APPENDIX.

PASSAGE OMITTED IN BOOK I. SECT. V. P. 119.

ON THE LAWS OF HARMONY.

Now that our disquisitions have led us to touch on this
subject, it seems desirable to give a fuller explanation of
its leading points. To do this in a manner befitting the
present occasion we must begin by observing, that a tone
is a sound with an assignable duration. When this is
repeated with the requisite degree of sharpness and
flatness without producing that effect which is the property
of harmony, it does not come within the scope of
the science of music; such scope being limited to tones
of such a character that their interval in regard to sharpness
and flatness, or the interval of the periods recurring
between them in regard to duration, contributes to a
harmonious or discordant relation; the first of which
divisions is termed harmony, and the second of which is
termed melody. Now when two tones are taken which
differ in sharpness and flatness, the difference between
them will necessarily be constituent of a relation either
harmonious or discordant. For if the difference be referable
either to *like in fact* or *like in effect*, it is harmony;
and if not, it is discord. The meaning of *like in
fact* is this, that the measure of the interval is equal to
the less; which may happen when one is double of the
other, like 4 and 2, 6 and 3; and this is termed the
*Diapason* interval. The meaning of *like in effect* is this,
that what is not like in fact may by duplication be rendered
so: which is of two sorts; one, where this property
resides in the difference, as with 6 and 4, which differ by
2, which by duplication becomes 4; and this is termed
the *Progressional* proportion: the other, where this property
resides in one of the differents, as with 6 and 2,
which differ by 4; whereas 2, which is one of the differents,
by duplication becomes 4; and this is termed the
*Multiple* proportion. Every proportion which proceeds
on these conditions, or is capable of being reduced to
them, is harmony; and every one repugnant to them is
discord. Thus all couples of tones not having a numeral
ratio, that is, tones whose ratio is an involved one with
peculiar properties for which numbers are not to be
found, are discordant; such as the tone produced by the
whole string, and that produced by such part of it as
bears to the whole the ratio of a square’s side to its
diagonal. And even if it be a numeral proportion, but
the smaller number divide not the greater, or the difference
between the two is not by a part having the
power of the greater, and it is not capable of being reduced
to the harmonious proportions by any of the
methods presently explained, it must still be discordant.
For instance, two tones of which one is greater than the
other by *Diapason* interval;
if greater, it is the *Multiple* relation.

Again, when the difference is by a part which divides
the greater number, if that part makes up a half or near
to half to one of the numbers, as a half or third, they call
it the proportion of *middle* intervals, which is reducible
to the same two; for if the difference is between 4 and 6,
the differential part forms half; and if it is between 7 and
5, it makes what is near to half. Of these middle intervals,
the first sort is termed the *Diapente* interval,
such as 2 and 3; and the second sort the *Diatessara* interval,
such as 3 and 4. And if the difference is by a part
which makes not the number half or near it, it is termed
the proportion of *Minor* intervals, or *Hypertessara.*

Now these different descriptions of harmony, which are all either the involution of one number in another, or else their differing by a part being a divisor of the greater number, are contemplated only as far as the difference can be perceived, and the human frame possesses the power of putting it forth. If the difference be of such sort that it is not the subject of sensation, or be excessively trifling, or such as the human frame cannot enunciate, it comes not within the limits of this science. For on the supposition of its escaping the perception, or having only an exceedingly minute expression in it, that agreeable sensation which is the object of joining sounds together does not result therefrom; and in the latter case, though it be possible to bring it forth from other instruments, yet being not on a scale with the physical demands of man, that is, with his own organic intonations, the human system finds no attraction in them, neither do they attain the height of being agreeable; whereas the science of music being placed in following out the highest, this is no part of its scope. It appears, then, that a proportion not formed on the scale of man’s organic intonations is no subject for our consideration. The limit of combination in organic tones (as actually effected) in the class of major intervals is that one should be double of the other’s double, as 4 and 1: in the class of minors, that one should be greater than the other by one thirty-sixth part, that is, one being 36 and the other 37: all beyond this is not had in contemplation.

