Kite's thoughts on enharmonic unisons in ups and downs notation
In any temperament, including any edo, any note or interval can be respelled by adding or subtracting an enharmonic interval, or EI for short. Often there are multiple possible EIs. This article presents the canonical EIs of edos and rank-2 temperaments for ups and downs notation.
EDOs
Overview
Sharp-1 edos, flat-1 edos and 7edo have only one EI. All others have an infinite number of EIs, but only two canonical EIs, which can be combined in various ways to create all the others. The canonical EIs are chosen to be easy to memorize and use. There are a few edos for which a third EI is sometimes useful.
The two canonical EIs are generally the A1 (aug unison) and the d2 (dim 2nd), upped or downed. The number of ups equals the negative sharpness and the positive dodeca-sharpness of the edo respectively. For example, 31edo's EIs are vvA1 and vd2. Some edos also use the downminor 2nd or the dudminor 2nd.
Adding an A1 to a note adds a sharp. So another name for 31edo's dudaug1 is dudsharp. Adding a dudsharp to an interval augments it and lowers it by 2 arrows.
Adding or subtracting a diminished 2nd sounds difficult, but it's what we routinely do when respelling notes in 12edo. For example, what's another name for D#? Did you have to think very hard to get the answer, Eb? Adding a d2 is that easy!
To add a d2, respell the note as its 12edo equivalent one degree higher. To subtract a d2, respell as one degree lower. Intervals are a little trickier. For diminished intervals, think of the interval as if from B. For augmented intervals, use F. For example, what's a dim 4th? A perfect 4th from B is E, so a dim 4th must be Eb. And Eb is also D#. And B-D# is a major 3rd. So d4 = M3.
To simplify a note name or an interval name, first add/subtract the dim2 to reduce sharps/flats and augs/dims, and perhaps to arrive at the desired degree. Then add/subtract the sharp to reduce arrows.
For example, the sum of two upmajor 3rds is a dup-aug 5th. If we're in 41edo, how can we respell that? 41edo's EIs are updim2 and quudsharp. ^^A5 + ^d2 = ^3m6, and ^3m6 + v4A1 = vM6.
19edo's EI is a double-dim 2nd. To add it, add the d2, then flatten. For example, C# becomes Db, which flattens to Dbb. For intervals, diminish: the aug 4th becomes the dim 5th, which diminishes to the double-dim 5th.
A survey of various edos by category and sharpness
Diatonic edos (except for sharp-1 edos) use the dim2 and the sharp:
- 12edo has the dim2 (C# = Db and A1 = m2)
- 17edo has the dudsharp (C = vvC# and ^1 = vA1) and the updim2 (C# = ^Db)
- 22edo has the trudsharp (C = vvvC#) and the dupdim2 (C# = ^^Db) and perhaps the downminor2 (C = vDb)
- 24edo has the dudsharp and the dim2
- 31edo has the dudsharp and the downdim2
- 41edo has the quudsharp and the updim2
Sharp-1 edos don't use ups or downs, so they have only one EI. Those other than 12edo don't use the dim2:
- 12edo has the dim2 (C# = Db)
- 19edo has the double-dim2 (C# = Dbb)
- 26edo has the triple-dim2 (C## = Dbb)
Pentatonic edos use the sharp and the minor2.
- 5edo has the minor2 (C = Db)
- 10edo has the dudsharp (C = vvC# and ^1 = vA1) and the minor2
- 15edo has the trudsharp (C = vvvC# and ^^1 = vA1) and the minor2
Sharp-0 edos (perfect edos) use the sharp and the perfect 2nd.
- 7edo has the sharp (C = C# and P1 = A1)
- 14edo has the sharp and the dud2 (C = vvD and ^1 = v2)
- 21edo has the sharp and the trud2 (C = vvvD and ^1 = vv2)
Superflat edos can be notated two ways. Melodic notation has major wider than minor, and sharp raises the pitch, as usual. Harmonic notation has major narrower than minor and sharp lowers the pitch. Melodic notation makes interval arithmetic very difficult, so it's not recommended. For harmonic notation, the three flat-1 edos use aug2 not dim2.
- 9edo has the aug2 (Cb = D, d1 = M2)
- 16edo has the double-aug2 (Cb = D#, d1 = A2)
- 23edo has the triple-aug2 (Cb = D##, d1 = AA2)
Flat-2 edos use dup-sharp and major2:
- 11edo has the dupsharp (C = ^^C#) and the downmajor2 (C = vD)
- 18b-edo has the dupsharp (C = ^^C#) and the dudmajor2 (C = vvD)
Very large eos use quip and quid (> and <) for quintuple arrows.
