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Sect. 1. Counterpoint, point-counter-point, note-against-note, is the art of constructing distinct melodies to be performed together, and may be best defined as melody against melody. A counterpoint is a melody to accompany another melody. For convenience of study, counterpoint is classed in five species, each of which may be separately practised in special exercises (explained in later chapters), but all of which are constantly exemplified in musical composition.
2. Antiphony signified, in Greece, the singing of a melody by boys or women, an 8th higher than it was sung at the same time by men, being thus, literally, sound against or with sound. In church use it signifies the alternation of opposite choirs in response, and so means sound answering, or after sound.
3. Descant seems to have been the art of improvising a melodic accompaniment to a fixed song, which was practised before the period of written counterpoint, and of the codification of its rules.
4. Diaphony preceded descant. It is alleged to have been the singing of one melody by two voices, or choirs of voices, at the interval of a 4th, or a 5th, or an 8th asunder.
It is incredible that simultaneous singing in 5ths or in 4ths can ever have been authorised ; and the appropriation of the term diaphony to singing in 8ths may be supposed to be erroneous.
It seems likely that the term may have meant alternation or response, and that the parts which, in ancient copies, stand one over another at the interval of a 5th, a 4th, or an 8th, were sung in succession and not together, their presentation in writing having no analogy to the modern idea of a score. This is but a conjecture, whose proof must rest with the antiquary; but it is based on the natural rule of reason, that progressions, which are now in the highest degree offensive to the ear, can never have been habitually performed or authoritatively sanctioned. If the conjecture be admissible, it will point to diaphony as the germ of the fugue, a melody having been, perhaps, firstly alternated by responsive choirs at the interval of a 5th, or a 4th, or an 8th higher or lower, and at an after
at the distance of an 8th and a 3rd, or of two 8ths and a 3rd. Not so the 8th below a an 8th,which distinguishes these from a 2nd and a 1st.
10. Semitone is the conventional name for the smallest interval on keyed
อ: instruments, and it signifies the distance from any note to the one nearest to it. The fallacy of the term is proved by the inequality of these two quoted semitones; the first (with all semitones of which one of the notes is on a line and the other in a space, or of which the notes have different alphabetical names) is diatonic or major; the second (with all semitones of which both notes are on a line or both in a space, or of which both notes have the same alphabetical name) is chromatic or minor. The higher note of a major semitone has 16 vibrations to every 15 of the lower; the higher note of a minor semitone has 25 vibrations to every 24 of the lower.(a)
11. A tone is an interval of a major semitone and a minor semitone, either of which may be above or below the other; and the tone is major or minor, according to the expanse of its minor semitone. The relative vibrations of the two notes of a major tone are 9 : 8, those of a minor tone are 10:9.
12. Inversion of intervals is the change of the relative position of their notes, placing the lower above the higher, or the higher below the lower.
As the central note belongs both to the interval and its inversion, and is therefore counted in each, the number of any interval within an 8th added to the number of its inversion makes nine, and of every interval within a 15th (or double-octave) makes sixteen.
(a) This is the accepted standard; but the discrepancy between the two notes of a chromatic semitone varies according to the position of the semitone in the chromatic scale, and the distinction is thus induced of a major from a minor tone.
(6) It is strangely remarkable, that, though men of science and musicians have spent elaborate attention upon other philosophical points, which may or may not link acoustical science to musical art, not one has openly discussed the phenomena that separate the 1st and the 5th, the 8th and the 4th, in character, effect, and treatment, from all other intervals. This is not the place to enlarge upon the very important subject, but the present opportunity may be utilised to suggest its scientific consideration, and to state a belief that any facts bearing upon it, which may be brought to light, will be of the highest possible interest and commensurate value.
14. The two notes of a perfect interval are of the same quality--that is, both are natural or sharp or flat, or double-sharp or double-flat; with the single exception of the intervals of which B and F, or F and B, of whatever quality, are the two notes.
15. 2nds and 3rds with their inversions, 7ths and 6ths, are major or minor, and differ from perfect intervals in this flexibility, retaining the character of discords or concords in either major or minor form; whereas, to raise or lower by a chromatic semitone either note of a perfect interval, changes the combination from a concord into a discord.
17. The inversion of perfect intervals produces perfect; the inversion of major intervals produces minor; the inversion of minor produces major; the inversion of diminished produces augmented; and the inversion of augmented produces diminished.
18. Here is a table of intervals and their inversions, with the number of semitones in each.
19. Concordant intervals are perfect lsts and 8ths, perfect 5ths (6) and 4ths, (c) and major and minor 3rds and 6ths.
20. Discordant intervals are, 4th from the bass, any notes next each other in alphabetical order, all diminished and augmented intervals from the bass, and the augmented 5th and its inversion in any parts. To these must be added the perfect 5th of the mediant (truly the last inversion of the dominant major 13th), the discordance of which is disputed, though examples of its good effect are referable to one or another rule for the treatment of discords, or else to the rules of sequence.
SCALES, MODES, AND KEYS.
21. A Scale is a succession of notes by regular but varying degrees. It is diatonic when the notes are in unbroken alphabetical order. It is chromatic when the seven diatonic notes are interspersed with the five inflected notes, and then comprises twelve sounds. It is pentaphonic when the 4th and 7th degrees from the key-note are omitted, and it has thus no semitone, but consists only of tones and minor 3rds.
22. A tetrachord is a scale of four notes, and was the basis of the Greek musical system. A diatonic tetrachord has a semitone, a tone, and another tone between the successive notes. The first tetrachord started from A, which subsequently became the middle note
Ꭷ; of the entire series. A second tętrachord was added to this, having the same A for its top instead of bottom note, and the two were called conjunct, because conjoined by the note that was
ก: common to both. The next advance was to have an 8th above the lowest note of the second tetrachord; but to make this addition true to the order of semitone and tones, LB was introduced; the interval from A to this B was called the diazeuctic tone, or tone of disjunction; and the new
5: tetrachord was called disjunct, because disjoined by B from the others. Then a tetrachord was added under the lower E; and then another over the higher E; and, lastly, a single note, A, at the bottom of all, to make euphony with the highest A, and this note was called Proslambanomenos. Here is the entire scale of the Greek greater perfect system.
(a) The 11ths and 13ths are here given to complete the table, but are unavailable in diatonic counterpoint. (6) The 5th of the mediant is exceptional (Sect. 20).
In the strict or diatonic style, 4ths are only concordant when-between upper parts, and not so when between the bass and any upper part.