12 Tone Equal Temperament

On a standard piano keyboard, one octave is divided into 7 whole tones: A, B, C, D, E, F and G. In between these tones are 5 further notes which can be called either sharps or flats: A# (Bb), C# (Db), D# (Eb), F# (Gb), G# (Ab). (whether it's a sharp of a flat doesn't really matter, the note has the same frequency, just a different name). This gives us a grand total of 12 notes in one octave.

If you were to measure the frequency of a note, then measure the frequency of a note exactly one octave higher, you would see that the higher note has exactly twice the frequency of the lower note. For example, middle C has a frequency of 261.626Hz, and the C above it has a frequency of 523.251Hz.

Additionally, if you were to measure the frequency of a note, and then divide it by the frequency of the next lowest note, you would always get the same result: 1.05946. For example, if we divide the frequency of middle C (261.626Hz) by the next lowest note, a B with frequency 246.942Hz, we get 1.05946. In other words, the ratio between all adjacent notes is always the same. The number 1.05946 is special because it is the twelfth root of 2. This means that if you were to multiply it by itself 12 times, the answer would be 2. If you began at middle C, and multiplied 261.626Hz by 1.05946 12 times in a row, you would end up with 523.251Hz, which is exactly one octave higher as we found out in the paragraph above!

Dividing octaves up in this fashion, with equal ratios between notes, is called equal temperament. And when there are 12 notes in the octave, it is called 12 tone equal temperament, or 12-TET for short. But there is no law saying you have to do this. In the demo above, we have divided the octave up into 19 notes. This means the ratio between each note is now the nineteenth root of 2, which is 1.0371.


Folksonomies: mathematics music

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Octave (0.982517): dbpedia | freebase
Musical tuning (0.952322): dbpedia | freebase
Piano (0.888039): dbpedia | freebase | opencyc
Note (0.797618): dbpedia | freebase
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Semitone (0.703833): dbpedia | freebase
Ratio (0.631241): dbpedia | freebase | opencyc
Multiplication (0.604504): dbpedia | freebase

 19 TET Keyboard
Electronic/World Wide Web>Internet Article:  academo.org, (2015), 19 TET Keyboard, Retrieved on 2016-02-05
  • Source Material [academo.org]
  • Folksonomies: mathematics music


    05 FEB 2016

     Mathematics of Notes and Octaves

    12 Tone Equal Temperament > Comparison > The Mind on Music