by Adam Ahuja
The modern quantum physicist finds himself in a moment of imperfection when studying particle physics. As soon as he comes to understand the position of a particle, he cannot measure its velocity. On the other hand, as soon as she measures the velocity, she knows not the particle’s position. It is a game of constant trickery.
Humanity’s relationship with harmony, in many ways, has been similar to the quantum quandary. There is an inherent complication when considering the tonal position in a passage of music and its related harmony. On one hand, harmonies and melodies can be understood as flowing forth as natural whole number ratios stemming from a single position, the key center. However, when the tonal center moves from one position to the next, the ratio of harmonies must then move to accommodate the new key, as we strive, on some level, to maintain mathematically purity from one key to the next. In theory, this doesn’t sound too complicated; we may just shift harmonies to accommodate their new tonal center. This does tend to be the case when performing music for vocals or instruments that are not bound to few existing pitches or intervals in the instrumental infrastructure. However, most modern instruments have been constructed with fixed pitches and intervals, as an attempt to quantify, communicate and simplify the complex infinite spectrum of tonality into a simple, approximated and recognizable formula. As a result, humanity has gradually developed, and continues to develop, tuning interpretations in order to organize and access the infinitesimally vast harmony in nature that exists between any given pitch and its octave self.
Across the western spectrum, the tuning systems which have developed may broadly be broken into various phases: pythagorean and the recognition of just tuning systems, meantone systems and the use of irregular temperaments, and the modern equal temperament system.
The Pythagorean System
The pythagorean system is the initial system of western tuning, wherein Pythagoras (580-500 BC) recognized that intervals in music are derived from natural whole number ratios of frequencies (Wright and Simms 10, 11). For example, 2:1 constitutes the octave, 3:2 constitutes the fifth, 4:3 constitutes the fourth, 5:4 constitutes the major third, and so on. Similarly, 1:2 constitutes the octave, 1:3 the perfect fifth, 1:5 the major third, and so on. From Pythagoras’s work, scales using natural ratios were constructed; most utilized a perfect fifth, perfect fourth, octave, and often whole tone intervals were used to construct basic major scales (Wright and Simms 10-11). Similarly, a series of perfect fifths would be used to form a scale in a single octave range. An interesting discovery however, was that when constructing the series of perfect fifths, the 12th tone in the series was slightly beyond the octave by a small fraction, referred to as a comma (Goodall). This created a challenge: the octave would not be naturally met in such a sequence, hence the system was a spiral, not a circle.
Music mainly existed within these Pythagorean intervalic boundaries throughout the Medieval Era. Music did not enharmonically modulate as a result, since the simple scale and ratios would not effectively accommodate key changes. Although, during this period, tonal systems were explored by Guido de Arrezzo, who organized a slight modulation to accommodate a few more key centers, namely C, F and G, thus incorporating a Bb (Wright and Simms 36). It should be noted that earlier Ancient Greeks had actually included many enharmonic intervals in their scales, but this knowledge was not carried forth at this stage in the development of Western tuning. A Greek enharmonic study was later quietly revived by Nicola Vicentino in the Renaissance (Wright and Simms 12, 196).
The following is an example of a Gregorian chant of the Medieval Era. Although this piece was sung vocally and not performed on an instrument, this piece would have been composed from the theoretical constructs of Pythagorean intervals:
A system of tuning which applies natural ratios as the basis for tonal selection would be considered the “just” tuning of such intervals (Musical temperament). For example, intervals in this Pythagorean system would be considered just when constructing intervals from natural ratios of seconds, fourths, fifths, and octaves from the key center. However, not all intervals within this spectrum were close to enough to their correspondent natural ratios, since some of the pythagorean intervals were constructed as derivative ratios of tones in a series of fifths, and not directly from the key center. In particular: there was no natural ratio for the major third in such a system when considering the key center (Duffin 32).
The Meantone System
At the time of the Renaissance, there was greater desire to hear major third harmonies in music (Wright and Simms 116). As this is one of the most basic ratios, it is naturally perceived to integrate an enhanced richness into music. In order to achieve this intervalic possibility within the tuning system, the meantone system was developed at this time. This system involved the ‘tempering’ or adjusting of fifths, away from their natural ratios, in order to achieve a major thirds that were closer (or in some cases exact) to their pure intervals when compared to the “Pythagorean fifth” derived third, which was much wider (in fact, it was wider than the modern equal temperament system’s major third). (Musical temperament)
The quarter-comma meantone system was the most widely used in the Renaissance Era. The system tempered the fifths to a degree that allowed the first four fifths in a series to reach a pure major third from the tonic. The term meantone refers to the fact that the pitch distance within a whole tone was half that of a major third in this system (Duffin 32). The following is an example of the quarter-comma meantone tuning on the oldest known playable harpsichord, built between 1520-1530. The piece performed is “Madame Vous Avez Mon Cour” by Marc Antonio Cavazzoni:
Utilizing meantone tunings allowed for richer harmony by using major thirds. However, when attempting to develop the rest of the tonal series by continuing the fifth in this sequence, a “wolf” interval resulted, which was an extremely large gap left between intervals, making certain keys highly imbalanced versus others (Montagu). This caused theorists to continue exploring and tempering fifths by various amounts in order to achieve a balance between the quality of the fifths versus major thirds. In some cases, theorists began devising systems of irregular temperament, meaning that some of the fifths would be tempered by different amounts in a single system to accommodate more keys and achieve a better balance between intervals (this, instead of tempering all of the fifths the same amount was the case in ‘regular’ temperament systems such as quarter-comma meantone). (Duffin 36-37).
