Ben Johnston's notation

From Xenharmonic Wiki
Jump to navigation Jump to search

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:

Johnston notation inflections
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.

External links