" 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.

If two strings, tuned to give forth the same number of vibrations per
second, are struck at the same time, the tone produced will appear to
come from a single source; one sweet, continuous, smooth, musical
tone. The rea