Prerequisite: music theory, physics, algebra 2
Most of us are taught that Db and C# enharmonic, or are the same notes in different notation. Quite evidently, they share the same key on keyboards and same frets on guitars. I suppose it is convenient to assume that they are practically the same.
Except they are theoretically different, or microscopically different. A lot of instruments are tuned with perfect fifths, which is the traditional way to do it. Actually, this causes the enharmonic notes to misalign by tiny microtones.
This difference between enharmonic notes is a particular nuisance to fixed pitch instruments such as the piano and guitar. It causes certain chords and intervals to become significantly muddy and out of tune. To solve this problem the 12-TET (Tone Equal Temperament) was made, also known as 12-EDO (Equal Division of Octave).
12-T means that there are twelve distinct notes in the system, which in our current case helps to eliminate the distinction between enharmonic notes. The E means that the perceived relationship between each neighbouring tone is equal. The final T, for temperament, is the system of note pitches that an instrument assumes. Most of our music is in 12-TET .
In 12-TET, the relation between two semitones is the twelfth root of two. To clarify "twelfth root of two":
The number two comes from the fact that humans perceive pitch logarithmically. Two notes that are one octave apart, having a frequency ratio of 1: 2, sounds "the same".
The frequency in Hertz of a microtone within an equal temperament can be expressed as such:
where m is the order of the microtone, n is the number of distinct microtones in the octave, and ƒ is the fundamental frequency or reference note. Which means, one octave can have as many or less equal tones as you like. You can have 5-TET, 17-TET, 144-TET, or set two 143-TET microtones as one microtone. Experiment to your heart's desire! Microtones do not have to be a nuisance.
Some fascinating microtonal albums:
Brendan Byrnes - Micropangaea
Cryptic Ruse - Chains of Smoke
The microtone is not a modern concept. In fact it used to be the norm, especially with ancient folks having a keener sense of pitch. Some people have engineered microtonal instruments such as the microtonal guitar and fluid piano for traditional music. It helps to play maqamat where microtones are often used to sharpen or flatten "corners", or to build special modes. They are not equal temperaments, but tones microtones of varying sizes are involved.
Go to Xenharmonic for further reading~
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