In Search of the Imperfect Fifth (a non technical explanation of modern tuning)
Temperament is the compromising of pure intervals of just intonation in the tuning of a musical instrument in order to satisfy broader requirements in the composition and performance of music. This was done to avoid problems caused by the Pythagorean comma, the difference in pitch between seven octaves (ratio 2:1) and twelve justly tuned perfect fifths (Ratio 3:2)
In the ancient Pythagorean system of tuning, every note was tuned with reference to a progression of perfect fifths. By the renaissance, musicians were frustrated by the limitations of this system and wanted to explore the use of major thirds, which under this system sounded noticeably dissonant even to non musicians and so were used sparingly.
The pitch ratio of a perfect fifth is 3:2, meaning that when a tonal note and its fifth are played on a musical instrument, the higher of the two notes will make three oscillations in the same time as two oscillations of the lower note. An untempered major third interval has a ratio of 5:4, that is the higher of the two notes will vibrate five times for every four times the lower note vibrates.
Modern Western music tends to make heavy use of triads, that is a note, it's third and it's fifth, what is called a major chord. On an instrument tuned in the Pythagorean way this does not sound good, prompting renaissance musical theorists to look at ways to mitigate the problems and expand the realm of possibilities in musical composition.
The main weakness of the Pythagorean system is that fifths will never map to a note that is a pure octave above the starting note, making it impossible to have an octave that is a closed loop. To understand why this is a serious problem, imagine a series of perfect fifths beginning on C. This would give: C, G, D, A, E, B, F sharp, C sharp, G sharp, D sharp, A sharp, E sharp, and B sharp. Using pure fifths, this B sharp is actually higher in pitch than C seven octaves above the starting note. This necessitated having eleven pure fifths and one small (wolf) fifth that sounded disturbingly dissonant.
In 1581, Vincenzo Galilei, the father of Galileo, advocated the 12 note equal temperament model which is now commonplace in modern Western music. An octave is divided into twelve semi tones of equal temperament, which is to say that the ratio of every note to its adjacent notes is the same, the twelfth root of two. The note A at 440Hz is commonly used as the reference point to tune to concert pitch because A has an integer number of vibrations per second. To a musician not particularly interested in theory, this would all boil down to the fact that each change in note is a diatonic half step and would sound evenly spaced in pitch interval.
Equal temperament does result in a fifth that is slightly smaller than a pure fifth and a major third that is slightly larger than a pure major third, but the error is small enough not to be disturbing. In an equal temperament system where pure fifths are not used in the circle of fifths, the last B sharp would be an enharmonic equivalent, that is equal in pitch, to C seven octaves above the initial starting note, giving a closed loop of fifths and octaves. To accomplish this, a detuning of 2 cents (a fiftieth of a semitone) from each fifth is required.
Is the two cent detuning a problem?
The ratio difference in an equally tempered fifth to a pure fifth is tiny. 1.4983 instead of 1.5 (3/2), a discrepancy of 0.0017. The largest discrepancy from the pure note is the major seventh, 1.8897 instead of 1.8333 (11:6), a discrepancy of 0.0564. These tiny errors are almost certain to be overshadowed by imperfections in the intonation of the musical instrument being tuned. Even in the most perfectly precisioned guitar, there is going to be some stretching of the string when the musician pushes down on the fret.
A singer who can pitch with uncommon precision will consistently pitch to within 5 cents, which is far higher than the 0.02 semitone correction factor on the perfect fifth. Most people do not find the small errors inherent in human vocal performances disturbing, in fact, most people don't notice them at all. As a point of interest, the human voice is not affected by the Pythagorean comma because a human vocal performer can instinctively avoid dissonance while pitching a note.
There was opposition to the equal temperament system of tuning, notably from J.S. Bach, who felt that compromising note intervals in this way was sullying the purity of the music being produced under the new system. Bach advocated a system called well temperament.
The various well temperament tunings tried to strike a balance between having or staying close to pure intervals and not having wolf intervals. A well temperament might have several justly tuned perfect fifths in it and some smaller fifths, but not small enough to be wolf intervals. In such systems, tuning would be noticeably different in each key, but every key would still be usable. Because of this tuning variance between keys, different keys in well temperament were considered to correlate to different colours and emotions.
While some musicians still advocate the use of well temperament, at least for compositions created with it in mind, in the modern day, the vast majority of musicians and theorists unequivocally accept equal temperament as the standard system for tuning musical instruments. Equal temperament is by far the more straight forward system, not requiring retuning for music that changes key. It is also far better suited to music that is chromatic or harmonically complex for the same reason.
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About Graeme Young
Graeme Young is a sound engineer based in the South of Scotland, starting his career after being presented with a Yamaha MD8 in the late 1990's by a friend who had despaired of figuring out how "the damned thing" worked. Brimming with enthusiasm as he learned his way around his new toy, he immediately set to work with gusto, creating some of the most abominable sound recordings of musical doodling ever committed to tape.
Nevertheless, the heady times of fun and friendship that were built up in the early days convinced Graeme to go back to college and expand his knowledge and skill set, meeting contacts and learning the tradesman's tricks from industry professionals. In the meantime student loans were spent on studio equipment to expand on the trusty MD8 while the O2R96 and the old StudioMaster at the college provided experience of working in both the analogue and digital domains.
Now, Graeme has gained years of industry experience working on a number of studio recordings and location recordings with professional musicians and in directing and editing video projects for professionals in various fields. Graeme has fronted two bands, the now discontinued Popping666Cherries alongside Gwen Smith, who had fifteen minutes of fame on the X-Factor before being told by Simon Cowell that she would "never do anything" and now his own Moonstruck Project, with a loose collaboration of musical friends.
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