The Music & Writings of Graham Jackson

What is Different About the Tuning?

There are two differences.

1. The piano is tuned to a certain lower pitch (A= 432 Hz or C= 256 Hz.) which is reputed to be more natural and healthful, and which has been used quite a bit in past centuries. There is an international movement, including some renowned musicians, to make this the standard concert pitch, replacing A= 440. The standard pitch (440), while more "brilliant", has been blamed for contributing to the already high stress levels of our time.
   
   
2. Instead of the usual equal temperament, this piano is tuned to the "twelve true-fifths tuning" discovered (or invented) by Maria Renold, which, having more natural intervals, allows for an increased resonance. 

How is it tuned?

 For the usual, equal-tempered tuning, one takes the interval of the perfect 5th, found in the overtone series, and starting from any note--say F--sets one 5th on top of another, continuing for twelve steps until one circles back to the original note. The resulting twelve notes constitute the 5 black and 7 white notes of the piano keyboard.

Unfortunately, they do not come back exactly to the same note, but overshoot slightly. Hence, since Bach's day, tuners make each of those 5ths slightly smaller--out-of-tune, one could say--to round off the circle neatly.

The difference from the natural 5th in this equal-tempered tuning is so slight that most people do not notice it consciously, although subconsciously they do, as it causes a slight dulling of the resonance. Hence string quartets, choirs, etc. often prefer slightly adjusting their pitches to the more natural consonances.

For the "twelve true-5ths tuning": you first set C at 256 Hz. Then you tune the 7 "white keys" by the circle of 5ths, using however natural 5ths. Then you divide the octave at C exactly in half (which can be done handily with a special tuning fork), and tune the 5 "black keys" by natural 5ths to that F#.

You end up with two series of natural 5ths--one of 7 notes and one of 5 notes, linked by an "unnatural" interval of an augmented 4th (which is actually the same augmented 4th found in the equal-tempered system).

Why does this work?

The musician will notice that you then wind up with two 5ths (Bb-F, and B-F#) which are not only unnatural, but smaller even than the tempered 5th. Does the ear not reject these?

The answer may seem rather roundabout. The frequencies of all valid musical intervals, used in any musical system, will be found to be based on one of three mathematical series: the arithmetic series, the harmonic series, or the geometric series.

The arithmetic series, e.g. 1, 2, 3, 4, 5; or 2, 4, 6, 8, 10, etc. is the basis of the familiar overtone series which produces the major triad as its basic sonority. The harmonic series, e.g. 1/1, 1/2, 1/3, 1/4, 1/5, or 3/2, 3/4, 3/6, 3/8, 3/10, etc. is the basis of the lesser known and disputed series--sometimes called the undertone series---which produces the mirror image of the overtone series, and features the minor triad.

The geometric series grows constantly at the same rate, e.g. 2, 4, 8, 16, 32, or 8, 12, 18, 27, 40 1/2. Musically it produces any series of equal intervals, including, for instance, octaves, the tempered circle of 5ths, the equal-tempered 12-tone scale, or the C-F# interval above produced by dividing the octave in half.

Our usual system is a kind of happy meeting--with some compromise--of all three: the overtone series, the undertone series and the equal 5ths of the circle of 5ths.

It is because of this that the equal-tempered scale works at all, but where intervals prominent in the overtone series (such as the 5th and major 3rd) are concerned, the overtone-resonance factor over-rides the other, and the string-quartet player will prefer to tune it to the overtone sound.

The player however will not mind the tempered minor 3rd, augmented 4th or major 6th (e.g. from C to Eb, F# or A) because the ear hears them as derived from an equal division of the octave into 2 or 4 parts-i.e. from a geometric series.

It seems to be a law that intervals derived from valid intervals in turn make valid intervals. Hence the unusual 5ths we made from Bb-F and B-F# are accepted by the ear.

Judge for yourself by listening to Transparencies or Transparencies Too.

Why I like the tuning

 My own piano (on which the recording was made) has been tuned this way for about 20 years. My first reaction after playing on it, was how clear and resonant it sounded. The notes seemed to float free in the room, whereas when I play a piano with equal-tempered tuning, the notes tend to feel glued to the keys, relatively speaking. My next reaction, after playing on it for half an hour, was that I somehow felt healthier.

After these years, my impressions, although less noticeable, have not changed. Certainly the tuning does not harm the music. Bach, Mozart, Chopin, not to mention my music, just sound even better.