Ben Johnston's notation
Ben Johnston's notation is a staff notation system for just intonation. Ben Johnston developed it up to the 31-limit, employed in his String Quartet No. 9, although intervals exceeding the 13-limit are encountered mostly incidentally in his music.
The base notes (white keys on the piano) are selected so that the chord F A C E G B D consists of three stacked 4:5:6 chords, i.e. F A C, C E G, and G B D are just major triads. Then the following symbols are used for inflections, all of which denote superparticular ratios or their reciprocals:
| Symbol | Ratio | Symbol | Ratio |
|---|---|---|---|
| + | 81/80 | − | 80/81 |
| ♯ | 25/24 | ♭ | 24/25 |
| 7 | 35/36 | 7 | 36/35 |
| ↑ | 33/32 | ↓ | 32/33 |
| 13 | 65/64 | 13 | 64/65 |
| 17 | 51/50 | 17 | 50/51 |
| 19 | 95/96 | 19 | 96/95 |
| 23 | 46/45 | 23 | 45/46 |
| 29 | 145/144 | 29 | 144/145 |
| 31 | 31/30 | 31 | 30/31 |
Johnston combines the symbols 7 7 ↑ ↓ with ♯ ♭ if symbols from both categories are present.
A circle of just fifths is given by ... D♭−− A♭− E♭− B♭− F C G D A+ E+ B+ F♯++ ..., with a plus or minus added for every loop around the ends of the core F A C E G B D sequence. The odd harmonic series up to 31 starting on C is given by C G E B♭7 D F↑ A♭13 B C♯17 E♭19 F+7 F♯+23 G♯ A+ B♭29 B31.
Johnston's notation sacrifices some mathematical purity compared to Helmholtz-Ellis notation, as it is based on 4:5:6 chords rather than Pythagorean tuning. This comes at the possible advantage of fewer inflection markers needed for music that emphasizes the 5-limit.