159edo notation

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Accidentals in the 9/8 space. First sketch by Aura.
Accidentals in the 9/8 space. Reworked version.

Helmholtz-Ellis-based notation

The first 159edo-notation system proposed by Aura modifies the Helmholtz-Ellis notation in such a way that undecimal quartertones play a central role alongside the traditional accidentals, with mainstream quartertone accidentals being used to represent them. This is because undecimal 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 the microtonal community proper have at least a passing familiarity with. It should be noted that most of the symbols listed in this section derive their functions from JI. In addition, it should be noted that accidentals are not just modifiers for single notes, but signs that indicate the change of both the tonal and modal base, and can thus appear in key signatures.

Available base glyphs

Quartertones and Syntonic commas

As in HEJI, classic accidentals and arrows for syntonic commas are combined. Beyond HEJI, quartertone accidentals are also used, being combined with syntonic comma arrows in the same way.

-30 -22 -15 -7 0 +7 +15 +22 +30
+6 Heji6.svg HeQd3-su2.svg Heji13.svg HeQd1-su2.svg Heji20.svg HeQu1-su2.svg Heji27.svg HeQu3-su2.svg Heji34.svg
+3 Heji5.svg HeQd3-su1.svg Heji12.svg HeQd1-su1.svg Heji19.svg HeQu1-su1.svg Heji26.svg HeQu3-su1.svg Heji33.svg
0 Heji4.svg HeQd3.svg Heji11.svg HeQd1.svg Heji18.svg HeQu1.svg Heji25.svg HeQu3.svg Heji32.svg
-3 Heji3.svg HeQd3-sd1.svg Heji10.svg HeQd1-sd1.svg Heji17.svg HeQu1-sd1.svg Heji24.svg HeQu3-sd1.svg Heji31.svg
-6 Heji2.svg HeQd3-sd2.svg Heji9.svg HeQd1-sd2.svg Heji16.svg HeQu1-sd2.svg Heji23.svg HeQu3-sd2.svg Heji30.svg

Septimal commas and darts

Glyphs for septimal commas and darts are also used, but are generally not combined with each other except in certain musical contexts such as modulation. That said, if this system were to be extended to the 13-limit, the septimal comma glyphs would likely be combined with the 13-limit glyphs due to both 7-limit and 13-limit intervals having close connections to trientones (one third tones).

The septimal comma glyph traditionally indicates a change by 64/63, and thus, in this particular system, it is bound to the equivalent step size, which is around 30 cents- slightly larger than normal.

Delta Glyph
+8 Heji39.svg
+4 Heji38.svg
-4 Heji37.svg
-8 Heji36.svg

The dart glyph in this system indicates a change by a single 159edo step (around 7.5 cents). As the dart has no traditional binding to a specific comma, it can also be used for other EDOs to indicate a single step in their corresponding notation systems.

Delta Glyph
+2 Dart-u2.svg
+1 Dart-u1.svg
-1 Dart-d1.svg
-2 Dart-d2.svg

Combined accidentals

The following table illustrates modification of a given tone by accidentals up to 30 steps of 159edo. The terms in the glyph name sections reflect absolute displacements, means down, + means up. Furthermore, symbols with multiple names will have both names sharing the same cell. The choice between the primary and secondary glyphs for any given step is often determined both by absolute TE error and by harmonic and melodic context for any given score, particularly as the primary and secondary glyphs usually reflect different intervals in Just Intonation. For reference, the just whole tone interval, 9/8, is mapped to 27 steps of 159 edo (or 9 steps of 53edo) which is (27*1200/159=) 203.77 cents (0.2 cents flat).

