This is an illustration of the Howe-Way music notation system from the cover of Book Two of the Howe-Way Music Method, by Hilbert A. Howe (year: 1964).
On the Howe-Way staff there are six vertical staff positions per octave: three lines and three spaces. Each line or space represents two of the twelve pitches of the chromatic scale. The two pitches appearing on a given line or space are distinguished by their noteheads. The lower pitch has a white/hollow notehead and the pitch a half-step higher has a black/solid notehead. (This pattern corresponds to the keys of a 6-6 piano keyboard, see below.)
In practice the music is apparently always written on a six-line staff, covering two octaves. The third line from the top is a heavier line (this is not so obvious in the image above). This line and the bottom line of the six-line staff are both C. For piano music, each hand has its own six-line staff.
The Howe-Way music method is designed for learning to play an isomorphic, 6-6 piano that has two rows of keys (like this, but with different coloring). The keys in the upper row are shaped like the black keys of a regular piano, grouped in two groups of three per octave: three black keys alternating with three red keys. All the keys in the lower row are white keys, shaped like the D key on a standard keyboard. C is in the lower row, as on a standard keyboard. C#, D#, and F are black keys in the back row; G, A, and B are red keys in the back row. This piano keyboard was built by Orville Wood of the piano company Pratt & Read and is mentioned here: http://orgs.usd.edu/nmm/KEYBOARD.HTM#experiment
The references to “black” and “red” in the image above do not refer to colors in the notation, but to the colors of the keys in the back row of the keyboard. The notation is black and white.
Pitches are named with numbers. Unlike most numerical nomenclatures, successive numbers refer to whole steps. One whole-tone scale (corresponding to the white keys of the piano) has regular numerals, and the other whole-tone scale has numerals with underlines. 1 is C, 1 with an underline is C#, 2 is D, etc. C# is referred to as “one and a half,” D# as “two and a half,” etc., but for speed they can be pronounced 1u, 2u, etc.
Registers are indicated with Roman numerals. Middle C is the beginning of register IV (matching the Acoustical Society of America and later the ISO standard).
Rhythmic notation is mostly traditional. The whole note has a stem, half of which rises from the notehead, and half of which descends from it. The half note has no stem. There is a note saying that the inventor hoped for a special typewriter key that would render a special notehead shape for a whole note, perhaps a diamond shape, in place of the aforementioned special stem on a regular round notehead.