"TWELVE TRUE-FIFTHS TUNING: Maria Renold (1917-2003) was a professional viola-player who, playing in string quartets, struggled with tunings for decades, until in 1962 she hit on this tuning, which some just call the Twelve Fifths Tuning. The point of it however is that in today's equal-tempered system, none of the 5ths are true natural 5ths as found in the overtone series, whose frequencies are exactly in the proportion 3/2. That is because a piano-tuner tunes each note a 5th above the previous note, in what is called the circle of 5ths. However, if he uses natural 5ths, he will find on returning after 12 steps to the beginning note, he has overshot a little bit. Hence he makes each 5th slightly smaller to make them fit. Thus we never hear the pure, ringing 5th on a keyboard instrument. Choirs and string players constantly adjust a little to make it possible.

Briefly, in this tuning, you tune the white notes on the piano—the 7 notes from F to B—in natural 5ths. You then find the exact middle of the C octave—F#/Gb—and tune the black notes—the next 5 notes—from there, again in natural 5ths. You thus have two series of natural 5ths, linked by a new interval, called a "formed 5th" between B-F#, and Bb-F.

This interval however satisfies because it is based on one of the three systems underlying all real scales in all cultures—the overtone series, the undertone series, or equal divisions; or in mathematical terms—arithmetic, harmonic or geometric proportion, respectively. Apparently, intervals derived from valid intervals are also valid. Thus, the equal division of the octave in half created these two valid bridging 5ths. Ultimately, our inner experience is the judge.

There is a further refinement, however, she discovered later, and which, when added, makes the tuning what its fans call "Renold II". That is, remarkably, that one can slightly expand the 5ths—and hence the octaves—slightly past the mathematical interval and find a satisfying interval that still works.

This tuning is based on an indication by Rudolf Steiner, on whom she based all her work."