Alternative symbols for ups and downs notation
This article discusses about an alternative set of symbols based on current practice of microtonal music with some experimental extrapolation. The notation is, in essence, ups and downs notation with a different look.
Alternative symbols may be useful for the following reasons:
- One may prefer a more conventional look of the score;
- The up and down symbols may not be quite accessible in computer-aided score typing.
All symbols proposed in this article are available in MuseScore 3, and is used by the n-Edo Retuner Plugin. (Feel free to address others.)
Detail
Symbol set
The symbol set comprises traditional accidentals (sharps, flats, and naturals), Stein-Zimmermann quartertone accidentals (semisharps and semiflats), and up and down arrows in Gould arrow quartertone symbols or part of Helmholtz-Ellis Just Intonation accidentals.
The traditional accidentals, as specified in ups and downs notation, modify the note by the sharpness value, which equals the steps of a chromatic semitone (apotome, 2187/2048).
The Stein-Zimmermann quartertone accidentals modify the note by half the sharpness value. They apply to ETs of even sharpness value only.
The Gould arrow quartertone symbols are up and down arrows attached to sharps, flats or naturals. They modify the note by one step. The syntonic comma in Helmholtz-Ellis Just Intonation accidentals looks very similar, and can be used interchangeably.
Below is a table showing the characteristics of each ET below 72 in the context of traditional fifth-generator heptatonic ups and downs notation. Each row represents the steps of a chromatic semitone. Each column represents the steps of a diatonic semitone (limma, 256/243), located between E–F and B–C.
| -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
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| -3 | 6b | ||||||||||
| -2 | 4 | 11 | 18b | ||||||||
| -1 | 2 | 9 | 16 | 23 | 30b | ||||||
| 0 | 7 | 14 | 21 | 28 | 35 | 42b | |||||
| 1 | 5 | 12 | 19 | 26 | 33 | 40 | 47 | 54b | |||
| 2 | 3 | 10 | 17 | 24 | 31 | 38 | 45 | 52 | 59b | ||
| 3 | 1 | 8 | 15 | 22 | 29 | 36 | 43 | 50 | 57 | 64 | 71b |
| 4 | 6 | 13 | 20 | 27 | 34 | 41 | 48 | 55 | 62 | 69 | … |
| 5 | 11b | 18 | 25 | 32 | 39 | 46 | 53 | 60 | 67 | … | |
| 6 | 23b | 30 | 37 | 44 | 51 | 58 | 65 | 72 | … | ||
| 7 | 35b | 42 | 49 | 56 | 63 | 70 | … | ||||
| 8 | 47b | 54 | 61 | 68 | … | ||||||
| 9 | 52b | 59 | 66 | … | |||||||
| 10 | 64b | 71 | … |
Usage guide for each sharpness value
A usage guide for ETs of sharpness value below 8 will be provided here.
Sharp 1
Sharp-1 ETs have a sharp that raises 1 step. ETs of this category include 5, 12, 19, 26, etc.
Traditional notation can be used since an up is exactly equivalent to a sharp.
| Step offset | +2 | +1 | 0 | -1 | -2 |
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Sharp 2
Sharp-2 ETs have a sharp that raises 2 steps. ETs of this category include 10, 17, 24, 31, etc.
As are commonly seen in 24-edo and 31-edo, semisharps and semiflats could be used for a substitution of the tilde.
| Step offset | +4 | +3 | +2 | +1 | 0 | -1 | -2 | -3 | -4 |
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Sharp-3
Sharp-3 ETs have a sharp that raises 3 steps. ETs of this category include 22, 29, 36, 43, etc.
This is where you really want to use ups and downs.
| Step offset | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
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Sharp-4
Sharp-4 ETs have a sharp that raises 4 steps. ETs of this category include 27, 34, 41, 48, etc.
A full combination of semisharps, semiflats, ups and downs looks very neat.
| Step offset | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
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Sharp-5
Starting with sharp-5, which includes the famous 53-ET, you'll have to use double ups and downs.
| Step offset | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
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Sharp-6
| Step offset | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
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Sharp-7
| Step offset | 17 | 16 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
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Sharp-8
| Step offset | 19 | 18 | 17 | 16 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
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| Flat symbol |
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Provided with more stacks of arrows, ETs of higher sharpness value can be notated in the same method as above.
The rest will be discussed below.
Flat-1
Flat-1 ETs have a sharp that lowers 1 step. ETs of this category include 9, 16, and 23. To have a sharp that actually lowers the tone can be counter-intuitive, yet reasonable for the system. Regardless, you could just flip it around.
| Step offset | +2 | +1 | 0 | -1 | -2 |
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Flat-2
Flat-2 ETs (virtually 11-ET only) have a sharp that lowers 2 steps. So besides the special flavor of the sharps and flats, there are also semisharps and semiflats to fill up the spaces between.
| Step offset | +4 | +3 | +2 | +1 | 0 | -1 | -2 | -3 | -4 |
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Zero
The zero row is even more special in that the traditional accidentals cannot raise or lower the pitch, so the note can only be modified by arrows. ETs of this category include 7, 14, 21, 28, and 35.
| Step offset | +3 | +2 | +1 | 0 | -1 | -2 | -3 |
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Limitations
Some notes cannot be reached in ETs of sharpness value higher than 9 due to MuseScore not providing the stacks of arrows required. ETs below 72 known to be impractical for this reason: 59, 66, 71. Some notes cannot be reached in ETs with a very flat fifth due to MuseScore not providing multiple sharps and flats. ETs below 72 known to be impractical for this reason: 21, 26, 28, 33, 35, 40, 45, 47, 52, 64. Therefore, such ETs are capped from its full strength.
Conclusively, this set of symbols still cover most ETs below 72.