Just Intonation - Staff Notation

Staff Notation

Originally a system of notation to describe scales was devised by Hauptmann and modified by Helmholtz (1877) in which Pythagorean notes are started with and subscript numbers are added indicating how many commas (81/80, syntonic comma) to lower by. For example the Pythagorean major third on C is C+E ( Play) while the just major third is C+E₁ ( Play). A similar system was devised by Carl Eitz and used in Barbour (1951) in which Pythagorean notes are started with and positive or negative superscript numbers are added indicating how many commas (81/80, syntonic comma) to adjust by. For example, the Pythagorean major third on C is C-E0 while the just major third is C-E−1.

While these systems allow precise indicate of intervals and pitches in print, more recently some composers have been developing notation methods for Just Intonation using the conventional five-line staff. James Tenney, amongst others, preferred to combine JI ratios with cents deviations from the equal tempered pitches, indicated in a legend or directly in the score, allowing performers to readily use electronic tuning devices if desired. Beginning in the 1960s, Ben Johnston had proposed an alternative approach, redefining the understanding of conventional symbols (the seven "white" notes, the sharps and flats) and adding further accidentals, each designed to extend the notation into higher prime limits. Johnston‘s method is based on a diatonic C Major scale tuned in JI, in which the interval between D (9/8 above C) and A (5/3 above C) is one syntonic comma less than a Pythagorean perfect fifth 3:2. To write a perfect fifth, Johnston introduces a pair of symbols representing this comma, + and −. Thus, a series of perfect fifths beginning with F would proceed C G D A+ E+ B+. The three conventional white notes A E B are tuned as Ptolemaic major thirds (5:4) above F C G respectively. Johnston introduces new symbols for the septimal ( & ), undecimal (↑ & ↓), tridecimal ( & ), and further prime extensions to create an accidental based exact JI notation for what he has named "Extended Just Intonation". For example, the Pythagorean major third on C is C-E+ while the just major third is C-E♮.

In the years 2000–2004, Marc Sabat and Wolfgang von Schweinitz worked together in Berlin to develop a different accidental based method, the Extended Helmholtz-Ellis JI Pitch Notation. Following the method of notation suggested by Helmholtz in his classic "On the Sensations of Tone as a Physiological Basis for the Theory of Music", incorporating Ellis' invention of cents, and following Johnston's step into "Extended JI", Sabat and Schweinitz consider each prime dimension of harmonic space to be represented by a unique symbol. In particular they take the conventional flats, naturals and sharps as a Pythagorean series of perfect fifths. Thus, a series of perfect fifths beginning with F proceeds C G D A E B F♯ and so on. The advantage for musicians is that conventional reading of the basic fourths and fifths remains familiar. Such an approach has also been advocated by Daniel James Wolf. In the Sabat-Schweinitz design, syntonic commas are marked by arrows attached to the flat, natural or sharp sign, Septimal Commas using Giuseppe Tartini's symbol, and Undecimal Quartertones using the common practice quartertone signs (a single cross and backwards flat). For higher primes, additional signs have been designed. To facilitate quick estimation of pitches, cents indications may be added (downward deviations below and upward deviations above the respective accidental). The convention used is that the cents written refer to the tempered pitch implied by the flat, natural, or sharp sign and the note name. A complete legend and fonts for the notation (see samples) are open source and available from Plainsound Music Edition. For example, the Pythagorean major third on C is C-E♮ while the just major third is C-E♮-arrow-down.

One of the great advantages of such notation systems is that they allow the natural harmonic series to be precisely notated.