The Nature of Sound.
Before we get into microtones, it helps to have a basic understanding of sound.
Sound is produced when moving air molecules hit your ear x number of times per second. These air molecules are moved by vibrations that upset the air and get the air to move. Some examples of things that cause vibrations: a string vibrating the air in a guitar body, the wood vibrations coloring the guitar sound, the vibrating skin of a drum you hit, the moving speaker cone in your speaker cab, or the impact of your feet on the floor when you jump up and down.
All of these cause vibrations, which you perceive as sound when they reach your ear.
Vibrations happen in cycles. Think of a speaker cone: forward and back, forward and back, or a vibrating string, up and down, up and down.
The number of times a cycle repeats, is called the frequency. Any note played on any instrument has its frequency. The A note following middle C for example, on every instrument vibrates 440 times per second.
The sound we hear when air molecules hit our ear 440 times per second, as a result of a string or speaker cone that is vibrating 440 times per second, is the sound of the note A. The more often the air hits your ear per second, the higher the pitch you perceive.
When the vibrating speaker cone moves more than 440 times in a second, we will hear a note that is higher than A.
Pythagoras Discovers The Math in Music
Pythagoras (6th century BC) is credited for being the one who first discovered mathematical relationships between notes.
One of the things he discovered, is that 2 strings of exactly the same thickness with exactly the same tension exerted on both strings, will sound exactly the same. This is called a unison in music theory.
He then found out something really interesting about the sound when he cut that string in half.
Turned out that when the string is half the length (taken into account that exactly the same force is exerted on the string), it produces the same note but sounding in the next higher register. “Register” is what we call all 12 notes before they start over again. When after note 12 we keep going up in pitch from there, we moved on into the next higher register. That next register starts again with note number one.
That 1, that we end back up on after walking up all 12 notes we have in music, is an “octave” above the 1 we started our 12-note ascend on. We call it an octave (from the Greek “Octo”, which means “eight”) because we only use 7 letters to name the 12 music notes. Letter number 8, is the same sound as letter 1, but in the next higher register.
When you cut a string in half, the string vibrates twice as quickly, because there is only half as much mass to move at half the string length.
When vibration doubles in speed, the note jumps up on an octave.
Conclusion: an octave is a 2:1 ratio.
When you play the A note an octave above the A that vibrates 440 times per second, your string vibrates 880 times per second. (The standard unit for “vibrations per second” is Hertz, abbreviated as Hz).
A doubling of the frequency (number of vibrations per second), produces a note an octave higher.
Pythagoras also discovered that when one string is two-thirds the length of the other, the note produced by the vibration of the shorter string, sounds a 5th ( a 5 letter distance: i.e. C D E F G) higher than the note you hear when hitting the longer string. (a 3:2 ratio)
From there on the space between unison (full-length string) and octave (half the length of the string), was later divided into 12 equal distances, using math to figure out how to evenly space all 12 notes in the space between the unison and octave.
This is called “tuning”.
It is these mathematical relationships that define the correct string lengths to produce the correct notes for each piano key, and the correct locations of the frets on your guitar.
Tuning: Scales With More Than 12 Notes.
The octave is fixed. Mother nature is in complete control over this.
People who study physics learn that an octave is mathematically always a 2:1 ratio. The octave is not something “we came up with, decided, or created”, but a natural phenomenon like wind, color, rain, or gravity.
However: we are not limited by the number of ways we can divide an octave.
From a vibration of x number of times per second to the vibration that moves twice as quickly (an octave), the space in between those 2 numbers can be divided into more divisions than 12.
Tuning the octave into 12 already proved more than difficult enough.
Not to turn this into a music history lesson on tuning or a book about tuning, but as turned out, the ratios and math involved to precisely divide an octave into 12 equal parts that work well in all keys, proved to be much more difficult than Pythagoras’ expected.
Given the straightforward 2:1 rato of the octave, Pythagoras and anybody who attempted to work this out in the centuries after Pythagoras expected that dividing an octave into 12 would be equally straightforward. Yet something always sounded out of tune somewhere.
It is Zhu Zaiyu (China) and Simon Stevin in my home country (Dutch/Flemish) who both around the same time in the 1580s, worked out the complex math problems that kept occurring when attempting to divide an octave into 12 equal steps.
Their mathematical solution is called the 12-tone equal temperament tuning.
This is abbreviated to 12-TET (tone equal temperament) or 12-ET (equal temperament) or 12-EDO (equal division of octave)
Most of the music in the world uses this 12-TET tuning and octave division.
Other equal temperaments exist that divide the octave into smaller steps.
These tunings are called “[insert number of notes] equal temperament”
When you divide the octave range into 15 equal steps, giving you a 15-note system, that is called the “15 equal temperament”. (also called 15-TET, 15-ET OR 15-EDO)
American musician Wendy Carlos uses 15-TET in the track “Afterlife” from the album “Tales of Heaven and Hell”.
Musicians call the extra notes in a temperament with more than 12 notes “microtones”.
Music written with these temperaments is called microtonal music.
Of course: the more notes there are in a temperament, the smaller the distance between the notes.
As a result of the smaller distance between the notes when you divide the octave into a higher number of notes, the music written with that tuning system, will theoretically sound “better in tune”.
Not being confined by the larger distances between the notes of TET-12, the availability of “in-between” notes provides a higher level of “intonation accuracy”. (not unlike a tape measure, for example, that will allow greater measuring accuracy when there are smaller subdivisions marked on the tape measure)
Oddly enough: because our ear and brain are so programmed to hearing all music in 12-TET, microtonal scales produce notes that we perceive as sounding “out of tune”.
Certain I am not, but it is my guess that your ear eventually would adjust and start hearing music with a higher number TET to sound more in tune than 12-TET music.
Other equal temperament tunings that composers and musicians have experimented with are TET-19, TET-23, TET-27, TET-29, TET-31, TET-24 (quarter-tone scale), or even TET-96. Imagine a guitar with 96 frets in 1 octave under each string: madness! It’s already hard enough managing just 12 notes.
Arabic music uses 24-TET.
In 12-TET, the smallest interval is called a half-step or semitone. With double the notes per octave, the smallest interval in 24-TET is half a half step. This is called a quarter tone.
Charles Ives is one of the various composers who experimented with music written in 24-TET tuning. One of the ways composers experimented with this, was by having 2 pianos that are tuned a quarter-tone apart.
Some Examples of Music in Different Temperaments.
This is where the fun really starts: hearing music in all these different tuning systems.
Dr. Warren Burt – Erv Wilson’s Impact on Future Directions in Music
Kraig Grady Interview – Microtonal Composer on Erv Wilson
Sevish – Moonopolis (15-tone microtonal music)
He explains19-TET realy well.
19-TET and 12-TET Comparison
Sevish – Gleam (22-tone microtonal 5/4 beat)
31-TET tuning Row Your Boat
Johann Pachelbel – Canon in D (31-ET)
Sevish – Droplet (53-tone microtonal music)
1200-TET, 72-TET and Other Fascinating Examples.
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