y are used more by the mass of players who play little but
popular music, which is mostly written in keys having flats in the
signature.

Upon the system table you will notice that the first five tones tuned
(not counting the octaves) are C, G, D, A and E; it being necessary
to go over these fifths before we can make any tests of the complete
major chord or even the major third. Now, just for a proof of what has
been said about the necessity of flattening the fifths, try tuning all
these fifths perfect. Tune them so that there are absolutely no waves
in any of them and you will find that, on trying the chord G-C-E, or
the major third C-E, the E will be very much too sharp. Now, let your
E down until perfect with C, all waves disappearing. You now have the
most perfect, sweetest harmony in the chord of C (G, C, E) that can be
produced; all its members being absolutely perfect; not a wave to mar
its serene purity. But, now, upon sounding this E with the A below it,
you will find it so flat that the dissonance is unbearable. Try the
minor chord of A (A-C-E) and you will hear the rasping, throbbing
beats of the too greatly flattened fifth.

So, you see, we are confronted with a difficulty. If we tune our
fifths perfect (in which case our fourths would also be perfect), our
thirds are so sharp that the ear will not tolerate them; and, if we
tune our thirds low enough to banish all beats, our fifths are
intolerably flat.

The experiment above shows us beautifully the prominent inconsistency
of our scale. We have demonstrated, that if we tune the members of the
chord of C so as to get absolutely pure harmony, we could not use the
chord of A on account of the flat fifth E, which did duty so perfectly
as third in the chord of C.

There is but one solution to this problem: Since we cannot tune either
the fifth or the third perfect, we must compromise, we must strike the
happy medium. So we will proceed by a method that will leave our
fifths flatter than perfect, but not so much as to make them at all
displeasing, and that will leave our thirds sharper than perfect, but
not intolerably so.

We have, thus far, spoken only of the octave, fifth and third. The
inquisitive student may, at this juncture, want to know something
about the various other intervals, such as the minor third, the major
and minor sixth, the diminished seventh, etc. But please bear in mind
that there are many peculiarities in the tempered scale, and we are