5-limit harmony extends upon the familiar 3-limit harmony by adding its own accidental, denoted by ^v or ↑↓. We name this new accidental up/down

The up/down accidental allows four factors of 3 to be exchanged for one factor of 5

For example, the interval E, or 81/64, may be simplified by using a "down" accidental to produce the interval Ev, or 5/4, pronounced "E down"

Ptolemy's intense diatonic is one scale which may be produced in this system:

C   D  Ev   F   G  Av   Bv

1 9/8 5/4 4/3 3/2 5/3 15/8


+-------------+
|Naturals     |
+---+---------+
| F |   4/3   |
| C |   1/1   |
| G |   3/2   |
| D |   9/8   |
| A |  27/16  |
| E |  81/64  |		      
| B | 243/128 |		      
+---+---------+		      

+---------------+
|Accidentals    |
+---+-----------+
| # | 2187/2084 |
| b | 2084/2187 |
| ^ |   81/80   |
| v |   80/81   |
+---+-----------+

+--------------------+
|Tone map            |
+------+------+------+
| A bv | F b  | Dbb^ |
+------+------+------+
| E bv | C b  | Abb^ |
+------+------+------+
| B bv | G b  | Ebb^ |
+------+------+------+
| F  v | D b  | Bbb^ |
+------+------+------+
| C  v | A b  | F b^ |
+------+------+------+
| G  v | E b  | C b^ |
+------+------+------+
| D  v | B b  | G b^ |
+------+------+------+
| A  v | F    | D b^ |
+------+------+------+
| E  v | C    | A b^ |
+------+------+------+
| B  v | G    | E b^ |
+------+------+------+
| F #v | D    | B b^ |
+------+------+------+
| C #v | A    | F  ^ |
+------+------+------+
| G #v | E    | C  ^ |
+------+------+------+
| D #v | B    | G  ^ |
+------+------+------+
| A #v | F #  | D  ^ |
+------+------+------+
| E #v | C #  | A  ^ |
+------+------+------+
| B #v | G #  | E  ^ |
+------+------+------+
| F##v | D #  | B  ^ |
+------+------+------+
| C##v | A #  | F #^ |
+------+------+------+
| G##v | E #  | C #^ |
+------+------+------+
| D##v | B #  | G #^ |
+------+------+------+