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 one approach to the n-Edo Retuner Plugin is based on this. (Feel free to address others.)
What It Resolves
How is the conventional notation for 31-ET related to that for 22-ET? Is there a logical unity despite the apparent differences?
Detail
Below is a table showing the characteristics of each ET in the context of heptatonic ups and downs notation.
Each row represents the steps of a sharp/flat (apotome, 2187/2048), to be called "sharp value" below. The sharp value is the basic criterion to determine the symbol set to be used.
Each column represents the steps of a small tone (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 | … |
The symbol set comprises traditional sharps, flats, semisharps, semiflats, and up and down arrows attached to sharps and flats. Semisharps and semiflats apply to even sharp values only.
Sharp Value 1 to 4
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 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 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 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 Value 5 to 8
Starting with sharp-5, which includes the famous 53-ET, you'll have to use double ups and downs.
There's a small version of the arrow symbol available that can be stacked twice or thrice as part of the Helmholtz-Ellis notation, termed raise/lower by one syntonic comma, for which the step size of 53-edo is a close approximation. These arrow symbols, possibly combined with semisharps and semiflats, could be used through sharp-5 to sharp-8 ETs.
It is basically another system, to be called the "complex system", in contrast to the "simple system" above. The complex system is compatible with lower sharp values. For example, you could use the sharp-8 68-ET for the notation of 34-ET, but the reason the former should not take priority is obvious. However you do, using arrow symbols of both systems should be definitely avoided, because they look too similar.
Sharp-5
| 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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The rest will be discussed below.
Negative Sharp Values
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 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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Sharp Value 0
The zero row is even more special in that the conventional meaning of the sharp or flat does not apply. As a result, this system is not a good solution and it virtually only works for 14-ET.
14-ET
| Step offset | +1 | 0 | -1 |
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Limitations
Sharp value higher than 9 cannot be notated at all. ETs below 72 known to be impractical for this reason: 59, 66, 71.
Sometimes the symbols available are not sufficient for every key of an ET. When the max step offset of an ET exceeds its accidental symbol set, some of the keys are capped from its full strength. This occurs with ETs that have a very flat fifth. ETs below 72 known to be impractical for this reason: 21, 26, 28, 33, 35, 40, 45, 47, 52, 64.
Conclusively, this set of symbols still cover most ETs below 72.