Ben Johnston's notation: Difference between revisions

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'''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 accidentals are used for inflections, all of which denote [[superparticular]] ratios or their reciprocals:

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:

{| class="wikitable center-all"

|+ Accidentals in ~~Ben~~ Johnston's notation

|+ Accidentals in Johnston's notation

|-

! Symbol !! Ratio !! Symbol !! Ratio

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Johnston combines the symbols 7 {{invert|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.

A circle of just fifths is given by:

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.

... 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.

== External links ==

@@ Line 3: / Line 3: @@
 '''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 accidentals are used for inflections, all of which denote [[superparticular]] ratios or their reciprocals:
+The natural notes (white keys on the piano) are selected so that the chord F&#x200A;–&#x200A;A&#x200A;–&#x200A;C&#x200A;–&#x200A;E&#x200A;–&#x200A;G&#x200A;–&#x200A;B&#x200A;–&#x200A;D consists of three stacked 4:5:6 chords, i.e. F&#x200A;–&#x200A;A&#x200A;–&#x200A;C, C&#x200A;–&#x200A;E&#x200A;–&#x200A;G, and G&#x200A;–&#x200A;B&#x200A;–&#x200A;D are just major triads. Then the following accidentals are used for inflections, all of which denote [[superparticular]] ratios or their reciprocals:
 {| class="wikitable center-all"
-|+ Accidentals in Ben Johnston's notation
+|+ Accidentals in Johnston's notation
 |-
 ! Symbol !! Ratio !! Symbol !! Ratio
@@ Line 33: / Line 33: @@
 Johnston combines the symbols 7 {{invert|7}} ↑ ↓ with ♯ ♭ if symbols from both categories are present.
-A circle of just fifths is given by ... D♭&#x2212;&#x2212; A♭&#x2212; E♭&#x2212; B♭&#x2212; 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.
+A circle of just fifths is given by:
-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.
+... G♭<sup>&#x2212;&#x2212;</sup> &mdash; D♭<sup>&#x2212;&#x2212;</sup> &mdash; A♭<sup>&#x2212;</sup> &mdash; E♭<sup>&#x2212;</sup> &mdash; B♭<sup>&#x2212;</sup> &mdash; F &mdash; C &mdash; G &mdash; D &mdash; A<sup>+</sup> &mdash; E<sup>+</sup> &mdash; B<sup>+</sup> &mdash; F♯<sup>+</sup><sup>+</sup> &mdash; C♯<sup>+</sup><sup>+</sup> ...
+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 &mdash; G &mdash; E &mdash; B♭<sub>7</sub> &mdash; D &mdash; F↑ &mdash; A♭<sub>13</sub> &mdash; B &mdash; C♯<sub>17</sub> &mdash; E♭<sub>19</sub> &mdash; F<sub>7</sub><sup>+</sup> &mdash; F♯<sub>23</sub><sup>+</sup> &mdash; G♯ &mdash; A<sup>+</sup> &mdash; B♭<sub>29</sub> &mdash; B<sub>31</sub>.
+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.
 == External links ==