- 159edo has triplequidsharp and trupdim2
Canonical EIs for edos 5-55
| EDO | written | spoken | 3rd EI | |||
|---|---|---|---|---|---|---|
| 5 | –– | m2 | ––– | minor2 | ||
| 6 | (12edo subset notation) | |||||
| 7 | A1 | –– | sharp | ––– | ||
| 8 | (24edo subset notation) | |||||
| 9 | –– | A2 | ––– | aug2 | ||
| 10 | vvA1 | m2 | dudsharp | minor2 | ||
| 11 | ^^A1 | vM2 | dupsharp | downmajor2 | ||
| 12 | –– | d2 | ––– | dim2 | ||
| 13b | ^3A1 | vM2 | trupsharp | downmajor2 | ||
| 14 | A1 | vv2 | sharp | dud2 | ||
| 15 | v3A1 | m2 | trudsharp | minor2 | ||
| 16 | –– | AA2 | ––– | dubaug2 | ||
| 17 | vvA1 | ^d2 | dudsharp | updim2 | ||
| 18b | ^^A1 | vvM2 | dupsharp | dudmajor2 | ||
| 19 | –– | dd2 | ––– | dubdim2 | ||
| 20 | v4A1 | m2 | quudsharp | minor2 | ||
| 21 | A1 | v32 | sharp | trud2 | ||
| 22 | v3A1 | ^^d2 | trudsharp | dupdim2 | vm2 | downminor2 |
| 23 | –– | A32 | ––– | tripaug2 | ||
| 24 | vvA1 | d2 | dudsharp | dim2 | ||
| 25 | <A1 | m2 | quidsharp | minor2 | ||
| 26 | –– | d32 | ––– | tripdim2 | ||
| 27 | v4A1 | ^3d2 | quudsharp | trupdim2 | vm2 | downminor2 |
| 28 | A1 | v42 | sharp | quud2 | ||
| 29 | v3A1 | ^d2 | trudsharp | updim2 | ||
| 30 | v<A1 | m2 | downquidsharp | minor2 | ||
| 31 | vvA1 | vd2 | dudsharp | downdim2 | ||
| 32 | <A1 | ^4d2 | quidsharp | quupdim2 | vm2 | downminor2 |
| 33 | –– | d42 | ––– | quaddim2 | ||
| 34 | v4A1 | ^^d2 | quudsharp | dupdim2 | vvm2 | dudminor2 |
| 35 | A1 | <2 | sharp | quid2 | ||
| 36 | v3A1 | d2 | trudsharp | dim2 | ||
| 37 | v<A1 | >d2 | downquidsharp | quipdim2 | vm2 | downminor2 |
| 38 | vvA1 | vvd2 | dudsharp | duddim2 | ||
| 39 | <A1 | ^3d2 | quidsharp | trupdim2 | vvm2 | dudminor2 |
| 40 | –– | d52 | ––– | quindim2 | ||
| 41 | v4A1 | ^d2 | quudsharp | updim2 | ||
| 42 | vv<A1 | ^>d2 | dudquidsharp | upquipdim2 | vm2 | downminor2 |
| 43 | v3A1 | vd2 | trudsharp | downdim2 | ||
| 44 | v<A1 | ^4d2 | downquidsharp | quupdim2 | vvm2 | dudminor2 |
| 45 | vvA1 | v3d2 | dudsharp | truddim2 | ||
| 46 | <A1 | ^^d2 | quidsharp | dupdim2 | ||
| 47 | – | d62 | ––– | |||
| 48 | v4A1 | d2 | quudsharp | dim2 | ||
| 49 | vv<A1 | >d2 | dudquidsharp | quipdim2 | vvm2 | dudminor2 |
| 50 | v3A1 | vvd2 | trudsharp | duddim2 | ||
| 51 | v<A1 | ^3d2 | downquidsharp | trupdim2 | ||
| 52 | vvA1 | v4d2 | dudsharp | quuddim2 | ||
| 53 | <A1 | ^d2 | quidsharp | updim2 | ||
| 54 | v3<A1 | ^>d2 | trudquidsharp | upquipdim2 | vvm2 | dudminor2 |
| 55 | v4A1 | vd2 | quudsharp | downdim2 | ||
Pergens
Overview
While most edos have two EIs, single-pair pergen notations have just one.
Some EIs are "tippy", meaning their direction can reverse depending on the tuning. Any EI that uses the dim 2nd is tippy. For example, (P8/2, P5) has ^^d2 if P5 > 700¢, but vvd2 if P5 < 700¢.
Some pergens use slashes (lifts and drops) as well as arrows, and thus have two EIs. Some pergens don't need slashes, but add them anyway, for flexibility of interval naming and chord naming.
Some pergens have multiple notations, each with its own set of EIs.
Canonical EIs for various pergens
| Pergen | written | spoken |
|---|---|---|
| (P8, P5) | (none) | |
| (P8/2, P5) | ^^d2 | dupdim2 |
| (P8, P4/2) | vvm2 | dudminor2 |
| (P8, P5/2) | vvA1 | dudsharp |
| (P8/3, P5) | ^3d2 | trupdim2 |
| (P8, P4/3) | v3A1 | trudsharp |
| (P8, P5/3) | v3m2 | trudminor2 |
| (P8, P11/3) | ^3dd2 | trupdubdim2 |
| " | v3M2 | trudmajor2 |
| (P8/4, P5) | v4d2 | quuddim2 |
| (P8, P4/4) | ^4dd2 | quupdubdim2 |
| (P8, P5/4) | v4A1 | quudsharp |
| (P8, P11/4) | (double-pair notation) | |
| (P8, P12/4) | v4m2 | quudminor2 |
| (P8/5, P5) | <m2 | quidminor2 |
| (P8, P4/5) | >d2 | quipdim2 |
| (P8, P5/5) | <A2 | quidaug2 |
| (P8, P11/5) | <A1 | quidsharp |
| (P8, P12/5) | >dd2 | quipdubdim2 |
| (P8, ccP4/5) | <M2 | quidmajor2 |