Here is an article by guest author Jean-Pierre Vial. I think you will find this introduction to microtonality and his own method to achieve consonance to be a worthwhile read.

Tuning systems reconciling

This article tends to reconcile and unify the various tuning systems for the twelve-note chromatic scale we use for Western music. (In the article, the author’s name hyperlink refers to his website. Other hyperlinks refer to selected web pages of Wikipedia.)

In the old ages, the Pythagorean system proceeded by fifths and octaves, which happens to generate a twelve-note scale, with some tuning approximation. More precisely, twelve successive fifths 3/2 result in the ratio 129.75 whereas seven successive octaves result in the ratio 128. It is a property unique to the twelve-note scale (or its subdivisions based on multiples of twelve notes – for instance, 24 or 36 notes).

The drawback of the Pythagorean system is that notes far from the reference (for instance, C#, Eb, G#, or Ab, when C is the reference) sound out of tune, generating much dissonant intervals. (Such a dissonant interval like C# to Ab or G# to Eb is known as a wolf interval.)

Then, in order to lessen the dissonances introduced by the Pythagorean system, a number of musicians theorized alternate tuning systems, or temperaments, based on various ratios. Gioseffo Zarlino was one who considered several systems acceptable, on the way to the equal temperament and to tuning systems reconciliation.

Johann Sebastian Bach was the great promoter of the equal temperament, where the twelve intervals are strictly equal (corresponding to the twelfth root of two), with the drawback that no intervals, except octaves, correspond to simple ratios.

Why looking for simple ratios? The simpler the ratio, the more consonant the interval. It is a matter of overtones and harmonics. Simpler ratios generate rich overtones and harmonics, which is literally the definition of consonance. As a matter of fact, any ratio would generate harmonics, but only simple ratios generate harmonics within the human hearing range.

My purpose consists of using maths and physics in order to reconcile the various tuning systems. By the way, doing so, I’m going to define another system (and potentially many more!), which must not be problematic so long as reconciliation can be achieved.

Starting with the equal temperament and looking for simple ratios, the maths solution consists of continued fractions, that is, series of rational numbers where the quotients provide the better approximations of the ratios to approximate, while using minimal values for numerators and denominators.

For instance, consider a major fourth, which corresponds to the ratio 2 ^1/3 (because an octave includes three major fourths), that is, approximately 1.259921049… Using continued fractions, the series of better approximations is 4/3, 5/4, 29/23, 34/27, etc. (Note: it is a property of continued fractions that approximations are alternatively higher and lower than the ratio to approximate.)

Well, we are now facing plenty of, better and better, tuning systems!

Don’t panic. No musician could probably tell the difference between 29/23 and 34/27. The reason is that the human hearing acuity normally prevents us from telling the difference in intervals lower than the fifth of a semitone. (Theorists call this interval a syntonic comma.)

Considering continued fractions, the best ratio in a series must be the first one where the difference between the approximation and the ratio to approximate is less than the human hearing acuity, that is, the fifth of a semitone as a standard.

This results in these ratios:

This is yet another tuning system but, as a matter of fact, it does not much differ from the one suggested by Gioseffo Zarlino.

Conclusion(s)

First, our twelve-note chromatic scale was not chosen at random. Only scales based on twelve notes (or subdivisions of those) can, with some tuning approximation, be generated by fifths and octaves, which makes that chromatic scale so useful for transposing and modulating.

As soon as the sixteen century, theorists considered several interval approximations and temperaments acceptable. Then, Bach promoted the equal temperament, thus giving up any and all simple ratios – except for octaves.

My main conclusion is that the human hearing process, accommodating some interval approximation, hears for consonance. Consonant intervals please our ears. Therefore, even if our ears hear intervals somehow out of tune, they search our hearing range for overtones and harmonics, thus unconsciously simplifying the interval ratio into a fraction.

Eventually, ears hear what they most like.

Jean-Pierre Vial,

composer,

former software designer

September 2019