Enharmonic unison
The notation of every temperament, including every edo, has at least one enharmonic unison, abbreviated as EU (with one rare exception, see below). An EU is by definition enharmonically equivalent to a perfect unison. (Enharmonically equivalent is used here in the modern sense of "the same exact pitch, merely named differently".) Any note or interval can be respelled by adding or subtracting an EU.
For example, in 12edo, A4 = d5 and F♯ = G♭. Such equivalences result from adding or subtracting a diminished 2nd, abbreviated as d2. But in 19edo, A4 = dd5 and F♯ = G𝄫. 19edo's EU is the dd2.
EUs are very useful for respelling notes and intervals less awkwardly. For example, in 12edo we can add a d2 to B♯ to convert it to C, or we can subtract a d2 from a diminished 4th to get a major 3rd.
Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of microtonal accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. EUs are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank.
| Tuning | Tuning's rank | Notation | Notation's rank without any EUs |
Minimum # of EUs needed |
|---|---|---|---|---|
| 19edo | rank-1 | conventional | rank-2 | 1 |
| 22edo | rank-1 | ups and downs | rank-3 | 2 |
| Meantone/Guti | rank-2 | conventional | rank-2 | 0 |
| Porcupine/Triyoti | rank-2 | ups and downs | rank-3 | 1 |
Just as a temperament can be defined by a list of primes (or more generally a JI subgroup) and a list of linearly independent commas, a notation can be defined by a list of accidentals and a list of linearly independent EUs. Furthermore, that notation implies a certain edo or pergen.For example, conventional notation (the usual 7 letters plus sharps and flats) with a dd2 EU must be 19edo, because that EU reduces the infinite chain of 5ths to a closed loop of 19 notes. Conventional notation plus ups and downs with a vvA1 EU must be (P8, P5/2), because P5 = vM3 + ^m3 = vM3 + ^m3 + vvA1 = vM3 + vM3.
Some notations have just one EU, others are multi-EU. A multi-comma temperament can be defined by various equivalent but different comma lists. Likewise, a multi-EU notation can be defined by various EUs. Some notations define a canonical list of EUs.
Notation-specific observations
There's one type of edo notation that does not produce any EUs: giving each note a unique letter. For example, an octave of 7edo is notated C D E F G A B C. The intervals are named 1sn, 2nd, 3rd, 4th, 5th, 6th, 7th, and octave, all perfect. There are no major or minor or augmented or diminished intervals. As long as one refrains from using sharps or flats, there will be one and only one name for each note and each interval. Because there is a finite number of possible note names, this notation is rank-1 not rank-2.
Likewise, if an octave of 8edo were notated as J K L M N O P Q J with no sharps or flats, there would be no EUs. Though, this type of notation is obviously only practical for small edos.
The usage of half-sharps and half-flats ( 
and 
) creates a rather obvious EU: A1.
Some notations, like ups and downs, notate all but the largest edos with only a single additional pair of accidentals. Other notations, like Sagittal and SKULO, notate edos using various commas such as 81/80, 64/63, and 33/32. Thus they notate an edo interval as a nearby JI interval, indicating the "feel" of the interval. For some edos, these notations use multiple such commas. For example, Sagittal notates 41edo using 81/80 and 33/32, and SKULO notates it using all three commas. Each comma used adds a pair of accidentals, and hence increases the minimum number of EUs needed, complicating the respelling of intervals.
Just intonation notations generally do not have EUs. Each ratio factors uniquely into primes, and each prime number above 3 has its own pair of accidentals, therefore each ratio is uniquely notated. But a JI notation that uses more accidental pairs than that (such as standard Sagittal, as opposed to prime-factor Sagittal) does have EUs. Furthermore, as a practical matter just intonation is audibly indistinguishable from certain microtemperaments that do have EUs.
EUs were first systematically investigated by Kite Giedraitis in 2018 while developing pergen notation.