Ben Johnston's notation: Difference between revisions

'''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 natural notes (white keys on the piano) are selected so that the chord {{dash|F, A, C, E, G, B, D|s=hair|d=med}} consists of three stacked 4:5:6 chords, i.e. {{dash|F, A, C|s=hair|d=med}}, {{dash|C, E, G|s=hair|d=med}}, and {{dash|G, B, D|s=hair|d=med}} 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

| + || [[81/80]] || − || 80/81

| ♯ || [[25/24]] || ♭ || 24/25

| ↑ || [[33/32]] || ↓ || 32/33

Johnston combines the symbols 7 {{invert|7}} ↑ ↓ with ♯ ♭ if symbols from both categories are present.

A circle of just fifths is given by:

... {{dash|G♭^{−−}, D♭^{−−}, A♭^−, E♭^−, B♭^−, F, C, G, D, A⁺, E⁺, B⁺, F♯⁺⁺, C♯⁺⁺|s=hair|d=long}} ...

with a plus or minus added for every loop around the ends of the core F A C E G B D sequence.

{{dash|C, G, E, B♭⁷, D, F↑, A♭¹³, B, C♯¹⁷, E♭¹⁹, F⁷⁺, F♯²³⁺, G♯, A⁺, B♭²⁹, B³¹|s=hair|d=long}}.

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.