Change in 1\159
Delta Primary Glyph Primary Glyph Name Secondary Glyph Secondary Glyph Name Remarks
0 Heji18.svg Natural
-1 Dart-d1.svg Dart down This is nothing more than a better version of the v marking from Ups and Downs Notation.
+1 Dart-u1.svg Dart up This is nothing more than a better version of the ^ marking from Ups and Downs Notation.
-2 Dart-d2.svg Double dart down This is two darts down fused into a single glyph for a more compact representation of vv from Ups and Downs Notation.
+2 Dart-u2.svg Double dart up This is two darts up fused into a single glyph for a more compact representation of ^^ from Ups and Downs Notation.
-3 Heji17.svg Natural –Syntonic comma HeQd1.svgHeji38.svg Quarter flat +Septimal comma, Demiflat +Septimal comma The intervals represented by the primary and secondary glyphs for this step differ by 385/384 in Just Intonation.
+3 Heji19.svg Natural +Syntonic comma HeQu1.svgHeji37.svg Quarter sharp –Septimal comma, Demisharp –Septimal comma The intervals represented by the primary and secondary glyphs for this step differ by 385/384 in Just Intonation.
-4 HeQd1-su1.svg Quarter flat +Syntonic comma, Demiflat +Syntonic comma Heji37.svg Septimal comma down The primary glyph overall represents a lowering by 55/54.
+4 HeQu1-sd1.svg Quarter sharp –Syntonic comma, Demisharp –Syntonic comma Heji38.svg Septimal comma up The primary glyph overall represents a raising by 55/54.
-5 HeQd1.svgDart-u2.svg Quarter flat with double dart up, Demiflat with double dart up
+5 HeQu1.svgDart-d2.svg Quarter sharp with double dart down, Demisharp with double dart down
-6 HeQd1.svgDart-u1.svg Quarter flat with dart up, Demiflat with dart up Heji16.svg Natural –2 Syntonic commas
+6 HeQu1.svgDart-d1.svg Quarter sharp with dart down, Demisharp with dart down Heji20.svg Natural +2 Syntonic commas
-7 HeQd1.svg Quarter flat, Demiflat This glyph represents a lowering by 33/32, much like its identical Helmholtz-Ellis equivalent.
+7 HeQu1.svg Quarter sharp, Demisharp This glyph represents a raising by 33/32, replacing its Helmholtz-Ellis equivalent on account of familiarity.
-8 HeQd1.svgDart-d1.svg Quarter flat, Demiflat with Dart down Heji36.svg 2 Septimal commas down
+8 HeQu1.svgDart-u1.svg Quarter sharp, Demisharp with Dart up Heji39.svg 2 Septimal commas up
-9 HeQd1.svgDart-d2.svg Quarter flat with double dart down, Demiflat with double dart down Heji13.svg Flat +2 Syntonic commas
+9 HeQu1.svgDart-u2.svg Quarter sharp with double dart up, Demisharp with double dart up Heji23.svg Sharp –2 Syntonic commas
-10 HeQd1-sd1.svg Quarter flat –Syntonic comma, Demiflat –Syntonic comma
+10 HeQu1-su1.svg Quarter sharp +Syntonic comma, Demisharp +Syntonic comma
-11 HeQd1.svgHeji37.svg Quarter flat –Septimal comma, Demiflat –Septimal comma Heji11.svgHeji38.svg Flat +Septimal comma The primary glyph overall represents a lowering by 22/21, and is the more simple of the two options presented here.
+11 HeQu1.svgHeji38.svg Quarter sharp +Septimal comma, Demisharp +Septimal comma Heji25.svgHeji37.svg Sharp –Septimal comma The primary glyph overall represents a raising by 22/21, and is the more simple of the two options presented here.
-12 Heji12.svg Flat +Syntonic comma
+12 Heji24.svg Sharp –Syntonic comma
-13 Heji11.svgDart-u2.svg Flat with double dart up
+13 Heji25.svgDart-d2.svg Sharp with double dart down
-14 Heji11.svgDart-u1.svg Flat with dart up
+14 Heji25.svgDart-d1.svg Sharp with dart down
-15 Heji11.svg Flat As this glyph represents a lowering by an apotome, and as the apotome is a complicated interval, this glyph is generally more likely to be seen as part of a key signature than as an accidental when not combined with other glyphs.
+15 Heji25.svg Sharp As this glyph represents a raising by an apotome, and as the apotome is a complicated interval, this glyph is generally more likely to be seen as part of a key signature than as an accidental when not combined with other glyphs.
-16 Heji11.svgDart-d1.svg Flat with dart down
+16 Heji25.svgDart-u1.svg Sharp with dart up
-17 Heji11.svgDart-d2.svg Flat with double dart down
+17 Heji25.svgDart-u2.svg Sharp with double dart up
-18 Heji10.svg Flat –Syntonic comma HeQd3.svgHeji38.svg Three quarter flat +Septimal comma, Sesquiflat +Septimal comma If accidentals for the 13-limit are added to this system, this will be a prime spot for one of them.
+18 Heji26.svg Sharp +Syntonic comma HeQu3.svgHeji37.svg Three quarter sharp –Septimal comma, Sesquisharp –Septimal comma If accidentals for the 13-limit are added to this system, this will be a prime spot for one of them.
-19 HeQd3-su1.svg Three quarter flat +Syntonic comma, Sesquiflat +Syntonic comma Heji11.svgHeji37.svg Flat –Septimal comma The secondary glyph represents a lowering by 243/224, and is the more simple of the two options presented here.
+19 HeQu3-sd1.svg Three quarter sharp –Syntonic comma, Sesquisharp –Syntonic comma Heji25.svgHeji38.svg Sharp +Septimal comma The secondary glyph represents a raising by 243/224, and is the more simple of the two options presented here.
-20 HeQd3.svgDart-u2.svg Three quarter flat with double dart up, Sesquiflat with double dart up
+20 HeQu3.svgDart-d2.svg Three quarter sharp with double dart down, Sesquisharp with double dart down
-21 HeQd3.svgDart-u1.svg Three quarter flat with dart up, Sesquiflat with dart up Heji9.svg Flat –2 Syntonic commas
+21 HeQu3.svgDart-d1.svg Three quarter sharp with dart down, Sesquisharp with dart down Heji27.svg Sharp +2 Syntonic commas
-22 HeQd3.svg Three quarter flat, Sesquiflat This glyph represents a lowering by a complex interval comprised of an apotome and a 33/32 quartertone, and as such, is generally more likely to be seen as part of a key signature than as an accidental when not combined with other glyphs.
+22 HeQu3.svg Three quarter sharp, Sesquisharp This glyph represents a raising by a complex interval comprised of an apotome and a 33/32 quartertone, and as such, is generally more likely to be seen as part of a key signature than as an accidental when not combined with other glyphs.
-23 HeQd3.svgDart-d1.svg Three quarter flat with dart down, Sesquiflat with dart down
+23 HeQu3.svgDart-u1.svg Three quarter sharp with dart up, Sesquisharp with dart up
-24 HeQd3.svgDart-d2.svg Three quarter flat with double dart down, Sesquiflat with double dart down Heji6.svg Double flat +2 Syntonic commas
+24 HeQu3.svgDart-u2.svg Three quarter sharp with double dart up, Sesquisharp with double dart up Heji30.svg Double sharp –2 Syntonic commas
-25 HeQd3-sd1.svg Three quarter flat –Syntonic comma, Sesquiflat –Syntonic comma
+25 HeQu3-su1.svg Three quarter sharp +Syntonic comma, Sesquisharp +Syntonic comma
-26 HeQd3.svgHeji37.svg Three quarter flat –Septimal comma, Sesquiflat –Septimal comma Heji4.svgHeji38.svg Double flat +Septimal comma
+26 HeQu3.svgHeji38.svg Three quarter sharp +Septimal comma, Sesquisharp +Septimal comma Heji32.svgHeji37.svg Double sharp –Septimal comma
-27 Heji5.svg Double flat +Syntonic comma
+27 Heji31.svg Double sharp –Syntonic comma
-28 Heji4.svgDart-u2.svg Double flat with with double dart up
+28 Heji32.svgDart-d2.svg Double sharp with double dart down
-29 Heji4.svgDart-u1.svg Double flat with dart up
+29 Heji32.svgDart-d1.svg Double sharp with dart down
-30 Heji4.svg Double flat This glyph represents a lowering by a stack of two apotomes, and as such, this glyph is more likely to be seen when combined with other glyphs, and key signatures using this glyph are likely to be considered remote and or exotic.
+30 Heji32.svg Double sharp This glyph represents a raising by a stack of two apotomes, and as such, this glyph is more likely to be seen when combined with other glyphs, and key signatures using this glyph are likely to be considered remote and or exotic.

