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. | '''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]] | 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, which makes {{dash|C, D, E, F, G, A, B, C|s=hair|d=med}} a justly-intonated Ptolemy–Zarlino [[Zarlino|"intense" diatonic scale]]. Then the following accidentals are used for inflections, all of which denote [[superparticular ratio]]s or their reciprocals: | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
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! Symbol !! Ratio !! Symbol !! Ratio | ! Symbol !! Ratio !! Symbol !! Ratio | ||
|- | |- | ||
| + || [[81/80]] || & | | + || [[81/80]] || − || 80/81 | ||
|- | |- | ||
| ♯ || [[25/24]] || ♭ || 24/25 | | ♯ || [[25/24]] || ♭ || 24/25 | ||
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Johnston combines the symbols 7 {{invert|7}} ↑ ↓ with sharps (♯) and flats (♭) if symbols from both categories are present. | Johnston combines the symbols 7 {{invert|7}} ↑ ↓ with sharps (♯) and flats (♭) if symbols from both categories are present. | ||
A | A chain of just fifths is given by: | ||
... {{dash|G♭<sup>& | ... {{dash|F♭<sup>−−</sup>, C♭<sup>−−</sup>, G♭<sup>−−</sup>, D♭<sup>−−</sup>, A♭<sup>−</sup>, E♭<sup>−</sup>, B♭<sup>−</sup>, F, C, G, D, A<sup>+</sup>, E<sup>+</sup>, B<sup>+</sup>, F♯<sup>++</sup>, C♯<sup>++</sup>, G♯<sup>++</sup>, D♯<sup>++</sup>, A♯<sup>+++</sup>, E♯<sup>+++</sup>, B♯<sup>+++</sup> |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. | with a plus or minus added for every loop around the ends of the core {{dash|F, A, C, E, G, B, D|s=hair|d=med}} sequence. | ||
The odd harmonic series up to 31 starting on C is given by: | The odd harmonic series up to 31 starting on C is given by: | ||
{{dash|C, G, E, B♭<sup>7</sup>, D, | {{dash|C, G, E, B♭<sup>7</sup>, D, F↑, A♭<sup>13</sup>, B, C♯<sup>17</sup>, E♭<sup>19</sup>, F<sup>7+</sup>, F♯<sup>23+</sup>, G♯, A<sup>+</sup>, B♭<sup>29</sup>, B<sup>31</sup>|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. | 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. | ||
Revision as of 18:54, 17 May 2024
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 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, which 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 the symbols 7 7 ↑ ↓ 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.