Ben Johnston's notation
Ben Johnston's notation is a staff notation system for just intonation developed by composer Ben Johnston, which supports prime harmonics up to and including 31. It is employed in his String Quartet No. 9, although intervals exceeding the 13-limit are encountered only occasionally 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. This in turn makes C–D–E–F–G–A–B–C a justly-intonated Ptolemy–Zarlino "intense" diatonic scale. Then the following accidentals 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 numeric accidentals (7, 7, ↑, ↓, 13, 13, etc.) with sharps (♯) and flats (♭) if symbols from both categories are present.
A chain of just fifths is given by:
... F♭−− — C♭−− — G♭−− — D♭−− — A♭− — E♭− — B♭− — F — C — G — D — A+ — E+ — B+ — F♯++ — C♯++ — G♯++ — D♯++ — A♯+++ — E♯+++ — B♯+++ ...
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 — F7+ — F♯23+ — G♯ — A+ — B♭29 — B31.
Johnston's notation sacrifices some mathematical intuition compared to Helmholtz–Ellis notation, as it bases the natural notes on 4:5:6 chords rather than Pythagorean tuning. This comes at the possible advantage of fewer accidentals needed for music that emphasizes the 5-limit.