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The following is my very short, simplistic attempt to explain why our (Western) musical scale is the way that it is. I do not claim this explanation is complete. For more information, you can examine the following websites about "equal temperament" and "Pythagorean tuning." I've obtained most of the information in this page from them, but any misunderstandings are my own.

Let's examine the key of C. It has seven whole notes (C, D, E, F, G, A, and B) and five sharps or flats: C#, D#, F#, G#, and A#. Remember B# = C and E# = F. If you look at the frequency of these musical notes (starting with C4, or middle C, which is formed by holding down the second string of the guitar, first fret), you get the result below. The names of the notes are in the second row, their frequencies (cycles per second) in the third row. The first row contains the numbers indicating the first, second, third, etc. -- as explained in the Key of C.

1 - 2 - 3 4 - 5 - 6 - 7 8
C C# D D# E F F# G G# A A# B C
261.63 277.18 293.66 311.13 329.63 349.23 369.99 392.00 415.3 440. 466.16 493.88 523.25

The far right "C" is C5, an octave higher than middle C. It has twice the frequency (or pitch) of middle C.

The Greek mathematician Pythagoras is the first individual credited with the formal study of music. This is the same man famous for his theorem about the sides of a right triangle. Now Pythagoras believed that all of the mathematical relationships in nature could be formulated in terms of ratios of whole numbers. In fact, he believed that relationships that could be expressed as ratios of small whole numbers (like 5:4 or 2:3 or 6:7) were the best, the most appealing, the most beautiful. The fact that some numbers cannot be expressed as the ratio of whole numbers -- numbers such as the square root of two (such numbers are called irrational) -- would have irritated Pythagoras no end. Apparently he wasn't aware of their existence.

If you want to divide an octave into twelve notes (C, C#, D, D#, etc. -- on up to B), you have to make the ratios of the frequencies of successive notes an irrational number, the twelfth root of two (approximately 1.059463). This is the number that, when multiplied by itself twelve times, gives two. (Pythagoras would really have been unhappy with this development.) So to raise the frequency of a note from C to C#, you multiply by this number. You multiply again to go from C# to D, by it again to go from D to D#, and so on. By the time you get back to C, you have doubled the original frequency. This type of scale is called equal temperament. It would be nice to make the ratios of successive notes small whole numbers, but you can't do this -- not exactly. The laws of mathematics just won't allow it. Musicians tried one scheme or another for hundreds of years in order to try to get around it. The websites I mentioned earlier give some of this history. The "equal temperament" site explains one system, of 31 notes, created by Christian Huygens, a mathematician who was a contemporary of Newton.

BUT -- the equal temperament scale is a way to make things come out approximately as ratios of small whole numbers. If we take the ratios of frequencies in the Table above, we find that E/C (329.63 divided by 261.63, the ratio of the third to the first) is close to 5/4. Then F/C (ratio of the fourth to the first) is close to 4/3. And G/C (fifth to first) is very close to 3/2. The ratio of A/C (sixth to first) is nearly 5/3. Remember from the Key of C that major chords use the first, third and fifth. Major sixth chords use an additional sixth, and suspended chords use a first, a fourth, and a fifth. Dominant seventh chords use a first, third, fifth, and a flatted seventh. The ratio of A#/C (flatted seventh to first) is roughly 16/9 -- more like medium whole numbers rather than small, but still fairly good. Major sevenths use a first, a third, a fifth, and a seventh. The ratio of B/C (seventh to first) is close to 17/9.

Thus the musical scale that we use (the equal temperament scale) does a pretty good job of forming notes which, when used to make the types of chords explained elsewhere on these pages, create ratios of small whole numbers. Note further that, as a result, the third (E) and fourth (F) are separated by the same interval that separates the seventh (B) and the eighth (C) -- and that both of these intervals are half the interval that separates the other whole notes. So the spacing from E to F and from B to C is the same as from A to A# or G to G#. The fact that E# = F and B# = C is thus a requirement of this type of scale, and the fact that we form chords using thirds, fifths, sixths, dominant and major sevenths, etc.
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