Week 26-27: What is temperament in music?

The past two weeks I have been looking into temperament. In the beginning I was rather ignorant about temperaments; isn't tuning an obvious thing you do to an instrument? And why not settle for one tuning that "sounds right"? There is, of course, more too it.

Actually, during the Baroque era, musicians and composers experimented with harmony and with different tunings. They started to invent new tuning methods, aside from the just and Medieval Pythagorean systems. This development was probably driven by the keyboard instruments and also the harp: instruments that are not easily tuned to all keys. Playing in a different key would cause dissonant intervals, called "wolf tones". A work around was to develop a tuning system that would - for lack of a better expression - spread the dissonance over the different keys. In that way music could be adapted to all the keys. Well tempered tuning was an important invention of the Baroque era, that did just that. But, of course, by using different tunings, the character of the music would change. So, tuning is an important dimension in music.

This subject can be treated rather mathmatically, discussing the various ratios that the temperaments are based on. I believ, for now, that is not particularly necessary. Instead I hope to capture the idea that tuning is not such an obvious matter. There are various tunings and they all were invented to solve a problem: how to transfer what can be played in one key to another without too much degradation in harmony, and, how can we work around the degradation and possibly exploit it add character to music. J.S. Bach succeede3d marvelously in creating higly varied compositions that explored the character of each key - minor and major - off the well tempered system.

What is a temperament? 

The twelve tone method of tuning we use today is a fairly recent invention, for in music history there have, in fact, been many others. The aforementioned system consists of an octave divided into equal 12 parts or semitones. This allows us to play the same piece of music in any key; everything is going to sound proportionally the same. However, this also inevitably lead to some of these intervals sounding slightly out of tune.

In this blog I shall discuss what I have learned about temperament in music. I must credit Nathan Nokes for giving such an concise and clear lecture on the important tuning methods in music history (Nokes, 2015).

A temperament is defined by The Oxford Companion to Music (2003) as "a method of tuning in which some concords are made slightly impure so that few or none will be unpleasantly out of tune." This became essential with the introduction of keyboard instruments like the spinet, clavicle and harpsichord. Voices and many other instruments can modify their notes as required, varying pitch slightly to keep in tune, but with keyboards all pitches are fixed (Latham, 2003). 

When a guitar or piano string is struck, it will start to vibrate, producing sound. This sound consists of the fundamental tone and a series of overtones. These are oscillations exactly the size of a fundamental higher. For instance if the fundamental C note oscillates at 100 hz, the next overtone will an oscillation at 200 hz (C').

Finding overtones (by Nathan Nokes, 2015)

In fact this is a pattern: we shall find an overtone by adding the size of the fundamental each time.

Since notes are vibrations and an interval is the distance between two notes, one can create mathematical ratios for any interval and its frequency (measured in Hertz) in music.

Overtones expressed as mathematical ratios (by Nathan Nokes, 2015)

Just intonation is a way of tuning so that each interval follows the overtone series as shown above. This creates a natural, clean and pure sound sound when you are in one key.

When one starts to play outside of that key, however, this will present a problem. For instance a just tempered chord, consisting of the 3rds 9:8, 7:6, and 5:4 sounds off. If we were to follow the just temperament overtone series on the piano, we would discern a difference between G# and Ab. In fact all of the sharps and flats will sound different from each other. Hence, just temperament is not an effective tuning system when it comes to creating harmonies.

In the Middle Ages, music theorists tried to develop a tuning system was that used mathematically pure tones, but at the same time provided a remedy for some of the imperfection.

The first medieval tempered scale was based on a solution by the ancient Greek mathematician Pythagoras, where every tone is a major tone and all fifths except one are pure, exactly in tune (Latham, 2003).

The tuning system was based entirely on perfect fifths, or the 3:2 ratio. If we were to have a starting frequency of 100hz, the next tone would be 100hz x 3/2 = 150hz, the next 5th would be 150hz x 3/2 = 225hz, and then 225hz x 3/2 = 337.5hz, and so on, until you get all the different pitches you can use for making music.

This Pythagorean system preserved the clean sound of the 5th, which was much appreciated at the time. Octaves and fifths are all perfect and preserved, but thirds would sound dissonant. The ratio of the major third according to the Pythagorean system would be 81:64, which is a lot more complicated than the just tuning of 5:4. The latter would be out of tune, but that was not so much a problem as major thirds in medieval times were viewed as to be avoided dissonances. This tuning system was used or some 1000 years.

If we were to use the pythagorean system of perfect fifths (x 3/2) , for instance from D, going up and down, we would eventually generate this sequence of pitches.

