Justonic Tuning Cube

The tuning of each note in a just scale is based on the relationship of that note to the first note, also called the key, root or tonic.

As the steps between each note differ, in order to play in natural harmony, a musical instrument with fixed tuning must be retuned whenever there is a change of key or root.

If you are a musician, you probably know that the equal tempered scale was invented a few hundred years ago to accommodate fixed-tone instruments like pianos or guitars. Pure harmony was sacrificed for the convenience of not having to retune the instrument if a different key was chosen.

Since the dawn of computers and electronic musical equipment, this accommodation is no longer necessary.

For those of you who are mathematicaly inclined, the rest of this description will be interesting. However, it is not necessary to understand it as the cube is built into the Justonic Pitch Palette software.

With our solution, a musician indicates a root change by playing that note on the MIDI tuning channel. That triggers a new MIDI octave tuning message to the synthesizer. The Pitch Palette provides the data in the format required by most popular synthesizers along with the MIDI Tuning Standard for Scale/Octave Real Time 2 byte resolution, which allows for tuning resolution of more than 16,000 steps per semitone.

Each cell in the 12 x 12 x 12 cube contains the multiple of the ratios for the key, root and interval for every octave in any musical scale. When the cube is connected to any MIDI musical instrument, it sends the tuning data to the synthesizer..

Arranging the pure harmonic ratios of a musical scale into a cube organizes the data necessary for choosing the correct harmonic tuning. Each cell in the cube is equal to the multiple of the ratios for the Key, the Root and the Interval. That value multiplied by the base pitch A440, in the Key of C (a minor third of A - 6/5) would create the note C as 440 x 6/5 = 528. In the Key of C, the Root G (a fifth of C - 3/2) is 440 x 6/5 x 3/2 = 792. The tuning of the octave is based on the frequency of the Root note. In the Key of C, Root of G, the frequency of G# is 440 x 6/5 x 3/2 x 16/15 = 844.80. The contents of that cell simply that the bvalue of 16/15 may be inserted into the equation, becoming (6/5 x 3/2 x 16/15 = 1.92 x 440 = 844.80.

Of the 1,728 values created in a cube of 12 x 12 x 12 ratios, many of the values in the same octave are equal to others but about 100 of them are unique, depending on the scale.

You can see the many duplicates below in the small example, using the Harmonic Classic scale.

• Using the above ratios applied to a Base Pitch of 440.00 Hertz, where K = Key and R = Chordal Root, these are the frequencies in Hertz:

In K1R1, Interval # 8 is the 5th chromatic note, so 3/2 x 440.00 = 660.00. From the Root of A, the 5th chromatic note is E.

In K1R2, the Root (A#) is 469.33 because 16/15 x 440 = 469.33. From the Root of A#, the 4# chromatic note is E so 469.33 x 7/5 = 657.07.

If this sounds complicated, rest assured, it is!

Our Pitch Palette does it all for you. After choosing a Scale and a Key, playing a single note on the tuning channel of any MIDI Controller will trigger a Root change, retrieving the 12 new Intervals from the cube, calculating the frequencies, converting them to cents, then to hexidecimal, and sending a MIDI tuning SysEx message to your synthesizer in milliseconds..

The author, Bill Gannon, invites your comments.

Reply to Bill Gannon to at Justonic Tuning Inc.