists, (think 318edo) the rastmic accidentals would not be used because 33/32 takes priority, and... well... because the half-apotome is not formed by tempering out the rastma in such EDOs. --Aura (talk) 15:24, 16 October 2020 (UTC)

Hey, Xenwolf, do you think we have time to work on this some more? --Aura (talk) 18:33, 27 January 2021 (UTC)

I'm asking because if you do, perhaps we can consider adding these accidentals to the mix...

I hope this makes sense. --Aura (talk) 18:47, 27 January 2021 (UTC)

- Currently I have not the time to work on that. One thing that just struck me: thin horizontal lines will be a problem between staff lines. --Xenwolf (talk) 16:20, 28 January 2021 (UTC)

Okay, I updated the picture of the second group of accidentals to change the thin, straight horizontal lines to thicker, more curvy ones. Does this make a difference? --Aura (talk) 06:17, 29 January 2021 (UTC)

- I think you should really give Inkscape a try, it's great for this kind of stuff, and there are a lot of tutorials around the web. --Xenwolf (talk) 19:09, 29 January 2021 (UTC)

Okay Xenwolf and Flora, after talking on Discord with other people about possible accidentals, I think I've arrived at a solution for a different set of accidentals for 159edo...

I hope you two like these. --Aura (talk) 22:52, 14 July 2021 (UTC)