Ben Johnston's notation: Difference between revisions

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~~Composer~~ '''[[Ben Johnston]]'s notation''' is a staff notation system for [[just intonation]]. Johnston ~~used it~~ up to the [[31-limit]].

'''Ben Johnston's notation''' is a staff notation system for [[just intonation]] developed by composer [[Ben Johnston]], which supports prime harmonics up to and including 31. It is employed in his String Quartet No. 9, although intervals exceeding the [[13-limit]] are encountered only occasionally 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 {{dash|F, A, C, E, G, B, D|med}} consists of three stacked 4:5:6 chords, i.e. {{dash|F, A, C|med}}, {{dash|C, E, G|med}}, and {{dash|G, B, D|med}} are just major triads. This in turn makes {{dash|C, D, E, F, G, A, B, C|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"

{| class="wikitable center-all"

|+ Johnston notation ~~inflections~~

|+ style="font-size: 105%;" | Accidentals in Johnston's notation

|-

! Symbol !! Ratio !! Symbol !! Ratio

|-

| + || [[81/80]] || - || 80/81

| + || [[81/80]] || − || 80/81

|-

| ♯ || [[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

|}

~~When both a~~ 7 ~~and a flat symbol are present~~, ~~Johnston combines them into a flat symbol with a line extending from the upper left~~. ~~This is not to be confused~~ with ~~a flat symbol~~ and ~~a minus sign~~.

Johnston combines numeric accidentals (7, {{invert|7}}, ↑, ↓, 13, {{invert|13}}, etc.) with sharps (♯) and flats (♭) 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 chain 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.

... {{dash|F♭−−, C♭−−, G♭−−, D♭−−, A♭−, E♭−, B♭−, F, C, G, D, A+, E+, B+, F♯++, C♯++, G♯++, D♯++, A♯+++, E♯+++, B♯+++|hair|long}} ...

with a plus or minus added for every loop around the ends of the core {{dash|F, A, C, E, G, B, D|med}} sequence.

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

{{dash|C, G, E, B♭7, D, F↑, A♭13, B, C♯17, E♭19, F7+, F♯23+, G♯, A+, B♭29, B31|hair|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.

== See also ==

* [[Johnston–Copper notation]]

* [[Helmholtz–Ellis notation]]

* [[Functional Just System]]

* [[Color notation]]

== External links ==

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* [http://tonalsoft.com/enc/j/johnston.aspx Johnston notation] on the [[Tonalsoft encyclopedia]]

* [http://www.marcsabat.com/pdfs/EJItext.pdf Marc Sabat - On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation]

[[Category:Just intonation]]

@@ Line 1: / Line 1: @@
 {{Wikipedia| Ben Johnston (composer) #Staff notation}}
-Composer '''[[Ben Johnston]]'s notation''' is a staff notation system for [[just intonation]]. Johnston used it up to the [[31-limit]].
+'''Ben Johnston's notation''' is a staff notation system for [[just intonation]] developed by composer [[Ben Johnston]], which supports prime harmonics up to and including 31. It is employed in his String Quartet No. 9, although intervals exceeding the [[13-limit]] are encountered only occasionally 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 {{dash|F, A, C, E, G, B, D|med}} consists of three stacked 4:5:6 chords, i.e. {{dash|F, A, C|med}}, {{dash|C, E, G|med}}, and {{dash|G, B, D|med}} are just major triads. This in turn makes {{dash|C, D, E, F, G, A, B, C|med}} a justly-intonated [[Zarlino|Ptolemy&ndash;Zarlino "intense" diatonic scale]]. Then the following accidentals are used for inflections, all of which denote [[superparticular ratio]]s or their reciprocals:
-{| class="wikitable"
+{| class="wikitable center-all"
-|+ Johnston notation inflections
+|+ style="font-size: 105%;" | Accidentals in Johnston's notation
 |-
 ! Symbol !! Ratio !! Symbol !! Ratio
 |-
-| + || [[81/80]] || - || 80/81
+| + || [[81/80]] || &minus; || 80/81
 |-
-| ♯ || [[25/24]] || ♭ || 24/25
+| &#x266F; || [[25/24]] || &#x266D; || 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
+| &uarr; || [[33/32]] || &darr; || 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
 |}
-When both a 7 and a flat symbol are present, Johnston combines them into a flat symbol with a line extending from the upper left. This is not to be confused with a flat symbol and a minus sign.
+Johnston combines numeric accidentals (7, {{invert|7}}, &uarr;, &darr;, 13, {{invert|13}}, etc.) with sharps (&#x266F;) and flats (&#x266D;) 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 chain 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.
+... {{dash|F&#x266D;<sup>&minus;&minus;</sup>, C&#x266D;<sup>&minus;&minus;</sup>, G&#x266D;<sup>&minus;&minus;</sup>, D&#x266D;<sup>&minus;&minus;</sup>, A&#x266D;<sup>&minus;</sup>, E&#x266D;<sup>&minus;</sup>, B&#x266D;<sup>&minus;</sup>, F, C, G, D, A<sup>+</sup>, E<sup>+</sup>, B<sup>+</sup>, F&#x266F;<sup>++</sup>, C&#x266F;<sup>++</sup>, G&#x266F;<sup>++</sup>, D&#x266F;<sup>++</sup>, A&#x266F;<sup>+++</sup>, E&#x266F;<sup>+++</sup>, B&#x266F;<sup>+++</sup>|hair|long}} ...
+with a plus or minus added for every loop around the ends of the core {{dash|F, A, C, E, G, B, D|med}} sequence.
+The odd harmonic series up to 31 starting on C is given by:
+{{dash|C, G, E, B&#x266D;<sup>7</sup>, D, F&uarr;, A&#x266D;<sup>13</sup>, B, C&#x266F;<sup>17</sup>, E&#x266D;<sup>19</sup>, F<sup>7+</sup>, F&#x266F;<sup>23+</sup>, G&#x266F;, A<sup>+</sup>, B&#x266D;<sup>29</sup>, B<sup>31</sup>|hair|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.
+== See also ==
+* [[Johnston–Copper notation]]
+* [[Helmholtz–Ellis notation]]
+* [[Functional Just System]]
+* [[Color notation]]
 == External links ==
@@ Line 41: / Line 55: @@
 * [http://tonalsoft.com/enc/j/johnston.aspx Johnston notation] on the [[Tonalsoft encyclopedia]]
 * [http://www.marcsabat.com/pdfs/EJItext.pdf Marc Sabat - On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation]
+{{Navbox notation}}
 [[Category:Just intonation]]