Perfection revisited

From time to time I puzzle about the numerological mysteries of musical scales and temperaments, with one picture of same to be found at reference 1 and one pondering to be found at reference 2. This by way of follow-up to the second: do all major scales sound the same? Do all minor scales sound the same?

On this occasion, my starting point was Wikipedia from where I got to reference 3, a book by a gentleman called Thomas Donahue, whom I think is an expert on organs and their tuning, a book which provides a lot of information about temperaments. For some historical, not to say evolutionary background, I also went back to references 4 and 5. The whole rounded off by the short and helpful book by a mathematician who sings at reference 6. Someone who has been inducted into the Barbershop Hall of Fame, that is to say the people at reference 7. And for the truly curious, there is a wealth of other material on the subject out there on the Internet, just waiting to be turned up by Bing or by Google.

An excursion taking several days, with what in retrospect appear to be the highlights being sketched below. Such a sketch is not going to replace a proper exposition of the subject, but it is offered as a sort of diagrammatic support for such an exposition, rather as I sometimes find it helpful to have diagrams to support the reading of novels, say detective stories.

Along the way, I was impressed by the way that elementary properties of the natural numbers impress themselves on music, certainly on western classical music. It is easy to see how otherwise sensible and able people became fascinated by, ensnared in the mysteries of numbers. With an important such property being the fact that no power of two is ever going to equal a power of three – although that fact that three to the power of twelve comes close to two to the power of nineteen is very important in the history of such music.

An octave is a musical interval in which the upper note has twice the frequency of the lower note – leaving aside the complication that a more precise definition would include the phrase ‘fundamental frequency’. Over the centuries, most people have agreed about the importance of octaves and about the need for scales, that is to say the systematic division of octaves into notes. But there has been plenty of disagreement about exactly what scales. Some people plumped for five interval scales while we, along with plenty of others, plumped for seven interval scales. Then what were these intervals to be? And these were not the only choices as there were then the questions of how many other notes were needed to support those scales and what those notes were to be.

One needs here to be mindful here of what was taught to me as the telegraph pole problem; the fact that twelve telegraph poles along the railway line define eleven intervals. As a scale of eight notes, the first an octave below the last, defines seven intervals. With the complication that some people work down rather than up, as we do.

Partly as a result of the work of Pythagoras of ancient Greece, we were keen on the interval called a fifth, and this leads to a twelve interval chromatic scale to underpin our seven interval diatonic scales. A scale rather spoiled by the existence of the Pythagorean comma, which might be thought of as a sort of rounding error – arising from the important property of the powers of three and two already mentioned above.

Following Donahue, I define a temperament as the process of tuning, of adjusting the twelve interval chromatic scale derived from fifths in such a way as to preserve most of the fifths while bending them to fit with the all-important octave.

But in the Middle Ages, we became keen on thirds and these caused other problems, as doing thirds and fifths on the same chromatic scale was tricky. Incommensurate even.

A bit later still, we became keen on keyboard instruments, culminating in the modern piano, which have more or less fixed tuning: certainly it is not very convenient to change the tuning as one goes along, possible to some extent with an instrument like the violin or the human voice. And along with keyboard instruments, we developed a taste for harmony and modulation, this last being the restless shifting from one key to another.

Much brain power was expended on how exactly to temper our twelve interval chromatic scale to meet these various requirements. Kings, Popes and other important people took an interest. Maybe some ladies got in on the act too. Donahue lists lots of these temperaments: most of them have both pluses and minuses and it looks as if some of them are still used – perhaps by lovers of 17th century music – although Donahue does not tell us much about that – or at least, if he does, I have not found that bit yet.

And the equal temperament, where the octave is divided into twelve equal intervals, eventually won the day. It had its problems, but it was the best show in town; a bit like the European Union. Which means that, unless you happen to have absolute pitch, all major scales sound pretty much alike and all minor scales sound pretty much alike.

But there seem to be at least three reasons why that is not quite the whole story.

The first arises from beats, the beats you get when two sounds with nearly the same frequency are played at the same time. And when one plays an equal tempered fifth, this is what happens. One overtone from one note beats with another overtone from the other note, and the frequency of this beat, detectable to the trained ear, varies with the frequency of the notes in question. To this extent, one equal tempered fifth might not sound quite the same as another.

The second arises from various hints, hints that pianists might ask for the tuning of their piano to be tweaked away from true equal temperament to better meet the needs of some particular piece of music.

The third arises from the fact that piano strings do not behave as we would like, their overtones are not quite what is desired, and for this reason the octaves at the two ends of the piano are stretched or squashed – away from equal temperament – so that they all sound the same to us. Maybe this interferes with the way that scales sound.

Work in progress. Maybe a next step is to actually listen to some scales!

References

Reference 1: http://psmv3.blogspot.com/2018/01/perfection.html.

Reference 2: https://psmv4.blogspot.com/2020/02/dorking-two.html.

Reference 3: The evolution of the art of music – Sir. C. Hubert H. Parry – 1905.

Reference 4: Temperament – Stuart Isacoff – 2001

Reference 5: A guide to musical temperament – Thomas Donahue – 2005. Inter alia, the source for the illustration above.

Reference 6: Mathematics and Music - David Wright – 2009.

Reference 7: https://www.barbershop.org/.

Reference 8: https://psmv3.blogspot.com/2017/01/virtual-pitch.html. A old by-way on this particular quest.