View source for Kite's thoughts on enharmonic unisons

{{Wikipedia| Enharmonic equivalence }}
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, {{nowrap|A4 {{=}} d5}} and {{nowrap|F&#x266F; {{=}} G&#x266D;}}. Such equivalences result from adding or subtracting a diminished 2nd, abbreviated as d2. But in 19edo, {{nowrap|A4 {{=}} dd5}} and {{nowrap|F&#x266F; {{=}} G&#x266D;&#x266D;}}. 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 {{nowrap|B&#x266F;}} to convert it to C, or we can subtract a d2 from a diminished 4th to get a major 3rd.

Just intonation does not have EUs, 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. 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.

{| class="wikitable"
|-
! Tuning
! Tuning's rank
! Notation
! Notation's rank<br>without any EUs
! Minimum # of<br>EUs needed
|-
| 19edo
| rank-1
| conventional
| rank-2
| 1
|-
| 22edo
| rank-1
| ups and downs
| rank-3
| 2
|-
| [[Meantone|Meantone/Guti]]
| rank-2
| conventional
| rank-2
| 0
|-
| [[Porcupine|Porcupine/Triyoti]]
| rank-2
| ups and downs
| rank-3
| 1
|}

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 (t and d) creates a rather obvious EU: dd#.

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.

EUs were first systematically investigated by [[Kite Giedraitis]] in 2018 while developing [[Pergen|pergen notation]].

== See also ==
* [[Interval arithmetic]]
* [[Enharmonic unisons in ups and downs notation#Canonical EUs for edos 5-55]]
* [[Enharmonic unisons in ups and downs notation#Canonical EUs for various pergens]]