The relation of the Octave to a Fundamental is expressed by  2/1
"     "         "  Fifth to a         "          "          3/2
"     "         "  Fourth to a        "          "          4/3
"     "         "  Major Third to a   "          "          5/4
"     "         "  Minor Third to a   "          "          6/5
"     "         "  Major Second to a  "          "          9/8
"     "         "  Major Sixth to a   "          "          5/3
"     "         "  Minor Sixth to a   "          "          8/5
"     "         "  Major Seventh to a "          "         15/8
"     "         "  Minor Second to a  "          "         16/15

QUESTIONS ON LESSON XIII.

    1.  State what principle is demonstrated in Proposition II.

    2.  State what principle is demonstrated in Proposition III.

    3.  What would be the vibration per second of an exact (not
    tempered) fifth, from C-512?

    4.  Give the figures and the process used in finding the vibration
    number of the exact major third to C-256.

    5.  If we should tune the whole circle of twelve fifths exactly as
    detailed in Proposition III, how much too sharp would the last C
    be to the first C tuned?

LESSON XIV.

MISCELLANEOUS TOPICS PERTAINING TO THE PRACTICAL WORK OF TUNING.

Beats.—­The phenomenon known as “beats” has been but briefly alluded to in previous lessons, and not analytically discussed as it should be, being so important a feature as it is, in the practical operations of tuning.  The average tuner hears and considers the beats with a vague and indefinite comprehension, guessing at causes and effects, and arriving at uncertain results.  Having now become familiar with vibration numbers and ratios, the student may, at this juncture, more readily understand the phenomenon, the more scientific discussion of which it has been thought prudent to withhold until now.

In speaking of the unison in Lesson VIII, we stated that “the cause of the waves in a defective unison is the alternate recurring of the periods when the condensations and the rarefactions correspond in the two strings, and then antagonize.”  This concise definition is complete; but it may not as yet have been fully apprehended.  The unison being the simplest interval, we shall use it for consideration before taking the more complex intervals into account.

Let us consider the nature of a single musical tone:  that it consists of a chain of sound-waves; that each sound-wave consists of a condensation and a rarefaction, which are directly opposed to each other; and that sound-waves travel through air at a specific rate per second.  Let us also remark, here, that in the foregoing lessons, where reference is made to vibrations, the term signifies sound-waves.  In other words, the terms, “vibration” and “sound-wave,” are synonymous.