The Music & Writings of Graham Jackson

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, 41/2. 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.