Ben Johnston's notation: Difference between revisions

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

{| class="wikitable center-all"

Line 35:

A circle of just fifths is given by:

... {{dash|G♭−−, D♭−−, A♭−, E♭−, B♭−, F, C, G, D, A+, E+, B+, F♯++, C♯++|~~spaces~~=hair|~~sep~~=~~mdash~~}} ...

... {{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.

Line 41:

The odd harmonic series up to 31 starting on C is given by:

{{dash|C, G, E, B♭7, D, F↑, A♭<~~sub~~>13</~~sub~~>, B, C♯<~~sub~~>17</~~sub~~>, E♭<~~sub~~>19</~~sub~~>, F<~~sub~~>7~~~~+, F♯<~~sub~~>23~~~~+, G♯, A+, B♭<~~sub~~>29</~~sub~~>, B<~~sub~~>31</~~sub~~>|~~spaces~~=hair|~~sep~~=~~mdash~~}}.

{{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.

@@ 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 natural notes (white keys on the piano) are selected so that the chord {{dash|F, A, C, E, G, B, D|spaces=hair|sep=ndash}} consists of three stacked 4:5:6 chords, i.e. {{dash|F, A, C|spaces=hair|sep=ndash}}, {{dash|C, E, G|spaces=hair|sep=ndash}}, and {{dash|G, B, D|spaces=hair|sep=ndash}} 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 {{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:
 {| class="wikitable center-all"
@@ Line 35: / Line 35: @@
 A circle of just fifths is given by:
-... {{dash|G♭<sup>&#x2212;&#x2212;</sup>, D♭<sup>&#x2212;&#x2212;</sup>, A♭<sup>&#x2212;</sup>, E♭<sup>&#x2212;</sup>, B♭<sup>&#x2212;</sup>, F, C, G, D, A<sup>+</sup>, E<sup>+</sup>, B<sup>+</sup>, F♯<sup>+</sup><sup>+</sup>, C♯<sup>+</sup><sup>+</sup>|spaces=hair|sep=mdash}} ...
+... {{dash|G♭<sup>&#x2212;&#x2212;</sup>, D♭<sup>&#x2212;&#x2212;</sup>, A♭<sup>&#x2212;</sup>, E♭<sup>&#x2212;</sup>, B♭<sup>&#x2212;</sup>, F, C, G, D, A<sup>+</sup>, E<sup>+</sup>, B<sup>+</sup>, F♯<sup>+</sup><sup>+</sup>, C♯<sup>+</sup><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.
@@ Line 41: / Line 41: @@
 The odd harmonic series up to 31 starting on C is given by:
-{{dash|C, G, E, B♭<sub>7</sub>, D, F↑, A♭<sub>13</sub>, B, C♯<sub>17</sub>, E♭<sub>19</sub>, F<sub>7</sub><sup>+</sup>, F♯<sub>23</sub><sup>+</sup>, G♯, A<sup>+</sup>, B♭<sub>29</sub>, B<sub>31</sub>|spaces=hair|sep=mdash}}.
+{{dash|C, G, E, B♭<sub>7</sub>, 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.