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

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:

{| class="wikitable center-all"

|+ Johnston notation ~~inflections~~

|+ Accidentals in Ben Johnston's notation

|-

! Symbol !! Ratio !! Symbol !! Ratio

Line 14:

| ♯ || [[25/24]] || ♭ || 24/25

|-

| 7 || 35/36 || ~~~~7~~~~ || [[36/35]]

| 7 || 35/36 || {{invert|7}} || [[36/35]]

|-

| ↑ || [[33/32]] || ↓ || 32/33

|-

| 13 || [[65/64]] || ~~~~13~~~~ || 64/65

| 13 || [[65/64]] || {{invert|13}} || 64/65

|-

| 17 || [[51/50]] || ~~~~17~~~~ || 50/51

| 17 || [[51/50]] || {{invert|17}} || 50/51

|-

| 19 || 95/96 || ~~~~19~~~~ || [[96/95]]

| 19 || 95/96 || {{invert|19}} || [[96/95]]

|-

| 23 || [[46/45]] || ~~~~23~~~~ || 45/46

| 23 || [[46/45]] || {{invert|23}} || 45/46

|-

| 29 || [[145/144]] || ~~~~29~~~~ || 144/145

| 29 || [[145/144]] || {{invert|29}} || 144/145

|-

| 31 || [[31/30]] || ~~~~31~~~~ || 30/31

| 31 || [[31/30]] || {{invert|31}} || 30/31

|}

Johnston combines the symbols 7 ~~~~7~~~~ ↑ ↓ with ♯ ♭ if symbols from both categories are present.

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.

@@ 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 symbols are used for inflections, all of which denote [[superparticular]] ratios or their reciprocals:
+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:
 {| class="wikitable center-all"
-|+ Johnston notation inflections
+|+ Accidentals in Ben Johnston's notation
 |-
 ! Symbol !! Ratio !! Symbol !! Ratio
@@ Line 14: / Line 14: @@
 | ♯ || [[25/24]] || ♭ || 24/25
 |-
-| 7 || 35/36 || <span style="display: inline-block; transform: rotate(180deg);">7</span> || [[36/35]]
+| 7 || 35/36 || {{invert|7}} || [[36/35]]
 |-
 | ↑ || [[33/32]] || ↓ || 32/33
 |-
-| 13 || [[65/64]] || <span style="display: inline-block; transform: rotate(180deg);">13</span> || 64/65
+| 13 || [[65/64]] || {{invert|13}} || 64/65
 |-
-| 17 || [[51/50]] || <span style="display: inline-block; transform: rotate(180deg);">17</span> || 50/51
+| 17 || [[51/50]] || {{invert|17}} || 50/51
 |-
-| 19 || 95/96 || <span style="display: inline-block; transform: rotate(180deg);">19</span> || [[96/95]]
+| 19 || 95/96 || {{invert|19}} || [[96/95]]
 |-
-| 23 || [[46/45]] || <span style="display: inline-block; transform: rotate(180deg);">23</span> || 45/46
+| 23 || [[46/45]] || {{invert|23}} || 45/46
 |-
-| 29 || [[145/144]] || <span style="display: inline-block; transform: rotate(180deg);">29</span> || 144/145
+| 29 || [[145/144]] || {{invert|29}} || 144/145
 |-
-| 31 || [[31/30]] || <span style="display: inline-block; transform: rotate(180deg);">31</span> || 30/31
+| 31 || [[31/30]] || {{invert|31}} || 30/31
 |}
-Johnston combines the symbols 7 <span style="display: inline-block; transform: rotate(180deg);">7</span> ↑ ↓ with ♯ ♭ if symbols from both categories are present.
+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.