What is Microtonality & 12 Equal Temperament?

What is microtonality?

It an extensive and exciting topic to get around for most musicians and amateur music theorists like myself.

Typically taken literally, microtonality is simply meaning having intervals smaller than a semitone. In a nutshell, it is everything in music that is not 12-edo -- and edo stands for equal divisions of the octaves; a synonym can also be used such as ET, 12ET which means 12 equal temperaments (Monzo, 2018). But not always does this happen, edo’s such as 5-edo and, 7-edo also pop up in microtonal music of various cultures of East-Asia and/or African music (Monzo, 2017) (Giedraitis, 2018, p. 162). Even though they do not contain intervals smaller than a semitone, such tunings are usually considered microtonal as well.

With my experience in microtonal for one year and a few months, I have realized that a bit off fractions, decimals, and logarithms knowledge can take you a long way (more of it will be discussed in later blogs). In 12-edo each semitone is 100 cents, where cents is a unit of interval size. So, a perfect fifth is 7 semitones from the root note size, like C-G; thus it would be 700 cents, this usually changes for other tunings and edo’s as well (Nave, 1999).

Well-temperaments make sure we have enharmonic equivalence (G# = Ab, etc.). Equal temperament does too, but well-temperaments were used in practice on keyboard instruments much earlier, and accurate equal temperament had to wait until 1900 at least before piano tuners knew how to tune it accurately (Jorgensen, 1991).

There were numerous mathematicians and musicians such as Zhu Zaiyu (Cho, 2010), Vincenzo Galilei (Barbour, 1951, p. 8), Aristoxenus (True, 2018, pp. 61–74), Marin Mersenne (Christensen, 2006, p. 207), Simon Stevin (Van De Spiegheling der singconst, 2015) (not in a specific order of date) that contributed to developing and popularizing an equal temperament as that we know it today.

Well-temperaments (let alone equal temperament) were not typical in Western music before the 19th century. Mozart, for example, taught his students to play G# lower than Ab which is a way of expressing that—that is, there was still meantones were still presented as standard during his era of composition and performance (Monzo, 2001). Meantone means G# is lower than Ab, C# is lower than Db, etc. Meantones are chains of fifths with size usually a bit cents below 700c --, the range can vary amongst microtonalists and xenharmonic musicians; usually, it can be from 692 cents to 700, 699 cents.

We use the chain of fifths (instead of chains of thirds or sevenths) because it can contain all the notes diatonically instead of thirds or sevenths; the chain of fifths was the basis of western music for centuries, it provided us with the diatonic modes such as Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian. If we were to use a different interval of the chain i.e. the chain of major thirds or minor sevenths, new (non-diatonic) modes or scales would be created introduced (Paul Erlich, personal communication, June 12th, 2019) This is a common path of exploration for modern microtonalists.

I consider music theory that we learn in schools to only be a speck in the realm of microtonality, and there is so much more to learn and find new notes and sounds.

According to my friend, mentor Paul Erlich, he has three reasons as to why some of us would venture into non-12-edo music, the reasons are (Paul Erlich, personal communication, June 11th, 2019):

There are three underlying reasons people from the 12-equal world; many of us live in often get interested in non-12-equal tuning systems:

1. Non-Western tuning systems as found in Thai, Indonesian, various African cultures', Maqam, etc. music;

2. Early Western music, before the establishment of well-temperament (or 12-equal) or there about as standard, so roughly before 1780-1825;

3. 20th-21st century experimental approaches based on ideas like the harmonic series / just intonation, or dividing the semitone equally to add new pitches, etc., that may involve aspects of (1) or (2) or be inspired by them, for example, unique musical temperaments, or for another example, "rationalizations" of "found" scales.

REFERENCES:

Barbour, J. M. (1951). Tuning and temperament: A historical survey. East Lansing: Michigan State College Press

Cho, G. J. (2010). The Significance of the Discovery of the Musical Equal Temperament In the Cultural History. Journal of Xinghai Conservatory of Music, 02. Retrieved from http://en.cnki.com.cn/Article_en/CJFDTOTAL-XHYY201002002.htm

Christensen, T. S. (2006). The Cambridge History of Western Music Theory (The Cambridge History of Music). United Kingdom: Cambridge University Press

Giedraitis, K. (2018). Alternative Tunings: Theory, Notation, and Practice: Part V – Alternative Frameworks. Retrieved from http://www.tallkite.com/words.html

Jorgensen, O. H. (1991) Tuning: Chapter 8 Well-Temperament. Michigan: Michigan State University Press

Monzo, J. (2001). Mozart's tuning: 55-edo and its close relative, 1/6-comma meantone. Retrieved from http://tonalsoft.com/monzo/55edo/55edo.aspx

Monzo, J. (2017). Equal-temperamanet. Retrieved from http://www.tonalsoft.com/enc/e/equal-temperament.aspx

Monzo, J. (2018). 12-tone equal-temperament. Retrieved from http://www.tonalsoft.com/enc/number/12edo.aspx

Nave, C. R. (1999). Cents. Retrieved from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html

True, T. M. (2018). The Battle Between Impeccable Intonation and Maximized Modulation. Musical Offerings, 9(2), 61–74. doi:10.15385/jmo.2018.9.2.2

Van De Spiegheling der singconst. (2015). Retrieved from https://adcs.home.xs4all.nl/stevin/singconst/singconst.html

APPENDIX:

I have developed a long chain of fifths that consist of numerous intervals Fbbb-Cbbb-Gbbb-Dbbb-Abbb-Ebbb-Bbbb-Fbb-Cbb-Gbb-Dbb-Abb-Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#-Fx-Cx-Gx-Dx-Ax-Ex-Bx-F###-C###-G###-D###-A###-E###

Note: While in Equal Temperament B# = C, this is not the case in other tuning systems.