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 going to have you fully and explicitly informed on every point, if you will be content to absorb as little at a time as you are prepared to receive.  While it may seem to us that the tempered scale is a very complex institution when viewed as a specific arrangement of tones from which we are to derive all the various kinds of harmony, yet, when we consider that the chromatic scale is simply a series of twelve half-steps—­twelve perfectly similar intervals—­it seems very simple.

Bear in mind that the two cardinal points of the system of tuning are: 

    1.  All octaves shall be tuned perfect.

    2.  All fifths shall be tuned a little flatter than perfect.

You have seen from Lesson VIII that by this system we begin upon a certain tone and by a circle of twelve fifths cover every chromatic tone of the scale, and that we are finally brought around to a fifth, landing upon the tone upon which we started.

So you see there is very little to remember.  Later on we will speak of the various other intervals used in harmony:  not that they form any prominent part in scale forming, for they do not; but for the purpose of giving the learner a thorough understanding of all that pertains to the establishing of a correct equal temperament.

If the instruction thus far is understood and carried out, and the student can properly tune fifths and octaves, the other intervals will take care of themselves, and will take their places gracefully in any harmony in which they are called upon to take part; but if there is a single instance in which an octave or a fifth is allowed to remain untrue or untempered, one or more chords will show it up.  It may manifest itself in one chord only.  A tone may be untrue to our tempered scale, and yet sound beautifully in certain chords, but there will always be at least one in which it will “howl.”  For instance, if in the seventh step of our system, we tune E a little too flat, it sounds all the better when used as third in the chord of C, as we have shown in the experiment mentioned on page 94 of this lesson.  But, if the remainder of the temperament is accurate, this E, in the chord in which E acts as tonic or fundamental, will be found to be too flat, and its third, G sharp, will demonstrate the fact by sounding too sharp.