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 unison, or EU for short. Often there are multiple possible EUs. This article presents the canonical EUs of edos and rank-2 temperaments for ups and downs notation.
EDOs
Overview
Sharp-1 edos, flat-1 edos and 7edo have only one EU. All others have an infinite number of EUs, but only two canonical EUs, which can be combined in various ways to create all the others. The canonical EUs are chosen to be easy to memorize and use. There are a few edos for which a third EU is sometimes useful.
The two canonical EUs are generally the A1 (aug unison) and the d2 (dim 2nd), upped or downed. The number of downs on the A1 equals the sharpness of the edo. The number of ups on the d2 equals the dodeca-sharpness of the edo. For example, 31edo's EUs 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 edosteps.
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 G♯? Did you have to think very hard to get the answer, A♭? 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, to avoid double flats, think of the interval as if from B. For augmented intervals, to avoid double sharps, use F. For example, what's a diminished 4th? A perfect 4th from B is E, so a dim 4th must be E♭. And E♭ 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 EUs are updim2 and quudsharp. ^^A5 + ^d2 = ^3m6, and ^3m6 + v4A1 = vM6.
19edo's EU is a doubly-diminished 2nd. To add it, add the d2, then flatten. For example, C♯ becomes D♭, which flattens to D𝄫. 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♯ = D♭, A1 = m2)
- 17edo has the dudsharp (C = vvC♯, ^1 = vA1) and the updim2 (C♯ = ^D♭)
- 22edo has the trudsharp (C = vvvC♯) and the dupdim2 (C♯ = ^^D♭) and perhaps the downminor2 (C = vD♭)
- 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 EU. Those other than 12edo don't use the dim2:
- 12edo has the dim2 (C♯ = Db)
- 19edo has the double-dim2 (C♯ = D𝄫)
- 26edo has the triple-dim2 (C𝄪 = D𝄫)
Pentatonic edos use the sharp and the minor2.
- 5edo has the minor2 (C = D♭)
- 10edo has the dudsharp (C = vvC♯, ^1 = vA1) and the minor2
- 15edo has the trudsharp (C = vvvC♯, ^^1 = vA1) and the minor2
Sharp-0 edos (perfect edos) use the sharp and the perfect 2nd.
- 7edo has the sharp (C = C♯, P1 = A1)
- 14edo has the sharp and the dud2 (C = vvD, ^1 = v2)
- 21edo has the sharp and the trud2 (C = vvvD, ^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. (But see 16edo#Chord names for a workaround.) For harmonic notation, the three flat-1 edos use aug2 not dim2.
- 9edo has the aug2 (C♭ = D, d1 = M2)
- 16edo has the double-aug2 (C♭ = D♯, d1 = A2)
- 23edo has the triple-aug2 (C♭ = 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 edos use quip and quid (> and <) for quintuple arrows.
- 159edo has triplequidsharp and trupdim2
Canonical EUs for edos 5-55
| EDO | Written | Spoken | 3rd EU | |||
|---|---|---|---|---|---|---|
| 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 EUs, single-pair pergen notations have just one.
Some EUs are "tippy", meaning their direction can reverse depending on the tuning. Any EU 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 EUs. 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 EUs.
Canonical EUs 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 |