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 4/7ths, as when one is 7 and the other 11, with a difference between them of 4/7ths, neither 7 the lesser will halve 11, nor will 4, which is the measure of the dif­ference. But where the less will divide the greater, the measure of the difference must be either equal to the lesser, or greater than it. If equal, it is the ratio of double and half, or, as it is termed, the 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 21/2, and 5 is the representative of 21/2; all which species they call concordant by the secondary concordance. And here the intelligent reader will perceive that the 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.

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.