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 | The base 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. This in turn makes {{dash|C, D, E, F, G, A, B, C|s=hair|d=med}} a justly-intonated [[Zarlino|Ptolemy–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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| + || [[81/80]] || − || 80/81 | | + || [[81/80]] || − || 80/81 | ||
|- | |- | ||
| | | ♯ || [[25/24]] || ♭ || 24/25 | ||
|- | |- | ||
| 7 || 35/36 || {{invert|7}} || [[36/35]] | | 7 || 35/36 || {{invert|7}} || [[36/35]] | ||
|- | |- | ||
| | | ↑ || [[33/32]] || ↓ || 32/33 | ||
|- | |- | ||
| 13 || [[65/64]] || {{invert|13}} || 64/65 | | 13 || [[65/64]] || {{invert|13}} || 64/65 | ||
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|} | |} | ||
Johnston combines | Johnston combines numeric accidentals (7, {{invert|7}}, ↑, ↓, 13, {{invert|13}}, etc.) with sharps (♯) and flats (♭) if symbols from both categories are present. | ||
A chain of just fifths is given by: | A chain of just fifths is given by: | ||
... {{dash| | ... {{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 {{dash|F, A, C, E, G, B, D|s=hair|d=med}} 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. | ||
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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, | {{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. | ||