Chapter V: What is a Tuning System?

So far in our review of the development of music throughout modern history we have seen a movement from simplicity to complexity, from an original purity that slowly gives way to the rise of human creativity. A single, universal and omnipotent sound, originally referred to as the OM unfolds into many Harmonics, the purest of which are used to create ancient music.

Over the course of several centuries, more and more harmonics were added gradually as the music scale slowly began to take the form we know today. The notes of the music scale had mathematically perfect, small-numbered, “pure” ratios to one another, which sounded pleasing to the human ear when creating intervals and chords.

Then came the Cultural Revolution known as the Renaissance, and with it, an explosion of creativity in all realms of human expression, including music. A very important development occurred during this artistic expansion: the movement away from Modal into Polymodal music.

Modal music was a type of music where only one mode or “mood” dominated throughout the whole duration of a song or instrumental piece. The traditional Greek scales, such as the Dorian, Phrygian and Lydian scales are all examples of Modal scales. Byzantine music is another example, also very similar to Gregorian Chanting, as practiced by monks since the middle Ages. In Modal music, only a single tonality is used, a fact which plays an essential role in how the interval ratios could be constructed.

As we learned in Chapter III, the mathematically pure interval ratios that originally formed the musical intervals were calculated with respect to the fundamental tone, or the tonic. This meant that the resulting interval worked perfectly, but only with respect to the tonic. As long as your song remained only in one tonality, all notes would always be perfect. This way to place the notes of the musical scale had no name, as it was the only plausible way to do it, but is nowadays referred to as Just Intonation.

With the new developments of music along came polymodal music, and with it the need for intervals of fixed tuning to work just as well beyond just a single tonality. A new challenge needed to be overcome regarding the tuning of instruments, since now the pure intervals would lose its perfection when the music jumped to another tonality, or when musicians wanted to transpose a song to a different tonality.

Here is an example that illustrates the problem that needed to be solved with the arising of polymodal music:


Suppose you were playing a modal song centered at the tonic of C. Your perfect fifth, G, would be exactly tuned with a ratio of 3:2, meaning that the frequency of the note G would be exactly 1.5 times that of the note C. If your instrument were tuned with a C = 256 Hz, your G would have to be tuned to 256 * 1.5 = 384 Hz. This dyad would sound in perfect harmony.

Suppose now, that you wanted to play a different tonality in the same song. Now you ventured to lead your song into the tonality of D with an interval to the note G, which would require building a perfect fourth with a ratio of 4:3. Your instrument would have a D already tuned at 284.44 Hz to build a perfect major second with respect to the tonic, C. But in order to have a perfect fourth from D to G you would require a G tuned at 284.44 * 1.33 = 379.26 Hz! Now you have a problem, because your G was tuned at 384 Hz and this tuning cannot be changed dynamically as your song changes modes.


The example above makes it clear, that something had to be changed in the way organs and flutes were built and strings tuned, otherwise it would be impossible to incorporate many modes into a single song, in a way that all intervals would still sound pleasing throughout the duration of that song. Surely enough, people came with a number of solutions to this problem, some through empirical adjustments performed patiently by instrument makers; other by way of mathematical analysis. In any case, this development in the history of music gave rise to what is known as alternate Tuning Systems.

Many attempts at creating the most adequate tuning system to accommodate multiple modes and transcriptions have been made throughout history, the oldest of which can be traced to the 4th century B.C. in the writings of Aristoxenus, a pupil of Aristotle. Allusions of an equal division of the musical scale appear as rudimentary inscriptions on pottery as old as the 27th century B.C. in ancient China, and are attributed to the mythical figure of Ling Lun.

From the practical point of view, one of the oldest solution attempts to this problem was called the quarter-comma meantone tuning system, where single notes where compromised to an average tuning, so as to accommodate most of the possible intervals in an acceptable tuning range. This was only a partial solution, since there remained still a number of oddly sounding intervals, named “Wolfs” in the seventeenth-century Austria. Each instrument maker had his own empirically and proudly developed tuning system, of which one of the most famous was the “Wohltemperierte Stimmung”, or well-tempered tuning developed by Andreas Werckmeister, a music theorist and organist of the Baroque era.

This challenge was ultimately to be solved analytically, basically by linearization of the exponential nature of pitch through the use of the logarithmic function. Since this approach involves an elegant mathematical solution, as opposed to an empirical search for favorable pitches, a system of strictly equally-spaced notes arises, known today as the Equal Tempered tuning system.

The first approximation for equal temperament was proposed by He Chengtian, a Chinese mathematician who lived around the 5th century A.D. Many other treatises and manuscripts in the West dating from the 16th and 17th centuries also describe such attempts, for example by Vincenzo Galilei, the father of Galileo Galilei, Hernicus Grammateus or Simon Stevin, to name only a few.

The first person to have deciphered the equal temperament mathematics fully correctly was Zhu Zaiyu, a Chinese mathematician, who spent 30 years of his life researching this topic and published his results in 1580. In the West, the German mathematician Johann Faulhaber published an accurate equal temperament tuning table, although he didn’t explain how he got the results. Yet all these studies were pure theory, and it wasn’t until the first decade of the 19th century, that musical instruments began to be built with the modern equal temperament tuning system as standard, as is still used today.


 

Note

Just Intonation

Equal Temperament

Do

1

1

Do#

1.06666...

 1.05946...

 Re

1.11111...

1.12246...

