Such is the advantage of a well-constructed language that its simplified notation often becomes the source of profound theories.
— Pierre-Simon Laplace
Welcome to the website of the Functional Just System (FJS)!
What is the FJS?
Watch the introductory video!
The Functional Just System (or FJS) is a new revolutionary notation system for music in Just Intonation (JI). It uses an extension of traditional staff notation to notate any Just Intonation based tuning system. With its beauty, ingenuity and simplicity, it outmatches existing notation systems for this purpose, including both Helmholtz-Ellis notation and Ben Johnston notation; it is my greatest invention and proudest accomplishment so far in the domain of microtonal music.
The Functional Just System:
- is a means of translation between the raw ratios of Just Intonation and the musical language understood universally by all musicians – staff notation;
- is extremely easy to learn to read and write, and its simplicity far surpasses that of other Just Intonation notation systems;
- provides a bijection or isomorphism between notation and sound, every tone can be notated in exactly one way and can be written and read unambiguously;
- can notate all of the infinity of Just Intonation, not only up to a certain fixed prime limit, and can hence express a superset of the meanings possible in other notation systems;
- has infinite resolution, thus each tone is notated “pitch perfect”, not only up to a certain fixed precision;
- is procedurally generated at every step by an algorithm, and so it is perfect for usage in computer music, and can be completely automated without the need for memorization or lookup tables. (This website contains several calculators for these purposes.)
What Is Just Intonation?
If you have to ask this question, then you are probably in the wrong place!
Most modern music relies on a tuning system known as twelve-tone equal temperament (or simply equal temperament), which divides the octave into twelve equal intervals. Just Intonation (JI) is a method of tuning which differs significantly from this mainstream tuning system. Its principles of tuning are based on the harmonic series and the natural acoustic properties of sound. Although JI is no longer used in conventional music, it remains important to music theory because it was commonplace in ancient cultures and the Middle Ages, is still widely used in traditional music of non-Western cultures, and has been employed in avant-garde music. (Not mentioning its aesthetic merit, of course.)
A basic level of familiarity with, and understanding of, Just Intonation is required (in addition to elementary music theory) to be able to understand and use the FJS effectively. In general, more experience with Just Intonation leads to higher proficiency and fluency in using the FJS.
How Simple Is the FJS?
The following seven rules govern everything required to notate an arbitrary piece of Just Intonation music in the FJS. That’s right – only seven rules.
All notes with conventional accidentals represent Pythagorean tuning. All octaves are 2/1 and all perfect fifths are 3/2.
Every prime interval above 3/2 (that is, 5/4, 7/4, 11/8, etc.) is given a unique Pythagorean approximation according to the FJS master algorithm. This algorithm finds the simplest Pythagorean approximation that is within the radius of tolerance of the target. The difference between them is assigned a unique accidental for that prime, and the prime number itself is used as the symbol.
All FJS accidentals are positive in the direction in which the Pythagorean approximation becomes the target prime interval, and negative otherwise. Positive is not always upward and negative is not always downward; positive is always otonal, while negative is always utonal.
Compound accidentals are represented by multiplying the otonal parts and multiplying the utonal parts, so a double +5 is +25. Otonal and utonal are always written separately. If the numbers are too large, the factors may be written out with commas (punctuation marks) in between.
For note names, the positive compound accidental is added as a superscript and the negative compound accidental is added as a subscript to the right of the conventional name.
For staff notation, the positive compound accidental is written first (if any), then the negative compound accidental with a stroke in front (if any), then any conventional accidental, then the note.
All else is shorthand.
FAQ: Is there a FAQ?
Why indeed, there is one!
- Go here for a crash course in the FJS. With a mediocre understanding of JI, learning the FJS takes mere minutes.
- Read a quick summary of the entire system.
- Read the complete formal description for a reference of the exact rules when in doubt.
- See a comparison of the FJS with two competing notation systems: Helmholtz-Ellis and Ben Johnston notation, and understand why the FJS exceeds both of them.
- Use the website’s calculators to automatically perform FJS-related calculations. Advice: bookmark these immediately.
- See many more examples of the FJS in usage.