Just Intonation

(W.I.P.) A method for notating the infinite myriad of tones available to musicians in just intonation has been sought after for centuries. Before the twentieth century, reconciling or controlling the natural harmonics of multiple roots in the Western twelve-tone system through notation was thought to be impossible. Many of the attempts to reconcile the clashing overtones and various methods of deriving (divining) harmonies was to smudge the accurate differences between the tones of a gamut in order to create a sort of greyscale inharmonicity to the fabric of music, otherwise known as temperament. 
With enough time, though, composers began to notice the true color of music and began attempting to notate these pure microtones and include them in performances. Such examples include Giuseppe Tartini, who noticed the phenomenon of difference tones. Many others came along and many theorists began exploring the concepts behind the overtone series. Helmholtz, Grisey, Debussy, Lizst, Partch, etc. 
One of the most remarkable composers to breathe new ideas into the field of just intonation was Ben Johnston. His notation system, created in the 1960's, was the first to accurately notate the differences between the many natural harmonics of the fundamental tones we are used to (plus many more). With this system, he wrote nine string quartets and various pieces for other ensembles that were tuned to the overtone series and its reciprocal: the undertone series. While this system was incredibly effective at notating all the necessary microtones, and reasonably usuable as proved by many other composers such as La Monte Young and Toby Twining, some felt that there were a few problems. To outline a few, Johnston does not separate the notation of tones built off the third partial from those built on the fifth partial. For context, if A as 440hz is our universal reference pitch, E using the third partial of that A is about 21.5¢ away from the E using the fifth partial of C. This difference is called the Syntonic Comma. Johnston felt that natural pitches should be derived from their triadic parents in their simplest compound periodicity: 4:5:6. This meant that between D and F, there would have to be a boundary at which the general pitch space needed to be raised in order to maintain the chain of triads, which johnston used a plus or a minus to indicate. This meant that D to A in Johnston's notation did not indicate a perfect fifth with the ratio of 3:2, but a rather dissonant fifth with the ratio 40:27. A pure fifth from D would be notated as A+. This seemingly arbitrarily occuring alteration to the pitch space has led many performers to have to ignore Johnston's pluses and minuses for performance purposes, even though they may be mathematically imperative. Another negative quality of Johnston's notation has to do with the aesthetic qualities of intervals that are exclusive to the fourth octave of the harmonic series and beyond. These are the prime partials 17 through 61. For these partials, Johnston indicates their respective schismas, commas, and dieses the number of the partial attached to the accidental. Using numbers to indicate such specific pitch colors is not very intuitive to performers as the glyphs used in our cyphered number system as so similar that they are not very immediately recognizable. One of the good things about Johnston's notation system is that he sought to use the simplest ratios as possible, especially when finding superparticular ratios to which each glyph comma would be assigned. Unfortunately, one of the reasons this was possible was because of the unnotated use of prime 5 in the ratios. Another good quality of johnston's notation was that the accidentals were combinable, which allowed for a more simplified reading experience for performers, even though there was more assumed information than in other just intonation systems, like Helmholtz-Ellis notation.