While the practices of narrowing or widening certain pitches aided the ability to shift keys more reasonably, this did not achieve exact duplication of intervals in all keys; some keys always sound different and were considered preferable to others. However, Johann Sebastian Bach’s “Well Tempered Clavier” books marked an important step ahead in this journey towards music in all keys; the tuning system he used for these works was devised from irregular temperaments and included pieces in all 12 keys. Each key doesn’t sound exactly the same, but all would be considered acceptable to the ear (Musical temperament).
The following is a performance by Bradley Lehman of Bach’s Fugue in F# Minor. The tuning is an attempt to recreate Bach’s original temperament. Lehman reached this result by interpreting the diagram written by Bach on the cover of his “Well Tempered Clavier:”
Unequal and Irregular Temperaments
Another interesting step that helped characterizes the development of unequal and irregular temperaments was the division of an octave into 55 parts, or commas. This was the sixth-commas meantone system by Pier Francesco Tosi in the Baroque Era which resulted in both agreeable sounding fifths and major thirds (Duffin 53, 56) (Montagu). The development of this system actually led to a well-adopted practice of splitting semitones into major and minor parts, each consisting of 5 and 4 commas respectively (Duffin 53, 56). The knowledge of major and minor semitones followed through the Classical and much of the Romantic Eras and was essentially recognized and taught by Mozart (Duffin 65). This practice to help perform sharps and flats with more distinction and accuracy, particularly useful on unfretted string instruments such as the violin.
The following video demonstrates the range of choices that violinist may take between pythagorean, just, and equal temperament choices in order to maximize a balanced sense of melody and harmony in the context of a given performance piece. This is similar to the type of considerations teachers in the Classical Era may have taken regarding the usage of major and minor semitones to optimize leading tones and harmonic choices:
The Equal Temperament System
Generally, tuning systems until the Romantic Era contained unequal distances between semi- tones; hence the terminology “unequal” temperament systems. Equal temperament (ET) is the practice of dividing the octave into equal parts, which may have been introduced as early as the Baroque Period but did not come to complete prominence until the 19th and 20th centuries (Duffin 138). At this time, ET became the absolute standard for tuning. The 12-tone ET system divides the octave into 12 equal parts, wherein the ratio difference between adjacent semitones is the 12th root of 2. However elegant this division, the fifth is slightly narrower than the natural or just ratio, and the major third is reasonably wider than its just counterpart (Equal temperament). Thus, the system widely employed today allows for perfect movement across keys, but it uses imperfect harmonies; in fact, no interval other than the octave itself maintains a just ratio, although some are closer to just than others.
The following is a demonstration of a piece performed in 12 tone ET, the standard modern tuning system. The composer is Arnold Schoenberg, who is known for writing in an atonal musical style that which was inspired and made possible in the 20th century by the development of ET. This work is his Piano Concerto op 42. We can hear the piano and the entire orchestra’s instruments, now formalizing around the equal temperament structure:
ET systems must not be limited to 12 tones; there also exists other divisions which make available various unique ratios within the tonal spectrum. In the 21st century, composers such as Dolores Catherino are experimenting with ambitious tuning systems such as the 106 tone ET, which she refers to as a form of polychromic equal temperament. Here is a recording of her piece, “Temporal Parallax:”
In the total view of tuning’s western history, we have seen an interesting evolution, finding beauty in unexpected places and making sacrifices for the price of harmonic freedom. Modern music now utilizes a system which is flexible for integration of complex harmony across keys, but it does not leave much possibility to accurately express the most natural intervals themselves, to which the system approximates. This leaves the door open for future innovation in instrumentation and tuning systems that will undoubtedly make use a host of intervalic possibilities and further access to rich harmonic resolutions. If history is the guide, then this will be an inevitable development for the progress of art, science, spirituality, and humanity.
“Arnold Schoenberg: Piano Concerto op. 42 (Excerpt).” Youtube. 19 February 2007. Web. 8
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“Bach: Fugue in F# minor, on harpsichord – Bradley Lehman.” Youtube. 10 May 2007. Web. 8
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“Howard Goodall’s Big Bangs 3 – The tempered system Spanish Sub.” Youtube. 27 March 2014.
Web. 8 November 2015. <https://www.youtube.com/watch?v=tTVpzxcZe9s>
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“Marc’ Antonio Cavazzoni (ca.1490-ca.1570) on Renaissance Harpsichord, Naples ca. 1525.”
Youtube. 12 December 2014. Web. 8 November 2015. <https://www.youtube.com/watch?
Montagu, Jeremy. “temperament.” The Oxford Companion to Music. Ed. Alison Latham. Oxford
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“Temporal Parallax’ polychromatic composition by Dolores Catherino.” Youtube. 3 January 2014.
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Wright, Craig and Simms, Bryan. Music in Western Civilization Media Update. Boston. Schirmer
Cengage Learning. 2006, 2010, Print.
Special Thanks to Professor Tom Rudolph and Berklee College of Music