 Re#

1.2

1.18921...

 Mi

1.25

1.25992...

Fa

1.33333...

1.33484...

Fa#

1.375

1.41421...

Sol

1.5

1.49831...

Sol#

1.6

1.58740...

 La

1.66666...

1.68179...

La#

1.8

1.78180...

 Ti

1.875

1.88775...

Do'

2

2

Table 2: Comparison of the interval ratios of two tuning systems. Note how the Just ratios are formed from small whole-numbers, as opposed to the equal tempered ratios.

 

Figure 3: Distribution comparison of the ratios used by three different tuning systems. Note how most intervals have quite similar ratios, with the exception of the Tritone.


 

Once the equal tempered tuning system became widely accepted, old instruments were found to be “out of fashion”, and so was the mathematically pure tuning system, which had prevailed for centuries, perhaps even millennia. The birth of equal temperament was a compromise between musical purity and artistic expressivity; a necessary step in the evolution of musical creativity, but with one negative consequence that still persists today: a general amnesia of the origins of pure harmony, and its subsequent total neglect.

This is not to say that we should retune all instruments to a Just or Pythagorean intonation, as this would result in odd intervals when playing modern polymodal music, some of which would be quite unbearable to the listener. But as it would be antiquated and even musical sacrilege to play Mozart using Just intonation, it is equally nonsense to play modal music using equal temperament, only because we forgot the true origins of the musical scale through the standardization and globalization of musical instruments and the music industry in general.

My position is that a great proportion of spiritually uplifting and meditative music is still modal, and as such it is more appropriate to revert to the purest possible harmonies to make it. In this way we charge the music, which was intended for our transcendence in the first place, with the purity and force of the mathematical archetype, instead of subduing its perfection to modern practicality and lack of consciousness.

 

A Side Note: The Relevance of the 12-Semitone Scale

So far we have explained how the intervals were built to construct the musical scales, but a logical question must be investigated prior to doing so, namely how many different tones should a scale have in the first place, regardless of the specific intervals used (equally tempered, just, etc)? As discussed at the beginning of this chapter, the number of notes used to make music increased historically from just a few to the 12 or more tones we use today throughout the world. We saw a transition from the 5 tones of Ling Lun's mystical flute, or the 5 or so notes employed by Gregorian Chanting, to today's tone-rich musical possibilities.

It is a common misconception to think that a 12-semitone scale is solely a western tradition. At least in China, the accounts of  He Chengtian and  Zhu Zaiyu prove that this eastern culture also placed emphasis on dividing the octave in 12 tones. Furthermore, other traditions like the Indian utilize the so-called western tones as well, but they use additional tones occasionally to introduce even more variety and further embellish their art. It just makes sense to use the universal intervals of the fifth, the fourth, the thirds, sixths and sevenths, since they occur naturally in the harmonic series.

But why twelve tones, and not more? Even the harmonic series has only 8 tones in its 4th octave and as many as 16 in its 5th, from which the Indian music draws the extra tones that sound so unfamiliar to western ears. A good lead to this question could be found in Pythagoras' Perfect Circle of Fifths. The details about this mechanism can be found here. In a nutshell, Pythagoras discovered that if you start with any arbitrary note and build successive perfect fifth intervals, exactly after 12 notes you land back at the starting note (a few octaves above, of course). Consider that the fifth would be the purest interval to use in such an exercise, since only the octave is purer, and if you use successive octaves you never leave the original tone. By doing this Pythagoras defined the 12 known notes in a "western" octave for the first time.

Another lead could simply be the sacredness attributed to the number 12. The number 12 is a universal archetype, appearing in many accounts across cultures and throughout history. Only to name a few: The number of hours in a day and night and the number of months in a year, the number of zodiac-signs. In Christianity, the number of Apostles and the twelve tribes of Israel. In Hinduism, the twelve Adityas from Vishnu Purana, the twelve Jyotirlingas mentioned in the Shiva Purana and the twelve most important deities: Brahma, Vishna, Shiva, Krishna, Rama, Hanuman, Lakshmi, Saraswati, Ganesh, Skanda and Surya. In China, Korea and Japan, the twelve Early Branches. In the Islamic culture, the twelve descendants of Ali. In the Hebrew tradition. the twelve fruits of the Tree of Life and the twelve gates of the Heavenly City. In the Celtic culture: the twelve Paladins or peers of Charlemagne and twelve knights of the Round Table at King Arthur's court. In astronomy, the number of full moons in one year. From the geometrical point of view, the number of spheres that fit perfectly around a sphere of the same size. 

The author also conducted an experiment to objectively compare the tones of equally tempered scales of various sizes with respect to the harmonic series observed in nature. Using scales of 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16 notes in the comparison, the 12-tone scale has the least average deviance in cents with respect to the corresponding tones of the harmonic series. In other words, if we want to construct an equal tempered scale to be able to play all kinds of music with it, choosing a 12-tone scale over the other scales of this experiment gives us the most pure tones.  

 

So far we have cleared the issue of which relationships the musical notes should have to one another. In a relative sense, that is, we know which intervals we ought to use in our chosen style of music. In an absolute sense, however, this floating scale of set intervals needs to be grounded, anchored somehow, in order to actually be able to tune an instrument and use it. This gives rise to the concept named Concert Pitch, which will be dealt with in the next chapter.

 

 

Read More: What is Concert Pitch?

Audio Samples: Tuning Systems

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