Johnston-Copper notation
Johnston-Copper notation is a musical notation system for just intonation based on Ben Johnston's notation but extended by means of key signatures. This is the notation used by William Copper in the works composed by the techniques of intonalism.
Naturals, sharps and flats
Ben Johnston takes as a just intonation basis the staff notation without accidentals (white keys on the piano) so that there are three 4:5:6 chords (F major, C major, and G major). Copper extends this system so that the same three 4:5:6 chords are transposed according to a key signature. In the key of C major, the Johnston and Copper systems are the same. In the key of one sharp, G major, the three 4:5:6 chords are changed so that the basis of just intonation becomes C major, G major, and D major. Any note found in this basis and not altered by a special accidental is tuned according to 5-limit just intonation: the third of the triad, or the '5' of the 4:5:6 chord, is a pure major third from the root; the fifth of the triad is a pure perfect fifth from the root.
Beyond the unaltered basis associated with any key signature, individual notes may be altered by means of accidentals. The standard sharp, flat, and natural signs have the same effect as in common practice notated Western music. A sharp put before a note will either take that note out of the diatonic system indicated by the key signature, or return a note into the diatonic system. For example in C major, the sharp before an F makes the note F#, out of the basis of the key. In G major, where the F is already sharp as defined by the key signature, the sharp may be used to indicate a return to the key signature if the prior measure of notated music used a different notation, or if the complexity of the music requires a cautionary indication that this note is indeed to be F#. All this is just as in common practice.
Other accidentals
When the music demands a note to be tuned differently than the 'basis' then special accidentals are used: an altered sharp sign, an altered flat sign, two altered natural signs (up and down) and two neutral signs (up and down). In Johnston-Copper Notation each such change is exactly a comma, the specific comma indicated by the ratio 81/80 and often called a syntonic comma.
Neutral accidentals
An example from Johnston shows the use of a form of a 'neutral' sign. Johnston Quartet 4 -01 and an example from Copper shows the same usage with a somewhat altered symbol.
In Johnston's hand-engraved score the neutral sign looks somewhat like a thick dash or hyphen; in Copper's notation, made with the Score computer engraving program (scor4) , it is an arrow with a flag at the top (used for the 'down one comma' neutral indication). Both are used for exactly the same purpose: in C major, Johnston's score, the note D if unaltered would sound too high, since it is tuned by default as a perfect fifth above the G according to the three 4:5:6 chords. in beginning as he does, he wants the D to be tuned lower and in fact, tuned exactly to the open D string. In D major, Copper's score, the equivalent note is an E , a perfect fifth above the A according to the same three 4:5:6 chords in D major; and for the same reason, the note E if unaltered would sound too high in bar 3 if not altered down by a comma. This gives a first orientation toward understanding how Johnston's notation is modified by Johnston-Copper notation: the addition of a key signature changes which notes need alteration. Copper's notation in C major is identical to Johnston's, with the minor difference of symbol designs.
To indicate a 'neutral' comma higher, Johnston uses two symbols: the same thick dash or hyphen plus a small arrow pointing up. In Johnston-Copper Notation, this symbol is condensed into the simpler flagged arrow pointing up. Johnston also uses (especially in later works, such as the 6th string quartet) a modified sharp sign with a down arrow; Johnston-Copper notation uses the same symbol to indicate a note that is 'sharp', according to the key signature, and tuned down by a comma. In a flat key, such a note is indicated by a natural sign modified to add a small down arrow on the lower 'leg' of the natural sign. When a note is modified by a flat sign and at the same time must be tuned higher by a comma it is notated as a flat sign with an additional up-pointing arrow head. In a sharp key, such a note is indicated by a natural sign modified to add a small up arrow on the upper 'leg' of the natural sign. It is an interesting feature of just intonation, pointed out by Mozart in one of his letters to his father, that "flat notes are tuned sharp and sharp notes are tuned flat". For this reason, there is very rarely a need to indicate a "sharp note tuned sharp" or a "flat note tuned flat". In these rare instances, the notation used is the combination of a sharp sign with a preceding up arrow sign, or a flat sign with a preceding down arrow sign. Almost always, such notation is found when the sharp or flat is simply used to return to the base key, but the music requires a high-tuned note in the base key (for the sharp) or a low-tuned note in the base key (for the flat).
Altered accidentals
To distinguish the cases when an altered sharp or a normal sharp is required, the same 5-limit basis as the diatonic scale dictates the usage. To stay in a chromatic key-based just intonation, we use a normal sharp sign for the sharp 4th of the key and the sharp root of the key. In C major, of course, these are F# and C#. Continuing higher by fifths (this process often called following the Pythagorean spine), the succeeding sharps must be identified by an altered sharp sign, the sharp with a small arrow on the lower right leg. Thus, the sharp 5th, sharp 2nd, sharp 6th, sharp 3rd, sharp 7th, etc. In C major, these are vG#, vD#, vA#, vE#, vB# using the usual convention when typing, with the preceding 'v' indicating a comma-low marking.
In the same way, the cases for an altered flat or a normal flat follow the 5-limit basis of the diatonic scale: the flat 7th uses a normal flat sign. From the flat third, continuing lower by fifths, the usual tuning will require an altered flat for the 3rd, 6th, 2nd, 5th, etc. In C major, these become Bb (ordinary flat) then ^Eb, ^Ab, ^Db, ^Gb, ^Cb, and the preceding '^' indicates a comma-high marking.
Because in practice music rarely stays purely diatonic or completely centered on the same harmonies, any particular individual moment of music may require a tuning that is different than the just intonation basis. When a comma higher than the altered sharp sign would indicate is needed, the notation uses a regular sharp sign. When a comma lower than the altered flat sign would indicate is needed, the notation uses a regular flat sign. It is very rare that a tuning is needed that is two commas higher than the altered sharp sign, or two commas lower than the altered flat sign, but in such cases an additional comma modifier may be used. It is not uncommon, however, for a musical moment to require a comma higher than the altered flat or lower than the altered sharp, and in the notation being described here the relevant accidental is 'decorated' with two small arrow heads, up in the case of the double high flat or down in the case of the double low sharp.
Just as with sharps and flats, natural signs perform the same functions in what might be called the opposite key types. That is, in a key with many sharps, the natural sign functions just like a flat sign and in a key with many flats, the natural sign functions as a sharp sign, in each case altered the indicated note by a half step. Therefore, the same modifications are used, and for a natural functioning as a flat, the 'decorations' are placed on the upper 'arm' of the natural sign and for a natural functioning as a sharp, the 'decorations' are placed on the lower arm (leg) of the natural sign.