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.

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Ben Johnston (composer) #Staff notation

The natural 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 accidentals are used for inflections, all of which denote superparticular ratios or their reciprocals:

Accidentals in Johnston's notation
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 sharps (♯) and flats (♭) if symbols from both categories are present.

A circle of just fifths is given by:

... G♭⁻⁻ — D♭⁻⁻ — A♭⁻ — E♭⁻ — B♭⁻ — F — C — G — D — A⁺ — E⁺ — B⁺ — F♯⁺⁺ — C♯⁺⁺ ...

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.

Ben Johnston's notation

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