Advantages and disadvantages

Among the advantages of this system is the fact that it keeps the traditional naturals, sharps, flats, double sharps and double flats, and assigns them to values based on 3-limit JI, resulting in this system having some level of familiarity for musicians from a traditional background. However, the disadvantages of this system are that this system relies on too many glyphs, a number of which need to be deciphered, and furthermore, the arrangement of combined and simple accidentals- as well as arrangement of the quartertone accidentals- is counterintuitive.

Ups-and-Downs-based notation

Another system proposed by TallKite involves simpler notation, with as few extra accidental pairs as possible. While one could do with only one extra pair, only ups and downs (that is, darts), but one would need at least septuple ups, and in practice octuple or more, rendering such a system impractical, so as a result, this system has two extra pairs of accidentals.

According to this system, 159edo would be notated with with a combination of ups/downs and lifts/drops- the latter (referred to as "slants" here) 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

This system uses both darts and slants for larger EDOs only when the EDO is multi-ring- that is, 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, and the drop ring is also a double-lift ring.

Advantages and disadvantages

Among the advantages of this system is the fact that it keeps the traditional naturals, sharps, flats, double sharps and double flats, and assigns them to values based on 3-limit JI, resulting in this system having some level of familiarity for musicians from a traditional background, while being simpler and more intuitive than the first system. However, among the disadvantages of this system are the lack of quartertone accidentals, and, the need for a prior understanding 53edo, which is often not the most straightforward to begin with in terms of notation, as noted by Wikipedia's article on 53edo. The net effect of these two disadvantages is that this system is likely to be less accessible to many microtonal musicians due to many beginners in particular being more likely to have experience with the more straightforward 24edo.

Sagittal notation

The following table shows sagittal notation for 159edo in ASCII form.

Todo: use smilies (i.e. graphical form).

Step raise 1 2 3 4 5 6 7 8
Accidental |( ~|( /| |) (|( / /| /|\ (|)