Now to apply this to the present subject, the proportion
of doubles, which they call the *like* proportion, is
the primary and most pure of all. One instance of its
exceeding purity and closeness to unity is this, that either
side may be substituted for the other without prejudice to
its harmony; that is to say, whether we employ the tone
duplicate or the reverse, the thread of connexion is not
broken, nor the tie of concord dissolved. For instance,
the tone represented by 8 being double that represented
by 4, if we substitute 8 for 4, and place it in union with
the tone represented by 3, a harmonious interval is obtained
from 8 and 3, although there is no primary concord
between them: their harmony being in this wise,
that 4, which is the half of 8, is in harmony with 3; or if,
looking to the 3, we say 3 is half of 6, between which
and 8 there is harmony, the same principle is educed;
and on either supposition it resolves itself into the *Diatessara*
interval. Or else, combining 5 with 3, a harmony
is obtained which reduces itself to the *Minor* interval; for
this reason, that by the minor interval there is a harmonious
relation between 5 and 6, and 3 is the representative
of 6. Or we might say it has the relation of
Minor intervals with 2*Diapente* interval is capable of
being reduced to the *Multiple*, or interval of 4; as likewise,
the *Diatessara* to the *Diapente.* For if, in the first
way, we take 2 as a representative of 4, it (the *Multiple*)
falls into the *Diatessara;* and if, in the second way, we
take 3 as the representative of 6, it falls into the *Diapente.*

Another instance of pure and radical properties in the
*Diapason* interval (the compartment of *like in fact*) is that
both the intervals by which it is divided are means as
well of arithmetical as of harmonical proportion. Now
the meaning of arithmetical mean is that it is intermediary
between two numbers, so that its ratio to both extremes,
in respect of proximity and distance, is alike; like 4,
which is intermediary between 6 and 2. And the meaning
of harmonical mean is that it is a number, the ratio of
whose excess over the lesser term is to the excess of the
greater term over it, as is the ratio of the lesser term to
the greater term; such as 4, which is harmonic mean
between 3 and 6; for the excess of 4 over 3 is 1, and the
excess of 6 over 4 is 2, and the ratio between 1 and 2 is
as the ratio between 3 and 6.

The application of the first is this: the ratio of 4 to 2
is the *Diapason* interval; and when 3, which is the numeral
mean, is introduced, two ratios arise; one between
2 and 3, which is the *Diapente* interval; the other between
3 and 4, which is the *Diatessara* interval. The
application of the second is this: the ratio of 6 to 3 is
the *Diapason* interval, and when 4, which is the harmonic
mean, is introduced, two ratios arise, one the ratio of 4 to
3, which is the *Diatessara* interval, the other that of 4 to
6, which is the *Diapente* interval. From which particulars
it is that we discover the reason of naming the
duplicate ratio a *Diapason* [or all-pervading] interval;
and that of naming the other two concordant ratios as
above.

From these preliminaries it is clear that all the harmonious
intervals come back to the proportion of similar
ratios. For in the *like in fact* the measure of difference
is like in reality; and in the *like in effect* it is
like in operation, either through the properties of one of
the two differing numbers, or by nature, or by mediary
connexion, as has been explained. The element of harmony
then is similarity, which is an image of unity.

Like | ||||||

In fact | In effect | |||||

Diapason | Progressional | Multiple | ||||

1 : 2 | 4 : 6 | Primary Concordance |
2 : 6 | |||

Major Intervals | ||||||

Diapente | ||||||

2 : 4 | Take 2 : 3 : 4 | Diatessara | Diapente | |||

4 : 3 | Secondary or Representative Concordance |
2 : 3 | ||||

Diatessara | ||||||

Diatessara | ||||||

3 : 6 | Take 3 : 4 : 6 | |||||

Diapente | Diatessara | |||||

2 : 3 | 2 : 3 | Doubly Representative |
4 : 3 | |||

Diapente | ||||||

Middle Intervals |

Minor Intervals 5 : 6 &c. Hypertessara

Called; Diapason (through all) because the interval is equal in all the terms, or because all the other ratios are involved in this one; Progressional, because the difference the lesser term and the greater term are in Arithmetical progression (2 : 4 : 6); Multiple, because the lesser term is contained in the greater one more than twice (2+2+2=6); Diapente, because a distribution of five (2+3=5); Diatessara, because 4 is the ruling number; Hypertessara, because the interval becomes smaller than can be expressed by a fourth, &c.

END OF VERSION.