enharmonicity

the recognition of two pitches or intervals, which are separated by a small pitch distance, as the same musical gestalt. This occurs mainly as a result of musical context.

As a trivial example, in the 12-EQ scale, since the interval between all degrees is exactly 1.00 Semitone [= 100 cents], any interval plus or minus 0.50 Semitone from one of the 12-EQ degrees, will be perceived as the closest 12-EQ degree. This example is for explanatory purposes only - in most actual music the situation is considerably more complex.

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(added 2001.1.2, adapted from YahooGroup Tuning posts:)

... What I meant was that I've seen Beethoven scores which have notated pitches with flats which are then tied to their enharmonically equivalent sharp, or vice versa, and that Paul [Erlich] has pointed out to me that Mozart wrote a few things like this also.

But Paul's point is that Mozart (and Beethoven, and Wagner) must have had in mind when they wrote these, some form of temperament in which what he calls a "commatic" unison-vector disappears, and I agree.

I put "commatic" in quotes because in this 5-limit case the only actual comma which might be involved is the Pythagorean comma. The diaschisma is another possibility and it is also comma-sized. But other unison-vectors which might vanish have very different sizes, both larger and smaller than a comma; these could be the skhisma, diesis, or possibly a few others.

Here's a section of the 5-limit rectangular lattice in which I've notated only the D#'s and Eb's. Observe that notes with the same notation are a syntonic comma apart. We're already assuming that that vanishes, because the notation of the composers in the standard late-romantic repertoire never distinguishes it.

     4  D# .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 
     3  .  .  .  .  D# .  .  .  .  .  .  .  .  .  .  .  . 
     2  .  .  .  .  .  .  .  .  D# .  .  .  .  .  .  .  .
     1  Eb .  .  .  .  .  .  .  .  .  .  .  D# .  .  .  .
5^y  0  .  .  .  .  Eb .  . n^0 .  .  .  .  .  .  .  .  D#
    -1  .  .  .  .  .  .  .  .  Eb .  .  .  .  .  .  .  . 
    -2  .  .  .  .  .  .  .  .  .  .  .  .  Eb .  .  .  .
    -3  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  Eb

       -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9
                            3^x

In this example, we're considering unison-vectors which connect any of the D#'s with any of the Eb's.

I decided to add numbers to each of the notes on my lattice, so that we can formulate an "algebra of enharmonicity" for the 5-limit:

      4  D#1.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 
      3  .  .  .  .  D#2.  .  .  .  .  .  .  .  .  .  .  . 
      2  .  .  .  .  .  .  .  .  D#3.  .  .  .  .  .  .  .
      1  Eb1   .  .  .  .  .  .  .  .  .  .  D#4.  .  .  .
 5^y  0  .  .  .  .  Eb2.  . n^0 .  .  .  .  .  .  .  .  D#5
     -1  .  .  .  .  .  .  .  .  Eb3.  .  .  .  .  .  .  . 
     -2  .  .  .  .  .  .  .  .  .  .  .  .  Eb4.  .  .  .
     -3  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  Eb5
 
        -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9
                             3^x
 

All the D#-Eb pairs on this lattice are: (note the last two columns)

 
                             x,y              higher        lower
                ratio    coordinates  ~cents   note subtract note
 
 Pyth.comma 531441:528244  (12, 0)      23.46   D#(x+3) -  Eb(x)
  skhisma    32805:32768   ( 8, 1)       1.95   D#(x+2) -  Eb(x)
 diaschisma   2048:2025    (-4,-2)      19.55   Eb(x)   -  D#(x+1)
   diesis      128:125     ( 0,-3)      41.06   Eb(x)   -  D#(x)
               648:625     ( 4,-4)      62.57   Eb(x+1) -  D#(x)
              6561:6250    ( 8,-5)      84.07   Eb(x+2) -  D#(x)
            531441:500000  (12,-6)     105.6    Eb(x+3) -  D#(x)
                           (16,-7)     127.1    Eb(x+4) -  D#(x)

(Observe that D# is the higher note of the pair for the skhisma and Pythagorean comma, whereas it is Eb for the rest.)

[from Joe Monzo, JustMusic: A New Harmony]

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