Generating a sequence of pitches using Pythagorean temperament (adapted from Nathan Nokes, 2015)

In modern times Ab and G# are spellings of the same note. If we put them together and compare them, they are, in fact, different tunings. 

Demonstrating the deviation between Pythagorean and Just temperament tuning (Nathan Nokes, 2015) 

Using one instead of the other, produces a chord so out of tune it howls like a wolf, hence the expression "wolf 5th" (Latham, 2003) or "wolf tone" (Nokes, 2015). Among all the perfectly tuned fifths, there will always be one fifth that is slightly larger (G#) or smaller (Ab). There were also four wolf thirds, wildly sharp, but these were kept in keys which composers took care to avoid (Alison, 2003). 

During the Renaissance, when harmony had evolved to the stage where almost any third was required, a new tuning method had to be devised, with all 3rds pure and all 4ths and 5ths as nearly pure as possible. This was Quarter-comma mean-tone temperament, as first discussed by Zarlino in 1571 (Alison, 2003). There are several forms of mean-tone temperament, but this is the most popular.

A "comma" in music is the difference between two tunings of the same note, such as the difference between the just system major third and the pythagorean system major third. On a piano keyboard, getting from C to E, one needs to stack four perfect fifth intervals (C-G, G-D,  D-A, A-E). Since all these fifth intervals are a little too large, this would create a cumulative deviance. The solution provided by quarter-comma mean-tone temperament is to shorten the fifths by a fourth of the difference between the just major third and pythagorean major third.

This fixes the dissonance of the major third interval at the expense of the perfect fifth, worsening the wolf tone. In fact wolves of some sort are inevitable in any temperament which uses the same correction all the way (known as "regular temperament"), like a cumulative error.

In mean tone temperament systems, not all keys will sound as pleasing as others because of the wolf tones. When one had the desire to play in another key, the instrument would need to be retune to chase the wolves away. To fix this bane, from the 17th century, different tuning methods were invented ("irregular temperaments"), like well-temperament. In this system just thirds are tuned for the first key, then altering the ratios to improve some of the keys more likely to be used. Keys less often used were then, as a consequence, degraded somewhat.

In well temperament, it is possible to play in all keys freely without the necessity of re-tuning. However, each key does have its own character, which can be utilized by the composer for creating an additional dimension to the music. Bach's keyboard music collection "The Well-tempered Clavier" showcases the possibilities of this temperament. The impressive body of 48 prelude-fugue pairs "goes round the horn", cycling through all 24 possible major and minor tonalities. Bach demonstrates the power of this tuning system "showing the purity of the better keys and others, with rapid passage-work, disguising the faults of the poor ones" (Latham, 2003). NYU and Montclair State University professor of Music Technology Ethan Hein (Quora, 2013) explains: "One key (usually the key of C major) sounds very pure, with almost no beats. As the music moves into adjacent keys, the purity of intervals successively diminishes. In equal temperament, the lack of purity is spread equally among all keys" (Quora, 2013).

I found this example on youtube. The consonances should sound degraded in well temperament, but the compositions are in fact never noticeably dissonant or harsh. The way that the strengths of this tuning system are beautifully demonstrated, while cleverly working around the weaknesses (dissonance) testifies to the genius of Bach

In 1584 Vincenzo Galilei (father of Galileo Galilei) proposed a twelve-tone equal temperament, which was generally favored by lutists and guitarists in the Baroque era. This system - also used much in modern times - in fact degrades virtually all the intervals, but does make it much easier for a musician to play in all keys without re-tuning. An interval in one key has the same frequency ratio as the same interval in any other key. 

Graphic indication of difference in intervals between various tuning systems (by Nathan Nokes, 2015)

Sources

Jacky Tran (2012), Bach - The Well Tempered Clavier Book 1 (Feinberg), Youtube, viewed 17th of July 2017, <https://www.youtube.com/watch?v=OzerJmdStq8&t=989s>.

Latham, A. (ed.) (2003) The Oxford Companion to Music, 3rd edition. Oxford: Oxford University Press (pages: 1262 - 1263).

Nathan Nokes (2015), Brief History of Western Tuning (Understanding Equal Temperament), Youtube, viewed 17th of July 2017, <https://www.youtube.com/watch?v=wUBkbrvCmGA&t=2s>.

Quora, 2014, What is the significance of The Well-Tempered Clavier? [online] Available at: <https://www.quora.com/What-is-the-significance-of-The-Well-Tempered-Clavier> [Accessed 20th of july 2017]