[Polyhedron intro]
As long as humans have been able to talk, they have been able to sing. Not well, though
[], so tuning systems were developed by almost every culture. Though each system is different,
all of them are based on certain principles of physics, and thus contain certain patterns,
intervals, and chords. It’s easy to take our current tuning system for granted, but
it wasn’t always this way.
TUNING SYSTEMS OF WESTERN MUSIC
The western chromatic scale
Before we explore the different tuning systems of western music, it’s important to discuss
what exactly we’re tuning. Though western music has tens, probably hundreds, of unique
scales, they all use only 12 unique notes. Together, these notes form the chromatic scale.
Looking at this diagram, you might be wondering how instruments like the piano can have more
than 12 notes. The key is that the notes repeat; the 13th note of the chromatic scale, if you
will, is the same as the first note, just ...higher. [This note is a C, and this note
is *also* a C]. The interval between these “same” notes is called an octave, and
it will become important later.
Now that that’s out of the way, let’s move on to our first tuning system, just intonation.
Just Intonation
Debately invented by Pythagoras, an ancient Greek mathematician and cult leader, just
intonation is possibly the most fundamental tuning system, and is based on the harmonic
series.
For those of you who haven’t taken a physics course recently, let me try to explain with
just intonation with *minimal* math. In order to make a sound, an object, like a string
or a column of air, must vibrate in a way that forms a standing wave. Humans perceive
the frequency of this vibration as pitch. The wavelength of the wave on the string is
inversely proportional to its frequency, meaning, more simply, that a longer wavelength creates
a lower pitch, In fact, this relationship can be defined by the simple equation “speed
equals frequency times wavelength”. Because the speed of sound is somewhat constant, you
can calculate the wavelength using the frequency, and vice versa.
Anyway, the simplest standing wave that a string can make has a node at each end with
the entire length of the string vibrating in between. In this configuration, the string
is half as long as the wavelength. This wave, and the pitch that it produces, is called
the fundamental, and is the lowest frequency that the string can produce. It is also by
far the loudest note, and is therefore the one associated with the string.
However, this string can produce many more notes, all of which are higher and much quieter
than the fundamental; a standing wave can have three, four, or really any number of
nodes. Because each of these configurations have different wavelengths, each one produces
a different pitch which relates to the fundamental by a simple ratio.
For example, the second harmonic produces a pitch twice as high as the fundamental because
the wavelength is half as long. Accordingly, the ratio between the two frequencies is two
to one. The third harmonic has a ratio of three to two to the second harmonic, and the
fourth harmonic has a ratio of four to three to the third harmonic. This pattern continues
forever, and, in aggregate, is called the harmonic series. When a string is plucked,
all of the notes of the harmonic series sound at different volumes. Though the upper harmonics
get drowned out by the fundamental, they give a string its distinct sound, or timbre.
Now, it’s important to note that the human ear tends to hear simple ratios between frequencies
as consonant and “in tune”, and more complex ratios as dissonant and “out of tune”.
Because all of the harmonic ratios are simple, by definition, they are all named intervals
in western music. The ratio of two to one is the octave that I mentioned before. The
ratio of three to two is called the perfect fifth. Next comes the perfect fourth, major
third, minor third, a few unnamed ratios, and finally, the major whole tone, with a
ratio of nine to eight.
The various methods of just intonated tuning use these intervals to construct every note
in the western scale using a root note as a starting point. For example, using C as
a root note, the frequency for G can be constructed by multiplying the frequency of C by a ratio
of 3 to 2, F by multiplying by a ratio of 4 to 3, E by multiplying by a ratio of 5 to
4, and so on.
Intervals that aren’t found within the harmonic series can be constructed by stacking two
harmonic intervals on top of each other. Mathematically, this is done by multiplying their frequencies.
It is also possible to construct every note by stacking perfect fifths; this is called
Pythagorean tuning.
Though this system still sees some use today, it has glaring issues that make it inapplicable
to most contexts. For example, because the frequency of each note is based on its relationship
to a single root note, it will only sound in tune in that note’s key. Take a listen
to this justly intonated melody in the root key [], and now, taken out of its tonal context
[]. Sounds kinda ...off, doesn’t it? In addition, some notes can be “constructed”
by multiple sets of ratios. This leads to phenomena like comma pumps, which constantly
inflate the pitch of melodies without technically going out of tune.
It became clear that a new tuning system was necessary. So, in 1584, 12 tone equal temperament
was invented.
12 tone equal temperament
Instead of being based on the ratios of the harmonic series, 12 tone equal temperament
is based on dividing the octave into 12 equally spaced semitones, hence the name *12*-tone
*equal* temperament. The ratio between any two adjacent semitones is always a constant
twelfth root of two. Because the twelfth root of two is irrational, no simple intervals
can exist within this system. except for one: the octave. Because twelve times the twelfth
root of two is just two, any and every pair of notes an octave apart will sound perfectly
in tune. All of the other intervals will be slightly off, but not enough for most people
to notice.
Now, because this system isn’t based on a root note or key, instruments capable of
playing in any key became possible. Instruments like the harpsichord, and later piano, were
invented and became widely used because of their pitch and dynamic range. The twelve
tone equal tempered system served as a base for western music from the baroque period
onward.
So far, I’ve only covered the tuning systems used in western classical music. It is important
to recognize that other cultures have their own tuning systems. For example, the Indonesian
Gamelan uses the Sléndro [] and Pélog [] scales. The Japanese Hirajōshi scale [] uses semitones
similar to those in western equal temperament. Finally, the Arab tone system uses a subset
of 24 equal temperament. This system can be thought of as adding a note in between every
note in the western chromatic scale.
Conclusion: The twentieth century, twenty-first century, and beyond
As of the writing of this script, 12 tone equal temperament is the solidly the most
widely used tuning system in western music. It’s unlikely that this will change anytime
soon: western music theory, and everything built atop it, is in turn built on 12 tone
equal temperament.
This hasn’t stopped experimentation, though. Microtonal music, usually using the 24 equal-tempered
or quarter tone system, has steadily been gaining popularity in western music. In the
early to mid twentieth century, composers like Bela Bartok, Charles Ives, and Pierre
Boulez experimented with quarter tones in their compositions. More recently, artists
like Jacob Collier have embraced quarter-tone tuning in the context of popular music. Some
artists have also started experimenting with other equal divisions of the octave, or EDO:
19 and 31 EDO are among the most popular.
It’s worth noting that music outside of 12 tone equal temperament is extremely difficult
to perform on traditional instruments. In fact, some instruments, like the piano, can’t
play microtonal music at all without serious modifications. For instruments with continuous
pitch ranges, like violin and voice, it’s still much harder to play in tune, and requires
a lot of practice. Because of this, most microtonal musicians record in 12 tone equal temperament
and use a digital audio workstation to change the pitch, or exclusively use virtual instruments.
Because of this, it’s unlikely that 24-tone equal temperament will become the dominant
tuning system, no matter how awesome that would be.