A keyboard for exploring our reaction to harmonic and rhythmic ratios

 

This is a design for an instrument to illustrate basic principles of harmony as described by Pythagoras, that is that simple ratios between the lengths of vibrating strings seemed to be attractive to the ear.  It can also demonstrate polyrhythms.

We now know that if the length of string is divided by a number, its vibration rate is multiplied by that number (inversely proportional).

Pythagoras only included ratios up to the number 4. The Pythagorian ratios are shown in the top unshaded rectangle.

The area shaded in red extends the ratios to include the number 6. This larger diamond is called Just Intonation.

The area shaded in blue extends the diamond to include the number 7 which enables the uneasy status of the 7th harmonic to be explored along with the ambiguity of the tritone and diminished 7th chord. The table could be extended as far as desired. One would have to create an 18x18 table to obtain an approximation (17/18) of the equally tempered semitone suggesting that the semitone is an interval of convenience arrived at because it can approximate the intervals on the 7x7 table.

In Intervallic mode the keys play frequencies as ratio of the base frequency (= 1/1 ) set by the user. The timbre of the note could also be chosen by the user.

In Polyrhythmic mode the keys play tempi as ratio to the base tempo (= 1/1)  set by the user. Various percussive timbres could be assigned to the keys.

In Combined mode the keys play repeated notes whose frequencies and tempi in ratio to the bases set by the user as 1/1.

The instrument is not yet made and I don't  have the necessary computing or electronic expertise. It could make an interesting design project for someone working in the field of musical electronics. It need not be a physical keyboard and the principles could be demonstrated first on a program activated by mouse or touch screen. There may be a market for such an instrument/program in music/physics/maths departments of schools, colleges and universities and for the hands-on rooms of museums. These ideas are public domain. The sounds could be MIDI generated (with adjustment of Equal temperament to obtain pure ratios) or generated from first principles.

The layout is a division table ,numerator as columns, denominator as rows, turned clockwise through 45 degrees so as to present the reciprocal of each ratio in left-right reflection.

The string diagram could be animated to show a vibrations. The length of string is inversely proportional to the rate of vibration (at least in an ideal world). The rate of vibration would have to be scaled down to be visible. Interestingly, our perception of pitch begins at 20Hz , similar to the rate at which we fuse video frames into continuous movement.

The controls could be keys on a computer keyboard (shift, space bar etc.) or pedals on a constructed instrument.

Keys on a constructed instrument could be touch sensitive.

The composer and instrument maker Harry Partch made numerous acoustic instruments along similar lines. 

A ratio engine using ideas from the above design has been created by the tabla player Marc Clayton. Two tempos that run parallel in simple ratios are a feature of advanced tabla technique. Here's the link. When open, hover over the right hand ? for an explanation of usage.

http://marcclayton.net/ratioEngine/ratioEngine.html