Kite's thoughts on enharmonic unisons
The notation of every temperament, including every edo, has at least one enharmonic interval, or EI for short (with one exception, see below). An EI is by definition enharmonically equivalent to a perfect unison, thus it's just a perfect unison under a different name. Any note or interval can be respelled by adding or subtracting an EI.
For example, in 12edo, A4 = d5 and F♯ = G♭. Such equivalences result from adding or subtracting a diminished 2nd, abbreviated as d2.
EIs are very useful for respelling notes and intervals less awkwardly. For example, in meantone temperament the diminished second corresponds to the diesis; as such, we can add a diesis to C𝄪 to convert it to D, and we can also subtract a diminished second from a diminished fourth to get a major third.
Just intonation does not have EIs, although as a practical matter just intonation is audibly indistinguishable from certain microtemperaments that do have them.
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. EIs 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 enharmonics 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 EIs |
Minimum # of EIs 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 |
Some notations have just one EI, others are multi-EI. A multi-comma temperament can be defined by various equivalent but different comma lists. Likewise, a multi-EI notation can be defined by various EIs. Some notations define a canonical list of EIs.
Notation-specific observations
The one type of edo notation that does not produce any EIs: 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 sharps or flats, and no major or minor or augmented or diminished intervals. Thus there is 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 EIs. Though, this type of notation is obviously only practical for small edos.
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 adds an EI. Respelling the sum of two intervals becomes much more complicated.